analysis of uncertainty in river water quality modelling - imperial
TRANSCRIPT
Analysis of Uncertainty in River
Water Quality Modelling
by
Neil McIntyre
A thesis submitted for the degree of Doctor of Philosophy from
the University of London
2004
Department of Civil and Environmental Engineering, Imperial
College London
Abstract The case is given for attention to the evaluation of uncertainty in water quality modelling,
in the contexts of new demands for assessment of risk to water quality status, and typical
paucity of supporting data. A framework for the modelling of water quality is outlined
and presented as a potentially valuable component of broader risk assessment
methodologies, and potentially useful methods of numerical uncertainty analysis are
reviewed and demonstrated. A selective library of dynamic models and numerical tools
for model solving and uncertainty analysis are compiled into novel software for model
uncertainty analysis and risk-based decision-support. This software is applied to a series
of case studies in an exploration of the underlying numerical problems and their relevance
to modelling and management objectives using relatively sparse data sets. Issues
examined in some detail are the importance of reconciling numerical solution tolerances
with overall model precision; relative effects of numerical approximations, data and
model structural biases on optimal design of field experiments and on prediction
reliability; and the value and limitations of extending established methods of uncertainty
analysis to decision-support. These investigations lead to discussions about priorities for
the water quality modelling research community, in the face of contemporary and
emerging numerical, technological and management problems. The main conclusion is
that the current generation of modelling software can make very limited contribution to
risk-based decision support, due to general absence of formal uncertainty analysis
capabilities. This restriction is becoming more important due to new, ambitious spatial
and ecological management challenges. Further research into numerical issues is needed
to provide tools that allow these new challenges to be met, as well as to resolve persistent
deficiencies in modelling capability. A more pressing concern, however, is that
practitioners and their clients begin to confront issues of uncertainty, and create a demand
for software that facilitates risk-based planning.
Declaration
The work presented in this Thesis is my own except where otherwise
acknowledged.
Neil McIntyre
Acknowledgements
Without diminishing the role of all my colleagues, friends and family in supporting my
work on this PhD dissertation, here I give specific acknowledgement to a special few.
I have benefited from working with outstandingly talented and dedicated hydrologists
whose enthusiasm and high professional standards have been a constant source of
inspiration. I owe special thanks to my supervisor Howard Wheater for the confidence he
has had in my potential, allowing me to conduct the research in my own time and in my
own way, and for his intelligent management of my work-load. I am also indebted to my
colleague Adrian Butler on this latter point, as well as for his mentoring and friendship
though challenging times. Steve Chapra was the key motivating figure behind my
decision to begin a PhD - his enthusiasm taught me that an academic career was not such
a bad idea. I had the pleasure to work alongside Thorsten Wagener and Luis Camacho for
three years, and I thank them for making the day-to-day PhD experience rewarding and
fun. I thank my co-authors on the papers which have emanated from the PhD research for
their expert advice – Howard Wheater, Thorsten Wagener, Steve Chapra, Matthew Lees,
Beth Jackson, and Zeng Siyu. My thanks also to Angela Frederick for her zealous
assistance in putting this and related documents together.
My strongest thanks are for my parents, Anne and Donald McIntyre. Only thirty-four
years of nurturing and encouragement from them have permitted the effort that has
resulted in this document. Finally, I acknowledge with gratitude the role of my partner
during the PhD years, Juliet Vuong, who pushed me out of a few low points and
celebrated my milestones.
Preface
Six of the chapters in this dissertation are based on works either published by Neil
McIntyre or awaiting publication (as listed below). Although these works have co-
authors, the substantial intellectual, research and writing effort was that of the first author,
and any specific contributions of co-authors have either not been presented in this
dissertation or have been duly acknowledged.
Chapter 1
McIntyre, N.R., Wagener, T., Wheater, H.S. and Zeng, S. 2003. Uncertainty and risk in
water quality modelling and management. Journal of Hydroinformatics 5(4), 259-274.
Chapter 2
McIntyre, N.R., Wheater, H.S. and Lees, M.J. 2002. Estimation and propagation of
parametric uncertainty in environmental models. Journal of Hydroinformatics 4(3),
177-198.
McIntyre, N.R., Lees, M.J. and Wheater, H.S. 2001. A review and demonstration of
methods of uncertainty analysis in numerical environmental modelling. Proceedings of
the 8th Europia International Conference - Advances in Design Sciences and
Technology, Delft, The Netherlands, 183-196.
Chapter 4
McIntyre, N.R. and Wheater, H.S. 2003. A tool for risk-based analysis of surface water
quality. Environmental Modelling and Software (In press).
Chapter 5
McIntyre, N.R., Jackson, B., Wheater, H.S. and Chapra, S. 2003. Numerical efficiency in
Monte Carlo simulations - a case study of a river thermodynamic model. ASCE
Journal of Environmental Engineering (In press).
Chapter 6
McIntyre, N.R. and Wheater, H.S. 2003. Calibration of an in-river phosphorus model:
prior evaluation of data needs and model uncertainty. Journal of Hydrology (In press).
Chapter 7
McIntyre, N.R., Wagener, T., Wheater, H.S. and Chapra, S.C. 2003. Risk-based
modelling of surface water quality - A case study of the Charles River, Massachusetts.
Journal of Hydrology 274, 225-247.
Table of Contents
1. Introduction....................................................................................................... 1 1.1 Introduction to analysis of uncertainty in surface water quality modelling 2
1.1.1 Motivation .................................................................................................. 2 1.1.2 Causes of uncertainty.................................................................................. 4 1.1.3 Analysis of uncertainty............................................................................... 5 1.1.4 Surface water quality modelling applications............................................. 8
1.2 Introducing a framework for risk-based surface water quality modelling 10 1.2.1 Risk in context .......................................................................................... 10 1.2.2 A framework outline................................................................................. 11 1.2.3 Technical considerations .......................................................................... 12 1.2.4 A tool for risk-based management of water quality ................................. 16
1.3 Background to the case studies ................................................................. 16 1.3.1 The Hun River characteristics .................................................................. 17 1.3.2 The Charles River characteristics ............................................................. 19
1.4 Explanation of the structure of the remainder of this dissertation............. 20
2. Estimation and propagation of parametric uncertainty in environmental models................................................................................................................... 22
2.1 Introduction ............................................................................................... 23 2.1.1 Background and scope of chapter............................................................. 23 2.1.2 The sources of uncertainty and their representation in the model ............ 24
2.2 Approaches to uncertainty-based model calibration ................................. 25 2.2.1 Definitions ................................................................................................ 26 2.2.2 Objective functions and likelihood functions ........................................... 26 2.2.3 The significance of model structure errors and data bias ......................... 29 2.2.4 Possibility theory and the HSY method.................................................... 30 2.2.5 Generalised Likelihood Uncertainty Estimation ...................................... 32 2.2.6 Model output versus data uncertainty....................................................... 33 2.2.7 Multiple objective analysis....................................................................... 33
2.3 Sampling and global optimisation techniques........................................... 35 2.3.1 Monte Carlo simulation ............................................................................ 35 2.3.2 Metropolis algorithm ................................................................................ 36
2.3.3 Genetic algorithms.................................................................................... 38 2.4 Example of calibration .............................................................................. 38
2.4.1 The model and data .................................................................................. 39 2.4.2 Sampling the data error distribution ......................................................... 39 2.4.3 GLUE using a likelihood function as an objective function..................... 42 2.4.4 Metropolis using weighted squared errors as an objective function......... 46 2.4.5 GLUE using a subjective GLUE likelihood as an objective function ...... 48 2.4.6 HSY using a possibilistic objective function............................................ 49 2.4.7 Summary of this demonstration of calibration ......................................... 50
2.5 Uncertainty propagation ............................................................................ 51 2.5.1 Monte Carlo methods ............................................................................... 52 2.5.2 First order and point estimate approximations ......................................... 52 2.5.3 Possibility theory ...................................................................................... 53
2.6 Propagation of the Streeter-Phelps model parameters............................... 54
2.7 Summary ................................................................................................... 58
3. An overview of river water quality modelling theory and commonly used modelling tools..................................................................................................... 60
3.1 Introduction ............................................................................................... 61
3.2 Developments ............................................................................................ 61
3.3 The components of a river water quality model........................................ 64 3.3.1 Hydraulic and routing models .................................................................. 65 3.3.2 Solute transport models ............................................................................ 68 3.3.3 Thermodynamics ...................................................................................... 70 3.3.4 Water quality processes ............................................................................ 75
3.3.4.1 Carbon and dissolved oxygen............................................................. 75 3.3.4.2 Photosynthesis .................................................................................... 78 3.3.4.3 Nitrogen and phosphorus cycles......................................................... 79 3.3.4.4 Organic toxins and oils ....................................................................... 81 3.3.4.5 Suspended solids ................................................................................ 82
3.4 Summary ................................................................................................... 84
4. Water quality risk analysis tool (WaterRAT) .............................................. 86 4.1 Introduction ............................................................................................... 87
4.2 The concept and structure of WaterRAT................................................... 88
4.3 Spatial and temporal resolution................................................................. 90
4.4 Boundary conditions, initial conditions and model parameters ................ 91
4.5 Calibration and optimisation ..................................................................... 93
4.6 Multi-objective analysis ............................................................................ 96
4.7 Sensitivity analysis .................................................................................... 97
4.8 Prediction uncertainty................................................................................ 99
4.9 Output ...................................................................................................... 100
4.10 WaterRAT review ................................................................................. 101 4.10.1 General limitations ............................................................................... 101 4.10.2 Critical comparison with alternative modelling tools........................... 102
4.11 Summary ............................................................................................... 104
5. Numerical efficiency in Monte Carlo simulations – a case study of a river thermodynamic model ...................................................................................... 106
5.1 Introduction ............................................................................................. 107
5.2 The thermodynamic model...................................................................... 108
5.3 Monte Carlo simulation........................................................................... 112
5.4 Numerical methods.................................................................................. 114
5.5 Results ..................................................................................................... 120
5.6 Discussion ............................................................................................... 124
5.7 Summary ................................................................................................. 126
6. Identification of a phosphorus mobilisation model: prior evaluation of data needs ................................................................................................................... 128
6.1 Introduction ............................................................................................. 129
6.2 Model Description................................................................................... 131
6.3 The data ................................................................................................... 135
6.4 The calibration, prediction and performance evaluation procedures ...... 138
6.5 Results, discussion and supplementary experiments............................... 140
6.6 Summary ................................................................................................. 150
7. Risk-based modelling of surface water quality: a case study of the Charles River, Massachusetts ........................................................................................ 152
7.1 Introduction ............................................................................................. 153 7.1.1 Motivation .............................................................................................. 153 7.1.2 Scope and objectives .............................................................................. 153 7.1.3 The case study ........................................................................................ 154
7.2 Model Structure and Methods ................................................................. 155 7.2.1 Specification of the model structure ....................................................... 155 7.2.2 Specification of prior parameters and uncertainty in observed data....... 160 7.2.3 Multi-objective model conditioning ....................................................... 163
7.2.4 Graphical model evaluation.................................................................... 165 7.2.5 Regional sensitivity analysis .................................................................. 166 7.2.6 Risk-based appraisal of intervention strategies ...................................... 169
7.3 Results and discussion............................................................................. 170 7.3.1 Preliminary model evaluation................................................................. 170 7.3.2 Sensitivity analysis (1996 conditions) .................................................... 174 7.3.3 Sensitivity analysis (eutrophication reduction) ...................................... 178 7.3.4 Appraisal of intervention strategies ........................................................ 179
7.4 Summary ................................................................................................. 182
8. Conclusions .................................................................................................... 184 8.1 Summary ................................................................................................. 185
8.2 Review of GLUE..................................................................................... 185
8.3 Prior and posterior knowledge ................................................................ 186
8.4 Monte Carlo sampling ............................................................................. 188
8.5 Review of WaterRAT.............................................................................. 190
8.6 Review of Chapter 5: numerical issues ................................................... 191
8.7 Review of Chapter 6: prior identification of data needs and assessment of model capability. ........................................................................................... 193
8.8 Review of Chapter 7: a framework for model conditioning, sensitivity and risk analysis ................................................................................................... 194
8.9 The Hun River case study........................................................................ 197
8.10 A look to the future................................................................................ 200
References .......................................................................................................... 205
Notation.............................................................................................................. 221
1
1. Introduction The aim of this Thesis is to assert a philosophy for river quality modelling, deliver and
demonstrate associated software, and set an agenda for future research and improved
modelling practice. This aim is pursued through eight chapters which 1) formulate the
motivation for the research, 2) describe the tools and software, 3) explore the key
modelling issues using case studies and a set of numerical analyses, and 4) critically
discuss the work’s significance. The focus is on modelling problems that are supported by
typically sparse data sets, and this is reflected in the chosen case studies. The dissertation
is written with a view that the reader has limited prior knowledge of numerical modelling
and statistical methods.
In this first chapter, the case is presented for increased attention to the evaluation of
uncertainty in water quality modelling practice. A framework for the modelling of water
quality is outlined and presented as a potentially valuable component of broader risk
assessment methodologies. Technical considerations for the successful implementation of
the modelling framework are introduced. The primary arguments presented are as
follows: 1) For a large number of practical applications, deterministic use of complex
water quality models is not well supported by the available data and/or human resources,
and is not warranted by the limited information contained in the results. Modelling tools
should be flexible enough to be employed at levels of complexity which suit the
modelling task, the data and the available resources. 2) Monte Carlo simulation has
largely untapped potential for evaluation of model performance, estimation of model
uncertainty and identification of factors (including pollution sources, environmental
influences, and ill-defined objectives) contributing to the risk of failing water quality
objectives. 3) For practical application of Monte Carlo methods, attention needs to be
given to numerical efficiency, and for successful communication of results, effective
interfaces are required. This chapter finishes with an introduction to the case studies used
in later chapters, and with an overview of the content of the dissertation.
2
1.1 Introduction to analysis of uncertainty in surface water quality
modelling
1.1.1 Motivation
In the European Community, the recently introduced Water Framework Directive (CEC
2000) requires that member states formulate River Basin Management Plans which
identify objectives for achieving good water quality status on a catchment-wide basis.
Similar standards apply to much of the world, for example catchment management in the
United States has been guided by the Environmental Protection Agency’s Water Quality
Criteria and Standards Plan (US EPA 1998), in Australia by the National Water Quality
Management Strategy (DAFF 2000) and in China by the Environmental Quality Standard
for Surface Water (PRC SEPA 1999). Simulation models are a central part of these basin
management plans because they can apply best available scientific knowledge,
conditioned by historical evidence, to predict water quality responses to changing
controls. For example, the development of the integrated catchment model BASINS is an
explicit part of basin management plans in the United States (US EPA 1998, 1999), and
the UK government has recently commissioned new tools for diffuse source modelling
(DEFRA 2002: p76).
The new high expectations for the aquatic environment, incorporated into the current
wave of directives and regulations, is prompting additional complexity with regard to
modelling spatial variability, micro-pollutants and ecological indicators (Somlyody et al.
1998, Thomann 1998). Facilitated by improved computational resources, there is a trend
for spatial discretisation to be increased, multi-media and multi-constituent models to be
developed (e.g. Havnø et al. 1995, Cole and Wells 2000, Neitsch et al. 2002), and for
traditional water quality determinants to be broken down into constituent species (Chapra
1999, Shanahan et al. 2001). As a consequence, the typical number of modelled
components has risen exponentially over the past years, and this growth is expected to
continue (Thomann 1998).
Despite the increasing expectations placed upon water quality models, contemporary
deterministic models, when audited, frequently fail to predict the most local and basic
biological indicators with a reasonable degree of precision (e.g. Jorgensen et al. 1986).
Even when models are claimed to be ‘reliable’ following audits, a very significant margin
of error is allowed (e.g. Hartigan et al. 1983). The application of modelling to the new era
of high ecological standards presents severe challenges, especially given that our
3
modelling experience is with relatively stressed ecological systems (Beck 1997, Shanahan
et al. 1998), and that the economic implications of model errors may be relatively serious
(Chapra 1997). While additional model complexity might be expected to improve the
precision of model results, this has proven to be unfounded in a variety of studies (e.g.
Gardner et al. 1980, Van der Perk 1997, Lees et al. 2000, also see Young et al. 1996).
Furthermore, future driving forces such as climate (Hulme et al. 2002) and distributed
pollution sources (Shepherd et al. 1999) are poorly defined and themselves cannot be
modelled with much precision. Clearly, identification of suitable water quality policy
must take account of the uncertainties associated with both the validity of the models and
the driving forces. However, as increased model complexity hinders the formal evaluation
of uncertainty, due to the large number of uncertain model components to be
simultaneously analysed, there is a danger that our ability to evaluate uncertainty will
decrease.
The challenges facing water quality modellers should be contemplated in the wider
perspective of risk-based decision support. Firstly, a high degree of model uncertainty is
not necessarily an undesirable outcome, and undoubtedly is preferable to no indication of
reliability at all. Secondly, uncertainty in environmental models should be viewed as a
source of risk, as is traditional in other fields of engineering (e.g. Tung 1996), and should
be used to establish and achieve an acceptable failure probability in terms of water quality
status, rather than be used to decry the modelling approach (Beven 2000a). Given that
risk is a concept that can be used to integrate external criteria such as economics and
safety, as well as integrating the model result over the relevant model responses,
expressing results as risk is potentially attractive and seems inevitable. Thirdly, it is worth
noting that, in the context of decision-support, we are not justified in investing resources
in modelling (including the identification of prediction uncertainty) unless this will be
instrumental to the decisions that need to be made (Beven 1993). Therefore, we should
keep sight of the modelling task, and accept that (very) approximate solutions may be
appropriate.
To allow intelligent use of complex simulation models, and to allow informed
interpretation and application of model predictions, it is essential that a new generation of
tools is developed and disseminated. These should be directed at evaluation of model
uncertainty, as well as its minimisation, with respect to the modelling tasks. For results to
be justified and interpreted properly, methods used for uncertainty analysis must be
theoretically or intuitively well-founded, and transparent to the modeller. For methods to
be practical for day to day use, they should be relatively easy and fast to implement.
4
These requirements are challenges which will be addressed in the rest of this dissertation,
beginning in this chapter with review of the factors contributing to uncertainty, brief
review of current practices in the water quality community, and a proposed outline of a
framework for risk-based water quality modelling. A tool for modelling of river and lake
water quality where supporting resources are restricted, is then introduced.
1.1.2 Causes of uncertainty
Uncertainty in a water quality simulation model is inevitable due to the difficulty of
identifying a single model (including grid-scale, process formulations and parameter
values) which can accurately represent the water quality under all required model tasks
(see the discussions of Beck 1987, Van Straten 1998, and Adams and Reckhow 2001).
Although we have extensive knowledge about water quality processes from laboratory
experiments, extrapolation of this knowledge to models of the real environment has
consistently proven to be difficult. This is partly because the modelling scale is different
to the laboratory scale, and the diversity of species and heterogeneity found in natural
environments must (to some degree) be modelled approximately using lumped state
variables. This means that formulations and parameter values identified at laboratory
scale can only be used as a starting point for model design, rather than as a definitive end
result. Nor is there yet any basis for regionalisation of water quality models. Therefore,
models identified for one case study cannot be used with any confidence for another.
Literature which describes established formulations and parameter values (Bowie et al.
1985, Thomann and Mueller 1987, Chapra 1997), is evidence of the wide range of models
which are equally justified prior to observing a system’s behaviour in detail, and that the
uncertainty associated with modelling water quality on the basis of prior knowledge is
extremely large.
Given that it is desirable to evaluate the performance of models with respect to observed
water quality data, the accuracy, frequency and relevance of the available data dictates the
attainable degree of certainty in the model. Unfortunately, water quality data can be
expensive to collect and analyse, often requiring special handling and analysis in
laboratories. This means that data to support model identification are generally sparse,
often coming from sampling programmes which are fixed in frequency and location for
regulation purposes, rather than designed to encapture the system’s dynamic responses as
required for successful model identification (Berthouex and Brown 1994). Also, water
quality data are susceptible to noise and bias due to sampling, handling and measurement
procedures (see Keith 1990). In addition, information about model boundary conditions,
5
such as sources of pollution, often suffers from the same short-comings, especially for
distributed variables which are difficult to measure (pollution runoff, sediment quality,
etc). In summary, lack of good quality data to support model identification is a major
cause of model uncertainty.
Closely related to the issue of data quality is model equifinality, whereby different
models appear equally justified at the model design stage, but may give widely different
realisations of the future. Equifinality is caused by interactions between model
parameters, and by the near-equivalence of different model structures at the stage of
model identification. This means that the same (or effectively the same within the context
of the data errors) response can be achieved using different models. Clearly, the problem
magnifies as both the number of interacting parameters increases, and as the precision of
the data decreases. The use of parsimonious models, i.e. models which only include
parameters which can be uniquely identified from the data, is one approach to avoiding
equifinality. A parsimonious model implies that model components that are inactive
during model identification are left out, and that strongly interacting components are
combined into one (Young et al. 1996, Wagener et al. 2001). The inevitable omission of
model components which are potentially relevant means that parsimonious models may
seriously underestimate the uncertainty in model forecasts (Reichart and Omlin 1996).
When the aim of the modelling is to investigate risks associated with proposed water
quality interventions or other disturbances, it is essential that the uncertainty arising from
previously unobserved behaviour is adequately allowed for, and so parsimonious models
may be inappropriate. Thus, it may be said that identifying a single optimal model may
not be a justifiable approach.
The problem of equifinality and uncertainty in modelling environmental systems is
inevitable and model predictions based on a single ‘optimal’ model will, in general, be
rather arbitrary, and of very limited value. For this reason, a number of investigators have
devoted their attention to rationalising the modelling problem, and redefining it as
essentially stochastic whereby a population of feasible models (and by implication, a
population of model predictions) are identified (e.g. Hornberger and Spear 1980, Van
Straten and Keesman 1991, Beven and Binley 1992, Reichart and Omlin 1996, Yapo et
al. 1998, Melching and Bauwens 2001).
1.1.3 Analysis of uncertainty
Identification of a population of feasible models can include both identification of
alternative model structures (grid-scales and process formulations) and corresponding
6
parameter distributions. Model structures should be of a complexity consistent with the
difficulty and scale of the modelling task, and the supporting information and resources.
They should be consistent with prior knowledge of how best to represent system
processes at the scale and complexity in question. Given adequate supporting data, they
can be assessed and amended using various identification techniques (e.g. Beck 1983,
Qian 1997, Wagener et al. 2002a,b).
If one structure can be demonstrated as the most suitable for a particular modelling task
(that is, for the particular system, and the particular information which the modeller aims
to retrieve) then it would be reasonable to use this structure exclusively. On the other
hand, if there are justified alternatives then ideally, from an analytical point of view, the
implications of these also should be considered (e.g. Gardner et al. 1980, van der Perk
1997). This raises two issues. Firstly, it may be that no structures can be identified as
‘suitable’. Then (as will be expanded upon in Chapter 2) either an improved structure
should be developed, or the stringency of the model assessment should be reviewed and
the parameter uncertainty increased. Secondly, analysis of more than one structure may
not be feasible given the available resources - such analysis will be costly, perhaps
requiring purchase of additional software. Even using tools which offer some flexibility
in the choice of water quality model structure, such as DESERT (Ivanov et al. 1996) or
RWQM1 (Vanrolleghem et al. 2001) exploring candidate structures can significantly add
to the burden on human and computer resources. In such a case (and this tends to be the
case) all the significant model uncertainty must be represented, as far as possible, as
parameter uncertainty within a single suitable structure. From a mathematical point of
view, this has implications for the reliability of predictions (Draper 1995), but in a
management context it is justifiable if it has relatively little bearing on the decisions being
supported. In summary, investigating the sensitivity of decisions to different model
structures is commendable, but may be neither viable due to resource constraints, nor
worthwhile due to over-riding uncertainty in boundary conditions and parameter values.
Given a model structure, the identification of feasible sets of parameter values can be
approached by conditioning (constraining) the prior population of parameter sets so that a
specified modelling objective is better achieved. The modelling objective at this stage is
generally to simulate observed data, and is generally expressed objectively as a function
of the model residuals (the distances between the model result and the observed data). In
traditional deterministic modelling, the response of this objective function (OF) to
changes in the model parameters is used to estimate an optimum set of model parameters.
This is achieved by manual perturbations of the parameters or, more suitably for complex
7
models, by automatic algorithms. For uncertainty analysis, a joint distribution of
parameters is identified rather than a single optimum, by recording the response of the OF
across the parameter space. Depending partly on the algorithm which has been used, this
joint distribution may be represented as a variance-covariance matrix, or as a discrete
distribution (point estimates of probability mass over the parameter space), or as a
population of feasible parameter sets. Methods of parametric uncertainty analysis for
environmental simulation modelling are reviewed in Chapter 2.
Selecting an objective function to use for the conditioning of an environmental model is a
difficult issue which involves a degree of speculation and subjectivity. This is because
statistically-based identification of the parameter uncertainty requires knowledge of the
combined error structure of the model, the data and the boundary conditions. However,
especially when data are sparse or unreliable and the model structure is complex, there is
little or no theoretical basis for estimation of the error structure (see Chapter 2). While
parameter conditioning is often based on statistical likelihood functions (e.g. van Straten
1983), the result is dependent on the simplifying assumptions made about the error
structure. As well as being difficult to justify from prior information, such assumptions
can lead to significant mis-representation of model uncertainty (Beven et al. 2001), in
which case the model will fail to adequately explain the real system. In particular, the
common assumption that the model and/or data are unbiased can lead to a serious
underestimation of parameter and prediction uncertainties (Chapters 2 and 6).
As an alternative to statistical measures, the conditioning of the model can be based on
subjectively derived rules, for example, “if the parameter set returns a model result that is
highly consistent with my belief of true system behaviour then I will associate a relatively
high weighting”, or some objective expression of this, for example, “the relative
probability of each parameter set will be equal to the proportion of the variance of the
observed data explained by the model”. Given that it is subjectively based, such an
approach allows some freedom in achieving a satisfactory description of uncertainty,
without the encumbrance of statistical rules and the long list of associated simplifying
assumptions. Such conditioning of an environmental model, with the OF transformed to a
probability without necessarily being related objectively to the error structure, was
promoted by Beven and Binley (1992) in the context of their Generalised Likelihood
Uncertainty Estimation.
Once the uncertainty in the model is estimated, it can be propagated to give predictions.
Methods of uncertainty propagation which are relevant to simulation modelling can be
8
classified as variance propagation methods, point estimate methods, and Monte Carlo
methods. Tung (1996) gives an overview of these methods, and a review and
demonstration is included in the next chapter. The choice of method partly depends on the
description of the parameter uncertainty, and partly on the computational resources, with
the Monte Carlo methods generally (but not always) being more reliable and
computationally demanding.
1.1.4 Surface water quality modelling applications
There is a variety of literature promoting understanding and application of uncertainty
analysis in surface water quality modelling (e.g. Beck 1983, Beck 1987, Reckhow 1994,
Adams and Reckhow 2001). However, the application of uncertainty analysis to surface
water quality modelling seems to be relatively scarce, especially in practical decision-
support.
In the most widely used river water quality models, formal investigation of model
uncertainty is very rare. Uncertainty identification in many contemporary models such as
WASP5 (Ambrose et al. 1993), MIKE11 (Havnø et al. 1995), CE-QUAL-W2 (Cole and
Wells 2000) and ISIS (Wallingford Software 2002) is difficult because they are relatively
complex, and often linked to computationally intensive hydrodynamic, among other,
modules. Although these models are well founded in theory and well established in
practice (see Ambrose et al. 1996), their usefulness is arguably limited by their high
demand on resources, and the unknown uncertainty in their predictions. The large number
of decision-support applications of these models which do not include analysis of
uncertainty (amongst many others, Gunduz et al. 1998 and Warwick et al. 1999) is
evidence of this. It is reasonable to assume that unpublished commercial applications of
such models also under-represent the significance of uncertainty.
The popular modelling tool QUAL2E-UNCAS (Brown and Barnwell 1987), which is a
river modelling component of the US EPA’s BASINS tool, has a built-in uncertainty
analysis option. Reckhow (1994) recognises QUAL2E-UNCAS as an especially useful
development, not only because it allows formal uncertainty analysis, but the associated
documentation promotes uncertainty analysis amongst a large body of decision-makers.
QUAL2E-UNCAS relies on estimation of prediction uncertainty through specification of
feasible parameter and boundary condition ranges, and does not include a tool for
conditioning the input uncertainties on observed data. Nor does the model allow
covariance of inputs to be considered, meaning that uncertainty may be significantly over
or under-estimated (Reckhow 1994, Brown 2002).
9
Further to his commentary on QUAL2E-UNCAS, Reckhow (1994) notes that regulators
in the USA tend to favour relatively simple water quality models, as complex models are
too demanding on human resources, in addition to their high data demands. The UK
Environment Agency have developed the relatively simple steady-state SIMCAT model
to support regulation of river water quality (UK Environment Agency 2001a). SIMCAT
is based on the recognition that model prediction uncertainties stem mainly from
limitations in the calibration and pollution load data, rather than from the assumptions
implicit to the model equations. SIMCAT was arguably a major step forward in the
practice of river water quality modelling, in that parameter uncertainty can be identified
from data sampling error by optimising the model parameters against different
realisations of the data. As the model formulations used in SIMCAT are simple and easily
solved, it is practical to use the computationally intensive sampling method. At the same
time, the simplicity of the model structure makes the model less suitable for some tasks,
such as extrapolation to changed boundary conditions, or simulation of dynamic events,
when the effects of model structural error are more likely to be significant.
The decision-support role of relatively simple models coupled with uncertainty analysis is
evident from the continuing practices of both the UK and US environmental regulators.
This contrasts with the popularity of complex, resource-intensive models such as
WASP5, MIKE11 and CE-QUAL. Accepting that both modelling approaches may have a
role depending on the degree of detail sought and the resources available, there is
arguably a benefit in providing tools that include a hierarchy of models. Supplementing
this with uncertainty analysis facilities allows the limitations of both approaches to be
evaluated for specific modelling tasks.
DESERT (Ivanov et al. 1996, also see Somlyody 1997) is a tool for catchment
management optimisation which provides a framework in which the user can design his
own one-dimensional river water quality model. DESERT allows parameter conditioning,
although the effect of parameter interactions cannot be included in application of the
conditioned model. Based on dynamic programming, DESERT identifies all the sets of
model inputs which conform to a series of constraints, which can include cost constraints
for pollution control interventions, as well as in-river water quality criteria. In these
respects, DESERT has the capacity for uncertainty analysis and flexibility of model
design which will be needed for future water quality management problems, and is a
valuable precedent for future developments.
10
1.2 Introducing a framework for risk-based surface water quality
modelling
Following introduction to the driving forces behind water quality modelling, the inherent
problems in this discipline, and previously proposed directions for addressing these
problems, an outline of a modelling framework (which will be developed in Chapter 4) is
now proposed and some desirable facets of a potential modelling tool are discussed.
Beforehand, it is worth reviewing the significance of the term ‘risk’ in the context of
surface water quality modelling.
1.2.1 Risk in context
In the present context, risk may be usefully defined as “a combined measure of the degree
of detriment to society or the aquatic ecosystem caused by a defined event (or
combination of events), and the probability of that event occurring”. Traditionally, in
surface water quality management, the degree of detriment is simplified to a series of
pass-fail criteria, each criterion representing a class of water quality (e.g. UK
Environment Agency 1998). Risk can then be evaluated as probability of failure to
achieve the target class. Modelling, then, has at least two potentially valuable roles – to
extrapolate point measurements of water quality so that spatial and temporal criteria can
be used in water quality classification rather than discrete, localised measurements of
concentration; and to predict the response of risk to changing controls, to allow objective
risk management.
This brief introduction to the role of modelling in risk-based water quality management
raises a few issues. Firstly, it is important to differentiate between the frequency of failure
that will actually occur due to system variability, and the modelled probability of failure,
which includes (or should include) the influence of the uncertainty in the model and in the
estimates of future boundary conditions. That is, there is a risk that any water quality
intervention will fail to achieve its objectives due to the limitations of the modelling
employed at planning stage. Consequently, where a modelling study implies a
management option to be high-risk, this may be mainly due to the limited information and
resources available for model and boundary condition identification, and a clear
management priority would be to invest in more research. Also, there may be
considerable risk associated with ill-defined objectives – that is, a water quality
intervention may fail to be successful because at the time of planning the objectives were
11
under-researched or impossible to clearly define. For example, while it is reasonable to
suggest that there will be lengthy debate over local and regional definitions of ‘good
ecological status’ (Definition 22 in CEC 2000), the planning required to achieve such
questionable status is already underway (e.g. UK Environment Agency 2001b). Finally
on the point of associating risk, it is useful to distinguish between the risk stemming from
anthropogenic system variabilities (for example diurnal variations in effluents) which are
generally manageable, and risk stemming from ‘natural’ system variabilities (for example
those due to meteorological influences) which are less manageable. In particular, if the
risk of failure is predominantly due to unmanageable natural processes then reviewing the
targets would be a logical way forward. With the capability of exploring reasons for risk,
modelling has an essential role in, not only appraising pollution intervention options, but
identifying sensible precursors to intervention.
1.2.2 A framework outline
Figure 1.1 outlines a general framework for risk-based modelling of water quality that
will be further justified, developed, demonstrated and reviewed in the course of this
dissertation. Using such a framework it is intended that water quality managers have
access to risk-based evaluation of surface water quality, and be able to respond to and
develop this evaluation by,
• Identification of the principal factors affecting risk to water quality status.
• Evaluation of risk associated with alternative pollution control strategies,
potentially with integration of external criteria, such as social and economic costs
of water quality improvements.
• Consideration of alternative modelling criteria, in terms of identifying feasible
water quality targets, and identifying acceptable compromises between non-
commensurate criteria (e.g. between water quality status and need for water
abstractions).
• Consideration of different models for forecasting water quality response to
pollution interventions (to reduce and evaluate risk associated with model
structure uncertainty).
• Establishing priorities for collecting more data with which to improve model
identification (reducing risk associated with data uncertainty),
Using modelling in this manner is consistent with more general risk assessment
guidelines and frameworks used by environmental regulators. For example, UK
environmental regulators (DETR et al. 2000) encourage proactive risk management using
12
a tiered framework of quantitative risk assessments, whereby models, monitoring, and
management options are reviewed as the analysis moves from risk screening to the
advanced stages. This includes analysis of how the different sources of uncertainty
contribute to the final risk estimate, and review of costs and benefits. Such a tiered
approach to risk assessment has been recommended for implementing the requirements of
the Water Framework Directive (UK Environment Agency 2002). In applying this
general risk assessment framework to management of water quality and aquatic ecology,
there is clearly scope for iterative, model-based risk analyses, such as that promoted by
Figure 1.1.
1.2.3 Technical considerations
In pursuit of a practical modelling tool that provides such a capacity for risk evaluation,
the following tool features are considered essential;
1. Accessibility (ease of use), flexibility and extensibility (to cover a range of
modelling tasks).
2. Efficiency of numerical techniques (to achieve the maximum benefit from Monte
Carlo simulation).
3. Sensitivity analysis and risk evaluation capabilities.
Although the former three stipulations are common goals in the design and development
of modelling tools in general, there are important implications in the water quality
modelling context which deserve further discussion.
The need for accessibility, flexibility and extensibility
Accessibility of results is an important issue, as major management decisions usually
must be supported using visually insightful reports, hence the benefit of an adequate
interface for the graphical reporting of results. The value of advanced modelling
techniques, for example Monte Carlo simulation, should not be diminished by perceptions
that they are not transparent to decision-makers and stakeholders; effective interfaces may
go a far way to avoid or resolve this concern. Furthermore, investigation of a variety of
potential sources of risk, possibly including a large number of pollution sources and other
system characteristics, requires careful attention to the thoroughness of model input
specification. This draws attention to the value of an effective interface for model
specification and data input.
13
Prediction and furthersensitivity analysis
Model conditioning, sensitivity analysis and
model evaluation
Specification of modelstructure, grid scale and prior parameter ranges
Risk evaluation
Pollution load and regulation
scenarios
Monitoringdata
Modelling task
Externalconsiderations
Prediction and furthersensitivity analysis
Model conditioning, sensitivity analysis and
model evaluation
Specification of modelstructure, grid scale and prior parameter ranges
Risk evaluation
Pollution load and regulation
scenarios
Monitoringdata
Modelling task
Externalconsiderations
Figure 1.1 A framework for risk-based modelling of water quality
The requirement for flexibility is applicable to a number of aspects of a risk-based water
quality modelling tool. Firstly, unavoidable subjectivity in estimating model uncertainty
means that some choice of estimator should be provided, which is illustrated in studies by
Freer et al. (1996) and Franks and Beven (1997). Application of multi-objective
optimisation and sensitivity analysis (e.g. Bastidas et al. 1999) also requires flexibility in
specification of model performance criteria. Central to the modelling procedure illustrated
in Figure 1.1 is the capacity to explore different model structures, depending on the
modelling task, data and computational resources available. If the model uncertainty is to
be adequately represented by parameter uncertainty, the modeller should have the
opportunity to identify a model structure which best allows this. In particular, the
modelling grid scale (the spatial and temporal resolution of model) must be selected
according to the water quality problem. Uncertainty introduced by spatial and temporal
aggregations should be explored. Extensibility is essential so that new model structures
and water quality determinands can be incorporated, and so that the tool can be linked to
new databases and other conjunctive software. In particular, as the directives driving
water quality modelling promote integrated catchment management, and as the challenge
14
of diffuse pollution management gathers pace, the increased use of Geographical
Information Systems (GIS) as interfaces and platforms for water quality models is
inevitable, and this might be borne in mind at the development stage, whatever the
immediate modelling applications.
The need for numerical efficiency
Monte Carlo simulation provides us with the capability to retrieve a large amount of
information about the sensitivity of model results to model inputs, which is extremely
advantageous given the current limitations in the practice of water quality modelling.
Although computational costs continue to diminish, the value of a Monte Carlo
simulation will always depend on how well the continuum of possible model
inputs/outputs is represented by a finite number of realisations. This would be especially
relevant, for example, in catchment-scale distributed GIS-based modelling, due to the
large amount of computation involved as well as the large number of spatially distributed
model inputs which may be included in the analysis. There is therefore a need to either
maximise the number of realisations achievable at a given computational cost, for
example by implementing efficient numerical solvers and specifying numerical tolerances
that are consistent with the overall reliability of the analysis, or to reduce the number of
realisations required for an adequate representation by using variance reduction
techniques (Cochran 1977). For example, a variance reduction technique which has been
found useful in water quality modelling applications is Latin hypercube sampling (LHS;
MacKay et al. 1979). LHS is, in the current context, designed to thoroughly sample the
univariate distribution of each model input while leaving the sampling of interactions to
chance. While some water quality modellers (e.g. Melching and Bauwens 2001) have
successfully employed LHS to enormously improve the efficiency of sensitivity analysis,
Press et al. (1988) note “if there is an important interaction between the design
parameters, then Latin hypercube sampling gives no particular advantage (over simple
random sampling)”.
Notwithstanding the merits of efficient sampling and solution schemes, more fundamental
precursors to successful Monte Carlo analysis are, 1) appropriate limitation of model
complexity, and 2) minimisation of the number of inputs to be sampled. Again, this draws
attention to the need to match the model complexity to the specific modelling task, and
the need to provide tools that offer some flexibility in model structure choice.
15
The need for sensitivity analysis and risk evaluation capabilities
Monte Carlo-based approaches to sensitivity analysis such as those implemented by
Hornberger and Spear (1980), Beven and Binley (1992) and Kuczera and Parent (1998)
have found wide application in environmental modelling, including a limited number of
applications to surface water quality modelling, as reviewed in Chapter 2. Incorporation
of these methods into water quality modelling tools is an essential part of implementing
the framework outlined in Figure 1.1. Firstly, they allow evaluation of the suitability of a
model, in terms of reviewing the ability of the model and the associated parameter
uncertainty to explain observed data. Thereafter, uncertainty in model forecasts can be
estimated (e.g. Van Straten and Keesman 1991), avoiding the need for unqualified ‘best
estimate’ forecasts. Monte Carlo methods not only have the potential to produce summary
statistics of model sensitivities (e.g. Spear and Hornberger 1980, Wade et al. 2001), but
can be used to evaluate risk to water quality status due to individual pollution sources and
system properties, and can be extended to incorporate uncertainties in water quality
criteria (see Chapter 7). Such evaluation has clear potential for risk-based decision
making, particularly under conditions where data for identification of model and
boundary conditions are limited. It also has the potential to be extended to simulating
ecological risks, including spatial and temporal exposure as well as probability of
occurrence.
Emphasis has been put on the value of Monte Carlo simulation because it is a relatively
straightforward way of analysing how water quality objective functions respond over all
feasible combinations of model inputs. This can be supplemented by alternative,
computationally less demanding techniques of sensitivity analysis and uncertainty
propagation. Using first order sensitivity analysis, the effect on a model response of
perturbing each input variable around a specified value, while keeping the values of all
other inputs fixed, is calculated. This has the advantage of allowing for the response
components to be associated with individual inputs in a simple manner (e.g. Melching
and Bowens 2001). However, the interactions between inputs are not explored and non-
linear responses are not estimated, so there is very restricted scope for exploring response
surfaces, and effects (for example on risk) of low-probability values of model inputs are
likely to be misrepresented. Also, the result will generally be dependent on the value
around which the input is perturbed, as well as on the fixed values of all the other inputs,
which may be quite arbitrary given the problem of model equifinality.
16
1.2.4 A tool for risk-based management of water quality
As part of a European Commission Project investigating the role of computational
methods in management of surface water quality in developing countries, where
supporting data are unavoidably sparse, a modelling tool called WaterRAT (Water quality
Risk Analysis Tool) has been developed. This tool is built around the methods and
principles outlined above, and is designed to be employed in the manner illustrated by
Figure 1.1. WaterRAT allows exploration of the uncertainties arising from all sources of
prediction error – field data, model parameters, boundary and initial conditions, model
structure, scale and numerical approximations. Model parameters, boundary and initial
conditions can all be input as distributions, and can be conditioned to field data or other
designed objectives using built-in algorithms. Four simultaneous objectives can be
specified, and Pareto-optimal trade-offs can be identified. Regional sensitivity analysis
using Latin hypercube sampling is complemented by factorial sensitivity methods.
WaterRAT allows the effects of output uncertainties to be evaluated in terms of risk of
failing water quality targets, and will plot risk of failure against any one input variable,
supporting, for example, risk-based management of pollution control. Additionally, the
water quality targets can themselves be assigned uncertainty, thus incorporating risk due
to poorly defined objectives. Dynamic models are solved using an adaptive time-step
procedure, with the temporal numerical tolerances pre-specified by the user. A
description of the WaterRAT tool is provided in Chapter 4.
1.3 Background to the case studies
Two case studies are used in the development of this Thesis. They are the Hun River in
Liaoning, northeast China, and the Charles River in Massachusetts, northeast USA. The
Hun River was the focus of the European Community-funded project, Total Pollution
Load Estimation and Management (TOPLEM). As a whole, the TOPLEM project was
aimed at developing pollution management decision-support software suitable for use in
developing countries, where data and resources are especially limited. The Imperial
College role was development of suitable river and lake water quality modelling tools,
which resulted in the research constituting this dissertation. The degree to which the
research could be based on the Hun River was limited due to unforeseen difficulties in
accessing the promised data and models, notably river flow data and pollution load
models. While unfortunate, the especial limitations affecting the Hun River study are
17
extremely relevant to the Thesis, and provide points for discussion in Chapter 8. Due to
the failure of the TOPLEM project to deliver quality data, the Charles River was
introduced as a study for which suitable information is available (with gratitude to Steve
Chapra of Tufts University, and Camp Dresser and McKee Inc. of Cambridge,
Massachusetts).
1.3.1 The Hun River characteristics
The Hun River is one of three major rivers in the Liao River Basin in Liaoning Province,
illustrated in Figure 1.2. Liaoning is a centre for heavy industry. It is rich in deposits of
oil, coal and iron, all of which are mined heavily. Other important industries are power
generation, production of steel, oil refining and petrochemical production. The largest
city in the Liao Basin, with a population of 5 million, is Shenyang. 70km up the Hun
River from Shenyang is another important industrial centre called Fushun, with a
population of approximately 3 million. As well as its industrial importance, Liaoning
produces substantial quantities of sorghum, soya and rice, among other crops. The Hun
River originates in the northeast of Liaoning, and for the first 110 km the river descends
southwest through highlands, before entering the large Dahuofang reservoir. Following
the reservoir the Hun enters flat lowland territory, passing through Fushun and Shenyang.
After Shenyang, the river enters its lower catchment which is intensively used for
agriculture and oil drilling. 250km downstream of the reservoir, the Hun joins the larger
Taizi River.
Liaoning’s climate is monsoonal, with a hot wet season which generally includes July,
August and September, and a freezing, dry winter from November until March. The
remaining months are termed ‘mid-season’ with variable climate. Figure 1.3 shows a
typical time-series of ambient air temperature and precipitation at Shenyang (for the year
1999-2000). The flow regime of the lower Hun River is dominated by a number of
artificial controls, starting with the dam of the large Dahuofang reservoir. Due to the
threat of drought, little or no compensation flow is released from the reservoir except
during especially wet periods. While the 90 percentile low flow upstream of the reservoir
is over 10m3s-1, at the lower end of the river, it is 2m3s-1 (Montgomery Watson 2001a).
The locations of the significant point pollution loads (and where quality and flow data
were collected from September 1999 until September 2000 as part of the TOPLEM
project) are illustrated in Figure 1.4. The uses of the lower Hun River (i.e. downstream of
the Dahuofang reservoir) are severely restricted due to high pollution levels. Insight into
pollution management problems in the Hun catchment can be found in Ma (2001) and the
references therein.
18
90°E
North KoreaBeijing
Hebei Province
Lioaning Province
Inner Mongolia
Fushun
Shenyang
Taizi River
Hun River
Liao River
Liaotung Gulf
Yellow Sea
Jilin ProvinceArea
of detail
45°N
30°N120°E
PR China
N100 km
90°E
North KoreaBeijing
Hebei Province
Lioaning Province
Inner Mongolia
Fushun
Shenyang
Taizi River
Hun River
Liao River
Liaotung Gulf
Yellow Sea
Jilin ProvinceArea
of detail
45°N
30°N120°E
PR China
N100 km
Figure 1.2 Location Plan for Hun River
Air
tem
pera
ture
(o C)
0
10
20
30
40
50
60
70
80
90
Dai
ly p
reci
pita
tion
(mm
)
20/0
6/00
20/0
7/99
20/0
8/99
20/0
9/99
20/1
0/99
20/1
1/99
20/1
2/00
20/0
1/00
20/0
2/00
20/0
3/00
20/0
4/00
20/0
5/00
-25
-15
-10
5
15
25
35
-20
20
30
10
-5
0
Date
Daily precipitation
Ambient air temperature
Air
tem
pera
ture
(o C)
0
10
20
30
40
50
60
70
80
90
Dai
ly p
reci
pita
tion
(mm
)
20/0
6/00
20/0
7/99
20/0
8/99
20/0
9/99
20/1
0/99
20/1
1/99
20/1
2/00
20/0
1/00
20/0
2/00
20/0
3/00
20/0
4/00
20/0
5/00
-25
-15
-10
5
15
25
35
-20
20
30
10
-5
0
Date
Daily precipitation
Ambient air temperature
Figure 1.3 Air temperature and precipitation during 1999-2000 at Shenyang
Pu River (RK 160)
Xi River (RK 125)
Bataipo River (RK 70)
Shenyang 5 large mixed-source sewer outfalls (RK 58-66)
Fushunsewer(RK 19)
Downstream boundary of simulation (RK 185)
Upstreamboundary ofsimulation(RK 44)
DahuofangReservoir
Monitoring Section
YinguanRiver(RK 52)
RK river kilometer from reservoir
N
Pu River (RK 160)
Xi River (RK 125)
Bataipo River (RK 70)
Shenyang 5 large mixed-source sewer outfalls (RK 58-66)
Fushunsewer(RK 19)
Downstream boundary of simulation (RK 185)
Upstreamboundary ofsimulation(RK 44)
DahuofangReservoir
Monitoring Section
YinguanRiver(RK 52)
RK river kilometer from reservoir
N
Figure 1.4 The main sources of pollution to the Hun River While the TOPLEM project considered the full extent of the Hun River and its catchment, the work presented in Chapters 5 and 6 focus on the river reaches at Shenyang and immediately upstream of the Dahuofang reservoir.
19
1.3.2 The Charles River characteristics
The headwater of the Charles River is located in the hills of eastern Massachusetts in the
USA. The river flows approximately 130km through the state, through numerous towns
and over a succession of dams, before discharging into Boston harbour. Water quality
problems associated with the Charles River in previous decades were industrial pollution
and combined sewer overflows which led, among other unwelcome effects, to nutrient
enrichment and eutrophication. Storm-water interceptions and other interventions in the
1990s have greatly improved the overall ecology and amenity value of the river, although
they have failed to control eutrophication satisfactorarily. Further measures are currently
being implemented by installing phosphorus stripping facilities at a number of wastewater
treatment plants (CRWA 2000). The study in Chapter 7 looks at the 40km length of the
Upper Charles River, between the Populatic Pond in Medway County and the Cochrane
Dam in Dover County. Figure 1.5 shows the path of the Charles, and the location of the
main point sources of pollution in this stretch.
Point sources
Charles River
Monitored sections
Model boundaries
Basin boundary
1
2
3
4
5
6
78
9
A B
C
D
E
F
G
1. CRPCD WWTP2. Mill River3. Stop River4. Medfield WWTP5. Bogastow Brook6. Sewall Brook7. Indian Brook8. Waban Brook9. Trout Brook
A. HeadwaterB. DS of Mill RiverC. Mill River + 4.5kmD. US of MedfieldE. US of Sewall BrookF. Sewall Brook + 4kmG. South Natick DamH. US of Trout BrookI. Cochrane Dam
HI
BostonHarbour
Scale:10km N
Point sources
Charles River
Monitored sections
Model boundaries
Basin boundary
1
2
3
4
5
6
78
9
A B
C
D
E
F
G
1. CRPCD WWTP2. Mill River3. Stop River4. Medfield WWTP5. Bogastow Brook6. Sewall Brook7. Indian Brook8. Waban Brook9. Trout Brook
A. HeadwaterB. DS of Mill RiverC. Mill River + 4.5kmD. US of MedfieldE. US of Sewall BrookF. Sewall Brook + 4kmG. South Natick DamH. US of Trout BrookI. Cochrane Dam
HI
BostonHarbour
Scale:10km N
Figure 1.5 Charles River model boundaries, point sources and monitoring locations
20
1.4 Explanation of the structure of the remainder of this
dissertation
The general objective of this dissertation is to progress the science and practice of
simulation modelling to reflect the needs and resource constraints of surface water quality
managers. More specifically, the dissertation aims to develop methodologies and tools
that will assist in identification of river water quality management priorities, through
evaluation of the risk that various uncertainties pose to the decision-making procedure.
The wide scope of this aim became apparent early in the course of the work and, rather
than pursue a very specialised tract, the following chapters reflect different but inter-
related aspects of the challenge. The contribution offered by the dissertation is, therefore,
not only individual theses contained within the chapters, but the recognition of the need
for integrated examination of the issues, and the delivery of a framework to do so.
Chapter 1 has justified the concept of the Thesis and introduced the principal methods of
uncertainty and sensitivity analysis that will be employed. In Chapter 2, the importance of
parameter uncertainty estimation and propagation is recognised, and approaches
previously used in environmental simulation modelling are reviewed in some detail, and
demonstrated using a simple water quality model of a hypothetical system. Some relevant
comparisons and contrasts between the different methods are drawn, and their role in the
Thesis is discussed. Chapter 3 is a review of the state-of-the-art of river water quality
modelling, focusing on review of the types of formulation used in the various models that
are introduced in Chapter 4. Chapter 4 expands upon the introduction to the WaterRAT
modelling tool given earlier in Chapter 1, describing the components and capabilities of
the tool. Although the shortest, Chapter 4 represents the main development effort, and
this is reflected in the cited, more comprehensive WaterRAT descriptions. Chapter 5
reflects some of the first difficulties that were encountered in modelling the Hun River,
i.e. achieving numerical stability when simulating high order systems, while reconciling
the numerical precision with the overall modelling uncertainty and the computational
demands of Monte Carlo simulation. The issues that Chapter 5 explores are
fundamentally important, and seem to be under-reported in previous literature. Chapter 6
again tackles an important aspect of uncertainty analysis which was raised early in the
TOPLEM project – the prior identification of data needs and the dependent issues of the
nature of input data, expectations of output data, perceptions of structural adequacy,
limitations of the calibration algorithms and cost constraints. Chapter 7 uses the Charles
21
River study to test the strengths and limitations of the WaterRAT tool and the methods it
employs, in using Monte Carlo methods to identify the various factors affecting decision-
making risk, and provides a basis for risk-based water quality management. Thus, the
case study chapters look in turn at 1) the numerical challenges, 2) the data collection and
experimental design challenges, and 3) the decision-support application challenges,
pertaining to the aim and objective of the dissertation. Chapter 8 critically reviews the
success of the case study chapters in achieving their aims and exposes the gaps that they
have left for further research. Chapter 8 also includes discussion of the partial failure of
the TOPLEM project, and cursory consideration of approaches that can be used when
data are especially sparse and unreliable. Finally, current and future directions for the
field of uncertainty analysis in water quality modelling are reviewed in light of the
dissertation.
22
2. Estimation and propagation of parametric uncertainty in
environmental models
The use of statistically-based likelihood functions as a basis for representing model
parameter uncertainty is introduced, and the difficulties of their application when
unknown model structure error and/or data bias may be significant is discussed. More
subjective approaches to estimating model uncertainty (GLUE and possibility theory),
which attempt to allow representation of the effects of model and data biases in the
parameter uncertainty, are described. An alternative to simple random sampling of
parameter distributions is described (the Metropolis algorithm), and the significance of
uncertainty derived using a multi-objective approach is compared with the traditional
method of lumping all data into one objective function. The chapter goes on to
demonstrate the estimation of uncertainty using a data error sampling approach, GLUE
and Metropolis using the Streeter-Phelps model of stream dissolved oxygen. It is also
shown that the three calibration methods converge the parameter distributions to
practically the same end result if consistent objective functions are employed, although
the significance of the objective function (whether it is a statistically-based likelihood
function or a more subjective GLUE ‘likelihood’) is evident. Methods of propagation of
parameter uncertainty are also reviewed. Rosenblueth’s two-point method, first order
variance propagation, Monte Carlo sampling and possibility theory are applied to the
Streeter-Phelps example. The first three methods are shown to be capable of producing
practically equivalent confidence limits on the model result, but again the significance of
the objective function is evident. The relative merits of the methods for more complex
modelling problems are discussed.
23
2.1 Introduction
2.1.1 Background and scope of chapter
Producing a reliable set of confidence limits on a model result is not difficult given ideal
circumstances. For example, to fit a linear model to observations which are normally and
independently distributed with constant variance requires standard regression techniques,
and derived confidence limits are theoretically sound (see Berthouex and Brown 1994).
However, the natural environment is very much non-linear and this biases parameter
estimates (e.g. Tellinghuisen 2000). Also, data generally carry sampling and
measurement errors, and are often unreliable, and, no matter how well behaved the data
are, if the structure of the model is fundamentally wrong then standard regression
techniques are flawed. Clearly then, extrapolation of the model into the future also
complicates the analysis, as the reliability of the model under new conditions is always in
question. The problem of model equifinality means that many different proposed models
may appear equally adequate when compared to the data but may give significantly
different results when extrapolated to new conditions.
This chapter is a review of methods of uncertainty analysis in environmental modelling.
This subject area has previously been reviewed elsewhere (Beck 1983, Beck 1987,
Melching 1995, Tung 1996, McIntyre et al. 2001, Adams and Reckhow 2001, Kavetski et
al. 2002) and the reader is directed to this literature for additional background and
discussion. This chapter complements these previous works by taking a demonstrative
approach to the review, aiming to give insightful comparisons between the methods using
simple examples and theory. As such, it is intended to be a practical guide to the available
methods, and to enable and encourage the modeller to implement them with forethought,
and to interpret the results properly. Notably, this review excludes methods of recursive
parameter estimation (see Beck 1987). The utility of those methods is evident when the
modelling objectives are relatively well defined by observations of the environmental
system (e.g. Whitehead and Hornberger 1984). Without diminishing the importance of
recursive parameter estimation, this chapter (and dissertation) is principally concerned
with methods most used for analysis of systems for which supporting observations are
relatively sparse.
24
2.1.2 The sources of uncertainty and their representation in the model
A definition of uncertainty analysis is ‘the means of calculating and representing the
certainty with which the model results represent reality’. The difference between a
deterministic model result and reality will arise from,
model parameter error,
model structure error (where the model structure is the set of numerical
equations which define the uncalibrated model),
numerical errors - truncation errors, rounding errors and typographical mistakes
in the numerical implementation,
boundary condition uncertainties.
As reality can only be approximated by field data, data error analysis is a
fundamental part of the uncertainty analysis. Data errors arise from,
sampling errors (i.e. the data not representing the required spatial and temporal
averages),
measurement errors (e.g. due to methods of handling and laboratory analysis),
human reliability.
Realising that an error-free model would equate to the error-free observations, the
relationship between the actual model result M and the actual observations O can be
summarised by,
7654321 εεεεεεε −−−=−−−− OM (2.1)
where ε1 to ε4 represent the model error arising from the four sources in the order listed
above, and ε5 to ε7 represent the data error arising from the sources listed above.
Representing the overall error on either side of Equation 2.1 is not generally a simple task
of adding the error variances together, as might be implied by the equation. This is
because the errors may be unknown, and/or not of a random nature (see below), and/or
the model output may be interdependent on the various sources of error in a manner that
precludes their simple addition.
It is the goal of the modeller to achieve, to within an arbitrary tolerance, an error-free
model by removal of ε1 to ε4. However, the modeller is generally neither in control of
model structure errors ε2, nor numerical errors ε3, nor boundary condition errors ε4.
Commonly, only the values of the model parameters are under the direct control of the
modeller. The aim would then become one of compensating as far as possible for ε2 to ε4
25
by identification of optimum effective parameter values. Central to this Thesis is the
argument that there is always some ambiguity in the optimum effective parameter values
caused by the unknown natures of, and inseparability of, ε2 to ε7, and that this ambiguity
can be represented by parametric uncertainty. As such, the model parameters are used as
error-handling variables, and are identified according to their ability to mathematically
explain ε2 to ε7. In most environmental modelling problems, significant bias in one or
more of these errors will inevitably lead to biased parameter estimates. While the ideal
solution would be to eliminate bias, for example by compensatory adjustments to data or
by model structure refinement, such measures are often not practical and never
comprehensive. In recognition of this, the potential importance of biased model
calibration will be illustrated in this chapter, and significant attention is given to methods
of uncertainty analysis which aim to deliver some robustness to bias.
The difficult task of identifying parameter uncertainty is generally approached using
methods of calibration which derive, from the pre-calibration (a priori) parameter
distributions, calibrated (a posteriori) distributions. In hydrological modelling, due to
lack of prior knowledge, the a priori distributions are often taken as uniform and
independent (e.g. Hornberger and Spear 1980). On the other hand, the a posteriori
distributions, constrained by the data, may be multi-modal and non-linearly inter-
dependent (Sorooshian and Gupta 1995). Inter-dependency arises when the model result
is simultaneously significantly affected by two or more parameters, such that the
distribution of each parameter must be regarded as conditional on the value of all inter-
dependent parameters. Therefore, it is necessary to refer to the joint parameter
distribution which is defined by a continuous function of all the parameters, and to
sampled parameter sets rather than individual parameter values.
2.2 Approaches to uncertainty-based model calibration
Calibration is the process of tuning the model by optimisation of the set of model
parameters. In traditional deterministic modelling, a single optimum parameter set is
found such that model results fit the data as closely as possible. The closeness of fit is
quantified by one or more objective functions (OFs), and a variety of automated
optimisation procedures are available (see Sorooshian and Gupta 1995). In an
uncertainty-based calibration, the modeller is interested in the response of the OF over the
entire a priori parameter space, i.e. the OF response surface (see Berthouex and Brown
26
1994). Definition and sampling of the OF, and interpretation and analysis of the
subsequent response surface are the means of deriving calibrated parameter distributions.
This discussion will critically review some different approaches to these tasks.
2.2.1 Definitions
Before beginning this review, some terminology must be defined. As stated above, an
objective function (OF) is a general term for a quantative measure of how closely a model
result fits to corresponding observed data, which may or may not have a probabilistic
basis. The term likelihood is used in the statistical sense, i.e. the probability of a set of
data given a model, while a likelihood function is a function measuring this probability,
i.e. a particular type of objective function. For use in the context of uncertainty estimation
using GLUE (see section 2.2.5), a GLUE likelihood measure is the measure of probability
of a sampled model (i.e. which may be either a subjective perception of probability or
based on a likelihood function).
2.2.2 Objective functions and likelihood functions
The method of maximum likelihood (see Ang and Tang 1975) is the traditional
mathematical route to model parameter calibration, and it is a necessary starting point for
this discussion. If the OF is defined as a likelihood function of the model then for each
trial model,
[ ] resi
ii NiPPP ,,4,3;)()()(OF 121121 KK =∩∩∩= ∏ −εεεεεεε (2.2)
where εi is the ith of Nres model residuals (i.e. the difference between the ith of Nres
available data points and the corresponding the model result), P(εi) is the probability
density of εi, and P(ε2ε1) signifies the probability of ε2 assuming ε1 has already
happened. If the Nres residuals are assumed to be independent and normally distributed
with zero mean and constant variance σ2, and there are Npar degrees of freedom (i.e.
parameters to be calibrated), then Equation 2.2 becomes,
( )( )
++−=
−= ∏
=
222
2122/21
2
2
2...
21exp
2
12
exp2
1OFresres
res
NN
N
i
i εεεσπσσ
ε
πσ (2.3)
If it can be assumed that ( ) ( )parresNres NN −++ 222
21 ...εεε is equal to 2σ then,
27
( )))(5.0exp(
2
1OF 22 parresN NNres
−−=πσ
(2.4)
Nres and Npar being constant during a model calibration, Equation 2.4 is reduced to,
( ) 2/2OFresN
Kσ
= (2.5)
where K is a constant. Therefore, assuming that the sum of the squared residuals divided
by an appropriate constant is equal to σ 2, the least sum of squared residuals maximises
the likelihood (Box and Jenkins 1970). Usually, one or more of the assumptions used in
the derivation of Equation 2.5 is not valid. For example, the assumption that the error
variance σ 2 can be accurately estimated using the sum of the squared residuals is not
tenable when the number of residuals is small, and Equations 2.3 to 2.5 would give a very
approximate likelihood. If more than one modelled variable is being included in the OF
then σ 2 cannot generally be taken as constant for all variables, and Equation 2.2 (from
Sorooshian and Dracup 1980) becomes,
( )∏=
=var
12/2OF
N
rN
r res
Kσ
(2.6)
where Nvar is the number of included variables each modelled and measured at Nres
locations (in time and/or space). For finding the maximum likelihood, this is equivalent to
minimisation of the sum of weighted squared residuals, assuming the responses are
independent. Similarly, if the variance changes in time and/or space with the magnitude
of the response then an appropriate weighting scheme may be used (Sorooshian and
Dracup 1980). For autocorrelated residuals, Romanowicz at al. (1994) describe a suitable
likelihood function.
The OFs in Equations 2.2 to 2.6 give the probability of a data set sample (say data
samplek) occurring given the model result. If this model result is defined by a set of
parameters (αi) sampled from the a priori joint parameter distribution, applied to a chosen
model structure (say model structurej),
( )[ ] ijikP OFstructure model,sample data =α (2.7)
28
Let us assume for now that model structure j correctly and uniquely describes the
modelled system, so that it drops out of the equation. Equation 2.7 may be manipulated
using Bayes theorem (see Ang and Tang 1975), to give the probability of the parameter
set given the data sample,
[ ] ( )i
i
kki P
PP OF
)(sample data
sample data =×α
α (2.8)
If only one data sample is considered, then P(data samplek) = 1. Furthermore, if it is
considered that all of Nsam sampled parameter sets have equal a priori probability so that
P(αi ) is equal to 1/ Nsam;
[ ]sam
iki N
POF
sample data =α (2.9)
The standardised objective function (so that all the discrete OFs total unity) is then an
estimate of probability mass from the posterior joint parameter distribution,
[ ]∑
=
=
samNll
ikiP
,1OF
OFsample dataα (2.10)
In practice, the evaluated probability of the model is conditional on the data sample
employed for calibration. This is generally important in river quality modelling because
subsequent data sets from the field, apparently drawn under the same conditions, often
result in quite different parameter distributions. Results using more than one data set can
be integrated into the joint parameter distribution. If it is assumed that alternative data
sets are sampled independently,
( ) [ ] [ ]( )∑=
×=datNk
kkii PPP,1
sample datasample dataαα (2.11)
where Ndat is the number of sampled data sets, ( )iP α here is the probability of iα given
these alternative data sets, and [ ]kP sample data would generally be considered constant
for all k. If it is considered that the error associated with the variability between data sets
(rather than the error variance within them) is the main source of uncertainty, then only
29
the maximum likelihood model for each data set realisation may be considered, with 2.12
expressed as,
( ) [ ]kk PP sample data' =α (2.12)
where 'kα is the maximum likelihood parameter set for the kth data set realisation.
2.2.3 The significance of model structure errors and data bias
In process-based river quality modelling, some structural error is inevitable because the
complexity of the aquatic environment (its physics, chemistry and biochemistry) has
always surpassed our ability to observe, understand and numerically represent it. In
particular, the behaviour of water quality systems is liable to shift when boundary
conditions are substantially altered (van Straten 1998), and so model structures that
appear to perform well during calibration may be structurally flawed when considering
intervention options. To what extent our model structures need to be correct is introduced
in Chapter 1, and a nominal example of the significance of structural error is given later
in Chapter 6. In this and the next two sections the implications of structural error for
objective function design and uncertainty analysis philosophy is discussed.
Equations 2.7 to 2.12 assume the correct model structure, and therefore the derived
parameter response surface becomes less relevant as the structural error becomes larger.
A particular danger that arises from structural error is that the parameter estimates
become biased and their uncertainty is underestimated, potentially causing misleading
predictions. Therefore, the structural error should somehow be confronted or integrated
into the OF response surface.
Confronting the error generally would require some inference about the nature of the
error from statistical or visual analysis of the model residuals. Confronting the error could
involve re-conceptualising the modelled system (perhaps calling upon new theoretical
knowledge), or by making empirical adjustments to the model output (although this
adjustment would be specific to conditions under which the model is assessed and
therefore potentially of less use for predictive purposes). Using recursive parameter
estimation techniques is another possible route to adjusting the model empirically, or
making inferences about faults in the model structure. An additional problem with trying
to confront the error by improving the model structure is that there are generally a large
30
number of “improvements” that would result in a better model fit, and so a “correct”
model (i.e. a single model that best describes the system) would still not be found.
Another reason why modifying the model structure may be problematic is the tendency of
water quality data to be significantly biased due to unknown sampling and measurement
errors. This means that it is often impossible to distinguish between residuals caused by
data errors and those caused by model inadequacy. In any case, as stated at the outset of
this chapter, we are interested in the more common case that data are not of good enough
quality to embark upon model structure modifications that can generally be inferred from
residuals. Therefore, we turn to methods of integrating the structural error into the
response surface.
The first method to consider is calibrating a response surface for each of a number of
alternative proposed model structures, so that the response surface of each would
integrate to give the a posteriori probability of that structure, and all response surfaces
together would integrate to 1. Model application would then consist of applying all
structures with non-zero probability. This is hardly ideal, as there is no way of knowing
that the proposed models are a representative sample from the population of plausible
models, and it would not be easy to assign prior probabilities to them. However, this is a
tractable and transparent method of averaging over competing, justifiable models.
Alternative methods aim to allow one prescribed model structure to be used, and the
additional uncertainty caused by model structure error to be represented notionally by
increased parameter uncertainty. Referring back to the discussion in section 2.1.2, this is
often the only viable approach in practice because modellers rarely have time to consider
changing the model code or using different model structures, even if they do have the
necessary expertise and code access. Two such approaches are reviewed next.
2.2.4 Possibility theory and the HSY method
One approach to improving robustness of the modelling exercise to sparse and possibly
bias data is to use possibility theory (Zadeh 1978, also see Wierman 1996). A possibility
distribution describes the perceived possibility of an event where the maximum
possibility is 1 and the minimum is 0. In possibility theory, the rules of union and
intersection differ from those in probability theory. For example, for independent model
residuals ε1 and ε2,
31
[ ])(yPossibilit,)(yPossibilitMinimum)(yPossibilit 2121 εεεε =∩ (2.13a)
[ ])(yPossibilit,)(yPossibilitMaximum)(yPossibilit 2121 εεεε =∪ (2.13b)
Applying possibility theory to model calibration requires a subjective measure of the
possibility of the outcome of each candidate model. Using Equation 2.13(a), for example,
the possibility of any model result is the model residual (out of all Nres model residuals)
perceived to be the least likely. Although the significance of the remaining Nres-1
residuals (apart from not being the largest) would be lost, the robustness to data bias
would be increased by avoidance of the multiplicative likelihood function.
Another particular appeal of applying possibility theory to model calibration is that it
provides a convenient basis for calibrating the model using subjectively defined support
criteria. While such reasoning can be based on interpretation of data it may also be
knowledge-based. That is, the possibility of any candidate model can be judged on the
basis of non-numeric (even non-documented) knowledge about the environmental system
rather than by “hard” data. This is important in water quality studies where there is often
more useful information in qualitative observations of water quality (e.g. observations of
algal blooms, fish kills and discolouration) than in a sparse set of spot samples. However,
model results will reflect the modeller’s subjective interpretation of the evidence and its
translation into possibility distributions (as well as any judgements used in formulating
the model itself).
Hornberger and Spear (1980) suggested a groundbreaking approach to calibration of
environmental models which has distinct parallels with possibility theory. In their
method, an a priori parameter set, applied to a given model structure, is considered to be
a possible model of the system if the corresponding model result lies wholly within a set
of characteristic system behaviour. The characteristic behaviour is defined by subjective
reasoning which may include analysis of available data. The result of this approach to
calibration is an a posteriori sample of equally possible parameter sets and a
complementary sample of impossible parameter sets. Van Straten and Keesman (1991)
demonstrate how the a posteriori sample of possible parameters can be propagated to a
range of possible results. Statistical comparison of the contents of these parameter sets
can robustly quantify model sensitivity to individual parameters (e.g. Spear and
Hornberger 1980, Chen and Wheater 1999), and so the method is often referred to as
Regional (or Global) Sensitivity Analysis. After Beck (1987), the method is referred to in
32
this dissertation as the Hornberger-Spear-Young (HSY) algorithm and a Monte Carlo-
based algorithm for implementation of this method is illustrated in Figure 2.1.
Yes No
Yes
No
Form an a-posteriori parameter pdffrom all possible parameter samples
Start
Randomly sample parameters froma-priori distributions and run model
Ready to propagate uncertainty
Define upper and lower limits which define the boundaries of the set ‘characteristic system behaviour’
possibility of that parameter
sample = 1
possibility of that parameter
sample = 0
Is the result wholly within the characteristic set ?
Has the a posterioripdf converged?
Yes No
Yes
No
Form an a-posteriori parameter pdffrom all possible parameter samples
Start
Randomly sample parameters froma-priori distributions and run model
Ready to propagate uncertainty
Define upper and lower limits which define the boundaries of the set ‘characteristic system behaviour’
possibility of that parameter
sample = 1
possibility of that parameter
sample = 0
Is the result wholly within the characteristic set ?
Is the result wholly within the characteristic set ?
Has the a posterioripdf converged?
Has the a posterioripdf converged?
Figure 2.1 HSY calibration procedure
2.2.5 Generalised Likelihood Uncertainty Estimation
Beven and Binley (1992) developed the HSY method into their Generalised Likelihood
Uncertainty Estimation (GLUE), so that every possible model was weighted with a
probability, called a likelihood in GLUE terminology. The GLUE likelihood measures are
interpreted as estimates of probability mass from the posterior joint parameter distribution
for that model, and predictions from alternative model structures with their own joint
parameter distributions can be combined. The GLUE methodology allows any (re-scaled)
objective function to be treated as the relative probability of the model if the modeller
considers this to meet the ends of the uncertainty analysis. Therefore, the modeller is able
to define how parameter uncertainty is measured and how it and model output uncertainty
should be interpreted. For example, some applications of GLUE use GLUE likelihoods
which are re-scaled values of the Nash-Sutcliffe efficiency (Nash and Sutcliffe 1970; see
Beven et al. 2001), with an arbitrary threshold defining what models are considered
behavioural. In that case, the modeller is making the statement that probability of the
33
model is equal to belief in the model, which is proportional to the measured relative
success of the model, without relating probability to the statistical properties of the
residuals. Therefore, it is emphasised that the estimated uncertainty depends largely on
the user’s design of the objective function, and the likelihood measure should not be
interpreted as a likelihood function, as used in Equations 2.2 to 2.6, unless it is
specifically designed as such (e.g. Romanowicz et al. 1994).
The obvious disadvantage of converting an objective function into a model probability
without relating it to the statistical properties of the residuals is that the model uncertainty
has no statistical significance, and model results will be arbitrary to some degree.
However, when model structural errors and data biases are unknown, and unknowable
given resource constraints, then the statistical assumptions underlying likelihood
functions will always be questionable, and the explicitly subjective judgement involved in
GLUE becomes more attractive.
2.2.6 Model output versus data uncertainty
If a likelihood function has been used, by definition the derived parameter uncertainty is
the uncertainty in the maximum likelihood parameter estimates. Hence model output
uncertainty will represent the uncertainty in the maximum likelihood result, rather than
the variance of the data. In practice, particularly for regulation purposes, the modeller
may want to predict the uncertainty in future measurements of water quality, and so he
would have to add the predicted data error variance onto the predicted model output
variance.
The GLUE framework allows the modeller to define the objective function (and the
behavioural threshold), so that the model is forced to encompass a satisfactory number of
the model residuals, without separately having to add the data error variance. For
example, the modeller could prescribe a value of Nres (Equations 2.3 to 2.6) which is less
than the number of data points, so increasing the parameter variance (e.g. Franks and
Beven 1997) to a visually satisfactory extent. This approach would have no statistical
basis, and arguably would lead to a cosmetic representation of model uncertainty, and can
only be justified when the objective function reflects the modeller’s belief in the sampled
models.
2.2.7 Multiple objective analysis
Using multiple OFs to measure model performance can add to the understanding of
uncertainty (by looking at what OFs are non-commensurate), and can provide an
34
alternative definition of uncertainty (by quantifying the disparity between equally
relevant modelling objectives). A well established method of multi-objective analysis,
which is emerging in hydrological modelling (e.g. Yapo et al. 1998), is to identify Pareto
fronts (Goldberg 1989). The Pareto front is the set of OF values (and corresponding
parameter values) whereby no further improvement can be made to any of the OFs
without unnecessary detriment to one or more of the others.
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10
OF2OF1
parameter x
OF
parameter x
0
0.10
0.20
0.25
0 2 4 6 8 10
para
met
er c
ombi
ned
likel
ihoo
d=
OF 1
×O
F 2
0.05
0.15
(b) (c)
Pareto front
OF2
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3OF1
Pareto front
(a)
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10
OF2OF1
parameter x
OF
parameter x
0
0.10
0.20
0.25
0 2 4 6 8 10
para
met
er c
ombi
ned
likel
ihoo
d=
OF 1
×O
F 2
0.05
0.15
(b) (c)
Pareto front
OF2
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3OF1
Pareto front
(a)
OF2
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3OF1
Pareto front
(a)
Figure 2.2 Demonstration of the significance of the Pareto set The significance of the Pareto front is demonstrated in Figure 2.2 based on a trivial
example. Two alternative performance criteria (measured by OF1 and OF2) exist for a
one-parameter (x) model – let these OFs be likelihood functions, and give the parameter
probability distributions shown in Figure 2.2(b). Figure 2.2(a) plots the relationship
between OF1 and OF2 and highlights the Pareto front. For this simple example, this
translates to all values of x between the two distribution peaks on Figure 2.2(b). All these
values of x are equally viable compromises between OF1 and OF2 and might therefore be
interpreted as a uniform distribution, shown on Figure 2.2(c) along with the distribution
obtained by multiplying OF1 and OF2 together, i.e. joint probability assuming
independence. Clearly, the difference between the two distributions in Figure 2.2(c)
35
increases as the peaks of OF1 and OF2 move closer together and/or the distribution
variances increase. If the objective functions have identical optima, the Pareto solution
has no uncertainty at all. This is perfect for applications where the performance criteria
are individually well defined and uncertainty arises only from conflicts of objective. Used
incorrectly in numerical modelling, the Pareto solution may imply certain predictions
irrespective of the magnitude of data and other errors. Also the Pareto front, by definition,
gives equal weight to each objective function, irrespective of the relative importance of
the objective function and of the quality and quantity of the contributing data. Additional
insight into Pareto optimisation is given in the less trivial example in McIntyre et al.
(2001).
2.3 Sampling and global optimisation techniques 2.3.1 Monte Carlo simulation
Almost all of the work in this dissertation is based on Monte Carlo simulation, where a
random sample of a set of input variables is taken from known (or assumed) joint
distribution, and the corresponding sample of the model output is deterministically
simulated. Following a large number of samples of inputs, the joint distribution of model
outputs can be approximated. In the context of model calibration, the posterior
distribution of model parameters can also be approximated, using the objective functions
described in preceding sections. This use of Monte Carlo sampling of the prior parameter
space is a fundamental step in the GLUE and HSY calibration procedures.
One appealing feature of GLUE and HSY is that the potentially complex nature of the
response surface (including multiple local optima and non-linear dependencies between
parameters) is implicitly recognised by the large number of parameter samples and
associated probabilities, and this may be kept intact when propagating uncertainty to
model predictions (see section 2.7), rather than summarising the response surface as a
covariance matrix. However, the number of parameter samples is fundamental to the
adequacy of the approximation (Cochran 1977, Kuczera and Parent 1998). The required
number of samples to achieve a certain quality of approximation can be mitigated by
numerous variance reduction techniques (Cochran 1977), for example Latin hypercube
sampling and other stratified sampling methods (e.g. MacKay et al. 1979).
Monte Carlo simulation can be used, as well as for simulating the effect of model input
variable uncertainty, for simulating the effect of different realisations of data sampled
36
from a known (or assumed) distribution. This will be demonstrated later, where we use
this method to estimate parameter uncertainty arising from data sampling error, using the
algorithm summarised by the flow chart in Figure 2.3. The limitations of this approach
will also be discussed later.
No
Using these properties, generate an alternative realisation of the available field data
Start
Calculate the distributionalproperties of the residuals
Ready to propagate uncertainty
Yes
Find maximum likelihood model result for the available field data
Optimise the model parameters with respect to the generated data, and find maximum likelihood parameters
Form an a-posteriori parameter pdf from all realisations of the maximum likelihood parameters
Has the aposteriori pdf converged?
No
Using these properties, generate an alternative realisation of the available field data
Start
Calculate the distributionalproperties of the residuals
Ready to propagate uncertainty
Yes
Find maximum likelihood model result for the available field data
Optimise the model parameters with respect to the generated data, and find maximum likelihood parameters
Form an a-posteriori parameter pdf from all realisations of the maximum likelihood parameters
Has the aposteriori pdf converged?
Figure 2.3 Estimating parameter uncertainty by Monte Carlo sampling from a distribution of data errors 2.3.2 Metropolis algorithm
Using Monte Carlo simulation of the parameters, a large number of sampled objective
function values can be used to approximate the continuous response surface. However,
many thousands of parameter samples may be required for an adequate approximation to
be made (e.g. Kuczera and Parent 1998). To improve the efficiency of the calibration,
attempts have been made to adapt the a priori distribution to an a posteriori form using
Monte Carlo Markov Chain algorithms (see Brooks 1998).
Here, a Monte Carlo Markov Chain model proposed by Metropolis at al. (1953) is
described. The algorithm uses a Markov Chain process (see Rutenbar 1989, Brooks 1998)
which, in essence, assumes that the current state of a system dictates the probability of
moving to any proposed new state. The Metropolis algorithm was originally developed to
simulate the stochastic behaviour of a system of particles at thermal equilibrium. Applied
to model calibration, it adapts the population of parameters until the OF (in this case to be
minimised) is sufficiently described by the distribution,
37
( )1i /OFexp1)( Mi KK
P −=α (2.14)
where K is a standardisation constant such that the total of all P(αi) is unity, KM1 is a case-
dependent constant and αi is the ith parameter set in the derived population. While the
distribution of the accepted OFs converges to the Gaussian form of Equation 2.14, the
distribution of the accepted parameter sets depends upon the relationship between the
model parameters and the OF. The algorithm starts from an arbitrary location in the a
priori parameter space. From then on, the probability of any sampled parameter set αi
being accepted into the population depends entirely on comparison of OFi with that of the
last accepted set, OFi-1. This probability is defined by Equations 2.15(a) and 2.15(b),
iiM
ii
i
iii KP
PP OFOFfor
OFOFexp
)()(
)( 11
1
11 <
−==→ −
−
−− α
ααα (2.15a)
iiiiP OFOFfor1)( 11 ≥=→ −− αα (2.15b)
Each parameter set is sampled at a random distance and direction from the previously
added set, subject to the a priori constraints and a specified maximum distance, KM2. The
result of the Metropolis algorithm (in this context) is a large sample of parameter sets
from the posterior response surface (note the distinction from the output of GLUE which
produces a large sample of parameter sets from the prior distribution each assigned a
posterior probability).
An implementation of the Metropolis algorithm is suggested in Figure 2.4. This algorithm
could be refined by allowing constants KM1 and KM2 to be updated at intervals, thereby
gradually increasing focus on the optima.
Mailhot et al. (1997) find the Metropolis algorithm to be useful for uncertainty analysis of
a storm sewer model. Kuczera and Parent (1998) compare the performance of the
Metropolis algorithm with GLUE for estimation of rainfall-runoff model parameter
uncertainty (see comments in Section 8.3 of this dissertation).
38
Add αi to the population of a posteriori parameter sets.
Start
Sample parameter set αi (i = 0) in a priori parameter space
Define objective function OF, constant KM1 and maximum step KM2
Ready to propagate uncertainty
Take random sample αi of a-prioriparameter space within KM2 of αi -1
Run model and calculate objective function OFi
i = i + 1
Generate random number P’ between 0 and 1to simulate acceptance / rejection decision
No
Yes
YesNo
Run model and calculate OFi (i = 0)
Reject αi andadd another
αi-1
Probability P of addition of αi to parameterpopulation = Minimum(1, exp(OFi-1 -OFi )/ KM1)
Has the aposteriori pdfconverged?
Is P > P’ ?
Add αi to the population of a posteriori parameter sets.
Start
Sample parameter set αi (i = 0) in a priori parameter space
Define objective function OF, constant KM1 and maximum step KM2
Ready to propagate uncertainty
Take random sample αi of a-prioriparameter space within KM2 of αi -1
Run model and calculate objective function OFi
i = i + 1
Generate random number P’ between 0 and 1to simulate acceptance / rejection decision
No
Yes
YesNo
Run model and calculate OFi (i = 0)
Reject αi andadd another
αi-1
Probability P of addition of αi to parameterpopulation = Minimum(1, exp(OFi-1 -OFi )/ KM1)
Has the aposteriori pdfconverged?
Is P > P’ ?
Figure 2.4 A Metropolis calibration procedure
2.3.3 Genetic algorithms
Genetic algorithms (Holland 1975) are global optimisation procedures that are commonly
used in hydrological modelling (e.g. Duan et al. 1993, Mulligan and Brown 1998) to find
a single optimum parameter set. Unless they are designed to converge to a meaningful
distribution, for example by introducing a Marko Chain element (Vrugt et al. 2003a) or
by defining a Pareto set of parameters (Fonseca and Fleming 1995, Vrugt et al. 2003b),
then they are of limited value for uncertainty-based calibration. A good introduction to
genetic algorithms is given by Beasley et al. (1998).
2.4 Example of calibration
This example aims to demonstrate some of the above approaches to the estimation of a
joint a posteriori parameter distribution using Monte Carlo simulation, and elucidate
some similarities and contrasts between them. The importance of the objective function
39
design, with respect to the interpretation of the output uncertainty, is illustrated. To make
the demonstration manageable, the model is simple and the data are idealised. Attention is
drawn to the last paragraph in this section which discusses the limitations of the example
in the context of more complex and practical problems.
2.4.1 The model and data
A steady state model of biodegradable organic carbon (Cc) decay and dissolved oxygen
(Cox) in a river can be described by the Streeter-Phelps equations (Streeter and Phelps
1925, also see Chapter 3),
( ) ( )
−=
uxkCxC occc exp0 (2.16a)
( ) ( ) ( )( )
−−+
−−
−
−−=
uxkCC
uxk
uxk
kkCk
CxC raosoxraococra
cocosox exp0expexp
0
(2.16b)
where koc is the Cc decay rate, kra is the oxygen aeration rate, x is the distance downstream
from a point pollution source, and Cox(0) and Cc(0) are the respective concentrations in
the river at x = 0, u is the average transport velocity and Cos is the uniform concentration
of Cox at saturation. Synthetic data are generated by the model using the parameter values
and boundary conditions in Table 2.1, and random errors are introduced in Cox (=εox)
from an N(0, 22) population, and in Cc (=εc) from an independent N(0, 102) population.
With 20 data locations spaced at 5km intervals along a 100km stretch of river, the
synthetic data are illustrated in Figure 2.5.
In the following demonstrations, a joint distribution of parameters koc and kra is derived
from this model and data set (and variations of it). The other parameters are fixed at the
values in Table 2.1.
2.4.2 Sampling the data error distribution
The first method uses the available data set together with the corresponding maximum
likelihood model output to approximate the data error distribution. The available 20
observations of Cox and Cc are used to find the maximum likelihood parameter set (koc,
kra) from 1000 random samples from within the bounds koc = 0.4-1.6 and kra = 2.0-8.0.
The statistics of the residuals (mean and standard deviation) around this model output are
40
given in Table 2.2, together with the uncertainty in these statistics, derived using the
equations in Ang and Tang (1975, p232 & p248).
Using the algorithm summarised in Figure 2.3, 200 alternative realisations of data are
drawn from the sample distributions described by Table 2.2, and the maximum likelihood
model (out of 1000 random samples of (koc, kra ) taken from the above-stated ranges) for
each is found. A posterior joint distribution of (koc, kra ) is built up from these alternative
realisations of the maximum likelihood.
Table 2.1 ‘True’ parameter values for Streeter-Phelps example
Parameter Value Unit
koc 1 s-1
kra 5 s-1
Cc(x=0) 75 mgO/l
Cox(x=0) 12 mgO/l
Cos 12 mgO/l
u 0.5 m/s
Table 2.2 Distributional properties of the Cc and Cox residuals
Property of residuals Population Sample Standard deviation of
sampled property
Mean µ = 0 m = 0 Std(m) = 2.17 Cc
Standard deviation σ = 10.00 s = 9.69 Std(s) = 5.39
Mean µ = 0 m = 0 Std(m) = 0.43 Cox
Standard deviation σ = 2.00 s = 2.17 Std(s) = 0.87
This exercise is repeated with different quantities of synthetic data (i.e. varying the 20
locations shown in Figure 2.5), with the data error population distribution kept the same.
The comparison of calibrated marginal distributions is shown in Figure 2.6. Figure 2.7
gives a similar comparison of the marginal distributions, this time changing the data
quality (i.e. varying the population standard deviations shown in Table 2.2). Note that the
posterior distributions of koc and kra are correlated (correlation coefficient = 0.31),
41
meaning that the model must be defined by the bi-variate distribution of koc and kra as
opposed to the marginal distributions shown in Figures 2.6 and 2.7.
0
5
10
15
20
0 20000 40000 60000 80000 100000
Distance from point source
Cox
(mgO
/l)
0
20
40
60
80
100
0 20000 40000 60000 80000 100000
Cc
(mgO
/l)
Cc data
maximum likelihood
Cox data
maximum likelihood
0
5
10
15
20
0 20000 40000 60000 80000 100000
Distance from point source
Cox
(mgO
/l)
0
20
40
60
80
100
0 20000 40000 60000 80000 100000
Cc
(mgO
/l)
Cc data
maximum likelihood
Cox data
maximum likelihood
Figure 2.5 Synthetic data for Streeter-Phelps model
Note also from Figures 2.6 and 2.7 that the ‘true’ values of koc and kra (1 and 5
respectively) do not necessarily correspond to the identified maximum likelihood values
(see especially the result for 5 data locations in Figure 2.6). This is because the available
data, upon which this calibration is founded, are only a sample of the true water quality.
The fact that the realisations of data error are not independent samples from the true error
distribution, but from an approximation based on one sample, means that the estimates of
parameter uncertainty are of limited significance. Ideally, a large number of independent
realisations of data error would be built up from the error population (e.g. by repeatedly
sampling the river when it is under the same boundary conditions), although doing this
42
effectively would be difficult in practice due to resource constraints and the difficulty of
achieving independent measurement errors.
Figure 2.7 shows that this method of calibration gives a significantly uncertain value for
the parameter kra despite perfect data, which is contrary to intuition. This implies that
adequate convergence of the joint parameter distribution has not been achieved using 0.2
million model runs. Whether this is primarily due to the inefficiency of the random
sampling as an optimisation procedure, or due to the limited number of realisations of the
data, is not investigated here. However, it is clear that the difficulty of achieving
convergence, even for a relatively simple problem such as this, contributes to the
approximate nature of the solution.
It is common in environmental monitoring that data are biased descriptors of the true state
of the environment (Keith 1990, Jarvie and Neal 2002). This may be because of
heterogeneity which is not recognised in the sampling programme, or because of repeated
laboratory errors, or simply because of physical constraints such as a lower bound of
zero. To explore the effect of this, the Cox data are raised by a random amount between
zero and 5mgO/l, and to a minimum of zero and maximum of Cos. The calibration is done
as before, with 20 data locations, and the calibrated parameter distributions are shown in
Figure 2.8. This shows that where significant bias is suspected but unknown, then this
approach to calibration has failed. Note that the parameter uncertainty associated with kra
is implied to be significantly reduced, contrary to what we would desire. The effect of
model structure error is similar to that of data bias (at least in this case), in that it biases
parameter estimates and causes inappropriate reduction in parameter uncertainty. In
practice, model structure error is particularly relevant to the Streeter-Phelps model,
because it neglects many of the complexities of pollution transport and decay.
2.4.3 GLUE using a likelihood function as an objective function
The preceding method has shown how Monte Carlo sampling of data error can be used to
derive calibrated parameter distributions. Now it is shown that using a likelihood function
within the GLUE framework offers the opportunity to reduce the computation required by
not explicitly accounting for the data sampling error. Here, GLUE is applied to the
previous Streeter-Phelps example using the data sample illustrated by Figure 2.5. The
likelihood function defined in Equation 2.6 is applied (whereby, for now, we are opting
not to explore the full generality of GLUE, instead maintaining a likelihood function
without any behavioural threshold) along with Equation 2.10 to derive posterior estimates
of probability for each sample of (koc, kra) using a total of 2000 random samples.
43
0.5
1.5
2.5
0
1
2
2 3 4 5 6 7 8
kra
0
2
4
6
8
10
0.4 0.6 0.8 1 1.2 1.4 1.6
koc
mar
gina
l pro
babi
lity
dens
ity
100 data locations
20 data locations
5 data locations
100 data locations
20 data locations
5 data locations0.5
1.5
2.5
0
1
2
2 3 4 5 6 7 8
kra
0
2
4
6
8
10
0.4 0.6 0.8 1 1.2 1.4 1.6
koc
mar
gina
l pro
babi
lity
dens
ity
100 data locations
20 data locations
5 data locations
100 data locations
20 data locations
5 data locations
Figure 2.6 Calibrated distributions with different sample sizes and constant residual variance
0
2
4
6
8
10
0.4 0.6 0.8 1 1.2 1.4 1.6koc
mar
gina
l pro
babi
lity
dens
ity
std Cox = 4; std Cc = 20
std Cox = 2; std Cc = 10
std Cox = 1; std Cc = 5
std Cox = 0; std Cc = 0
0.5
1
1.5
2
2.5
3
2 3 4 5 6 7 8kra
std Cox = 4; std Cc = 20
std Cox = 2; std Cc = 10
std Cox = 1; std Cc = 5
std Cox = 0; std Cc = 0
0
2
4
6
8
10
0.4 0.6 0.8 1 1.2 1.4 1.6koc
mar
gina
l pro
babi
lity
dens
ity
std Cox = 4; std Cc = 20
std Cox = 2; std Cc = 10
std Cox = 1; std Cc = 5
std Cox = 0; std Cc = 0
0.5
1
1.5
2
2.5
3
2 3 4 5 6 7 8kra
std Cox = 4; std Cc = 20
std Cox = 2; std Cc = 10
std Cox = 1; std Cc = 5
std Cox = 0; std Cc = 0
Figure 2.7 Calibrated distributions with different residual variances and 20 data locations
0
0.5
1
1.5
2
2 3 4 5 6 7 8
krakoc
0
2
4
6
8
10
0.4 0.6 0.8 1 1.2 1.4 1.6
mar
gina
l pro
babi
lity
dens
ity
biased data
unbiased data
biased data
unbiased data
0
0.5
1
1.5
2
2 3 4 5 6 7 8
krakoc
0
2
4
6
8
10
0.4 0.6 0.8 1 1.2 1.4 1.6
mar
gina
l pro
babi
lity
dens
ity
biased data
unbiased data
biased data
unbiased data
Figure 2.8 Effect of Cox error bias (20 data locations)
44
The probability equi-potentials of the derived point estimates are shown in Figure 2.9. For
comparison with the results of the previous method (Section 2.4.2), the marginal
distributions of koc and kra are illustrated in Figure 2.10. Repeated for other data scenarios,
the results are summarised in terms of calibrated parameter variances in Figure 2.11.
2
3
4
5
6
7
8
0.4 0.6 0.8 1 1.2 1.4 1.6
0.0001
0.0020.004
0.006
0.008
koc
k ra
2
3
4
5
6
7
8
0.4 0.6 0.8 1 1.2 1.4 1.6
0.0001
0.0020.004
0.006
0.008
koc
k ra
Figure 2.9 Equi-potentials of point estimate probabilities using GLUE with a likelihood function
The similarity of the results from GLUE and those from the data error sampling (Figures
2.10, 2.11) is striking, considering that the GLUE method does not explicitly account for
data sampling error, and has reduced the computation from 0.2 million to 2000 model
runs. The theoretical basis for the similarity can be demonstrated at a simple level.
Equation 2.3 is re-expressed as,
( )[ ])(5.0exp
)(2
115.022 parresN
mGLUEGLUE NN
KL
res−−
+=
δσπ (2.17)
where LGLUE is the probability of any parameter set, δ 2 is the variance of the
corresponding model result around the maximum likelihood result, σm2 is the error
45
variance around the maximum likelihood result, and KGLUE is a standardisation constant.
In the data error sampling method, LR is the probability of any parameter set, but δ 2 is the
variance of the maximum likelihood result for any data realisation around the result for
the available data sample. Approximating the standard error of the maximum likelihood
as normally distributed with variance σm2/Nres (assuming that σm is accurate, see Ang and
Tang 1975) gives,
( )[ ]22
5.02
5.0
/5.0exp2
1mres
m
res
RR N
NK
L σδπσ
−= (2.18)
As it is known that the integrals of Equations 2.17 and 2.18 are both unity, to prove that
they give the same result for all parameter samples only requires that the ratio LGLUE : LR
is proven to be the same for all δ,
( )( )
[ ][ ]225.05.022
5.02
/5.0exp
)(5.0exp
)(2
2
mres
parres
resN
m
m
GLUE
R
R
GLUE
N
NN
NKK
LL
res σδδσπ
πσ−
−−
+= (2.19)
Amalgamating all terms which are independent of δ into one constant K,
( ) resN
m
m
R
GLUE KL
L5.0
22
22
)(/exp
+=
δσσδ
(2.20)
Expanding the exponential term into a MacClaurin series, and neglecting terms higher
than quadratic, gives,
resres
res
Nm
Nm
N
m
m
R
GLUE KK
LL
σδσ
σδσ
=+
+
=5.022
5.0
2
22
)( (2.21)
which is constant for all δ. Thus it is shown that Equations 2.17 and 2.18 are describing
the same probability distribution if δ4/σm4 and higher order terms can be neglected. These
terms may not be negligible if Nres is very low, but in such cases the assumptions
underlying Equations 2.17 and 2.18 are not justifiable anyway. Nevertheless, the theory
presented here supports the experimental results in Figures 2.10 and 2.11, and suggests
46
that using the likelihood function replicating the data sampling results by neglecting
higher order uncertainties.
0
1
2
3
4
5
6
0.4 0.6 0.8 1 1.2 1.4 1.6
koc
mar
gina
l pro
babi
lity
dens
ity
GLUE
data error resampling
0
0.2
0.4
0.6
0.8
1
2 3 4 5 6 7 8
kra
GLUE
data error resampling
0
1
2
3
4
5
6
0.4 0.6 0.8 1 1.2 1.4 1.6
koc
mar
gina
l pro
babi
lity
dens
ity
GLUE
data error resampling
0
0.2
0.4
0.6
0.8
1
2 3 4 5 6 7 8
kra
GLUE
data error resampling
Figure 2.10 Comparison of calibrated parameters using GLUE with a likelihood function and data error sampling
0
0.05
0.1
0.15
0.2
0 5 10 15 20
standard deviation of Cc data population
stan
dard
dev
iatio
n of
k oc
0
0.05
0.1
0.15
0.2
5 20 35 50 65 80 95
number of data locations
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
stan
dard
dev
iatio
n of
k au
0
0.2
0.4
0.6
0.8
1
1.2
5 20 35 50 65 80 95
GLUE
Metropolis
GLUE
data errorresampling
Metropolis
GLUE
Metropolis
GLUE
Metropolis
data errorresampling
data errorresampling
data errorresampling
0
0.05
0.1
0.15
0.2
0 5 10 15 20
standard deviation of Cc data population
stan
dard
dev
iatio
n of
k oc
0
0.05
0.1
0.15
0.2
5 20 35 50 65 80 95
number of data locations
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
stan
dard
dev
iatio
n of
k au
0
0.2
0.4
0.6
0.8
1
1.2
5 20 35 50 65 80 95
GLUE
Metropolis
GLUE
data errorresampling
Metropolis
GLUE
Metropolis
GLUE
Metropolis
data errorresampling
data errorresampling
data errorresampling
Figure 2.11 Comparison of calibrated parameter variances using GLUE, data error sampling and Metropolis
2.4.4 Metropolis using weighted squared errors as an objective function
The Metropolis algorithm (Figure 2.4) is observed to further increase the efficiency (in
terms of time for convergence of (koc, kra) covariance matrix) of the Streeter-Phelps
47
calibration by up to 60%. The OF is defined as the sum of the variance-weighted squared
errors, i.e.
∑∑==
+=res
c
c
res
ox
ox
N
iCi
Cm
N
iCi
Cm 1
2,2
,1
2,2
,
11OF εσ
εσ
(2.22)
where oxCm,σ and
cCm,σ are the error population standard deviations (from Table 2.2), and
oxCi,ε and cCi,ε are the ith residuals of Cox and Cc respectively, and Nres is 20 as before.
Then, the probability of selecting parameter set αi pursuant to αi-1 is given by Equation
2.15. KM1 is specified as 2, and the maximum permitted step, KM2 is specified individually
for koc and kra as (KM2, koc = 0.05, KM2, kra = 0.25). The data set illustrated by Figure 2.5 is
used. The converged koc and kra distributions are almost identical to those obtained using
the data error sampling method (Figure 2.10) and Figure 2.11 supports this result under a
range of data conditions. From Equation 2.14 it is clear that the Metropolis result is
sensitive to KM1, and it is not a coincidence that this choice of KM1 almost replicates the
data error sampling result. Idealising Equation 2.22 by considering a single response, and
using the definitions for Equations 2.14 and 2.17,
( )
−
−=
+−=
−=
12
2
12
2
12
22
1
expexp1
exp1exp1
Mm
resm
Mm
res
MET
Mm
resm
METM
i
METMET
KN
KN
K
KN
KKOF
KL
σσ
σδ
σσδ
(2.23)
Equating this with Equation 2.18 gives,
( )
−=
−
− 2
2
5.02
5.0
22
2
12
2 5.0exp
2
1expexp1
m
res
m
res
RMm
resm
Mm
res
MET
NNKK
NKN
K σδ
πσσσ
σδ
(2.24)
and equating the exponents with the δ 2 terms gives KM1 = 2. The specification of KM1 and
the OF used here is generally applicable to approximation of the standard error of a
maximum likelihood model result assuming a large number of independent Gaussian
errors. As σm is not generally known a priori, updating of KM1 within the algorithm may
be useful. While Metropolis is an adaptive search, and therefore potentially superior to
48
GLUE for finding the maximum likelihood and variance, the number of samples it retains
from extreme values is, by definition, relatively small.
2.4.5 GLUE using a subjective GLUE likelihood as an objective function
A simple demonstration is now given of how the parameter uncertainty can be increased,
to safeguard against underestimating uncertainty due to the presence of structural or data
bias, using a subjective GLUE likelihood employed within GLUE. The Nash-Sutcliffe
efficiency (Nash and Sutcliffe 1970) is a widely used objective function that measures the
proportion of the variance of the data about the mean of the data that is explained by the
model, which may be regarded as an sensible (although subjective) measure of the
relative belief in alternative models. In this example, the average value of the Nash-
Sutcliffe efficiency for the Cc data and that for the Cox data is used as the objective
function,
( ) ( )
−−+
−−=
∑
∑
∑
∑
=
=
=
=
res
res
c
res
res
ox
N
icc
N
iCi
N
ioxox
N
iCi
CCCC1
2
1
2,
1
2
1
2,
115.0OFεε
(2.25)
where Cox and Cc represent the model results for dissolved oxygen and oxygen demand
respectively, and oxC and cC the corresponding observations, and Nres = 20 as before.
All values of this (2000 random samples are used) equal to or below 0.4 are considered
non-behavioural and are discounted. The values greater than 0.4 are divided by a constant
so that they total one, and these are considered to be values of relative probability.
The equi-potentials of probability in the parameter space are shown in Figure 2.12. These
indicate that the parameter uncertainty is substantially greater than that in Figure 2.9
derived using the likelihood function. The 0.4 threshold is used arbitrarily in this
example; higher values reduce parameter uncertainty (e.g. a value of 0.75 resulted in less
parameter uncertainty than the likelihood function method) and lower values increase it
(e.g. a value of zero meant that almost the whole a priori parameter space was
behavioural).
In the Metropolis algorithm, to produce a similarly contrived posterior distribution, KM1
and the OF could be modified from the values used in Section 2.4.4.
49
2
3
4
5
6
7
8
0.4 0.6 0.8 1 1.2 1.4 1.6
koc
k ra
0.00060.00065
0.0007
0.00075
0.0008
0.0000
2
3
4
5
6
7
8
0.4 0.6 0.8 1 1.2 1.4 1.6
koc
k ra
0.00060.00065
0.0007
0.00075
0.0008
0.0000
Figure 2.12 Equi-potentials of point estimate probabilities using GLUE with a GLUE likelihood based on the Nash-Sutcliffe efficiency. Note: the zero contour is the boundary between behavioural and non-behavioural parameter sets.
2.4.6 HSY using a possibilistic objective function
Now consider the HSY method of Hornberger and Spear (1980). A set of characteristic
system response is defined, with the sampled parameter set given a possibility of 1 (P(δ)
= 1), if the model result falls wholly within pre-specified lower and upper limits. For the
Streeter-Phelps example, those limits are of Cox and Cc (Coxl Coxu and Ccl, Ccu respectively),
i.e.
ucclcuoxoxlox CCCCCCP <<∩<<= if1)(δ at all Nres locations (2.26a)
resultsotherallfor0)( =δP (2.26b)
For example, if the upper and lower limits are taken to be the 90% confidence limit of the
data sample (i.e. 1.28 × the standard deviation in Table 2.2) around its maximum
likelihood model result (denoted here by 'oxC and 'cC ) then, for all Nres data points,
50
17.228.1' ×−= oxlox CC (2.27a)
17.228.1' ×+= oxuox CC (2.27b)
69.928.1' ×−= clc CC (2.27c)
69.928.1' ×+= cuc CC (2.27d)
The corresponding possible set of (koc, kra) is represented in Figure 2.13. This was derived
using 10000 random samples from the parameter space. Note that, as opposed to Figures
2.9 and 2.12, the set limits defined in Figure 2.13 are not smooth due to the discontinuous
nature of Equation 2.26. The HSY method is potentially more robust to model error and
data bias than likelihood functions because results such as those illustrated in Figure 2.8
can be avoided by increasing parameter uncertainty using appropriate specification of the
upper and lower bounds of characteristic response. Of course, improvement in robustness
is at the expense of a less informative, more subjective description of uncertainty.
2
3
4
5
6
7
8
0.4 0.6 0.8 1 1.2 1.4 1.6
koc
k ra
2
3
4
5
6
7
8
0.4 0.6 0.8 1 1.2 1.4 1.62
3
4
5
6
7
8
0.4 0.6 0.8 1 1.2 1.4 1.6
koc
k ra
Figure 2.13 The set of possible (koc, kra)
2.4.7 Summary of this demonstration of calibration
Notwithstanding its demonstrative limitations (see below), this example compares data
error sampling, GLUE and Metropolis and shows that these methods are not
51
fundamentally different in so far as they can produce the same calibration results, given
consistent objective functions. It has also been shown that using more subjectively-
founded objective functions, specifically possibilistic measures and the Nash-Sutcliffe
efficiency, leads to a different magnitude and nature of uncertainty than achieved using
likelihood functions, and those results are sensitive to the subjective judgements
employed.
With regard to more complex and more realistic environmental modelling, the above
example has several important limitations. Firstly, it only has two inter-dependent
parameters, while many models have significantly more. In such cases converging the
posterior joint distribution would be expected to be much more difficult, perhaps
requiring many thousands of model runs (e.g. Thyer et al. 1999) depending on the
strength and nature of the interactions. Secondly, the response surface, which is illustrated
by Figure 2.9, is well behaved. Many practical problems involve multi-modal responses
together with discontinuities derived from the discontinuities in the model structure, again
increasing the difficulty of convergence. Thirdly, Equation 2.16 is an analytical solution
to the Streeter-Phelps model, which is solved easily and quickly, which facilitates Monte
Carlo methods. Models of the environment are more often in the form of systems of
differential equations to which approximate numerical solutions are required and
computational demands are relatively high (see the discussion in Chapter 5). While
computer power is continuously increasing and parallel processing facilities are available,
computation time remains a limitation in model calibration and uncertainty analysis.
Lastly, the data have been synthesised from a normal population of residuals which are
uncorrelated and have zero mean. Only a nominal look at the effects of data bias has been
included.
2.5 Uncertainty propagation
Uncertainty propagation in this context means propagating the calibrated parameter joint
distribution to a stochastic result. Methods of propagating probability distributions can be
classified as variable transformation methods, sampling methods, point estimation
methods and variance propagation methods. An alternative to probability theory is the
theory of possibility (Zadeh 1978). Each of these approaches except variable
transformations (although see the demonstration of the Mellin transform by McIntyre
2000) are discussed here.
52
2.5.1 Monte Carlo methods
Monte Carlo (MC) simulation applied to uncertainty propagation means generating
discrete parameter sets according to their probability or possibility distribution, and
running a simulation using each set. Alternatively, the parameter set samples and
associated probability masses which were derived during calibration can be recalled,
thereby avoiding the need for assumptions regarding the form of the distribution. The
results of multiple simulations give a close approximation to the analytical form of the
probability density function (PDF) using frequency analysis, and any model can be easily
included in such a framework with minimal input from the modeller. For these reasons,
MC is a well-used method of uncertainty propagation. The main disadvantage of MC is
that a great number of model runs may be required to reliably represent all probable
results especially when there is a number of random variables. Methods of estimating a
preferred number of samples are available (e.g. Cochran 1977), although this also
depends on the convergence or divergence during propagation and therefore is case
specific (Tellinghuisen 2000). Stratified random sampling and Latin hypercube sampling
(see MacKay et al. 1979) are often used to improve efficiency.
2.5.2 First order and point estimate approximations
First-order variance propagation is the most common method of uncertainty propagation
(Beck 1987). If a function Y=f(X), where Y= y1, y2, … yNvar and X=x1, x2, … xNpar, is
approximated by a first-order Taylor series expansion around the expected X, µX, then,
( )XY µfµ = (2.28a)
)()()(2 YXY TY ∆Ψ∆=σ (2.28b)
where ∆(Y) is the Npar × Nvar matrix of derivatives of Y with respect to X; ψ(X) is the Npar
× Npar covariance matrix of X, and µY and σY2 are the Nvar × 1 vectors of mean and
variance of Y. This is a linear approximation of uncertainty propagation which is only
completely reliable for linear models. The accuracy of this method for non-linear models
can be improved by using a higher order Taylor series expansion, but this becomes
computationally demanding, especially if the derivative values are calculated
numerically. Variance propagation is a useful method for models which can
approximated by quasi-linearisation (e.g. Kitanidis and Bras 1980), i.e. a series of
localised linear functions.
53
Rosenblueth’s point estimation method for symmetric and non-symmetric variable
distributions (Rosenblueth 1981) aims to reduce the computational demands of variance
propagation by eliminating the calculation of derivatives. The PDF of each random input
variable is represented by a number Np of discrete points, located according to the first,
second and third moments of the PDF. The joint PDF of Npar parameters is represented by
the array of projected points. Therefore, parNpN points are used. Each point is assigned a
mass according to the third moment and the correlation matrix. All points are propagated
discretely to parNpN solutions and the first moment is the weighted average; the second
moment, that of the squares; and the third moment, that of the cubes. Most usually a 2-
point scheme is used whereby parN2 points are required. For symmetrical distributions for
Npar > 2, the number of evaluations can be reduced to 2Npar by using Harr’s point
estimation method (Harr 1989). Harr’s method is a useful improvement on Rosenblueth’s,
but is limited by the necessity of symmetrical distributions. A similar approach which
allows for skewed distributions but not correlations is described by Hong (1998).
Protopapas and Bras (1990) have applied Rosenblueth’s 2-point method to a rainfall-
runoff model and Yeh et al. (1997) have similarly applied Harr’s method, and a useful
review of all these point estimate methods is given by Christian and Baecher (1999).
It is worth noting that some Monte Carlo-based methods of calibration, for example
GLUE (Beven and Binley 1992), are randomised point estimate methods. In GLUE,
many random samples of the a priori parameter space are assigned GLUE likelihoods,
then these become point estimates for the uncertainty propagation stage. Unlike
Rosenblueth’s method, it is generally assumed that there are enough points to derive the
model output PDF, not just the lower moments.
2.5.3 Possibility theory
Possibility theory (Zadeh 1978) offers a robust alternative to propagation of probability
distributions. To illustrate this, let f(x1, x2) be a function which is strictly increasing or
decreasing with respect to both variables x1 and x2, and let x1 and x2 be independent have
possibility distributions which rise internally to a single peak or plateau. Then, using the
rules of union in Equation 2.13(b), the two values of x1 and the two of x2 with possibility
P (called the P-level α-cut of x1 and x2) define the two values of f(x1, x2) with possibility
P. For example, if 1dxdf is positive for all x1 and 2xdf ∂ is negative for all x2 the
upper and lower bounds of the function, with P = 0 are calculated from,
54
( )luu xxfxxf 2121 ,),( = (2.29a)
( )ull xxfxxf 2121 ,),( = (2.29b)
This method can be extended to problems with many uncertain parameters so long as the
aforementioned ‘increasing-decreasing’, ‘single peak’ and independence conditions are
met. While an infinite number of α-cuts are required for the exact solution to a non-linear
problem (Wierman 1996), an approximation of the propagated possibility distribution can
be made with a small number of computations. The associated difficulties and limitations
should be recognised. Firstly, special attention must be given to the method of calibration
in order to derive meaningful parameter possibility distributions. Secondly, the possibility
is greater than probability at all points (Zadeh 1978), and so the former is a less specific
descriptor of uncertainty. Thirdly, if parameter α-cuts are to be used, prior knowledge of
the sensitivity of results to the parameters is required. Lastly, there remains the problem
of parameter inter-dependence which, as in probability theory, complicates the analysis,
generally requiring that the output possibility distribution be defined by taking a large
sample of parameter sets.
2.6 Propagation of the Streeter-Phelps model parameters
The joint (koc, kra) distribution previously identified using GLUE with the likelihood
function (Figure 2.9) is propagated to give spatially varying distributions of Cc and Cox.
Again, the boundary conditions defined in Table 2.1 are used. Firstly, each of the 2000
samples of (koc, kra) and corresponding probability mass is propagated through the
Streeter-Phelps model, then the first-order variance and Rosenblueth 2-point methods are
applied, using the covariance matrix of (koc, kra) derived from the same parameter set
probabilities. It is observed that the three alternative methods give (practically) identical
results for the first three moments and the same 90% confidence limits on the output of Cf
and Cox against x (Figure 2.14). This similarity indicates the numerical efficiency of the
first-order variance and Rosenblueth 2-point methods, despite the apparent non-linearity
of the model with respect to koc and kra. In fact, the model is only significantly non-linear
at low values of Cox, and the performance of the first-order method deteriorates with
either increased Cc loading or increased data uncertainty.
55
5
10
15
20
40000 60000 80000 100000Distance from point source
Cox
(mgO
/l)
0
20
40
60
80
100
0 20000 40000 60000 80000 100000
Cc
(mgO
/l)
Cc data
modelled 90%confidence limits
Cox data
modelled 90%confidence limits
00 20000
90% confidence limiton data error variance around maximum likelihood
90% confidence limiton data error variance around maximum likelihood
5
10
15
20
40000 60000 80000 100000Distance from point source
Cox
(mgO
/l)
0
20
40
60
80
100
0 20000 40000 60000 80000 100000
Cc
(mgO
/l)
Cc data
modelled 90%confidence limits
Cox data
modelled 90%confidence limits
00 20000
90% confidence limiton data error variance around maximum likelihood
90% confidence limiton data error variance around maximum likelihood
5
10
15
20
40000 60000 80000 100000Distance from point source
Cox
(mgO
/l)
0
20
40
60
80
100
0 20000 40000 60000 80000 100000
Cc
(mgO
/l)
Cc data
modelled 90%confidence limits
Cox data
modelled 90%confidence limits
00 20000
90% confidence limiton data error variance around maximum likelihood
90% confidence limiton data error variance around maximum likelihood
Figure 2.14 Propagated uncertainty using GLUE, first order analysis, and Rosenblueth’s two-point method (all give the same 90% confidence intervals). The uncertainty in the model parameters was derived using a likelihood function.
Using the likelihood function of Equation 2.6 has meant the modelled 90% confidence
limits in Figure 2.14 represent the uncertainty in the maximum likelihood solution, and
not the variance of the data error. Therefore, the posterior models derived from the
likelihood function do not reproduce the derived 90% confidence limits on the data error,
illustrated in Figure 2.14. Figure 2.15 shows the different modelled 90% confidence
limits that are obtained when using the Nash-Sutcliffe efficiency GLUE likelihood (i.e.
the set shown in Figure 2.12), and Figure 2.16 shows the limits of possibility obtained
using the possible set (shown in Figure 2.13). It is seen that the GLUE
likelihood/possibility measures used to obtain Figures 2.15 and 2.16 can produce wider
confidence limits than those based on likelihood functions, and the modeller might
56
consider contriving such objective functions to allow for the additional uncertainty due to
structural error or data biases. For example, the objective function could be designed to
force the modelled confidence limits to include a visually satisfactory proportion of the
data. The effect of calibration data bias on predicted confidence intervals is illustrated by
McIntyre et al. (2001), and the effect of model structural error is examined further in
Chapter 6 of this dissertation.
Also, it may be noted from Figures 2.14, 2.15 and 2.16 that model results are constrained
by the fixed boundary conditions (e.g. Cc = 12mg/l), irrespective of parameter
uncertainty. Therefore, if they are not precise, the modeller must treat the boundary
conditions as random variables as well as the parameters.
5
10
15
20
40000 60000 80000 100000
Distance from point source
Cox
(mgO
/l)
0
20
40
60
80
100
0 20000 40000 60000 80000 100000
Cc
(mgO
/l)
Cc data
modelled 90%confidence limits
Cox data
modelled 90%confidence limits
00 20000
90% confidence limiton data error variance around maximum likelihood
90% confidence limiton data error variance around maximum likelihood
5
10
15
20
40000 60000 80000 100000
Distance from point source
Cox
(mgO
/l)
0
20
40
60
80
100
0 20000 40000 60000 80000 100000
Cc
(mgO
/l)
Cc data
modelled 90%confidence limits
Cox data
modelled 90%confidence limits
00 20000
90% confidence limiton data error variance around maximum likelihood
90% confidence limiton data error variance around maximum likelihood
Figure 2.15 Propagated uncertainty using GLUE. The uncertainty in the model parameters was derived using a GLUE likelihood based on the Nash-Sutcliffe efficiency.
57
The example of the Streeter-Phelps model has illustrated that alternative methods of
propagation of parametric uncertainty can lead to practically the same result. However,
the example is too simple to fully show the limitations of the reviewed methods. There
may be strong non-linear dependency of parameters which must be approximated by a
covariance or correlation coefficient in Rosenblueth’s method and the first-order variance
method, leading to a poor approximation of prediction uncertainty. Practical
environmental models often include cyclic, non-continuous, systems of ODEs, or
otherwise non-linear mathematics which will test all methods more severely than was
attempted here.
5
10
15
20
40000 60000 80000 100000Distance from point source
Cox
(mgO
/l)
0
20
40
60
80
100
0 20000 40000 60000 80000 100000
Cc
(mgO
/l) Cc data
modelled limits of possibility
Cox data
modelled limits of possibility
00 20000
90% confidence limiton data error variance around maximum likelihood
90% confidence limiton data error variance around maximum likelihood
5
10
15
20
40000 60000 80000 100000Distance from point source
Cox
(mgO
/l)
0
20
40
60
80
100
0 20000 40000 60000 80000 100000
Cc
(mgO
/l) Cc data
modelled limits of possibility
Cox data
modelled limits of possibility
00 20000
90% confidence limiton data error variance around maximum likelihood
90% confidence limiton data error variance around maximum likelihood
5
10
15
20
40000 60000 80000 100000Distance from point source
Cox
(mgO
/l)
0
20
40
60
80
100
0 20000 40000 60000 80000 100000
Cc
(mgO
/l) Cc data
modelled limits of possibility
Cox data
modelled limits of possibility
00 20000
90% confidence limiton data error variance around maximum likelihood
90% confidence limiton data error variance around maximum likelihood
Figure 2.16 Propagated uncertainty where using possibilistic combination of parameter sets.
58
2.7 Summary
Imprecision in environmental modelling stems from the approximate nature of the
models, and from the inevitable difficulty of identifying a single ‘best’ model given the
limitations in our prior knowledge and in the information retrievable from field data. In
general, it may be said that the natural environment is too complex, with too many
heterogeneities and apparently random influences, to be usefully described without
including some estimation of uncertainty. The inclusion of uncertainty analysis adds to a
conventional modelling exercise in two main ways. Firstly, the calibration of model
parameters involves identification of parameter distributions rather than single parameter
values. Secondly, the parameter distributions (and alternative model structures if used)
are propagated to stochastic rather than deterministic results.
Estimation of parameter uncertainty is traditionally achieved through Bayesian
manipulation of likelihood functions which measure the probability of a data sample
occurring given a model. This is an attractive approach as it gives a result based on
assumptions about error distributions that can be clearly defined and are often statistically
auditable. However, in river quality modelling the inevitable presence of model structure
error and data biases complicates the analysis. When these errors are unknown and
neglected due to the lack of prior knowledge, lack of field data and lack of resources to
improve upon this situation, then using likelihood functions will generally underestimate
the true model uncertainty. In that case, alternative, more subjective measures of
uncertainty become attractive, including possibilistic measures (related to fuzzy set
theory) and Generalised Likelihood Uncertainty Estimation (GLUE). These tools allow
an objective function to be designed as a measure of the relative belief in alternative
models. The subjectivities involved mean that estimated uncertainty may be seen as
specific to the modeller rather than specific to the modelled system. It is therefore
important that the chosen objective function, as a basis for the estimated uncertainty, is
made explicit so that it is open to review and results can be properly interpreted.
This chapter has introduced these methods of uncertainty-based model calibration, and
has demonstrated their various similarities, differences and limitations with a simple
model of dissolved oxygen with synthetic data. This demonstration and discussion has
59
provided a background which will allow later results (Chapters 5, 6 and 7), which are
largely based on GLUE, to be properly interpretated and critically reviewed.
60
3. An overview of river water quality modelling theory and commonly used
modelling tools
The main components of water quality models are identified as transport models,
thermodynamic models, and water quality process models. For each, the theories and
alternative concepts used in model development are summarised, and their application to
some commonly used modelling tools is noted. A compendium of previous reviews on
modelling developments, tools and theory is given.
61
3.1 Introduction
The purpose of this chapter is to give context for the design of the water quality models
used in this Thesis. The current state of the art of water quality modelling, and its
development throughout this century are reviewed by Orlob (1992), Ambrose et al.
(1996), Chapra (1997) and Rauch et al. (1998). Currently applied theory and
implementations are described in detail by Bowie et al. (1985), Thomann and Mueller
(1987) and Chapra (1997). Somlyody et al. (1998) and Thomann (1998) review how the
state of the art may be developed in the future. Here, a brief history of the subject and a
summary review of current theory and practice, as pertinent to this Thesis, are given.
3.2 Developments
Early in the 20th century, two fundamental realisations were made with regard to human
impact on river water quality; 1. a reasonable quality of life for the metropolitan
population cannot be sustained without some formal wastewater treatment; 2. the optimal
design of wastewater treatment installations depends heavily on the natural assimilative
capacity of the receiving environment. With this design-orientated motivation, Streeter
and Phelps (1925) produced the first significant model of dissolved oxygen and organic
carbon in rivers. This model is based on the assumption that the organic carbon
decomposes aerobically at a first order rate koc, so that the concentration of easily
biodegradable organic carbon Cc (in units of oxygen demand) within any volume of water
(which has no flux of organic carbon across its boundaries), is described by the
differential equation,
cocc Ck
dtdC
⋅−= (3.1)
As the aerobic decomposition consumes oxygen, there is a corresponding decrease in the
concentration of dissolved oxygen. It is also assumed that there is oxygen exchange with
the atmosphere at a rate proportional to the oxygen deficit, where the deficit is the
saturation concentration Cos minus the modelled concentration Cox,
62
( )oxosracocox CCkCk
dtdC
−+⋅−= (3.2)
where kra is the reaeration rate. These equations can be solved analytically to give the
Streeter Phelps equation for river dissolved oxygen (Equation 2.15(b)). The fundamental
assumption used in the derivation of this model is that there is no flux of organic carbon
across the boundaries of each unit volume of water, nor any of oxygen except that due to
re-aeration at rate kra. Firstly, this implies that there are no sources of Ccf or Cox at all non-
zero x which means the model is limited to a single point source. Secondly, this means
that flow is steady and uniform, with no dispersion in the direction of x. Another
important assumption is that koc and kra are constant parameters, uninfluenced by dynamic
environmental conditions.
Despite the limitations of the original Streeter-Phelps model, its simple analytical solution
and proven validity as an approximate model, allowed it to be usefully applied for many
decades. However, since the 1920s (at least in ostensibly developed countries),
motivations arose for models which address the limitations of Streeter-Phelps, and which
give more widely applicable and more accurate results;
New pollutants emerged, and new knowledge about their social and environmental
significance meant that they could not be neglected. Consequently, the standards for
river water quality in developed countries became more specific.
Wastewater treatment technology became available to meet such standards, but at a
price which had to be justified before the investment by the expected environmental
or social improvement.
Drinking water standards increased, as did the sophistication and expense of water
treatment plants. This meant new demand for models which could predict the water
quality at the abstraction point as a function of upstream loading.
Improvements in monitoring methods meant that a direct measure of performance of
pollution control, and of water quality models, was possible.
National bodies were formed and made responsible for the regulation of river water
quality, including the setting of discharge consents which recognised the natural
assimilative capacity of the river system.
In the early 1960s, computer technology reached a stage where numerical solutions were
feasible. Orlob (1992) commentates on the developments from then. The Streeter-Phelps
model was extended to decay rates which varied spatially and with temperature. Non-
63
linear differentials using Monod kinetics were introduced, heat exchange models, and the
coupling of hydrodynamic models. In the early 1970s, the US EPA funded a model called
QUAL2, which could simulate systems of rivers at steady or unsteady flow, and allowed
for nitrogen oxygen demand. The 1970s also brought numerous attempts to model the
eutrophication process, after the environmental state of many lakes was severely damaged
by excessive algae growth. Since then, eutrophication has continued to be a major
problem in developed countries. The challenges of eutrophication modelling (e.g. food
chain interactions; heterogeneity in space, time and species; the importance of non-point
pollution sources; and the inevitable error in measurement procedures) have played a
major role in motivating water quality modelling research in the last 20 years.
In the 1980s and 1990s, the utility of water quality models has been fully recognised and
applied by governmental bodies and commercial organisations. Improved graphical and
menu-driven user interfaces have made models more marketable, and modelling tools for
a variety of specific applications are available e.g. QUAL2E (Brown and Barnwell 1987),
WASP5 (Ambrose et al. 1993), DESERT (Ivanov et al. 1996), QUASAR (Whitehead et
al. 1997a), OTIS (Runkel 1998), SIMCAT (UK Environment Agency 2001a), MIKE11
(DHI 2000), RWQM1 (Shanahan et al. 2000), ISIS (Wallingford Software 2002) and CE-
QUAL-W2 (Cole and Wells 2000). A comparative review of most of these models, and
some others not mentioned here, is given by Ambrose et al. (1996). Although they have
common elements, each of these models has specific features aimed at developing the
state of the art of water quality modelling. Separate modelling developments, not
generally integrated into models, have included sediment-water interactions (e.g. Di Toro
and Fitzpatrick 1993), micro-pollutant modelling (e.g. Chapra 1991), oil slick modelling
(e.g. Shen and Yapa 1993), inclusion of river ice processes (e.g. Shen and Chaing 1984,
Lal and Shen 1993), and inclusion of the higher food chain (e.g. Thomann 1989). Such
progress has been achieved through research into the physical processes occurring in the
river, and numerical representation of these processes in mechanistic models. As a result
of elaborated methods, increasing demands on the models and increased computer power,
the number of dependent variables and the spatial refinement of the models has increased
dramatically since the 1960s. This is illustrated in Figure 3.1 (based on a number of
modelling exercises in the USA, adapted from Thomann 1998).
64
1
10
100
1000
10000
100000
1000000
10000000
1920
1940
1960
1980
2000
B = number of spatial
compartments
A = number ofdependent
state variables
number of interactive components = A x B
1
10
100
1000
10000
100000
1000000
10000000
1920
1940
1960
1980
2000
B = number of spatial
compartments
A = number ofdependent
state variables
number of interactive components = A x B
Figure 3.1 Increasing sophistication of water quality models (from Thomann 1998)
A somewhat contrary development since the late 1970s was the realisation that the
multitude of physical processes which potentially affect the water quality cannot be
comprehensively identified and measured, and that resources to support use of complex
models are often in practice not available (Reckhow 1994). Therefore, as argued in
Chapter 1, a more realistic approach is to aggregate the many complex processes into a
limited number of model equations and parameters which represent the modeller’s
concept of a simplified river environment.
The remainder of this chapter reviews the physical and conceptual representations of the
river environment which are presently in use in river water quality modelling.
3.3 The components of a river water quality model
It is recognised that river water quality is strongly dependent on the river flow, the water
depth and the water temperature (Thomann and Meuller 1987). Therefore, in general,
river water quality models have the following distinct sub-models;
the hydraulic model
the thermodynamic model
the water quality process model.
The three sub-models can be idealised as forming a serial structure with the water quality
model at the end (affected by both the thermodynamic and hydraulic models), the
65
thermodynamic model in the middle (affected by only the hydraulic model), and the
hydraulic model at the start (physically independent of the other two). This is illustrated
in Figure 3.2. Possible exceptions to this one-way series of dependencies are the
calculation of evaporation as a function of water temperature, and the interaction of ice
growth with the hydraulic regime (Ashton 1986).
Secondary interactions which are generally neglected
Thermodynamicmodel
Primary interactions forming basis of water quality models
Hydraulic model
Water quality model
Secondary interactions which are generally neglected
Thermodynamicmodel
Primary interactions forming basis of water quality models
Hydraulic modelHydraulic model
Water quality model
Figure 3.2 The basic sub-models which make up a water quality model (Adapted from Thomann and Meuller 1987)
3.3.1 Hydraulic and routing models
An extensive review of the state of the art of river hydraulic and solute transport models
is given by Camacho (2000).
An adequate characterisation of the hydraulic state of the river is fundamental to the
success of the water quality model. The hydraulic state of the river strongly affects
various thermodynamic and kinetic processes. Examples of important hydraulic variables
are listed below;
flow rate (affects dilution),
solute and solids retention time (affects mass loss or gain due to various processes),
water surface velocity and area (affect aeration and heat exchange),
infiltration rates (affects mass losses),
turbulence (affects dispersion),
river bed shear velocity (affects sediment resuspension).
Models generally divide the water body into segments. In estuary, offshore and some lake
applications, where there is a significant flow element in more than one direction, it can
be valuable to use two or three dimensional segmentation (Watanabe et al. 1983). Rivers
tend to be relatively well-mixed over their depth and width, and the commonly adopted
66
approach is to segregate the river only lengthwise, i.e. to use a one-dimensional model.
An exception is when a sewage discharge or river confluence is to be studied in some
detail, for which near-field mixing models are available (see Rutherford 1994).
Routing through control volumes(conceptual)
Flow & momentum balance(physically-based)
Quasi steady
Linear
Non-linear
Advection-dispersion
Pure advection
Aggregated dead zone
Transient storage
Kinematic wave
St. Venant
Transfer function(empirical)
Diffusion wave
Linear
Non-linear
Flow modelling options Solute transport options
Fully mixedcells in series
Routing through control volumes(conceptual)
Flow & momentum balance(physically-based)
Quasi steady
Linear
Non-linear
Advection-dispersion
Pure advection
Aggregated dead zone
Transient storage
Kinematic wave
St. Venant
Transfer function(empirical)
Diffusion wave
Linear
Non-linear
Flow modelling options Solute transport options
Fully mixedcells in series
Figure 3.3 Options for modelling the longitudinal flow and solute transport in a river
Options for modelling the longitudinal flow and solute transport in a river are summarised
in Figure 3.3. Modelling the hydraulics on a physical basis, assuming that the only forces
are affecting the hydraulics are in the longitudinal dimension, requires a discretised
solution of the mass and momentum balance equations known as the Saint Venant
equations. Such solutions allow unsteady flow conditions to be accurately simulated if
there is adequate supporting data and/or measurements of the channel characteristics.
These solutions are most useful when the simulation of unsteady flow conditions is
essential to the success of the water quality model, e.g. in pollution spills or in dynamic
runoff events. However, such models are not computationally easy to solve especially
when there are complex boundary conditions (for example hydraulic structures within the
studied length of river), or when numerical stability and accuracy criteria require a very
refined discretisation. Depending on the nature of the problem, the acceleration terms
may be neglected, giving the diffusion wave, kinematic wave, or gradually varied flow
equations (Chapra 1997: 251). Recent work has shown that unsteady flow conditions can
be adequately simulated using more efficient conceptual routing models (Camacho 2000).
67
Using these, the river is divided into a series of reaches, with each reach generally sub-
divided into cells1. This is similar to the spatial discretisation used for the Saint Venant
solution. However, instead of using momentum balance and mass balance to route the
flow, the conceptual models employ one or both of the following methods;
1. lag the flow between cells by some time α,
),1(),( α−−= jiji QQ (3.3)
(where Q(i,j) is the flow in cell number i at time step j; this convention of subscripts i
and j for cell number and time-step is used throughout this dissertation) then calculate
the water depth, velocity and hydraulic retention time using either a physically-based
formula, e.g. Manning’s equation (Chow 1959), or an empirical stage-discharge
relationship.
2. attenuate the flow by regarding the cells as reservoirs with the flow out of them
dependent on the water volume,
tQtQVV jijijiji ∆−∆+= −−−− )1,()1,1()1,(),( (3.4a)
2),(1),(q
jiji VqQ = (3.4b)
(where V(i,,j) is the volume of water in cell i at time j and ∆t is the specified time-step,
and q1 and q2 are empirical routing parameters) then calculate the water depth, velocity
and hydraulic retention time directly from Q and V, together with the channel shape
parameters.
In the first option, the simplest case is not to explicitly apply a lag, but use that which is
implicit to the numerical method. For example, in a forward time, backward space finite
difference solution (see Appendix 1) this would simply be one model calculation time-
step for each cell in the reach. If this time-step is very small, any change in the boundary
condition flow is more or less instantly applied to all downstream reaches, and so the
hydraulic model can be described as quasi-steady-state. The second case is founded on
established methods of hydrological flood routing, which attempt to simulate the
1 The division of the river into static cells is also called the “method of control volumes” whereby the states in neighbouring cells are treated as boundary conditions rather than as fully interacting states.
68
attenuation of the flood wave as it proceeds downstream. The combination of the two
methods provides a numerically efficient routing model, which can be shown, in theory
and practice, to match the accuracy of the Saint Venant solution if backwater effects are
not significant. Camacho and Lees 1999 describe a ‘discrete lag cascade’ model which
applies this method.
A flow routing sub-model is necessary to accurately simulate the short-term response of
water quality to periods of high flow. However, in many modelling applications the
hydrodynamic response of the river is far faster than the response period of interest. For
example, the modeller may be interested in the week to week variability of the water
quality, whereas a normal flood event may pass down the river in a few hours. While the
water quality is likely to have some ‘memory’ of the flood, the justification of an accurate
routing model (and the associated data requirements) becomes increasingly dubious as the
disparity in time-scales increases. Flow dynamics which occur on, for example, a weekly
time-scale can arguably be described as ‘gradually varied flow’ and modelled using
quasi-steady methods (i.e. simplification of the Saint Venant solution to neglect
acceleration and pressure terms and assume that energy gradient equals bed slope) which
are far more computationally efficient2. Because the case-specific time-scale determines
the optimum method, some modelling tools allow a choice of steady, quasi-steady and
unsteady hydraulic models (e.g. DESERT, ISIS and MIKE11).
3.3.2 Solute transport models
Solute transport in rivers is widely recognised as not simply a case of advection according
to the average water velocity. There are four common explanations for this;
1. Non-uniform velocity over the width and depth of the river. This causes
downstream sections to respond to a pulse of solute sooner than would be
predicted using the average velocity, and for the solute to take longer to pass.
2. Fickian-type dispersion where solute disperses at a rate directly proportional to its
spatial concentration gradient due to turbulent eddies.
3. Dead zones where effects such as eddies behind obstructions which entrap
solutes, causing the pollutograph peak to lag behind the hydrograph peak, and the
pollutograph tail to lengthen.
4. Transient storage zones where effects such as temporary solute sorption to plants
and sediments which has a similar effect to dead zones.
2 This is not necessarily to say that the modeller can neglect shorter periods of high flow, rather that he may be justified in representing them with a quasi-steady model
69
Solute transport models generally represent one or two of these mechanisms,
conceptually encompassing the influence of them all. The traditional method is to neglect
(1), (3) and (4), and to represent mechanism (2) through discretisation of the advection-
dispersion equation (Taylor 1954, from Camacho 2000),
2
2
dxCdD
dxdCu
dtdC
+−= (3.5)
where C is concentration of an arbitrary solute, v is average water velocity, D is
dispersion, t is time and x is distance downstream. This can be incorporated into any of
the hydraulic models mentioned previously. However, the advection-dispersion model
has been shown to have limited success in describing solute transport in natural channels,
particularly the tails of pollutographs (Young and Wallis 1993). For this reason, transient
storage models have been added (e.g. Bencala and Walters 1983, Lees et al. 2000),
( )'
''2
2
dxCCD
dxCdD
dxdCu
dtdC
x−
++−= (3.6)
where the additional term represents Fickian dispersion Dx’ to a conceptual off-stream
store of solute of concentration C’ over a mixing length dx’, illustrated in Figure 3.4.
Ac
CC’D’
dx’
main channel u = Q / Ac
off-stream store of solute which does not contribute to flow and causes solute to lag flow
Ac
CC’D’
dx’
main channel u = Q / Ac
off-stream store of solute which does not contribute to flow and causes solute to lag flow
Figure 3.4 The transient storage concept
While the transient storage model has proven useful, there are a total of four parameters
(u, D, Dx’ and dx’ or equivalent) per river reach, which must be estimated or calibrated
from the data. Beer and Young (1983) developed the aggregated dead zone model (ADZ)
as a parsimonious alternative. In continuous time form, the ADZ model is,
70
( )),(),1()()(
1jiji
iiCC
TdtdC
−−= −− α (3.7)
where α is the solute time delay (i.e. the travel time of the leading edge of the
pollutograph from cell i-1 to cell i) and T is the dead zone residence time. Thus, by
assuming that Fickian dispersion, transient storage effects and non-uniform velocity
effects are negligible compared to the dead zone effect, the total number of parameters is
reduced to two (T and α) per river reach.
While modelling the lag and attenuation of solute in a river has primary application in
unsteady solute loading conditions, it has also been shown to be important in steady-state
applications. For example, Chapra and Runkel (1998) show that allowance for dead zones
and transient storage significantly affects the location of an oxygen sag downstream of a
steady-state point source of BOD. However, the ADZ model cannot simulate upstream
dispersion of solute which is common in estuaries and slow-moving rivers. While there is
an extensive knowledge-base and empirical formulae for the estimation of the ADE
parameters (i.e. the dispersion coefficient D and Manning’s n or equivalent), there is less
basis for uncalibrated estimates of ADZ and TS parameters (although recent work by
Camacho (2000) and Lees et al. (2000) has addressed this limitation). The use of standard
water quality data is generally insufficient for their calibration because of the interaction
effect of pollutant decay, and carefully planned and executed conservative tracer tests are
generally required (Wagner and Harvey 1997).
3.3.3 Thermodynamics
The thermal regime of a river affects the water quality because;
the water temperature affects the saturation concentration of oxygen and multi-phase
pollutants,
rates of biological and chemical activity generally depend upon temperature,
evaporation losses can be significant,
river ice affects water temperature, aeration, atmospheric heat exchange and flow
rates.
Water temperature is a variable in all but the most simple water quality models. Water
temperature might be prescribed by the modeller as a constant (e.g. HERMES), or as a
constant for each reach (e.g. an option in QUAL2E) or, for dynamic models, as a time-
71
series of data (e.g. WaterRAT; see Chapter 4). This is appropriate if it gives an
approximation of temperature which is adequate for the task at hand. For example, such a
method would not be appropriate if seasonally averaged temperature data are used
whereas the model is to be used for diurnal studies, nor if extrapolations of climate or
thermal loads were to be investigated. Alternatively, the temperature can be implicitly
calculated using a thermodynamic model. Modelling the thermal regime of rivers is
covered in detail by Ashton (1986) and Bras (1990). An overview is given here.
The temperature of the river may be significantly affected by;
bulk transfer by advection, dispersion, extractions and pollution sources, fb (Js-1),
long wave radiation to / from the atmosphere, sky and surrounding land, fl (Jm-2s-1),
effective short-wave radiation from the sun, fs (Jm-2s-1),
convection to and from the atmosphere, fc (Jm-2s-1),
evaporation losses and condensation gains, fe (Jm-2s-1),
conduction to and from the river bed, fsw (Jm-2s-1),
rainfall and snowfall, fp (Jm-2s-1),
conduction to and from ice, fiw (Jm-2s-1).
The first five of these processes are usually included in dynamic water quality models
(including options in QUAL2E, WASP5 and MIKE11, although in QUASAR only the
bulk transfer is accounted for). River bed conduction and friction effects are included in
some more specialist thermodynamic models (e.g. Evans et al. 1998). The first seven
processes are illustrated in Figure 3.5 and their implementation is summarised by
Equation 3.8, in which it is assumed that there is no ice cover.
( ) sswpcelsb AfffffffdtdJ
++++++= (3.8)
where As is the surface area of the water (m2) and J is the total heat in the river reach
(Nm), linked to the water temperature Tw (oC) by,
www sdV
JT⋅
= (3.9)
where dw is the density of water and sw is the specific heat capacity of water .Note that
Equation 3.8 is not analytically soluble, with each of the terms non-linearly dependent on
72
Tw. The use of relatively complex, physically-based thermodynamic models for the
derivation of the terms in Equation 3.8 is often justifiable because the theory is well
founded and validated in application to river modelling, and does not require a large
number of prior assumptions. Air temperature, humidity and daylight hours, which are
normally the primary factors affecting the water temperature (Ashton 1986), are generally
reliable and available on a daily basis. And lastly, water temperature is relatively easily
and accurately measured. Therefore, calibration and verification of the thermodynamic
model is not necessarily complicated by large data error.
advection and dispersion
bulk pollution source
extractions & bed leakage
surface convection short wave radiation evaporation
long wave radiation
ICE
ICE
ICE ICE
conduction through ice
T=0
T=air temp.
conduction through bed
T=water temp.
T=deep bed temp.
surface convection
advection and dispersion
bulk pollution source
extractions & bed leakage
surface convection short wave radiation evaporation
long wave radiation
ICE
ICEICE
ICEICE ICEICE
conduction through ice
T=0
T=air temp.
conduction through bed
T=water temp.
T=deep bed temp.
surface convection
Figure 3.5 The main processes affecting the water temperature in a river
The presence of ice on a river signifies a local water temperature of close to zero3.
However, thermal loads such as cooling water discharges or sewage discharge can cause
large heterogeneity which will raise the average water temperature significantly above
zero, despite the widespread presence of ice. Various approaches have been employed for
the modelling of ice on rivers. The simplest is the degree-day method (Shen and Chaing
1984) where ice thickness and cumulative degree-days below freezing are linked by a
simple empirical formula,
5.0ZKHi = (3.10)
3 While a somewhat specialised field of river water quality simulation, the modelling of ice is given significant attention here due to its perceived importance in the Hun river study.
73
where K is an empirical constant and Z is the cumulative degree-days below freezing. A
degree-day is determined by calculating the mean daily air temperature for the day and
subtracting it from a base temperature, in this case zero. The main limitation of this
approach in dynamic river water quality modelling is the assumption that ice thickness
only depends on the air temperature, therefore neglecting the influences of thermal
pollution, radiation, sediment heat and the retardation effect of flow turbulence. Shen and
Chaing (1984) suggest a heat balance approach which assumes linear heat gradients
between the air and the ice, the ice and the water, and the water and the river-bed, as well
as the usual heat exchanges (first five in the previous list). Neglecting the terms already
presented in Equation 3.8, the water heat gain can be modelled by (adapted from Shen
and Chaing 1984 and Lal and Shen 1993),
( ) ( ) swsswsfwiiw ATTkATTkdtdJ
−+−= (3.11a)
( ) ( )
−+−
+=
−
wiiwaiaii
i
ii
i TTkTTkk
Hlddt
dH1
11 (3.11b)
where kiw, ksw and kai are the heat transfer coefficients between the water and the ice, the
water and the sediment, and the ice and the air respectively (thereby, for each, the thermal
conductivity and boundary layer thickness are combined into one parameter); Ti is the ice
temperature at the water-ice interface which equals zero; Ts is the sediment temperature;
di and li are the density and latent heat of ice respectively; and ki is the thermal
conductivity of ice. This model simulates the lag between Tw and Ta. However, the
accuracy of results is limited because the response of the terms in Equation 3.8 to the
presence of ice is necessarily simplified due to lack of present knowledge. Particular
sources of error are the insulative effect of snow cover on the ice and the reduction in
evaporative losses during periods of ice cover. In arctic regions, it is often important to
model the ice progression and melt in detail. Shen (1979) suggests a framework for
incorporating the physical processes of frazil and floe formation, illustrated in Figure 3.6.
Numerical implementations of these processes are described in Maunula (1992) and Lal
and Shen (1993).
74
Cooling of water to 0oC
If U < 0.6 m/s then ice sheets form on surface
If U > 0.6 m/s then frazil production
Sheet bridges riverSheetsdevelop to floes
Floes accumulate
Frazils float and flocculate
Frazil pans
UndercoverdepositionThermal growth
Cooling of water to 0oC
If u < 0.6 m/s then ice sheets form on surface
If u > 0.6 m/s then frazil production
Sheet bridges riverSheetsdevelop to floes
Floes accumulate
Frazils float and flocculate
Frazil pans
UndercoverdepositionThermal growth
Cooling of water to 0oC
If U < 0.6 m/s then ice sheets form on surface
If U > 0.6 m/s then frazil production
Sheet bridges riverSheetsdevelop to floes
Floes accumulate
Frazils float and flocculate
Frazil pans
UndercoverdepositionThermal growth
Cooling of water to 0oC
If u < 0.6 m/s then ice sheets form on surface
If u > 0.6 m/s then frazil production
Sheet bridges riverSheetsdevelop to floes
Floes accumulate
Frazils float and flocculate
Frazil pans
UndercoverdepositionThermal growth
Figure 3.6 A framework for an ice model (Ashton, 1979)
Water quality models which allow for the effects of ice are rare. This may be because, in
Europe and the USA, the critical water quality periods are generally in the warm and dry
seasons, when dilution is low and oxygen-depleting decay rates are high. However, in
some climates, the cold season is also the dry season. Ranjie and Huiman (1987) simulate
the dissolved oxygen during winter in a river in northern China, and suggest that
decreased aeration due to the ice cover significantly affects the dissolved oxygen levels.
Their model is a steady-state model which assumes that the river is fully covered with ice
and that the average water temperature is zero. The QUAL2E documentation (Brown and
Barnwell 1987) recommends a factor of between 0 and 1 to adjust oxygen re-aeration
rates to allow for ice cover but does not allow dynamic representation of ice.
Table 3.1 Classes of water quality determinands commonly modelled Class of determinant Example model variables Significance
Carbon and oxygen Organic and inorganic carbon, dissolved oxygen ecology, aesthetics, WT
Nitrogen Nitrate, ammonia, organic nitrogen eutrophication, toxicity, WT
Phosphorus Orthophosphates eutrophication Organic toxins Phenols, pesticides toxicity, WT Oils Petrol, lubricants, fats toxicity, nuisance, WT Suspended solids Inorganic and organic solids aesthetics, WT, sorption Metals Mn, Fe, Hg, Ca toxicity, REDOX reactions Pathogens E.-coli, giardia disease, WT
WT = water treatment costs
75
3.3.4 Water quality processes
“Water quality processes” refers here to all the physical, chemical and biochemical
transformations of the water quality determinands. Due to the wide range of application
of water quality models, a variety of determinands may be included as state variables, and
these are broadly classified in Table 3.1. Thomann and Mueller (1987) and Chapra (1997)
review the approaches to modelling the processes transforming the state of these
determinands. Chapra also reviews recent methods of simulation of sediment-water
interactions, toxic substances, sorption, phytoplankton stoichiometry and bio-
accumulation. Apart from these subject areas, and some specialist fields (e.g. oil slick
modelling, Lal and Shen 1993) the basics of process modelling has not significantly
changed in the last 20 years, as it is recognised that the utility of the models is limited by
other factors (e.g. the difficulties of representing heterogeneities and of model
identification). Here, an overview is given of the state-of-the-art methods which are
pertinent to this Thesis, which includes only the first six classes in Table 3.1; carbon,
nitrogen, phosphorus, organic toxins, oils and suspended solids.
3.3.4.1 Carbon and dissolved oxygen
The traditional significance of organic carbon in a river is the impact that it has on
dissolved oxygen. Therefore, it is generally measured using its oxygen demand, most
commonly the chemical oxygen demand (COD) or carbonaceous biochemical oxygen
demand (BOD). These determinands are reasonably convenient to measure, and because
the methods are well-practiced, they allow regional and global comparisons. This means
that the data available for model calibration tends to be BOD and/or COD, and either the
model must be parameterised to give comparable results or the data must be adjusted. The
previously mentioned models are parameterised to simulate the carbon oxygen demand
via either the ultimate BOD or the 5-day BOD. Carbon has a fundamental role in water
quality processes which cannot be identified by BOD alone (Connolly and Coffin 1995,
Chapra 1999). Firstly, the fraction of carbon which will settle depends (among other
things) on the fraction which is solid. Without such data, settlement and sediment
processes cannot be modelled mechanistically, and the modeller must assume, neglect or
calibrate effective settling velocities of BOD. Secondly, the lumped approach neglects the
individual fates of the different fractions of carbon so, for example, there can be no
mechanistic model of hydrolysis or complex carbohydrates and long-chain hydrocarbons.
Thirdly, a mechanistic model of toxin sorption processes is not possible without an
estimate of the particulate and dissolved fractions of carbon (e.g. Tye et al. 1996).
76
In the reviewed models, carbon–dissolved oxygen processes are based on numerical
solutions of the original Streeter-Phelps model, and one or more of these improvements;
implicit calculation of aeration rate (QUASAR, QUAL2E, CE-QUAL-W2,
RWQM1),
allowance for anaerobic conditions (QUASAR, RWQM1),
temperature effects for both BOD decay and aeration (ISIS, QUASAR, QUAL2E,
WASP5, MIKE11, CE-QUAL-W2, RWQM1, SIMCAT),
some representation of sediment oxygen demand (ISIS, DESERT, QUASAR,
QUAL2E, WASP5, CE-QUAL-W2, RWQM1),
phytoplankton photosynthesis (ISIS, QUASAR, QUAL2E, WASP5, MIKE11, CE-
QUAL-W2, RWQM1),
unsteady transport models (OTIS, ISIS, WASP5, MIKE11, DESERT, CE-QUAL-
W2, RWQM1).
The aeration rate is widely recognised to be directly proportional to the dissolved oxygen
deficit, i.e. the saturation level minus the actual level. There are a variety of formulae (see
Bowie et al. 1985) for the estimation of the dissolved oxygen (Cox) saturation level Cos;
one of the most commonly used is that derived experimentally in Greenberg et al. (1992)
(from Chapra 1997) which relates it to water temperature and salinity (Csa in µg/l),
+×
++
−×−
+×
−+×
++
−+
+−
=−
2
32
4
11
3
10
2
)273(101407.2
)273(754.10107674.1
)273(1062195.8
)273(102438.1
)273(66423080
)273(1575703.139
exp
wwsa
wwwwos
TTC
TTTTC (3.12)
In most freshwater applications, Csa can be taken as zero. The aeration rate kra (m s-1) is
widely recognised to also depend on the water temperature through an Arrhenius type
relationship,
( )20
20−×= Tw
rara kk θ (3.13)
where kra20 is kra at Tw = 20oC and θ is called (at least in this work) the Arrhenius constant
which for aeration is typically taken as 1.024 (Chapra 1997). kra20 is generally calibrated,
although in QUAL2E and QUASAR, for example, it can be calculated using an empirical
or physically-based formula (see Cole and Wells 2000: Appendix B) which relates it to
77
the hydraulic state of the river. Cole and Wells also list formulae which model aeration at
hydraulic controls, and Gulliver et al. (1998) evaluate a number of such models.
Allowance for anaerobic conditions is made in QUASAR by setting the organic carbon
decay rate koc to zero when the Cox level reaches zero. This is not allowed for in many
models, making them potentially unreliable for highly polluted rivers. Strictly, in well
mixed rivers at Cox = 0, the aerobic koc should not exceed the aeration rate (Chapra 1997).
Alternatively, it is suggested that a Michaelis-Menton relationship,
( )ochsox
oxoc kC
Ck+
∝ (3.14)
where kochs is the dissolved oxygen half saturation constant for organic carbon decay, may
be more accurate (Chapra 1999), as it simulates inhibition of aerobic bacteria during
anoxia (i.e. Cox < 2mg/l). The Michaelis-Menton relationship is important in water quality
modelling generally because it is applied to limitation of microbial growth by, for
example, Cox, minerals, prey, or light.
Carbonaceous sediment oxygen demand (CSOD) is commonly represented as a zero
order process (Chapra 1997: 452) - for example in QUAL2E CSOD is input as a constant
by the user. This has been shown to be justified in some cases (e.g. Chen et al. 1999). An
alternative assumption is some simple quasi-steady relationship (for example, in
QUASAR, CSOD is directly proportional to the water Cox) or that CSOD is directly
proportional to the rate of settlement of organic carbon. CSOD involves degradation and
mixing processes which can cause Cox to lag the organic carbon load by days or weeks
(Harremoes 1982, Boyle and Scott 1984). The main processes leading to this effect are
illustrated in a simplified manner in Figure 3.7. To represent these processes
mechanistically (notwithstanding the problems of spatial heterogeneity and sediment
transport) requires a large number of differential equations and parameters (e.g. McIntyre
1998) although such formulations have been shown to be effective where supporting data
from the sediment are available (e.g. Di Toro and Fitzpatrick 1993). Simpler
conceptualisations have also been shown to simulate the effect on overlying water quality
(e.g. Li and Chen 1994).
78
SOLIDCARBON
SOLIDCARBON
DISSOLVEDCARBON
DISSOLVEDCARBON
DISSOLVEDOXYGEN
MINERAL CARBON
DISSOLVEDOXYGEN
SOLIDCARBON
DISSOLVEDCARBON
ANAEROBIC SEDIMENT
AEROBIC SEDIMENT
RIVERWATER
SOLIDCARBON
SOLIDCARBON
DISSOLVEDCARBON
DISSOLVEDCARBON
DISSOLVEDOXYGEN
MINERAL CARBON
DISSOLVEDOXYGEN
SOLIDCARBON
DISSOLVEDCARBON
ANAEROBIC SEDIMENT
AEROBIC SEDIMENT
RIVERWATER
Figure 3.7 Schematic illustration of sediment-water carbon interactions
3.3.4.2 Photosynthesis
Photosynthesis can significantly raise the Cox levels and decay of phytoplankton can
significantly reduce it, causing marked spatial and temporal variations in river Cox. A
multitude of case-specific models have been developed to simulate photosynthesis. While
there are exceptions (Reckhow and Chapra 1983a: 201-314, Whitehead and Hornberger
1984, Whitehead et al. 1997b), the general approach is to mechanistically model the mass
transport and sedimentation, together with the limiting effects of light, temperature and
nutrients on phytoplankton growth. Excessive organic carbon load or presence of
toxicants can inhibit photosynthesis and so, in heavily polluted rivers, additional
inhibition factors may be required. Also, grazing and predatory zooplankton can be
included if first order phytoplankton death is not a useful assumption. In rivers, transport
of phytoplankton significantly affects the spatial distribution of photosynthetic oxygen
production. Therefore, it is often useful to distinguish between phytoplankton, and fixed
photosynthesisers, and a number of modellers have done this (e.g. Howarth et al. 1996,
McIntyre 1998, Park and Lee 2002, Wade et al. 2002). For a detailed description of
numerical modelling of eutrophication processes and for reviews of alternative
approaches, refer to Reckhow and Chapra (1983a,b), Bowie et al. (1985: 279-365),
Chapra (1997: 519-621).
The popular models ISIS, QUAL2E, MIKE11, WASP5, CE-QUAL-W2 and RWQM1
can model the role of phytoplankton in the carbon and oxygen cycles. MIKE11 also has
the option to model macrophytes. QUASAR simulates photosynthetic oxygen production
but requires that the user inputs chlorophyll-a concentrations. The transport-orientated
OTIS neglects photosynthesis, while DESERT allows a user-defined representation.
79
3.3.4.3 Nitrogen and phosphorus cycles
Typically, it is the main inorganic forms of nitrogen, ammonia, ammonium and nitrate,
which are used as indices of nitrogen pollution. Partly this is for measurement
convenience, partly because there are particular health risks associated with ammonia and
nitrates (WHO 1996), and partly because they are often important controls on
eutrophication. Nitrogen modelling, however, requires some representation of the organic
state of nitrogen as this generally makes up a significant part of the pollution load
(Metcalf and Eddy Inc 1991) and some fraction of it will mineralise in the river. Clearly
there is the same problem of lumping the solid and aqueous fractions of organic nitrogen
together that is encountered in modelling carbon. In cases without data for organic
carbon, the modeller must approximate concentrations via an knowledge-based
relationship with the inorganic forms (e.g. Metcalfe and Eddy Inc 1991). Ammonia and
ammonium are generally modelled together. They exist in an equilibrium which depends
on pH and temperature (see the description of Whitehead et al. 1997a), so can only be
modelled separately when pH is also an input or modelled variable. Nitrite is generally
lumped into nitrate, as it exists in much smaller quantities. A notable exception is
QUAL2E which models nitrites explicitly. For the models used later in this dissertation
the following notation for nitrogen concentrations is used: organic nitrogen (Cns);
ammonia plus ammonium (Cna); nitrite plus nitrate (Cni).
The fundamental difference between carbon modelling and nitrogen modelling is that the
inorganic forms of nitrogen are of interest whereas those of carbon generally are not. The
mechanisms of the transformations of nitrogen are well documented (e.g. Sawyer et al.
1994), and are generally modelled using similar methods to those used for the decay of
organic carbon. That is, first order decay rates are used; the Arrhenius equation is used to
account for temperature variable decay rates; and oxygen limitation models such as
Equation 3.14 are used in the oxidation reactions. Perhaps the most important feature of
nitrogen (and phosphorus) modelling is the importance of the loads from distributed
sources. For example, an estimated 70% of the nitrate load in the UK is from distributed
sources (DEFRA 2002) because of nitrates in the rainfall, and the runoff of animal
detritus and excess chemical fertilizers. The importance of distributed sources has
motivated integrated runoff-water quality models, e.g. HSPF (Bicknell et al. 1997) and
INCA (Whitehead et al. 1998).
Denitrifying bacteria (responsible for the loss of nitrogen mass to the atmosphere) are
known to flourish on the aerobic surface layer of the sediment (e.g. Kusuda et al. 1994).
Therefore, for models without implicit representation of the active sediment area and
80
sediment nitrogen, there is a problem of how to conceptualise denitrification. QUASAR
assumes that denitrification is directly proportional to the nitrate concentration in the
water column (with a temperature correction), while QUAL2E neglects denitrification
altogether.
Although certain pesticides containing phosphorus are known to be toxic (Sawyer et al.
1994), there is no significant oxygen demand associated with phosphorus, nor any direct
detriment to the environment or to human health of commonly found inorganic forms.
Therefore phosphorus modelling is motivated by eutrophication, and is included in all of
the reviewed river eutrophication models. Modelling the phosphorus cycle is complicated
by the numerous and inter-related fractions of phosphorus which are relevant to the
modeller (see Figure 3.8 adapted from Chapra 1997; note that sediment fractions are
lumped together and sediment-water interactions are not shown). The various fractions
are generally represented by two mutually exclusive conceptual fractions, for example
organic P (Cps) and dissolved P (Cpo) (e.g. in QUAL2E). Alternatively, only one fraction
is assumed to be relevant at the time-scale in question (e.g. orthophosphates in WASP5).
An additional complication in phosphorus modelling is that inorganic forms tend to
absorb to solids, which can make the phosphorus unavailable as a nutrient (e.g. Tate et al.
1995). Of particular importance are the level of iron hydroxide in the sediment, because it
strongly absorbs orthophosphates, and the dissolved oxygen level, which reduces the
hydroxides thus releasing orthophosphates (Weber 1996). While these processes can be
modelled successfully under certain conditions, it is clear that adequate sediment data are
required.
Particulate unavailableorganic P
Particulateunavailable inorganic P
Non-particulateunavailable
organic P
Non-particulateunavailable inorganic P
Solublereactive
phosphorus
Unavailable for photosynthesisers
Available for photosynthesisers
Solid or attached
Dissolved
sediment P
Particulate unavailableorganic P
Particulateunavailable inorganic P
Non-particulateunavailable
organic P
Non-particulateunavailable inorganic P
Solublereactive
phosphorus
Unavailable for photosynthesisers
Available for photosynthesisers
Solid or attached
Dissolved
sediment P
Figure 3.8 Fractions of phosphorus relevant in a photosynthesis model
81
As with nitrogen, distributed runoff of fertilisers and animal waste is a major source of
phosphorus in many rivers. For example, an estimated 40% of the phosphorus load to UK
rivers originates from distributed sources (DEFRA 2002).
3.3.4.4 Organic toxins and oils
Organic toxins pose some special difficulties for the modeller;
They tend to accumulate and persist in sediments, and associate readily with other
organic material (Chapra 1999). This means that concentrations are sensitive to the
amount and nature of the organics present in the water and sediments.
There are a large number of types, subject to a large number of kinetic processes
(biodegradation, volatilisation, evaporation, photolysis, hydrolysis, sedimentation –
see Chapra 1997). This means that physically-based models are complex, and that
lumped conceptual models will give predictive results of limited accuracy, especially
when the predictive task differs markedly from the conditions of calibration.
Small concentrations may be hazardous, which increases the need for accurate
models, and increases the significance of spatial and temporal heterogeneity in the
river.
However, the implications of toxins mean that they have been relatively well studied. As
they are largely man-made, there is interest in the topic from the producers as well as
from environmentalists, regulators and engineers. Therefore, given supporting data, there
is motivation and scope for detailed mechanistic toxic models (Thomann 1998, also see
Chapra 1997 for a review of research and numerical implementations). Models of organic
toxins (e.g. those in WASP, MIKE11 and the specialist steady-state model EXAMS
(Burns 2000)) are generally complex with a large number of state variables. Simplified
approaches, which assume zero decay or first order decay, are available and may be
useful where the transport is considered the primary factor (e.g. applications of the OTIS
model).
The term ‘oil’ generally includes a wide variety of organic substances which are
dissolved in certain solvents (Sawyer et al. 1994: 603). This includes hydrocarbons,
esters, natural oils and fats, waxes and long-chain fatty acids. Modelling oil poses the
same difficulties as organic toxins with the specific problem that some fractions of oil are
hydrophobic and some of these are less dense than water. This means that many oils will
82
not disperse evenly in the water, instead forming globules which may float on the water
surface and these will tend to form distinct pools due to wind effects. Such heterogeneity
is difficult to represent in lumped models, in particular the transport processes of the
surface layer of oil are different to those assumed using previously mentioned methods.
Traditionally, river oil models are directed at modelling the transient effect of single-
source spills under gradually varied or steady flow, and this allows Lagrangian type
numerical schemes to be applied (a review of river oil models is given in Yapa and Shen
1995). Another feature of existing oil models is that the fraction of spilled oil is known,
and physically-based parameters are known with confidence. Modelling multiple sources
of oil for purposes of management of day to day pollution requires an alternative
modelling approach. In particular, industrial and municipal pollution sources include
numerous fractions of oil which are lumped together using standard methods of
measurement (Greenberg et al. 1992:5-25), therefore there is no scope to model the
fractions individually. Consequently some average, conceptual parameter values must be
used to represent the oil processes (e.g. Sherif 2000).
3.3.4.5 Suspended solids
The concentration and nature of suspended solids in a river is important because;
they control the transport and fate of absorbed pollutants, especially toxins,
suspended solids are a nuisance in water treatment, causing erosion of plant and
requiring pre-treatment sedimentation tanks,
sedimentation and erosion can be a nuisance, e.g. siltation behind weirs and erosion
of bridge piers,
sediment nature and distribution affects sediment-water interactions,
turbid water can be unaesthetic.
In lakes and deep rivers, a common conceptualisation of suspended solid behaviour is to
lump the solids into one variable, Css, and to assume an effective settling velocity, vss,
which is calibrated (e.g. Chapra 1991);
w
ssss
HCv
dtdC ×
= (3.15)
A problem with this model is that suspended solids will not be uniformly distributed over
the water depth (for example, hindered settling will occur towards the bottom) and so the
83
sedimentation will not generally be first order with respect to the average suspended
solids concentration. Also in such systems, the total suspended solids may be dominated
by bio-production and this also is a function of depth (due to light extinction) and time
(due to the changing physiology of the microbes). On the other hand, in shallow rivers,
which are relatively well mixed over their depth, the concentration of suspended solids is
dominated by local eddies, turbulence and sediment resuspension characteristics (Yang
and Molinas 1982) therefore the above simple model is also of limited value. To
overcome this limitation, specialist solids transport models have been developed.
The modelling of suspended solids in rivers can be regarded as comprising seven
elements; loading, sedimentation, bed-transport, flow-transport, resuspension, bio-
accumulation and bio-degradation. Sediment transport models are traditionally aimed at
the problems of erosion and excessive sedimentation, where the effect of organic solids is
relatively small. Therefore, sediment transport models generally neglect bio-degradation
and bio-accumulation. Even neglecting these processes, sediment transport is notoriously
difficult to simulate. This is shown by Nakato (1990) who compares the results of eleven
sediment transport models, finding large differences in results and that empirical models
are least reliable, and the most successful are those which conceptually relate sediment
transport (and the concentration of suspended solids) to the hydraulic energy dissipation
and the size classification of the sediment.
It may be concluded that accurate modelling of suspended solids and their effect on other
water quality variables requires model sophistication and data which are beyond the
resources of most water quality studies. Many standard water quality models (e.g.
QUASAR and QUAL2E) do not attempt to model the sediment, thus are limited in their
representation of sediment-water interaction. Other models, which include such
interaction, adopt the simple approach of Equation 3.15 (e.g. Ivanov et al. 1996).
Recognising the limitation of this, MIKE11 incorporates a choice of suspended solids
models (e.g. Engelund and Fredsoe 1976) for cohesive and cohesionless sediment
transport, and these have been applied with some success in data-rich studies (e.g.
Enggrob 1997). However, McNeil et al. (1996) note that such models are not reliable
under many high flow conditions because they do not account for heterogeneity in the
depth of the sediment.
84
3.4 Summary
1. Numerous river water quality models are freely available and are being widely
applied for design, management and research purposes.
2. Arguably, all water quality models can be described as conceptual because they
represent the river environment in a meaningful but greatly simplified way, and use
model parameters which generally cannot be measured, but must be estimated using
expert knowledge or calibrated using reference data.
3. Pollutant transport models are a fundamental part of a river water quality model as
they dictate the magnitude and location of pollution. Hydrodynamic models are
computationally expensive, and more efficient routing or quasi steady-state models
are used for most applications.
4. Water temperature models are a fundamental part of a river water quality model as
they dictate the rate of most water quality processes, and therefore the pollution
levels. Physically-based thermodynamic models can simulate the water temperature
in a well-mixed river if the air temperature and various other atmospheric conditions
are known. Ice coverage is more difficult to simulate accurately, and ice presence is
likely to reduce the accuracy of the water temperature model during the thaw.
5. The fate of pollutants in a river is simulated using the principle of conservation of
elemental mass. That is, nitrogen, carbon, etc. cannot be destroyed, only transformed
in location and in species. In essence, the task of the dynamic model is to simulate the
location and species at all times. This is complicated by the various interacting
species of nitrogen, carbon and phosphorus (among others) of potential interest.
Furthermore, the nitrogen, carbon and phosphorus fates are all inter-linked by the
critical role of oxygen in the aquatic life-cycle.
6. Phytoplankton have an important role in the carbon and nutrients cycles, especially
because they effectively fix carbon (and sometimes nitrogen) from the atmosphere.
They are difficult to model because of spatial and temporal heterogeneity in their
growth, due to the important influence of the higher food chain, and due to their
sensitivity to nutrients, light, water temperature and other environmental variables
that are not easy to accurately model or measure.
7. ‘Traditional’ pollutants are regarded as organic carbon and nutrients. Recently, the
value of water quality models in simulating toxic substances has been recognised.
There are various difficulties in simulating toxics which mean that toxic models are
necessarily more complex than those of the traditional pollutants.
85
8. Sediments have a recognised role in river pollution as they can store pollutants and
rapidly release them during scour, or slowly release them diffusively. However,
accurately modelling sediment transport is extremely difficult and is outwith the
scope of most models.
9. Different modelling tasks require different models because of the number of
potentially relevant pollutants, and the supporting expertise and data which is
available. This raises a difficulty with application of available models. Some of these
try to cover more pollutants with add-in modules, while others encourage the user to
specify his own differential equations for the model state variables if necessary.
86
4. Water quality risk analysis tool (WaterRAT)
This chapter summarises the river modelling and uncertainty analysis components of
WaterRAT (Water quality Risk Analysis Tool), developed by the author. WaterRAT
includes a number of methods of evaluation of model and prediction uncertainty, and a
library of river and lake models of different complexities to suit the predictive task, the
characteristics of the natural system, the available data and computational resources. This
analytical capability is designed to encourage the modeller to explore prediction
uncertainty fully, and hence make properly informed recommendations for water quality
management (including water quality objectives, interventions, monitoring and model
development).
87
4.1 Introduction
The WaterRAT software implements the modelling framework proposed in Chapter 1
using methods and ideas introduced in Chapters 2 and 3. In summary, WaterRAT aims to
allow:
1. Responsiveness of modelling approach to the user’s data and resource constraints.
2. Indication of the principal factors affecting various determinands of water quality,
under current and speculated conditions.
3. Evaluation of the risk of failure associated with alternative pollution control
strategies, and indication of the key sources of decision-making risk.
4. Identification of trade-offs and viable compromises between non-commensurate
modelling and management objectives.
5. Iterative review of management objectives, modelling approaches, and database
and model development priorities.
These aims are pursued by including the following characteristics:
1. A choice of model structures and modelled determinands, and a framework
within which additional models may easily be added.
2. A framework within which (almost) all model inputs may be treated as uncertain,
and optimised to any chosen combination of target outputs.
3. Incorporation of GLUE for model uncertainty estimation, supplemented by a
deterministic genetic algorithm and Pareto (multi-objective) optimisation.
4. A selection of objective functions allowing flexibility in derivation of parameter
uncertainty using the GLUE methodology.
5. Efficient numerical methods allowing Monte Carlo methods to be as effective as
possible, and first order approximations to supplement Monte Carlo methods.
6. An easy-to-use interface for model specification and result analysis.
As well as pursuing a generic modelling framework, WaterRAT has been tailored to the
specific objectives of the TOPLEM project, and in particular this dictated the modelled
determinands of water quality, and the user interfaces.
Descriptions of the equations used in the simulation models, and in the conditioning and
analysis modules have been omitted from this chapter, as they are lengthy and not all
88
immediately relevant. Instead, reference is made to the WaterRAT documentation
(McIntyre and Zeng 2002), and specific descriptions are given where needed in Chapters
5, 6 and 7. This chapter concentrates on describing the functionality of WaterRAT in
terms of its general structure, its inputs, model library, analytical modules and outputs.
Following these descriptions, the novelty and limitations of WaterRAT are reviewed.
4.2 The concept and structure of WaterRAT
WaterRAT is a spreadsheet-based modelling tool that includes a library of surface water
quality models of varying complexity, presently including a choice of one-dimensional
river models and two-dimensional lake models, as described further below. Alternative
numerical solution methods are provided, plus a limited choice of pre-processing models
for filling in missing boundary condition data. There is flexibility over spatial scale, and
the input-output time-step may be anything above one minute. With the exception of
meteorological boundary conditions, all values of input data and parameters may be
treated as uncertain variables, and their effects included in calibration and reliability
analysis. The currently included models are limited to individual lake and river water
bodies plus interacting sediments, rather than of the wider catchment, and lake-river
systems cannot presently be modelled as a whole. However, additional models to suit new
problems can be added to the library.
WaterRAT is built within MS Excel 2000, so that WaterRAT’s own data processing
modules can be supplemented by those of Excel. The input and output is via a series of
Excel spreadsheets and model specifications are made via Visual Basic (VB) menus and
dialogue boxes. The library of simulation models comprises a series of Dynamic Link
Libraries (DLLs), which minimises processing time and allows Monte Carlo techniques
to be efficiently applied. The interactions between the user interface, VB modules and the
core DLLs are illustrated in Figure 4.1.
WaterRAT has a library of model structures for its pollution transport, water temperature
and water quality modules. This library gives the modeller a choice of different model
complexities, and determinands which he wishes to model. This includes the capacity to
model total organic carbon, biochemical oxygen demand, chlorophyll-a, dissolved
oxygen, various nutrients, a toxic substance, floating and suspended oil, and total
suspended solids. This is supported by sediment models which include biochemically and
physically-driven sediment-water interactions. A thermodynamic model is available
89
which models heat fluxes from the atmosphere and sediment, and simulates ice thickness
and cover. This thermodynamic model may be bypassed by prescribing a time-series of
water temperature. For the river models, pollution transport is modelled using the one-
dimensional advection-dispersion equation supported by two alternative hydraulic models
(a quasi-steady friction formula and a non-linear store). The lake models use the
advection-dispersion model supported by level-pool routing, with the option of including
prescribed periods and strengths of thermal stratification. The model components
presently available are summarised in Figure 4.2. The transport, sources, losses and
changes of state of mass and energy are represented by systems of differential equations,
a detailed description of which is found in McIntyre and Zeng (2002).
Excelinterface
Visual Basic
Dynamic Link Libraries
Data input and model configuration
Data processor
River and lake simulation models
Result display
Result processor Visual Basic
Routing models
Thermodynamic model(optional)
Water quality models
Sediment quality model(optional)
Excel interface
Excelinterface
Visual Basic
Dynamic Link Libraries
Data input and model configuration
Data processor
River and lake simulation models
Result display
Result processor Visual Basic
Routing models
Thermodynamic model(optional)
Water quality models
Sediment quality model(optional)
Excel interface Figure 4.1 WaterRAT’s software components.
Total carbonTotal nitrogen
Total phosphorusTotal solids
Core model
Optionalcomponents
Routing Thermodynamics Water quality Sediment quality
FlowWater depth
Water temperature
Ice thicknessIce cover
Dissolved oxygen
Phytoplankton
Toxin
Floating oilSuspended oil
CODBOD
Organic NNitrates
Ammonia
Organic PInorganic P
Toxin
Oil
CODBOD
Organic NNitrates
Ammonia
Organic PInorganic P
Total carbonTotal nitrogen
Total phosphorusTotal solids
Core model
Optionalcomponents
Routing Thermodynamics Water quality Sediment quality
FlowWater depth
Water temperature
Ice thicknessIce cover
Dissolved oxygen
Phytoplankton
Toxin
Floating oilSuspended oil
CODBOD
Organic NNitrates
Ammonia
Organic PInorganic P
Toxin
Oil
CODBOD
Organic NNitrates
Ammonia
Organic PInorganic P
P = phosphorus; N = nitrogen; COD = chemical oxygen demand; BOD = biochemical oxygen demand. Figure 4.2 Model component options currently available in WaterRAT.
90
4.3 Spatial and temporal resolution
Appropriate spatial and temporal modelling scales depend on the resolution of the data
and boundary conditions available for model identification, the time available for
achieving results, and the scale at which model forecasts are required.
For river modelling, the river is represented as a series of well-mixed control volumes
(called ‘cells’) between which pollution transport processes are simulated. This concept is
illustrated in Figure 4.3. Each cell must be prescribed certain spatially-varying parameters
which depend on the transport model selected. For this purpose, adjacent cells may be
grouped together into reaches, within which the spatially-varying parameters are taken to
be constant. The lengths of cells and reaches, and their properties, are input to a
spreadsheet. The downstream boundary of each cell is specified in terms of kilometres
downstream from a datum. The lake models work on the same control-volume principal,
except that they can be two dimensional, able to represent the vertical variation in water
quality due to effects of thermal stratification as well as length-wise variations (Figure
4.4). The lake models’ spatially varying parameters are specified for each cell.
The output time-step is defined by the user, and may be anything greater than one minute.
The available input data will be automatically interpolated to this time-scale, using either
linear interpolation, a cubic-spline or a step-function (whereby the input will not change
until that time when the next data point is available), as chosen by the user. In general,
whatever interpolation model is used, the temporal resolution of the output will be
restricted by that of the input data. The time-domain of the simulation is specified by the
modeller, constrained only by computer memory and the available time-domain of the
boundary conditions.
The numerical integration in the time domain uses a Fehlberg adaptive time-step scheme
(see Chapra and Canale 1998). This ensures near-optimum speed of computation and a
numerical error in the time domain which is guaranteed to be below a specified
maximum. This is an important feature in the Monte Carlo simulation, where randomly
sampled inputs lead to numerical stability and accuracy criteria which can vary widely,
both over the time-domain and from one model realisation to the next (e.g. Chapter 5).
The user can vary the numerical tolerance so that precision is not inordinately high given
the overall model uncertainty, and so that the solution speed is not inordinately low given
91
the computational constraints. Spatial numerical errors and numerical dispersion are not
handled automatically - see the discussion in section 5.4.
Headwater
Sources& losses
Additionalcells todownstreamboundary
Advection-dispersion
Sedimentinteraction
Headwater
Sources& losses
Additionalcells todownstreamboundary
Advection-dispersion
Sedimentinteraction
Figure 4.3 Cells-in-series concept of a river with sediment interactions.
Headwater Sources& losses
Additionalcells todownstreamboundary
Advection-dispersion
Sedimentinteraction
Epi-limnion
Hypo-limnion
Segment 1 Segment 2 Segment 3 Segment 4
Headwater Sources& losses
Additionalcells todownstreamboundary
Additionalcells todownstreamboundary
Advection-dispersion
Sedimentinteraction
Epi-limnion
Hypo-limnion
Segment 1 Segment 2 Segment 3 Segment 4 Figure 4.4 Cells-in-series concept of a stratified lake with sediment interactions.
4.4 Boundary conditions, initial conditions and model parameters
Dynamic boundary conditions include the meteorological, source and abstraction data.
All meteorology time-series (rainfall, evaporation, dew-point, air temperature, wind
speed, and surface light intensity are needed as inputs to various alternative models) are
input on a single spreadsheet, and are assumed to be uniform over the river or lake. Any
number of pollution/flow sources can be input (subject to computer memory). Each is
input on a separate spreadsheet with the river kilometer (for river models) or the cell
number (for lake models) specified. The format of this input is illustrated in Figure 4.5
(where the flow, BOD and ammonium are being treated as uncertain – see below). If
negative flows are entered, then sources are taken as losses, and any associated pollution
92
loads are neglected. For the river models, distributed sources may also be specified,
whereby the loads are evenly distributed between specified upstream and downstream
river kilometers.
Static boundary conditions are specified for each cell. For the river models these are:
channel cross-section shape; a leakage rate that is specified as constant or proportional to
water volume in that cell; sediment oxygen demand or active sediment area (depending
on whether the sediment-water interaction module is being used); and hydraulic or
routing parameters depending on which solute transport model is being used. For the lake
models, the bathymetry is defined by a volume-level relationship for the lake, plus the
ratios of volumes (assumed to be constant) of the lake’s conceptual cells. Hydraulic or
routing parameters are not needed for the level-pool lake routing method.
Initial conditions can be either be entered via a spreadsheet as a model input, or they can
be estimated using a specified ‘warm-up’ period. During this period the dynamic
boundary conditions are assumed steady-state at those of the specified start time of the
simulation. For systems where the response time is significantly smaller than the
specified output time-step, this latter option is likely to be a sufficient approximation.
Model parameter values are entered on another spreadsheet. Templates are given for each
alternative model structure, making it clear which parameters are relevant for each option.
Figure 4.5 A typical format of time-series data input.
All model parameters, initial conditions, static boundary conditions, and point and
distributed loads can be considered as uncertain inputs. Prior to running the model, the
user signifies that an input is uncertain by specifying a maximum and minimum value
instead of an assumed value. For the time-series inputs, all entries in each time-series are
assumed to have the same level of uncertainty, and this may be specified as either
absolute or relative. For example, in Figure 4.5, all entries in the flow time-series have
93
prior uncertainties of ±30%, and all entries in the BOD time-series have prior
uncertainties of ±25%. Specified maximum and minimum values define uniform prior
distributions which are assumed independent of each other. Each distribution is
propagated to prediction uncertainty (Section 4.7), or included in the calibration or
sensitivity analysis. This means that the model calibration and predictions need not be
conditional on the precision and reliability of input data, and that the relative significance
of uncertainties in parameters and in other inputs can be revealed through sensitivity
analysis (Section 4.6).
4.5 Calibration and optimisation
WaterRAT has a number of options for automatic model calibration. There are three
alternative algorithms – Monte Carlo using Latin Hypercube or stratified random
sampling (see MacKay et al. 1979), a Monte Carlo Markov Chain algorithm (based on the
Metropolis algorithm -see Chapter 2), and a genetic algorithm (based on the descriptions
of Beaseley et al. 1993). Each of these alternative methods has advantages depending on
the nature of the calibration task, and depending on whether uncertainty or sensitivity
analysis is required.
An important feature of WaterRAT is its capability to estimate the uncertainty in the
calibrated, optimal parameter set by defining a posterior parameter response surface (i.e.
values of probability mass for a large number of parameter set samples), using the Monte
Carlo algorithms. From this response surface, marginal probability distributions of
parameters and their co-variance matrix can easily be derived, and used for regional
sensitivity analysis and risk evaluations (e.g. Portielje et al. 2000). Figures 4.6a and 4.6b
give examples of bi-variate response surfaces of parameters n and Hs, and ks and τcr from
the phosphorus model calibration in Chapter 6.
In addition to the choice of calibration algorithms, WaterRAT provides flexibility in
choice of the objective function used to define the likelihood response surface. This
includes selection of which determinands, which cells and which time-periods are to be
incorporated into the objective function. Objective function definition also includes
alternative likelihood functions (see Sorooshian and Gupta 1995) based on assumptions
about the nature of the data errors, and more subjective estimators of likelihood based on
94
the HSY method of Hornberger and Spear (1980) and the GLUE methodology of Beven
and Binley (1992).
0.05
0.05
0.10
0.15
0.150.10
0.20
Manning’s n
Sedi
men
t dep
th H
s
0.00030
0.00027
0.00023
0.00020
0.00017
(a)
0.05
0.05
0.10
0.15
0.150.10
0.20
Manning’s n
Sedi
men
t dep
th H
s
0.00030
0.00027
0.00023
0.00020
0.00017
(a)
0.000250.00023
0.00022
0.00020
0.00019
10 15 2520 30
0
1.0
0.8
0.6
0.4
0.2
Scour rate ks
Cri
tical
shea
r st
ress
τcr
0.00020
(b)
0.000250.00023
0.00022
0.00020
0.00019
10 15 2520 30
0
1.0
0.8
0.6
0.4
0.2
Scour rate ks
Cri
tical
shea
r st
ress
τcr
0.00020
(b)
Figure 4.6 Example of bi-variate response surface output of WaterRAT (taken from calibration described in section 6.4). The contours are interpolations of point values of probability mass.
Ideally, the objective function used to calculate the GLUE likelihood should reflect the
perceived data and model structure error so that, for example, a higher data error will lead
to an objective function which discriminates less between sets of factors. As discussed in
Chapter 1, achieving this objectively is difficult due to the complex and largely unknown
nature of the errors and, whatever approach is taken, some post-conditioning appraisal of
95
the OF specification is needed. This appraisal should take account of the number of sets
of factors which the OF has defined to be successful, and the performance of the
conditioned model with respect to the available data (i.e. are enough of the data explained
by the estimated uncertainty in the results?). On both these accounts, the adequacy of the
OF is dependant on the adequacy of the model structure and the data, and review of these
is a parallel part of uncertainty description and reduction. On this basis, an iterative
approach to model structure choice, data interpretation and OF design is suggested in
Figure 4.7.
Calibration dataavailable?
Redesign model structure
GLUE
Yes
No
Retrievable fault in model
structure ?YesNo conditioning
NoYes
No
Yes
No
Yes
No
OF* design anddata interpretation
* OF = objective function
Synthesisesome data
Start
End
Knowledge-basedoutput constraints?
Adequate stochasticdescription of data ?
Adequate number of valid parameter sets?
Calibration dataavailable?
Redesign model structure
GLUE
Yes
No
Retrievable fault in model
structure ?YesNo conditioning
NoYes
No
Yes
No
Yes
No
OF* design anddata interpretation
* OF = objective function
Synthesisesome data
Start
End
Knowledge-basedoutput constraints?
Adequate stochasticdescription of data ?
Adequate number of valid parameter sets?
Figure 4.7 An iterative approach to model conditioning with WaterRAT
In WaterRAT, calibration data (and their error bounds if these are entered) from the
monitoring points are entered into spreadsheets (one spreadsheet per monitoring point).
During automatic calibration, WaterRAT will search these entries for data that is included
in the specified objective functions. The relevant observed data (plus any specified error
96
bounds) are included in the graphical report of calibrated model output, so that the
success of the calibration can be visualised.
Using the calibration algorithms introduced above, pollution sources and other boundary
conditions can be optimised to meet water quality targets, defined as “pass or fail”
objective functions. Using the uncertainty analysis capability allows the risk of failing to
meet these targets (due to uncertainty in the other inputs) to be evaluated for different
intervention options. This is demonstrated in Chapter 7.
The observed water quality data on which the model is to be conditioned are input as
time-series in a similar manner to the pollution load inputs (although no synchronisation
of this data is required). To allow for the uncertainties introduced by sampling error, and
other errors, the data uncertainty can be specified. It is specified individually for each
data point as a relative or absolute error bound, so that the errors are taken as uniform and
independent. This uncertainty can then be employed in the calculation of the objective
function used to condition the model (see sections 7.2.2-7.2.3).
Unless a series of sample values has been provided in the appropriate spreadsheet, the
modeller is required to prescribe distributions or mean values for the model parameters.
The parameters pertinent to the chosen model structure are listed in a reference
spreadsheet together with default upper and lower bounds which are based on the
modelling literature (see McIntyre and Zeng 2002).
4.6 Multi-objective analysis
WaterRAT allows up to four objective functions to be simultaneously calculated during
calibration. Multi-objective analysis also allows the sensitivities of different modelling
criteria to be simultaneously evaluated and compared (e.g. Bastidas et al. 1999). Also, it
has been shown that using multiple objective functions for calibration can indicate model
equation error and the resulting prediction uncertainty (e.g. Gupta et al. 1998). For
example, in calibration of a model representing interactions between biochemical oxygen
demand (BOD) and dissolved oxygen (DO), this could reveal the disparity between the
optimum model of DO, and that of BOD (McIntyre et al. 2001). Such a disparity would
indicate a fault in the model’s representation of the interactions between the two.
Similarly, a marked difference between optimal parameter values identified using
97
seasonally-exclusive objective functions, would indicate a misrepresentation of seasonal
dynamics.
Following calibration using multiple objective functions, WaterRAT can filter out the
Pareto-optimal solutions. These are the solutions that provide a valid compromise
between the alternative objectives, and the variability of the Pareto-optimal solutions
represents the disparity of the objectives. If the objectives should be commensurate (as in
the BOD-DO model calibration example above) then disparity indicates an error either in
the data or in the model equations. On the other hand, assuming errors are relatively
minor, WaterRAT can be used to expose the necessary trade-offs between different
management objectives, and to explore acceptable compromises. For example, the trade-
off between maximising abstractions and minimising risk to downstream chemical status
(both of which can be formulated in a simple manner into objective functions) can be
assessed. WaterRAT, therefore, has potential application to catchment management
planning (e.g. UK Environment Agency 2001b, 2002).
4.7 Sensitivity analysis
Sensitivity analysis is implicit to model conditioning (in which the sensitivities of the OFs
to the inputs are explored). However, using sensitivity analysis to its full benefit, for
example identifying the key causes of pollution, requires that measures of sensitivity are
evaluated and reported explicitly. Also, sensitivity analysis can be used as a screening-
level approach, whereby the parameter responses need not be rigorously evaluated, but
can be used to give useful indications of the main driving forces of the system.
WaterRAT contains a number of complementary approaches to sensitivity analysis.
The simplest option is first order sensitivity analysis. Using this, the effect of perturbing
each uncertain input between its specified upper and lower bound is reported for each
output variable as an absolute value, displayed in tabular form for any chosen date and
cell number within the domain of the simulation. Since the results are absolute measures
they give an approximation of the variations which might be observed given the input
perturbations, although this result is local to the chosen mean values of all other inputs.
To make the analysis more robust to local effects, it can be extended to a factorial
analysis (Henderson-Sellers and Henderson-Sellers 1993) which allows for two-factor
interactions.
98
The Monte Carlo-based calibration methods evaluate the response of an objective
function over the possible combinations of values of the input variables. Therefore these
methods can be used to report sensitivities which take account of the high order
interactions between variables which the factorial methods cannot, and which are not
centred around an arbitrary point in the parameter space. Furthermore, the data upon
which the objective function is based may be synthesised, so that sensitivity of
speculative or regulation-based objectives to the various model inputs can be evaluated.
For example, the Monte Carlo methods can be used to give an indication of which input
uncertainties are most likely to cause failure of future chemical status objectives.
As part of the Monte Carlo procedure, values of the in-river (or in-lake) calibration data
can be randomly sampled from within their specified error bounds. In essence, this means
that WaterRAT can measure the sensitivity of the objective function value to how well
that objective has been defined by the target data. As well as indicating priorities for
collection of calibration data (e.g. Chapter 7), this allows the significance of uncertainty
in future water quality objectives to be evaluated. For example, should there be
uncertainty in the level of dissolved oxygen needed to secure good ecological status, this
could be represented as error bounds on that target. The sensitivity analysis would then
indicate whether refining the definition of this objective (i.e. improving the chemico-
ecological model) would be a research priority.
One option for reporting the results of Monte Carlo-based sensitivity analysis is through
comparison of the post-calibration covariance matrix with the uncalibrated equivalent
(where it would generally be expected that a parameter to which the objective function
was sensitive would reduce in variance during calibration). WaterRAT also allows the
user to summarise the results of Monte Carlo-based calibration by the Kolmogorov-
Smirnov (KS) statistic. In this context, the KS statistic is the maximum distance
separating the calibrated marginal cumulative distributions of the factor values, and the
uncalibrated, uniform marginal distribution.
It should be noted that review of KS statistics and variance-covariance matrices gives a
summary of the results of a Monte Carlo-based sensitivity analysis, which does not make
full use of the information available. To supplement this summary, scatter plots
(projections of the point estimates of OF onto a plane) can be used to view in detail the
variation of the OF over the range of each factor (e.g. Beven and Binley 1992, Freer et al.
1996). For example, this can be used to review skewness, peakedness and multi-modality
of the univariate response. Additionally, bi-variate plots of OF values allow response
99
surfaces to be visualised, potentially highlighting the non-Gaussian natures of posterior
distributions and non-linear dependencies between parameters. As examples, Figures 4.6a
and 4.6b show bi-variate distributions of parameters n and Hs, and ks and τcr from the
phosphorus model calibration in Chapter 6.
4.8 Prediction uncertainty
Whereas sensitivity analysis can highlight which uncertain inputs are most likely to
influence the model results, prediction of space and time-series is needed to show where
and when this influence is significant. WaterRAT offers three basic methods of
propagating uncertainty to model predictions, each of which can be applied to either the
uncalibrated or calibrated distributions of inputs. Using the first order-second moment
method (see Tung 1996) and Rosenblueth’s two-point estimation method (Rosenblueth
1981, see Tung 1996), the propagation is based on the prior or calibrated covariance
matrix and mean input values. Using Monte Carlo sampling, a specified number of
samples are taken either from the prior uniform distributions of inputs, or from the sets
pre-sampled during calibration. In this latter case, the relative probability of the model
result obtained from each sample is the relative probability of that sample (as calculated
during calibration). In the case that multiple measures of posterior probability have been
used during calibration, one may be chosen to define the uncertainty propagated to
predictions, or two or more may be combined in Bayesian or possibilistic manner.
Alternatively, all Pareto-optimal solutions may be regarded as equally likely and
propagated on this basis.
The aforementioned methods of uncertainty propagation use probability theory to derive a
probability distribution of each determinant at each time-step and cell. Using the first
order-second moment method, only the first and second moments are computed, and
confidence limits are then calculated assuming either a normal or log-normal distribution.
Using Rosenblueth’s method, the same distributional assumptions are employed, although
the first three moments are computed, and if the skewness is found to be negative, then an
inverted log-normal assumption may be used. An obvious limitation of these assumptions
is that estimates of the extreme percentiles may be unreliable, and that the constraint of
non-negative concentration is neglected. Using Monte Carlo-derived results, no
assumptions need to be made about the probability distribution of results. Instead, a
histogram approximating the true shape of the probability distribution at each time-step
and cell is derived, and used to compute percentiles (although, depending on available
100
computer memory, there may be limitations on the number of time-steps and cells at
which the full set of Monte Carlo output data can be stored).
The use of probability theory for the derivation of output confidence limits assumes that
the calibration has produced valid estimates of probability mass for each sampled set of
model inputs. This may be in doubt, considering the inevitable subjectivity and
assumptions used in defining the likelihood function or the GLUE likelihood measure,
and limitations in sampling frequency (especially true for the Pareto-optimal set of
solutions which may be a very sparse sample of the Pareto-optimal population – see
Fonseca and Fleming 1995). Given these limitations, a less ambitious and more liberal
representation of uncertainty may be preferred. WaterRAT offers this, using the rules of
possibility (Zadeh 1977, Wierman 1996) applied within the Monte Carlo procedure.
Using these, only a possible range of outputs at each time-step and cell are reported,
defined by the maximum and minimum values (for each time-step and cell) recorded
during the Monte Carlo-based uncertainty propagation. In general, this gives much more
significance to extreme values. Again, this method may be used pre or post calibration.
4.9 Output
Output data are stored in text files, processed by WaterRAT’s Visual Basic modules and
viewed in a series of spreadsheets and graphs. Results can be displayed as,
• Time-series for any state variable at any cell, showing the mean, and upper and
lower percentiles (or possible ranges) plus any relevant observed data.
• Spatial variation of any state variable on any date, showing the mean and upper
and lower percentiles (or possible ranges) plus any relevant observed data (Figure
4.8 is an example from a river modelling study).
• The estimated probability density function and cumulative density function for
any variable, date and cell.
• For each determinant, the modelled probability of failing to meet a specified
water quality target within a specified stretch of river (integrating the uncertainty
and variability over time).
• For any determinant, the modelled probability of failing to meet a specified water
quality target, plotted against the value of any uncertain input.
101
• A list of the sampled values of all uncertain inputs following calibration, with
associated relative probabilities. This list can be used to illustrate parameter
response surfaces (e.g. Figures 4.7a and 4.7b).
• The covariance matrix of calibrated inputs derived using one selected objective
function.
• A list of the KS statistic, defined by up to four different objective functions, for
all uncertain inputs.
• A list of the Pareto-optimal solutions following Monte Carlo-based calibration
using multiple objective functions.
0
10
Cum
ulat
ive
dens
ity
0
1
Prob
abili
tyde
nsity
Chainage 5km
0
10
0
1Chainage 15km
0
20
0
1Chainage 25km
Cum
ulat
ive
dens
ity
Prob
abili
tyde
nsity
Cum
ulat
ive
dens
ity
Prob
abili
tyde
nsity
Probability distributions
mean90% confidence limitsdata
Spatial variation of nitrate 15th October
Cni
(mgN
/L)
0
10
Cum
ulat
ive
dens
ity
0
1
Prob
abili
tyde
nsity
Chainage 5km
0
10
0
1Chainage 15km
0
20
0
1Chainage 25km
Cum
ulat
ive
dens
ity
Prob
abili
tyde
nsity
Cum
ulat
ive
dens
ity
Prob
abili
tyde
nsity
Probability distributions
mean90% confidence limitsdata
Spatial variation of nitrate 15th October
0
10
Cum
ulat
ive
dens
ity
0
1
Prob
abili
tyde
nsity
Chainage 5km
0
10
0
1Chainage 15km
0
20
0
1Chainage 25km
Cum
ulat
ive
dens
ity
Prob
abili
tyde
nsity
Cum
ulat
ive
dens
ity
Prob
abili
tyde
nsity
Probability distributions
mean90% confidence limitsdata
Spatial variation of nitrate 15th October
Cni
(mgN
/L)
Figure 4.8 Example of graphical output of WaterRAT
4.10 WaterRAT review
4.10.1 General limitations
The overriding limitation of WaterRAT is that it is restricted to evaluation of single rivers
and lakes rather than systems of rivers and the wider catchment. Groups of pollution
sources must be represented by a point or distributed load at the river boundary, rather
than discriminating between types of origin, and so scope for pollution management is
restricted. For example, without an integrated pollution load model, only the relative
significance of different sewers and tributaries can be assessed rather than the pollution
load components. While various pollution load models were developed as part of
TOPLEM (Qinghua University 2001), they were not integrated with WaterRAT, so that
the model identification and sensitivity analysis methods described above could not be
applied to the ‘whole’ system.
102
A further limitation of WaterRAT is the restricted number of river water quality models
available. While the provided selection of model structures gives some flexibility in
approach and allows model structure uncertainty to be explored somewhat, and the choice
of determinands allows a variety of tasks to be considered, there is a range of river
models that could be added to extend and strengthen these virtues. This could include
both more complex formulations, for example to look at new transient transport models
(e.g. Lees et al. 2000, Sincock and Lees 2002), and empirical formulations, for example
statistical models, static regression models (e.g. Robson and Neal 1997) and time-series
models (e.g. Whitehead et al. 1997b). For application to UK catchment management
problems, new state variables would be justified, for example coliforms and pesticides
(see DEFRA 2002). Even for the Hun River study, around which the tool was designed,
the importance of metals and organic toxins (Xianxin and Yongjiu 1991) is not reflected
in the nominal toxic substance model. The framework has been developed envisaging that
the library of models will be extended to include improved models – for example priority
is to include improved pollutant transport using the aggregated dead zone model and/or
transient storage model (see Young and Wallis, 1993) – and to meet the demands of
future modelling tasks,for example higher dimensionality, coupling of groundwater
models and GIS interfaces.
Another potentially valuable improvement to WaterRAT would be the provision of a
more powerful genetic algorithm, which could be used to estimate parameter
uncertainties (e.g. Vrugt et al. 2003), as opposed to the existing version which is designed
to converge to a global optimum with no significant representation of uncertainty.
Development of an uncertainty analysis capability of WaterRAT’s genetic algorithm is
considered by Lai 2002. Although WaterRAT includes the Monte Carlo Markov Chain
that was demonstrated in Chapter 2, its value relative to GLUE was not proven within this
research.
4.10.2 Critical comparison with alternative modelling tools
WaterRAT is now compared with three other, prominent tools for water quality
modelling, model uncertainty analysis and risk-based pollution management: QUAL2E-
UNCAS (Brown and Barnwell 1987), SIMCAT (UK Environment Agency 2001a) and
DESERT (Ivanov et al. 1996, de Marchi et al. 1999). These tools were introduced in
Chapters 1 and 3. All of them are more advanced than WaterRAT in that they have
relatively advanced windows interfaces, and are able to simulate systems of rivers. All are
the same as WaterRAT in that they do not extend to modelling catchment runoff,
103
groundwater or urban sewerage systems but represent their effects as point or distributed
sources to the river(s). All use Monte Carlo simulation to represent uncertainty in some
model parameters and pollution loads.
QUAL2E-UNCAS has various model structure options, including two choices of
hydraulic model (similar to those in WaterRAT), light extinctions models and aeration
formulae (for both of which there is no choice in WaterRAT). It does not have sediment-
water interaction, oil or ice modelling options but has all the other state variables of
WaterRAT, and in addition it models nitrites and coliforms. It allows a slightly more
restricted representation of channel cross-sectional shape than WaterRAT (see Chapter 5).
QUAL2E-UNCAS can only model steady-state loads, although it can model water quality
dynamics due to meteorological diurnal variability. Although it allows propagation of
uncertainty using Monte Carlo or first order methods, it does not have a facility for
automatic sensitivity analysis, conditioning or optimisation algorithms. It does not allow
task-specific models to be developed and inserted into its framework, nor does it allow
user-specified numerical tolerance.
SIMCAT has no choices for model structure, instead using fixed, relatively simple
formulations. It models carbonaceous BOD, ammonia, a conservative substance and
dissolved oxygen under steady-state conditions. Effects of sediment are not explicitly
considered. It has an advanced calibration routine which automatically balances flow in
each reach by adding distributed sources/losses, and which identifies parameter values on
a reach-by-reach basis. At each reach it uses sampled observations to identify samples of
parameter values, thus representing uncertainty in parameters due to the effects of data
sampling error. The key assumption is that all the uncertainty in the model can be
represented by the parameter uncertainty arising from the calibration data sampling error.
It also estimates uncertainty in calculated percentiles. Like WaterRAT, SIMCAT allows
pollution load variability to be represented by sampling from distributions (log-normal or
normal in SIMCAT; uniform in WaterRAT), or as a series of samples. Like QUAL2E-
UNCAS, SIMCAT does not enable automatic optimisation of pollution sources, nor user-
specified models to be used, nor does it allow user-specified numerical tolerance
DESERT has taken the extreme approach to providing choice of water quality model
structure and determinands, by providing an interface in which the modeller can write the
formulations. A number of pre-written models are also provided, and an extensive choice
of hydrodynamic/transport modules are also provided, from fully mixed steady-state
reaches increasing in complexity to a diffusion wave model. The modeller can also write
104
cost functions for pollution load reductions and optimise these against water quality
constraints using dynamic programming. Automatic calibration in DESERT is performed
using the HSY algorithm introduced in Chapter 2 (i.e. the same as WaterRAT’s
possibilistic use of GLUE), although uncertainty in boundary conditions is not
represented during the calibration, the calibration must be done in a steady-state period,
and the dependencies between parameter distributions are not considered. It is not clear
from the documentation whether the optimisation using dynamic programming takes into
account parameter uncertainty.
It may be said that each of these tools has its own merits – QUAL2E-UNCAS has ready-
made mechanistic models and basic uncertainty analysis modules, SIMCAT focuses on
ease of use and robustness to data uncertainty, and DESERT has extensive modelling
flexibility and optimisation capabilities. WaterRAT was an attempt to integrate such
features in a manner consistent with the remit of the TOPLEM project and the particular
needs of the end-users in Shenyang. The discussion of possible developments of
WaterRAT in Chapter 8 will draw from the examples set by these other tools.
4.11 Summary
A surface water quality modelling tool has been described which provides a framework
for the extensive analysis of uncertainty and associated risk. This includes established
methods of uncertainty estimation including first order methods, Regional Sensitivity
Analysis, Generalised Likelihood Uncertainty Estimation, and multi-objective
optimisation. Using these methods, the model can be conditioned to include the effect of
all sources of error, and the uncertainty can be propagated to stochastic forecasts.
Additionally, the sensitivity of model outputs to inputs can be explored thoroughly to
indicate the key driving forces (e.g. pollution sources) behind water quality. This can be
done using first order, factorial or Monte Carlo methods, at a screening level or part of a
more rigorous scenario investigation. Given a stochastic forecast, the modeller can
explore the risk of failing to achieve regulatory water quality criteria.
The tool includes a library of semi-distributed one-dimensional river models allowing
some flexibility in specifying model structure, and some capacity to explore the
uncertainty associated with model structure. Within the library, the tool has the capability
to model organic pollution, chlorophyll-a, dissolved oxygen, various nutrients, a toxic
substance, floating and suspended oil, and total suspended solids. This is supported by
105
sediment models which include biochemical and physical sediment-water interactions. A
thermodynamic model is available which models heat fluxes from the atmosphere and
sediment, and includes simulation of ice. Pollution transport is modelled using the
advection-dispersion equation supported by two alternative hydraulic models (a quasi-
steady friction formula and a non-linear store). The framework has been developed
envisaging that these libraries will be extended to meet the demands of future modelling
tasks, for example higher dimensionality, coupling of groundwater models, dead-zone
analysis etc.
An important limitation of this tool is that the measurement of uncertainty, and therefore
the risk that should be associated with using a chosen model, is based partly on the
subjective judgement of the modeller with respect to the relative reliability of the model
and the observed data. This is because unknown and irresoluble model structure error and
data bias always exist to some extent in practical water quality modelling problems.
Giving general guidelines for uncertainty definition does not seem possible given the
variety of prospective case studies.
106
5. Numerical efficiency in Monte Carlo simulations – a case study of a
river thermodynamic model
Trade-offs between precision of numerical solutions to deterministic models of the
environment, and the number of model realisations achievable within a framework of
Monte Carlo simulation, are investigated and discussed. A case study of a model of river
thermodynamics is employed. It is shown that the tractability of Monte Carlo simulation
relies on adaptation of the numerical solution time-step, giving results with a guaranteed
error in the time domain as well as near-optimum speed of calibration under any chosen
accuracy criteria. Time-step control is implemented using two adaptive Runge-Kutta
methods - a second order scheme with first order error estimator, and an embedded
fourth-fifth order scheme. In the case study, where the effects of sparse and imprecise
data dominate the overall modeling error, both the schemes appear adequate. However,
the higher order scheme is concluded to be generally more reliable and efficient, and has
wide potential to improve the value of applying the Monte Carlo method to
environmental simulation. The problem of reconciling spatial error with the specified
temporal error is discussed.
107
5.1 Introduction
Despite the increasing availability of parallel processing facilities, the computing restraint
is an important limitation in the practice of Monte Carlo simulation due to the large
number of model and data realisations which may be required. Consequently, the
numerical efficiency of the simulation can be a key factor in the overall feasibility and
value of the modelling exercise. In particular, the trade-off between numerical error (due
to the numerical approximations used in the simulation) and computing time should not
be arbitrary; rather it should be made objectively and in light of the particular modelling
task. Arguably, Monte Carlo simulation loses much of its analytical value if insufficient
attention is paid to this trade-off.
This chapter focuses on the need for effective Monte Carlo simulation when model
parameters must be calibrated to a sparse set of data, and on the benefits to be gained by
awareness of underlying numerical issues. This is pursued through studying dynamic
river water temperature simulation of the Hun River. This study provides a good setting
for the investigation because of the high a priori uncertainty in many of the model
parameters, the limited amount of supporting data, and because of the arduousness of the
numerical solution to the model’s partial differential equations. Numerical efficiency of
the Monte Carlo-based calibration procedure is investigated through comparison of
alternative solution schemes with respect to spatial and temporal discretisation.
As part of the Hun River water quality modelling programme, a thermodynamic model is
required to simulate the day to day fluctuations in water temperature. The parameters of
this model are to be calibrated to water temperature data. The water temperature in the
Hun River was measured at four river cross-sections on one day every month from
October 1999 until March 2000 and again in June 2000. The frequency of measurement
was thus restricted due to resource constraints and difficult field circumstances. The four
sections, marked on Figure 1.4, are at river kilometers 44 (i.e. 44km downstream of the
reservoir dam); 72, 135 and 185. For each measurement location and time, the
temperature was taken at one-third of the water depth at each of the quartiles of the river
width, then these three measurements were averaged. To represent the diurnal
temperature variation, the width-averaged temperature at river kilometer 72 was
measured every 4 hours on each monthly sampling day. For this numerical investigation,
the most interesting and challenging period is the winter period when warm wastewater
discharging to the Shenyang reach contrasts with sub-zero air temperatures, causing
exchange of heat which is rapid and of high numerical order.
108
5.2 The thermodynamic model
The Hun model is a one-dimensional dynamic model, that is the water temperature is
assumed constant over the depth and width of the river, and only the longitudinal and
temporal variations are simulated. The following heat transfer processes are considered
(see Ashton 1986, Chapra 1997); advection and dispersion of the river flow; point
pollution sources; long wave radiation to and from the atmosphere, sky and surrounding
land; short-wave radiation from the sun; convection and conduction to and from the
atmosphere; conduction to and from the river bed; conduction to and from ice;
evaporation losses and condensation gains.
The transport of heat in the river is modelled using the control volume approach (Chapra
1997: p.192) whereby the river is conceptualised as a series of instantaneously mixed
cells each of uniform temperature, flow and depth. The flow in the i th cell at time-step j,
Q(i,j), is assumed equal to the flow entering it from the upstream cell, Q(i-1,j), plus the
concurrent sources, Qs(i,j), minus the concurrent losses, Ql(i,j), and evaporation, Qev(i,j), in
that cell,
),(),(),(),1(),( jievjiljisjiji QQQQQ −−+= − (5.1)
in which all terms have units m3s-1. The temperature of the water leaving each cell is
assumed equal to that within the cell, and so the rate of increase in temperature, due to
advective transport of heat only, is given by,
( ) [ ] [ ] [ ] [ ] ),(),(),(),1(),(
jiwljipsjiwjiwji
w TQTQTQTQdt
TVd⋅−⋅+⋅−⋅=
⋅− (5.2)
where the specific heat capacity and the density of the river and the pollution sources are
assumed constant; V is the volume of water in the cell (m3); Tw is the water temperature
(oC) in the cell; and Tp is the temperature of the pollution source (oC). The dispersive and
convective heat transfer between cells is assumed to be directly proportional to the
difference in temperature,
109
( ) [ ] [ ]),(),1(),1(),(),1(),(),(
'' jiwjiwjijiwjiwjiji
w TTDTTDdt
TVd−+−=
⋅++− (5.3)
where D’ (m3s-1) is a dispersion coefficient, calculated as a function of water velocity and
depth (Chapra 1997: 245).
The other heat exchange processes are modelled to act uniformly across each cell, and are
based on the descriptions of Bras (1990) and Chapra (1997). The short-wave radiation
reaching the water (or ice) surface, fs (Wm-2) is calculated by,
)65.01( 2casf ss −⋅= (5.4)
where s (Wm-2) is the component of the daily average of the short-wave radiation which
is incident on the outside of the atmosphere above the subject site in a direction radial to
the earth; c is the effective cloud cover as a ratio of the area of open sky; and as is an
atmospheric transmission coefficient, the calculation of which is described by Bras (1990:
p.35). The calculation of long-wave radiation reaching the water (or ice) surface, fl (Wm-
2) is based on the Stefan-Boltzmann law,
( )( ) ( )445.0 273273031.0 +⋅−++= waairll TmBTeaBf (5.5)
where B is the Stefan-Boltzmann constant (Wm-2K-4); eair is the vapour pressure of the air
(mmHg); al is a transmission coefficient (dimensionless); Ta is the air temperature some
meters above river level (oC); m is a coefficient of emissivity of water = 0.97
(dimensionless); and Tw is the average water temperature (oC). The net heat gain due to
conduction and convection at the water-air interface, fc (Wm-2) is calculated as,
( )waWc TTkbf −⋅= (5.6)
where b is Bowen’s coefficient (0.47 mmHgK-1) and kW is a wind dependent heat transfer
coefficient (Wm-2mmHg-1) which is assumed to be described by the empirical
relationship,
295.019 Wkw += (5.7)
110
where W is the average wind speed (ms-1) measured 7m above the water surface. Heat
loss due to evaporation (or gain due to condensation), fe (Wm-2) is calculated by Dalton’s
law, or when the air temperature is below 0 oC, by the Russian winter formula (Ashton
1986),
( ) 0for >−= Taeekf airsatWe (5.8a)
( )( )( ) 0for95.2263.004.6 ≤−+−+= aairairsawe TeewTTf (5.8b)
from which Qev in Equation 5.1 is calculated as,
ww
eev ld
fQ = (5.9)
where eairs is the saturation vapour pressure of the air (mmHg); dw is the density of water
(kgm-3); and lw is the latent heat of evaporation of water (Jkg-1). Conduction from the ice
surface fiw (Wm-2) is calculated using an effective heat transfer parameter, kiw (W m-2K-1)
(from Ashton 1986),
( )wiiwiw TTkf −= (5.10)
where Ti is the temperature of the ice at the water-ice interface = 0oC. Calculation of the
conduction from the sediment fsw (Wm-2) conceptualises a layer of sediment through
which the temperature varies linearly from that of the underlying sediment, Ts, to that of
the water, Tw. This concept is a simplification of the more theoretical sediment heat
gradient through a homogeneous sediment (see Hondzo and Stefan 1994). The heat flux
from the sediment is,
( )wsswsw TTkf −= (5.11)
where ksw (Wm-2K-1) is an effective heat transfer parameter lumping together the
sediment-water transfer, the sediment thermal conductivity and the conceptual sediment
layer thickness.
Exchanges of heat directly between the atmosphere and the water are assumed to occur
only over the area of open water area in the river cell. That is, in frozen conditions the
111
water is insulated from the air, and the ice reflects or absorbs all radiation. Conversely,
the exchange between the ice and the water only occurs over the iced area in that cell.
Thus, the rate of temperature increase by cell i at discrete time j is,
( ) ( )( )( ) ( )[ ]
[ ] [ ] [ ] [ ][ ] [ ]),(),1(),1(),(),1(),(
),(),(),(),1(
),(),(
''
111
jiwjiwjijiwjiwji
jiwljipsjiwjiw
jiwswiwiwisecwlswwji
w
TTDTTD
TQTQTQTQ
AfAAfAAffRffsddt
TVd
−+−+
⋅−⋅+⋅−⋅+
++−−+−+=⋅
++−
− (5.12)
where Aw is the surface area of the water; Ai is the area of ice as a fraction of Aw; sw is the
specific heat capacity of water (Jkg-1K-1); and Rw is the water reflectance (dimensionless)
which is assumed the same for both short and long-wave radiation.
To predict the periods, locations and extent of ice cover, a simple ice model was included.
This is based on a heat balance approach which assumes linear heat gradients between the
air and the ice, and the ice and the water. Adapted from Shen and Chaing (1984) and Lal
and Shen (1991), the rate of increase of ice thickness is modelled as,
( ) ( ) ( )( )
−+−−+−
+=
−
ilswiiwaiaii
i
ii
i RffTTkTTkk
Hlddt
dH111
1
(5.13)
where Hi is the thickness of ice (m); di is the density of ice (kgm-3) and li is the latent heat
of melting of ice (Jkg-1); ki is the conductivity of ice (Wm-1K-1); kai is an air-ice heat
exchange rate (Wm-2 K-1); Ri is the reflectance of the ice. Ice growth is assumed to be not
initiated in cells with flow of velocity above 0.6 ms-1 (Ashton 1979). This simple ice
model does not explicitly represent a number of physical ice processes such as ice floe
transport, and frazil ice formation and deposition (see Ashton 1986). In particular, the
latter omission means that the modelled water temperature may fall slightly below zero,
as heat loss to the sub-zero atmosphere due to Equation 5.12 is not instantly transformed
into frazil ice production; rather there is a lag determined by parameter kiw through which
it is gradually transformed into increased ice thickness. The spread of ice across each
river cell cannot be represented conceptually because of the one-dimensional limitation,
so instead an empirical relationship is proposed. For Hi > 0,
−=
i
Aicei H
kA exp (5.14)
112
where kAice (m) is an empirical parameter, so that Ai rises to a maximum of unity as the ice
thickness becomes high.
5.3 Monte Carlo simulation
The model parameters included in Equations 5.4 to 5.14 may be classified according to
knowledge of their values prior to model calibration i.e.; 1. physically based parameters
of which the values are known, or can be measured with adequate certainty (includes m,
b, B, di, li, ki, dw, sw, lw, al); 2. conceptual or spatially averaged parameters of which the
values cannot be precisely measured but some range of possible values, based on
measurement or prior modeling experience, can be used to constrain the a priori
parameter space (includes c, Rw, Ri); 3. conceptual or empirical parameters of which there
is no prior knowledge, and of which the prior ranges should be made wide enough so they
do not constrain the a posteriori parameter space (includes kai, kiw, ksw, kAice). According to
this classification, the parameters and their assumed values or ranges are listed in Table 1
(from Ashton 1986, Lal and Shen 1993, Chapra 1997). For the purpose of this study, it is
assumed that the reach characteristics used for calculating water velocity v and water
surface area Aw, and the formula used for calculating dispersion D’ (Chapra 1997: 245)
are accurate.
A simple automatic method of model calibration and uncertainty analysis is random
sampling of the a priori parameter ranges, i.e. Monte Carlo simulation. Using this
method, the 'goodness of fit' (of model results to observed system response) which is
associated with each sampled parameter set is measured according to a pre-specified
objective function. If this function is suitably specified, the array of objective function
values obtained from the calibration can be regarded as point values of probability mass
from the a posteriori parameter distribution, and of the associated model result.
Confidence limits on the model parameters and the model outputs can then be derived.
Such an approach to model uncertainty estimation is justified by Beven and Binley (1992)
in the context of their Generalised Likelihood Uncertainty Estimation (GLUE). The
GLUE approach is used in the current study, using the objective function defined in
Equation 5.15:
113
( )rK
mmkwwk TTKL
−
=
−= ∑
63,1
2, (5.15)
Table 5.1 Thermodynamic parameter values and a priori ranges Parameter Description Value or range Units Ref. Parameter classification 1 - Values assumed known with certainty m emissivity of water 0.97 none C
B Stefan-Boltzmann constant 56.63 × 10-9 Wm-2K-4 C
b Bowen’s coefficient 0.47 mmHgK- C
di density of ice 917 kgm-3 L
li latent heat of melting of ice 0.334 × 106 Jkg-1 L
ki conductivity of ice 2.24 Wm-1K-1 L
dw density of water 1000 kgm-3 L
sw specific heat capacity of water 4182 Jkg-1K-1 L
lw latent heat of evaporation of water 2.5 × 106 Jkg-1 L
al long-wave attenuation coefficient 0.6 none C
Parameter classification 2 – Possible ranges supposed on some prior basis c cloudiness 0 - 1 none
Rw reflectivity of water 0 – 0.1 none A
Ri reflectivity of ice 0.2 - 0.75 none A
Parameter classification 3 – Possible ranges supposed without prior basis kai transfer from air to ice 5 - 50 Wm-2K-1
kiw transfer from ice to water 20 - 200 Wm-2K-1
ksw transfer from sediment to water 20 - 200 Wm-2K-1
kAice ice coverage coefficient 0.05 – 0.15 none C = Chapra 1997; L = Lal and Shen 1993; A = Ashton 1986
where Lk is the posterior probability of the kth sampled parameter set, ( ) mkww TT ,− is the
mth of the 63 residuals between observed water temperature wT and modelled water
temperature wT obtained from the kth sampled parameter set, Kr is a root constant which
defines the variance of the a posteriori distribution (taken as equal to 1 for this
application), and K is a standardisation constant such that the sum of all Lk values is equal
to unity. Equation 5.15 defines a statistically-based likelihood function which assumes
that the data errors are unbiased, normally distributed, constant and uncorrelated over
time and space; and that the model equation errors are small in comparison. The latter
assumption significantly simplifies the uncertainty analysis because we can neglect the
hypothesis that Equations 5.1 to 5.14 are wrong in structure, and define a series of trial
114
models by sampling from the a priori joint parameter distribution. There is no allowance
for numerical truncation error in the uncertainty analysis, as our aim is to manage this
error to be as large as possible without having overall significance to model reliability.
The a priori uncertainty in seven model parameters is defined by their ranges in Table
5.1; no assumption is made about their correlation. 6561 samples from this distribution
are taken using stratified random sampling (MacKay et al. 1979) (6561=38, three
stratifications for each parameter), and for the purpose of this investigation, it is assumed
that this gives adequate coverage of the parameter space.
5.4 Numerical methods4
For a finite difference model which includes both temporal and spatial variability (such as
our river model), accuracy and efficiency depend on the temporal and spatial grid sizes,
and the use of appropriate integration schemes. The simplest and most common approach
to implementing a finite difference solution is to specify a fixed grid size prior to the
simulation. However, consistently adequate accuracy demands that the fixed grid size be
designed such that the approximate solution is sufficiently close to the actual solution in
its most varying part, irrespective of the smoother regions where a much larger grid size
would achieve the required accuracy. Therefore, within any realisation of a model, the
preferred grid size is likely to vary significantly over the domain of integration. The
problem is compounded when using the Monte Carlo method, as the dynamics of the
simulation depend largely on the randomly sampled parameter values, and so the
preferred grid sizes are inherently random from one realisation to the next. To ensure
convergence of the solution using a single grid size for all realisations, the grid size will
need to be inordinately small. This often leads to a demand on computational resources
that is several orders of magnitude above the optimum. Such inefficiency is likely to be
compounded if there are discontinuities in the model structure (e.g. Equation 5.8), or in
the driving forces, since these generally require minute grid sizes to maintain the required
accuracy. Furthermore, since it is not usually known a priori, determination of an
appropriate fixed grid size may require additional human and computer effort. An
informal trial and error approach to this can leave us in some doubt as to the reliability of
final results.
4 The text of much of this section (5.4) is adapted from the work of Bethanna Jackson, in McIntyre et al. (2003).
115
All of these difficulties can be overcome by the use of an adaptive scheme. Such a
scheme automatically reduces the grid size when the truncation error is undesirably high,
and increases it when this error is unnecessarily low. A good implementation of an
adaptive scheme will not only guarantee a solution to a specified accuracy, but also
achieve this in a near-optimum time period. The step control mechanism should recognise
and handle any discontinuity in time, since the crossing of such will register as a large
error. It then recursively lowers the grid until it lands sufficiently close to the
discontinuity. We note that automatic step size control has been marked by many
numerical analysts (e.g. Gear 1971, Gustafsson 1993) as being the single most important
means of making an integration method efficient.
Since the accuracy of the river model is dependent on both the spatial and temporal
discretisation, a completely adaptive scheme should monitor and vary the grid in both
time and space. However, the difficulties associated with producing reliable and
representative measurements over the space dimensions for processes such as this make
adapting the spatial grid-lengths problematic. A second, more general concern is the
complex inter-relationships between different points in the spatial mesh. These tend to
produce errors of higher magnitude when a grid varies in space (Carver 1976). Due to
these complexities, it seems practical to have a predetermined spatial grid, and apply a
method that is adaptive in time only. This is achieved through transformation of the
system of partial differential equations to a system of ordinary differential equations
which step forward in time. This approach is generally known as the “method of lines”
(Berezin and Zdidkov 1965), commonly applied in the context of river modelling as the
“method of control volumes” (Chapra 1997: p.192). One complication with the method of
control volumes, solved using a backward-space difference method (implicit in Equation
5.12), is the presence of numerical dispersion, which tends to smooth out differences in
concentrations or temperature from one cell to the next (Chapra 1997: p.201). In this
example, we have chosen neither to calibrate a dispersion parameter nor to adjust the
calculated dispersion D’ to allow for numerical dispersion. Therefore it would be
expected that a compensation for numerical dispersion would be reflected in the
distribution of calibrated parameters.
To pre-determine an efficient solution scheme, we need a measure of computational cost
for comparative purposes. A convenient measure of this is the total number of function
evaluations, which, for the thermodynamic river model, is a constant quantity per time-
step. Therefore, the obvious aim is to minimise the number of time-steps. The maximum
permissible error (tolerance) must be specified and suitably related to the step size and
116
actual error per step. This relation gives rise to an expression we refer to as the step-
controller. Its aim is to monitor the error over each step, and estimate a new step length to
carry the integration as far forward in time as is possible within accuracy constraints.
The checks and step size predictions rely on manipulation of the lowest order error terms.
Since a numerical method of order p integrates a solution exactly up to the pth order, the
error term contributed over any step is a combination of all terms of higher order, that is,
the truncated terms in the Taylor’s series expansion of the function (Butcher 1987). If we
take one step forward to find the value of dependent variable β at time tj, β (j) = β (j-1)+∆t(j)
given a solution at β (j-1), the exact error contribution jζ over the jth step (the local error)
is,
∑∞
+=
− ∆Ω=
1
)()(
)1()( !pi
ij
iji
j itβ
ζ (5.16)
where each Ωi is a problem dependant error constant, and )()1(
ij−β denotes the ith derivative
of β at time step j-1.
If the step size ∆t is sufficiently small, the lowest order term remaining will dominate the
higher order terms to such an extent that they can be treated as negligible. This first term
is called the principal truncation error, and is directly proportional to the (p+1)th
derivative of the solution of α, and the (p+1)th power of the step size. The local error can
therefore be approximated by,
1
)()1(
)1(++
− ∆Ω= pj
pjj tβλ (5.17)
where Ω = Ωp+1/(p+1)! from Equation 5.16. Taking an arbitrary value for the maximum
permissible error over each step (denoted by ξ), the optimum step size at time t(j) is
approximately that for which ε(j) = ξ. Denoting this “near-optimum” step size by ∆t’(j), we
find from Equation 5.17 that
11
)()(
')(
)(
1)(
)1()1(
1)( )'(
++
+−
+
∆=∆⇒
∆=
Ω=∆
p
jjj
j
pj
pj
pj tt
tt
λξ
λξ
βξ (5.18)
117
The step controller takes the error estimate ε at the end of each step, and checks that this
is within the specified tolerance bounds (-ξ, ξ). If the estimate is outside this bound, the
step is rejected and recalculated with a lower step size. On acceptance, the error estimate
is assumed to be equal to the principal truncation error. It is then assumed that β (p+1) does
not change significantly between t(j) and t(j+1). The optimal value of ∆t can then be
estimated by calculating the step size giving λ(j) = ξ subject to these assumptions. The
step controller becomes,
11
)1()1()(
+
−−
∆=∆
p
jjj tt
λξ (5.19)
This is generally multiplied by a safety factor η to reduce the chance of overestimating
the maximum permissible step size, and performing a rejected step. The resultant step
controller is
11
)1()1()(
+
−−
∆=∆
p
jjj tt
λξη (5.20)
Since the necessity for a numerical solution precludes ready availability of the true
solution, a cheap, sufficiently accurate estimation of the truncation error is sought.
Generally, this means finding a second numerical solution which is considerably more
accurate than the first, so that the difference between the two is a good approximation of
the truncation error. A viable approach involves using methods of order p and (p+1)
which share the same function evaluation points. The cost then reduces to the difference
between obtaining the order (p) solution alone, and calculating the (p+1) solution along
with it. This error estimate can be combined with a suitable tolerance, and used by the
step-controller to govern the solution.
It should be noted that the tolerance should be carefully chosen such that misleading
accuracy is not generated. With an exact spatial representation, a robust adaptive scheme
should be capable of integrating a solution to arbitrarily high accuracy (subject only to the
computer’s machine precision). However, our time integration is along a solution path
which may have been significantly perturbed by spatial errors. Since the adaptive
temporal scheme is following the perturbed solution path rather than the exact one, any
118
precision surplus to that of the spatial representation places a purposeless burden on
computational resources.
1234567891011121617
444855606568728090100110120130140150160170185
River kilometer defining reach boundaries
River reach number15 14 13
Point thermal loadsPoint thermal loads
Upstreamboundary
Down-streamboundary
1234567891011121617
444855606568728090100110120130140150160170185
River kilometer defining reach boundaries
River reach number15 14 13
Point thermal loadsPoint thermal loads
Upstreamboundary
Down-streamboundary
Figure 5.1 Spatial grid
For the Hun River study, the spatial grid is made up of 17 reaches of river shown in
Figure 5.1 and each reach has four cells, giving 68 control volumes in series. Following
the discussion above, the grid will be automatically adapted only in the time domain. The
derivatives defined by Equations 5.12 and 5.13 are numerically integrated using a first-
second order adaptive scheme and a fourth-fifth order adaptive scheme (henceforth
referred to as (1,2) and (4,5) schemes respectively) for comparison. The second order
approximation is commonly known as Heun’s method (see Chapra and Canale 1998:
p.688),
)1(),()1,(
)1,(),( 5.0 −−
− ∆
++= j
ji
w
ji
wjiwjiw t
dtdT
dtdT
TT (5.21)
where ∆tj-1 is the adapted step size computed for time-step j-1; the first derivative is
defined by Equations 5.12 and 5.13 derived at time-step j-1, and the second is similarly
derived at time-step j using the approximation Tw(i,j) = Tw’(i,j) given by,
)1()1,(
)1,('
),( −−
− ∆
+= jji
wjiwjiw t
dtdT
TT (5.22)
The truncation error over any step is then estimated by the magnitude of the difference of
Equations 5.21 and 5.22. This is divided by Tw and maximised over all i to give the worst
relative error, λj, over the all N river control volumes at time-step j,
119
−=
= ),(,1)(
'ABSmax
jiw
ww
Nij T
TTλ (5.23)
λj is then used as a basis for adapting ∆t using the integral controller described previously,
and the calculations of Equations 5.21, 5.22 and 5.23 are repeated until the desired
tolerance ξ is achieved. Note that the estimated truncation error is that of the first order
solution, and we are guaranteed improved accuracy because the second order evaluation
of Tw is adopted. The (1,2) algorithm is illustrated in Figure 5.2.
Start End
Yes
( ) ( )( )
( )11,
1,'',, allFor −∆
+=
−− jt
dtdTTTi
ji
wjiwjiw
( )1,evaluate , allFor
−
ji
w
dtdTi
Set boundary conditions for t(j)
( )( )jiw
ji
w Tdt
dTi ,,
'' usingevaluate , allFor
( ) ( )( ) ( )
1,1,
1,'
, 5.0, allFor −∆
++=
−− jt
dtdT
dtdTTTi
ji
w
ji
wjiwjiw
( )( ) ( )
( )
−= '
,
'',
',
allfor ABSmax
jiw
jiwjiw
ij T
TTλ
( ) ? Is ξλ ≤j
( ) ( )'
,, , allFor jiwjiw TTi =( ) ( )( )
)1/(1
11 9.0+
−−
∆=
p
jjj t∆t
λξ
( ) ( ) ( )11 −− ∆+= jjj ttt
1+= jjSet boundary conditions for t(j-1)
( ) ( ) ( ) ttTTttj jwjwj ∆=∆=== −−− arbitrary : initial : initial:1 111
( ) ( )( )
)1/(1
19.0+
−
∆=
p
jjj t∆t
λξ
( ) 000189.0 Is >jε
( ) ( )14 −∆= jj t∆t
No
No Yes
Finished simulation ?Yes
No
( ) ( )( )prjj ttt ∆∆=∆ ,min
Start End
Yes
( ) ( )( )
( )11,
1,'',, allFor −∆
+=
−− jt
dtdTTTi
ji
wjiwjiw
( )1,evaluate , allFor
−
ji
w
dtdTi
Set boundary conditions for t(j)
( )( )jiw
ji
w Tdt
dTi ,,
'' usingevaluate , allFor
( ) ( )( ) ( )
1,1,
1,'
, 5.0, allFor −∆
++=
−− jt
dtdT
dtdTTTi
ji
w
ji
wjiwjiw
( )( ) ( )
( )
−= '
,
'',
',
allfor ABSmax
jiw
jiwjiw
ij T
TTλ
( ) ? Is ξλ ≤j
( ) ( )'
,, , allFor jiwjiw TTi =( ) ( )( )
)1/(1
11 9.0+
−−
∆=
p
jjj t∆t
λξ
( ) ( ) ( )11 −− ∆+= jjj ttt
1+= jjSet boundary conditions for t(j-1)
( ) ( ) ( ) ttTTttj jwjwj ∆=∆=== −−− arbitrary : initial : initial:1 111
( ) ( )( )
)1/(1
19.0+
−
∆=
p
jjj t∆t
λξ
( ) 000189.0 Is >jε
( ) ( )14 −∆= jj t∆t
No
No Yes
Finished simulation ?Yes
No
( ) ( )( )prjj ttt ∆∆=∆ ,min
Figure 5.2 First-second order adaptive time-step algorithm
The (4,5) scheme is an embedded Runge-Kutta, or Fehlberg method developed by Cash
and Karp (1990) (also see Chapra and Canale 1998: p.713). Methods of this type are
widely accepted as among the most competitive methods for many differential equation
systems. In principle, it is similar to the method of the (1,2) scheme, using two separate
numerical approximations to gain an error estimate: in this case, the absolute difference
between fourth and fifth order solutions of Tw. Since both estimates share the same
function evaluation points, the second estimate is of negligible additional cost. This (4,5)
method has the benefit of providing a more reliable estimator of the true temporal
truncation error than the lower order scheme. Its added reliability along high-order
120
solution paths makes larger time-steps permissible, but for each of these, there are four
more intermittent derivative evaluations. The preferred scheme in this application
depends on the relative importance of the high order processes in Equations 5.12 and
5.13, which will be investigated by experiment.
5.5 Results
The model calibration is first done using the (4,5) scheme using a tolerance, or specified
maximum temporal truncation error, of 0.2. This value is a ratio of the absolute estimated
truncation error to either the high order evaluation of temperature or 0.1oC, whichever is
higher (the lower limit avoids convergence to inappropriately low tolerances at
temperatures near zero). While this specified maximum error may seem relaxed, it is
quite justifiable for various reasons, as will be seen later. The posterior probability
associated with each of the 6561 parameter sets was used to derive confidence limits for
the water temperature at the Hun Gate (river kilometer 72), shown in Figure 5.3. These
confidence intervals were derived from the variance of the 6561 model results at each
time-step assuming a normal distribution. Although Figure 5.3 does not validate the
model (because the same data were used for calibration), it shows how the uncertainty
analysis is used to represent the variability in the data.
5
10
15
20
25
Wat
er te
mpe
ratu
re (
o C)
0
Observed water temperature90% confidence limitMaximum likelihood result
20/1
0/99
20/1
1/99
20/1
2/99
20/0
1/00
20/0
2/00
20/0
3/00
20/0
4/00
20/0
5/00
20/0
6/00
5
10
15
20
25
Wat
er te
mpe
ratu
re (
o C)
0
Observed water temperature90% confidence limitMaximum likelihood result
20/1
0/99
20/1
1/99
20/1
2/99
20/0
1/00
20/0
2/00
20/0
3/00
20/0
4/00
20/0
5/00
20/0
6/00
20/1
0/99
20/1
1/99
20/1
2/99
20/0
1/00
20/0
2/00
20/0
3/00
20/0
4/00
20/0
5/00
20/0
6/00
Figure 5.3 Time-series of modelled and observed water temperature at the Hun Gate
121
Consider the significance of the adaptive numerical scheme in obtaining these results.
Figure 5.4 is a scatter-plot of the simulation run-times during the calibration using the
higher order scheme (the cyclical trend in the run-times is due to the procedural ordering
of the stratifications used in the sampling procedure). Figure 5.5 shows the time-series of
daily average time-steps for the most time-consuming simulations.
0
10
20
30
40
50
0 1000 2000 3000 4000 5000 6000Simulation number
Tim
e ta
ken
for s
imul
atio
nTi
me
take
n fo
r sim
ulat
ion
(sec
s)
50
40
30
10
20
0
Simulation number10000 2000 3000 4000 5000 60000
10
20
30
40
50
0 1000 2000 3000 4000 5000 6000Simulation number
Tim
e ta
ken
for s
imul
atio
nTi
me
take
n fo
r sim
ulat
ion
(sec
s)
50
40
30
10
20
0
Simulation number10000 2000 3000 4000 5000 6000
Figure 5.4 Scatter of run-times during calibration by stratified sampling
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Date
Day
ave
rage
d tim
e-st
ep (s
ecs)
First-second order
Fourth-fifth order
20/10
/99
20/1
1/99
20/1
2/99
20/0
1/00
20/0
2/00
20/0
3/00
20/0
4/00
20/0
5/00
20/0
6/00
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Date
Day
ave
rage
d tim
e-st
ep (s
ecs)
First-second order
Fourth-fifth order
20/10
/99
20/1
1/99
20/1
2/99
20/0
1/00
20/0
2/00
20/0
3/00
20/0
4/00
20/0
5/00
20/0
6/00
Figure 5.5 Profile of time-steps during the most time-demanding simulation
These illustrations suggest that, due to the range of time-steps required, the use of an
adaptive scheme is fundamental to the feasibility of the calibration. For example, if the
122
minimum time-step from Figure 5.5 of 0.006 days were used (and this may be regarded as
a liberal estimate because it is a day-averaged value, not the actual minimum) the 6561
simulations would have taken over 10 days instead of under 12 hours. Alternatively, if a
feasible constant time-step of 0.1 days is used, which is considered practical for
performing 6561 simulations, then the results of a large number of realisations are ruined
by numerical instability, mainly during the numerically onerous period of freeze. If a
constant time-step of 0.006 is used, but the number of parameter set samples is reduced to
500, then stability is achieved, but comparing repeated results with those obtained using
more comprehensive sampling implies that 500 samples is inadequate to give reliable
confidence limits. Again, it is noted that the adequacy of 6561 samples is not investigated
in this chapter.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.2 0.4 0.6 0.8 1Specified maximum relative truncation error
Ach
ieve
d tru
ncat
ion
erro
r
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1Specified maximum relative truncation error
Tim
e ta
ken
for s
imul
atio
n (s
ecs)
First-second order scheme
First-second order schemeFourth-fifth order scheme
Fourth-fifth order scheme
(a) (b)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.2 0.4 0.6 0.8 1Specified maximum relative truncation error
Ach
ieve
d tru
ncat
ion
erro
r
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1Specified maximum relative truncation error
Tim
e ta
ken
for s
imul
atio
n (s
ecs)
First-second order scheme
First-second order schemeFourth-fifth order scheme
Fourth-fifth order scheme
(a) (b)
Figures 5.6(a) and 5.6(b) Comparison of the performance of the alternative adaptive schemes
Using the (1,2) scheme with the same specified tolerance, there was no significant
difference in the derived confidence limits, nor in the total time required for the
calibration. Although smaller time-steps were generally required for the (1,2) scheme,
only two derivative evaluations are required per time-step as opposed to six in the higher
order scheme. However, close inspection of the performance of the schemes under a
variety of tolerance criteria implies that the higher order scheme is potentially much more
efficient. Figures 5.6(a) and 5.6(b) show how the achieved tolerance and the time taken
for the simulation vary against the specified tolerance. These results are based on a single
simulation using the set of maximum likelihood parameter values, and use a specified
maximum tolerance of 0.0001 to approximate the numerically ‘true’ solution. In terms of
accuracy, the (4,5) scheme outperforms the (1,2) scheme by an order of magnitude for all
specified tolerances. The former is also faster for all specified tolerances above 0.2. Note
123
that the actual errors are up to 20 times lower than the specified tolerances for the higher
order scheme, implying that processes higher than fifth order are numerically
insignificant. In contrast, for the (1,2) scheme, the actual error is in cases larger than the
specified tolerance, highlighting the scheme’s limitations, and the importance of the step-
controller, both in terms of the robustness of the control mechanism itself, and its reliance
on a good error estimate. Since the error estimate of the (1,2) scheme is only first-order
accurate, it is substantially more vulnerable to changes in the smoothness of the solution
than the Cash-Karp estimate. The flattening of the curves in Figure 5.6(a) is because of
the upper limit to the computation time-step of 1 day.
Such exploration of numerical performance has been extremely valuable in improving the
functionality of the river temperature model, and in allowing reliable calibration and
uncertainty analysis to be performed. For example, there clearly is no justification in
specifying the tolerance as anything less than 0.2 when using the (4,5) scheme. Any lower
tolerance will be generating excess accuracy in the time domain, overshadowed by the
errors generated by the spatial grid, and those due to the variability and sparseness of the
calibration data. This accuracy demand is comparatively relaxed, and accordingly we find
that the (1,2) scheme appears to give adequate results at this tolerance when compared to
the overall uncertainty in model results. This (1,2) scheme can be regarded as a simple
and easily implemented method of avoiding numerical instability but does not give any
useful guarantee of truncation error. It is also expected to become increasingly less
competitive as other errors are reduced, for example by extension to the data set or by
refinement of the spatial grid, and increased demands are made on the temporal precision.
The higher order scheme is more robust and reliable in general, at least as fast, and has
wider applicability in modelling of high order processes.
In the river model described here, and many dynamic environmental models, accuracy is
limited by spatial as well as temporal truncation errors. In order to achieve a broad
understanding of the model’s numerical error, the spatial discretisation should be subject
to critical investigation. The spatial grid used in this study is variable in resolution along
the length of the river. In theory, this variation could be automated in a similar manner to
the temporal scheme, but the computation required to estimate the truncation error over
the whole time domain is not practical. Instead, an appropriate grid is identified prior to
calibration by the modeller by successively sub-dividing the grid and running individual
tests. For example, consider how the grid used above (4 cells per reach in Figure 5.1 = 68
cells in series) performs in comparison to a less refined grid (one cell per reach = 17 cells
in series) and a more refined grid (16 cells per reach = 272 cells in series). The model
124
parameter values which required the smallest time-step are used. The spatial model
results (up to the Hun Gate) for the water temperature on the 20th January are shown in
Figure 5.7(a), and Figure 5.7(b) shows the reach-averaged water temperatures. At the
Hun Gate it is found that the refinement of the spatial grid did not make a significant
difference to results. On the other hand, from the same figures, the spatial truncation error
is significant in the reaches further upstream, which contain the Shenyang point sources
of heat. While the model can assign confidence limits to all results, these are based on the
performance of the model at locations significantly downstream of the thermal loads,
where truncation error is low (e.g. the Hun Gate), and cannot reliably account for high
spatial truncation errors elsewhere. Thus, the stochastic model in its 68-cell form is not a
reliable predictor of the temperature profile within the Shenyang reaches during the
winter. However, even in these reaches, the adaptive temporal scheme is useful in
ensuring maximum efficiency given the spatial grid, and allows investigation of spatial
error without the complication of error interactions. While the model gives a reasonable
prediction (numerically at least) of the reach-averaged value in all reaches, whether or not
this is adequate for the task of supporting the Hun River water quality model is a matter
for further research.
-1
0
1
2
3
4
5
6
44 49 5459 64 69
River kilometer
Wat
er te
mpe
ratu
re (o C
)
-1
0
1
2
3
4
5
6
49 5459 64 69
River kilometer
Wat
er te
mpe
ratu
re (o C
)
72 44 72
(a) (b)
Reach-averaged TwCell-specific Tw
1 cell per reach4 cells per reach16 cells per reach
1 cell per reach4 cells per reach16 cells per reach
-1
0
1
2
3
4
5
6
44 49 5459 64 69
River kilometer
Wat
er te
mpe
ratu
re (o C
)
-1
0
1
2
3
4
5
6
49 5459 64 69
River kilometer
Wat
er te
mpe
ratu
re (o C
)
72 44 72
(a) (b)
Reach-averaged TwCell-specific Tw
1 cell per reach4 cells per reach16 cells per reach
1 cell per reach4 cells per reach16 cells per reach
1 cell per reach4 cells per reach16 cells per reach
1 cell per reach4 cells per reach16 cells per reach
Figures 5.7(a) and 5.7(b) The limitation of the spatial accuracy in the Shenyang reaches.
5.6 Discussion
The relevance and benefits of attention to numerical efficiency have been illustrated by
this case study, specifically in the derivation of the temperature time-series shown in
Figure 5.3 through use of an adaptive numerical grid. The derivation of the confidence
limits in Figure 5.3 employed random sampling of the wide range of parameter values
listed in Table 5.1, resulting in a numerical grid requirement which was highly
125
inconsistent over the domain of the sampling, illustrated by Figure 5.5. Thus it is argued
that the adaptive grid is beneficial, if not necessary. Furthermore, Figure 5.6 clearly
illustrates that the numerical accuracy should be specified conservatively with respect to
the overall model uncertainty, otherwise the simulation is much more likely to be
inhibitively computationally expensive. While this chapter has focused on the calibration
stage, the benefit of the specification of the adaptive grid and the required accuracy
applies also to predictive application of Monte Carlo simulation.
It is argued for the case study that the high a priori uncertainty in the model parameters,
as well as the high order dynamics of the modelled system, have led to the high
variability in required time-step during the model calibration. However, if the overall
model uncertainty is relatively low, a higher numerical accuracy is justified, and
variability of the time-step may be equally as high or, noting the exponential rise in
computation time for low truncation errors in Figure 5.6(b), even higher. This would be
the case when using an a posteriori parameter distribution which has been significantly
constrained by the calibration, i.e. in relatively well-defined environmental systems.
Furthermore, it may be noted that the results presented here have significance beyond
simple Monte Carlo simulation. A variety of evolutionary methods of parameter
calibration are in common use in environmental modelling, for example genetic
algorithms (e.g. Mulligan and Brown 1998), Markov chain modelling and shuffled
complex evolution (e.g. Thyer et al. 1999). All these employ an automatic semi-random
exploration of the parameter space which leads to unpredictable numerical behaviour.
The derivation of reliable stochastic model results is an extremely challenging task,
especially in the field of complex environmental modelling. For the conciseness of this
work, it has been necessary to make assumptions which, in general, would be detrimental
to the reliability of results. One already mentioned assumption is that 6561 stratified
random samples gives adequate coverage of the parameter space. Other limiting
assumptions are that the set of Equations 5.1 to 5.14 are correct, that the model boundary
conditions (e.g. air temperature) are errorless, and that the data errors are independent,
unbiased and normally distributed. However, inspection of Figure 5.3 shows that on 20th
March, all 6 data points lie to one side of the model result (and the lower 90% confidence
limit), implying that their errors are not independent or that the model structure is
incorrect. This present work is regarded as an essential preliminary step in addressing
such difficulties.
126
For deterministic environmental models that are considerably larger than the case study
model, perhaps with 2 or 3 spatial dimensions and hundreds of parameters and state
variables, the need for attention to efficiency in Monte Carlo simulation is especially
relevant. Whereas it might intuitively be expected that more complex models should
provide more accurate deterministic predictions, hence reducing the motivation for Monte
Carlo simulation, many investigations imply that this is untrue and that best-estimate
results of such models can be ambiguous (see, for example, the discussions of O’Connell
and Todini 1996, Reichart and Omlin 1996). Given that important decisions and policies
are supported using complex environmental models, one of the primary challenges facing
numerical modelers is the improved representation of uncertainty. As such models are
relatively costly to solve, a fundamental aspect of this challenge is the application of
efficient solution schemes and tolerances that are consistent with the overall prediction
accuracy. While, in some cases, the pursuit of this will be less straightforward than it was
for the case study, and the achievable computational cost will be higher, this study has
sought to demonstrate the degree of improvements that may be possible.
5.7 Summary
Monte Carlo-based methods of uncertainty estimation are central to the value of
environmental simulation modelling because data are generally too sparse and imprecise
to usefully identify a single representative model. Using a study of river water
temperature simulation, it has been shown that the value of Monte Carlo analysis, in
terms of its ability to explore the feasible models, depends heavily upon the selection of
appropriate numerical solution schemes and tolerances. In particular, the implementation
of an efficient adaptive time-step procedure has considerable benefits in handling the
variability of the time-step requirements over the time domain of individual realisations,
and in handling the inherent randomness of this variability from one model realisation to
the next. For the case study, identification of model parameter uncertainty was carried out
over-night as opposed to the several days it would have required with a fixed time-step
for practically equivalent numerical precision. The case study also illustrated the
potentially large benefits in conservative specification of the numerical precision as the
achieved precision may be significantly more than that specified, and may be completely
adequate given the uncertainty stemming from limitations in supporting data. The spatial
truncation error arising from the fixed spatial grid is noted as restricting the value of
adaptive time-step schemes in solution of space-time partial differential equations
127
problems, such as the case study. Further work is needed to rationalise spatial errors in
the context of overall modelling error.
128
6. Identification of a phosphorus mobilisation model: prior evaluation
of data needs
A simulation model of in-stream phosphorus mobilisation and transport, which was
developed to predict monthly phosphorus export from the upper Hun River, is described.
In order to evaluate alternative programmes for collection of in-river calibration data, a
set of a priori computational experiments are devised. Initially assuming that the model
and boundary condition data are error-free, daily in-river phosphorus concentration data
are synthesised, representing an idealised system response. Scenarios of error in the data
and model structure are then improvised, as a hypothetical representation of the
conditions under which the model will actually be calibrated and evaluated. The model is
calibrated under the various conditions of error, and its predictive reliability is tested
using an independent idealised data set. These controlled experiments allow evaluation of
sampling programme design, of other controls on model reliability, and of needs for
additional a priori investigations. The results indicate that the value of the calibration
data is seriously compromised by the presence of error in the pollution load data, error in
the model structure, and inherent parameter equifinality. While these controls were not
very detrimental to model reliability under calibration conditions, in cases they caused
serious misrepresentation of forecast phosphorus export. For the case study, it is
recommended that sampling should initially be restricted to the first major storm event of
the year, and that other resources are directed at collecting improved data on phosphorus
sources to reduce model input error. Also, in light of the limited resources, expectations
of model performance should be reviewed, and a more robust approach to model
uncertainty estimation adopted.
129
6.1 Introduction
The phosphorus budgets of rural catchments are often dominated by runoff-induced
mobilisation driven by rainfall events which occur on hourly or daily time-scales
(Kronvang et al. 1997). Such system responses are too localised in time to be
comprehensively observed and formulated into a model using typically available data - if
undertaken at all, weekly or monthly monitoring of in-river phosphorus is normal. As
illustrated later, the resulting structural error and bias in estimated parameter values may
lead to very misleading predictions of phosphorus export, even when they are moderated
by uncertainty analysis. In such circumstances, investment in supplementary field
experiments is clearly important. However, it is unreasonable that investment should be
made without prior cost-benefit analysis. Even if a modelling exercise has a guaranteed
budget, the wide range of expenditure options (e.g. land-use surveys, collection of rainfall
data, in-river phosphorus data, and sediment data) means that some objective
prioritisation of data requirements and resource allocation is beneficial, if not essential.
The design of a field experiment is not easy prior to detailed understanding of the system
– nevertheless a working hypothesis of the system is needed for preliminary appraisal of
options. This leads to the view that some preliminary a priori modelling is a necessary
starting point for field experiment design. The objective of this chapter is to explore the
frequency and quality of data needed to identify, with a sought degree of reliability, the
structure and parameters of a one-dimensional conceptual model of river sediment
phosphorus mobilisation and transport.
One aim of the TOPLEM project was to estimate the monthly nutrient budgets of the Hun
River (under both current conditions and proposed intervention strategies) into the
Dahuofang reservoir. In the present study, we look at an upper reach of the river, from the
river-reservoir boundary to the Beikouqian gauging station, 40km upstream (Figure 6.1).
The catchment of this reach covers 2750km2, approximately 60% of which is forested,
15% is arable, 6% is pastoral, and the remainder is either urban or unfarmed moorland. It
is estimated that, in an average year, 800 tonnes of phosphorus from agricultural sources
is washed through this reach of river (Qinghua University 2001). The hydrograph for
1998 at Beikouqian, shown in Figure 6.2, illustrates that the wet season from July until
September dominates the flow regime. The maximum headwater flow is 239 m3s-1 on the
24th of August, with three other distinct runoff peaks, on the 16th of July, the 6th of August
and the 12th of August. The average headwater flow is 19m3s-1 and the mode is 2m3s-1.
The nutrient budget is known to be dominated by these storm events (Qinghua University
2001), ostensibly due to the mobilisation of nutrients stored in the soil (e.g. Kronvang et
130
al. 1997, Daldorph et al. 2000), and the scour of the river sediments and subsequent
release of nutrients (e.g. House and Denison 1998).
Southeast to Yellow Sea
Shenyang City
Fushun City
Dahuofang reservoir
Hun River
Beikoqiangauging station
N25 km
Beijing
NORTHEASTCHINA
KOREA
JAPANShenyang City
Hun River
Modelled length
Southeast to Yellow Sea
Shenyang City
Fushun City
Dahuofang reservoir
Hun River
Beikoqiangauging station
N25 km
Beijing
NORTHEASTCHINA
KOREA
JAPANShenyang City
Hun River
Modelled length
Figure 6.1 Location of modelled length of Hun River
As the processes controlling phosphorus fluxes occur predominantly in the wet season,
intensive monitoring of river phosphorus concentrations throughout this period is
expected to return valuable information about an appropriate model structure and set of
parameter values. However, as in most research projects, the Hun River study has
resource constraints, and sampling more frequently than weekly would be at the expense
of other important project goals. Therefore, rationalisation of the benefits and costs of
alternative phosphorus sampling programmes is needed. Similarly, spatial sampling errors
and other measurement errors could be reduced by focussing resources, but with benefits
which are not easy to justify a priori. In planning event-based monitoring there is also a
need to offset the risk of missing the key features of the event against the cost of
improving response times of personnel and equipment (at least in studies like that of the
Hun River, where automatic sampling is not practicable). This chapter describes
preliminary computational experiments which lead to hypotheses about the potential
value of investing in extra and/or improved data collection in the Hun River.
The computational experiments are based on incomplete, prior knowledge of the system
dynamics, so it cannot be expected that an optimum, or even near-optimum, monitoring
plan will be achieved. Rather, the objective is to indicate the risks of wasting resources by
either over-investing or under-investing in data collection, given the limitations of prior
knowledge and inevitable model identification problems. These indications will allow this
risk to be managed by considering strategies where scope for losses (unreturned
131
investment) is limited, while offsetting this conservatism against the desire for rapid
results. For example, a minimalist sampling programme could be used as a preliminary
study of system response leading, in conjunction with additional modelling experiments,
to refinement of the monitoring programme design before the next wet season. Promoting
interactions between the monitoring and data analysis procedures in such a way is widely
regarded as fundamental to progress in water quality modelling and management
(Somlyody 1995). This study is restricted to looking at frequency and quality of
measurements at one river cross-section (at the entry-point of the river to the Dahuofang
reservoir). In other cases, the approach presented here could be extended to spatial-
temporal investigations.
At all stages in the investigation, synthetic phosphorus data are used so that errors in data
and in the model structure are known a priori. Thus, the purpose is not to validate the
particular model employed. Instead, controlled experiments are used to elucidate the
significance of hypothetical yet credible scenarios of errors. Firstly, an error-free model
and error-free data given at the same frequency as the model output (i.e. daily) are
employed, whereby the only sources of parameter error are, 1) the limitations of the
automatic calibration algorithm in identifying optimum parameter values (see below), and
2) the interactions between parameters leading to equifinality of parameter sets (Beven
1993). The effects of these errors on the ability of the model to replicate the calibration
data, and to replicate a second independent set of error-free data, are illustrated. The
calibration data are thinned out to two-daily, then to the baseline frequency of weekly, so
that the bias introduced to parameters and model predictions can be reviewed, and the
value of using event data (i.e. from only the distinct storm events within the wet season)
is also evaluated. Unbiased Gaussian error is introduced to the calibration data and to the
phosphorus load, and then a component of the model is simplified and the significance of
this known structural error (as a nominal representative of other unknown structural
errors) is evaluated. Finally, the value of including sediment phosphorus data is
examined.
6.2 Model Description
The description of the model given below is brief, giving the minimum information
needed in the current context - a comprehensive model description is given in McIntyre
and Zeng (2002).
132
A key concept employed in the proposed model is that phosphorus exchange between the
sediments and the main body of river water (which we will refer to simply as the ‘water’)
is driven by two mechanisms. First, there is a diffusive exchange, whereby flux of
phosphorus per unit wetted area of sediment Fsp (gm-2s-1) is proportional to the difference
between the reach-averaged sediment concentration (Sp/Hs) and the reach-averaged water
concentration Cp,
−= p
s
pdsp C
HS
kF (6.1)
where Sp (gm-2) is the sediment concentration, Hs (m) is the average depth of the
responsive sediment layer, and kd (ms-1) is the diffusion rate. Secondly, there is net
release Rsp (gm-2s-1) comprised of resuspension which is initiated if a critical shear stress
at the channel bed τcr (gm-1s-2) is exceeded, and a first order sedimentation term,
τ for τ1ττ crcr
>−
−= pppssp CvSkR (6.2a)
τ for τ cr≤−= ppsp CvR (6.2a)
where τ is shear stress at the channel bed (gm-1s-2), ks is the scour rate (g-1m2s) and vp is
the sedimentation velocity (ms-1). This is based on the resuspension model proposed by
Lick (1982), also see Blom and Aalderink (1998). The shear stress is calculated using
Manning’s friction formula (Chow 1954: p.201),
3/1
22
rnugdw=τ (6.3)
where dw is the density of water (1000 gm-3); g is the gravitational constant (9.8 ms-2); u is
the water velocity at the sediment-water interface (ms-1), which is assumed equal to the
average water velocity over the river cross-section; n (sm-1/3) is Manning’s coefficient and
r (m) is the hydraulic radius.
The in-river transport model is based on the one dimensional advection-dispersion
equation, with water quality averaged over the river’s width and depth. The river is split
133
longitudinally into a series of reaches between which the P advects and disperses. Then,
the rate of change of phosphorus in the water in reach i at time-step j is given by5,
[ ] [ ] [ ][ ] [ ]
),(),(),(),(),1(),1(
),(),1(),(),(),1(),(
'
')(
jipjiwjispspjipjipji
jipjipjijipjipji
p
ARFCCD
CCDCQCQdt
CVd
Φ+++−
+−+⋅−⋅=⋅
++
−− (6.4)
where V (m3) is the time-variable volume of water in each reach , Q (m3s-1) is the flow, Aw
(m2) is the surface area of water in the reach, Φp (gs-1) represents external loads of P, and
D’ (m3s-1) is a river diffusion coefficient calculated as a function of the water velocity and
depth (see Chapra 1997: p.245). The cross-section of the river channel is represented by a
composite section with sloping sides (Figure 6.3), so that the water surface area is a
function of the flow, allowing reasonable simulation of Aw over the wide range of flows
observed in the Hun River. Backwater calculations are based on a quasi-steady model,
where flow reaches steady-state at every computational time-step, hence transient effects
on phosphorus transport and resuspension are not fully accounted for.
Date01
-Jan-9
8
01-F
eb-98
01-M
ar-98
01-A
pr-98
01-M
ay-98
01-Ju
n-98
01-Ju
l-98
01-A
ug-98
01-S
ep-98
01-O
ct-98
01-N
ov-98
01-D
ec-98
01-Ja
n-99
0
50
100
150
200
250
300
350
Flow
(cum
ecs)
0.00
0.05
0.10
0.15
Tota
l pho
spho
rus
conc
entr
atio
n
Flow (measured)
Total phosphorus concentration(simulated)
Date01
-Jan-9
8
01-F
eb-98
01-M
ar-98
01-A
pr-98
01-M
ay-98
01-Ju
n-98
01-Ju
l-98
01-A
ug-98
01-S
ep-98
01-O
ct-98
01-N
ov-98
01-D
ec-98
01-Ja
n-99
01-Ja
n-98
01-F
eb-98
01-M
ar-98
01-A
pr-98
01-M
ay-98
01-Ju
n-98
01-Ju
l-98
01-A
ug-98
01-S
ep-98
01-O
ct-98
01-N
ov-98
01-D
ec-98
01-Ja
n-99
0
50
100
150
200
250
300
350
Flow
(cum
ecs)
0.00
0.05
0.10
0.15
Tota
l pho
spho
rus
conc
entr
atio
n
Flow (measured)
Total phosphorus concentration(simulated)
Figure 6.2 Beikoqian 1998 daily flow (measured) and total phosphorus (simulated) used as inputs to the river model.
Intuitively, and evidently from Equation 6.2, the flux of phosphorus from the sediment to
the water depends on the concentration of phosphorus stored in the sediment, Sp, which is
also simulated.
5 In all other differential equations in this chapter, all terms should be taken to have subscripts i and j.
134
( ) wspspp
s ARFdt
dSA −−= (6.5)
where Aw (m2) is the area of submerged sediment. The distinction between the area of
sediment As (time-invariant) and the area of submerged sediment Aw (time-variable) is
non-trivial. Sediment-water interactions will occur over Aw, while the overall sediment
store of phosphorus extends over a potentially much larger area (illustrated in Figure 6.3).
This allows dry-bed storage to be accounted for in an approximate manner.
Low flow –net sedimentation
High flow –net resuspension
Sediment store of phosphorus:constant volume, dynamic concentration
b2 = 100m
b1 = 10m
d2 =5m
d1=1m
Water surface
Water surface
Low flow –net sedimentation
High flow –net resuspension
Sediment store of phosphorus:constant volume, dynamic concentration
b2 = 100m
b1 = 10m
d2 =5m
d1=1m
Water surface
Water surface
Figure 6.3 Channel cross-section and sediment-water interaction concepts. The composite cross-section, defined by effective parameters b1, b2, d1 and d2, allows a wide range of flow conditions to be simulated with restricted model complexity. The sediment store concept aims at representing dry-bed storage, but only wet-bed sediment-water interactions.
For this investigation, the 40km length of the Hun River is divided into ten reaches each
of 4000m length. The set of ordinary differential equations given by Equations 6.4 and
6.5 are solved using the Fehlberg scheme that was described in Chapter 5.
In summary, this model is based on the concepts that 1) sedimentation is directly
proportional to the concentration of total phosphorus in the water, 2) resuspension will
occur at a rate proportional to the submerged area of sediment, the energy (above a
threshold) provided by the movement of water, and the concentration of phosphorus
within the sediment, and 3) the availability of sediment phosphorus diminishes in times of
net resuspension, and increases in times of net sedimentation. Clearly, the realism of the
model is flawed, not least because the parameters u, ks, and τcr are assumed to spatially
and/or temporally constant, and variables Cp, Sp and τ are lumped into reaches. Among
other simplifications, the responsive area As and depth Hs of sediments do not respond to
the resuspension process (instead the concentration of Cp within a conceptual sediment
volume decreases) and the model in no way discriminates between the fates and
135
interactions of different fractions of phosphorus (see for example Boers et al. 1998).
There is also the issue of whether the quasi-steady hydraulic model is an adequate
replacement for transient approach (e.g. Zeng and Beck 2001). Such conceptualisations
have allowed a reasonably elegant and intuitive a priori model of phosphorus
mobilisation in the Hun River to be formed – but one which, due to its limited physical
basis, relies heavily on calibration for identification of representative parameter values.
6.3 The data
The average channel slope (0.1%) and cross-sectional shape are applied to each of the ten
reaches that make up the river model. The downstream boundary of the modelled length
of river (i.e. the reservoir) is represented as a constant water level of 4m above bed level.
The upstream boundary is defined by a daily time-series of Q and Cp (Qu and Cpu) which,
for 1998, are illustrated in Figure 6.2. These P data are simulated using a modified
version of the integrated catchment phosphorus runoff model of Daldorph et al. (2000)
(the land-use and daily rainfall data used in this runoff model were estimated from the
records of Shenyang Institute of Environmental Science, Liaoning). The use of daily Qu
and Cpu data dictates that our idealised system response is daily. The effect of uncertainty
in the Qu and Cpu data on the value of the downstream calibration data will be
investigated. As well as the upstream boundary, a diffuse load of Q and P (Qd and Cpd),
distributed uniformly over the 40km length, is applied. The total distributed load is
assumed to be one half of the headwater contribution, on the basis of the relative
catchment area of the 40km length of river.
Table 6.1 Prescribed feasible ranges of parameters and true values Parameter Notation Units Range True value
Scour rate (Eq. 6.2) ks ms-1 10-30 20
Scour rate (Eq. 6.10) ks ms-1 0.005-0.5 0.3
Critical shear stress τcr Nm-2 0.001-0.1 0.01
Critical velocity ucr ms-1 0.1-0.7 0.2
Sediment depth Hs m 0.05-0.15 0.1
Sedimentation velocity vp ms-1 0-2 1
Manning’s coefficient[1] n sm-1/3 0.05-0.15 0.1
Initial sediment conc.[1] Sp0 % ±25[2] 0 1Same value is used for each river reach. 2Deviation from simulated value on 30th June.
136
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
2-Jul
12-Ju
l
22-Ju
l1-A
ug
11-A
ug
21-A
ug
31-A
ug
Error-free calibration dataSample of calibration data with error
“True” model result
(a) daily calibration data (d) daily event calibration data (all events)
2-Jul
12-Ju
l
22-Ju
l1-A
ug
11-A
ug
21-A
ug
31-A
ug
(e) daily event calibration data (first event)
(b) 2-daily calibration data
(c) weekly calibration data
CP
(gm
-3)
CP
(gm
-3)
CP
(gm
-3)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
2-Jul
12-Ju
l
22-Ju
l1-A
ug
11-A
ug
21-A
ug
31-A
ug
Error-free calibration dataSample of calibration data with error
“True” model result
(a) daily calibration data (d) daily event calibration data (all events)
2-Jul
12-Ju
l
22-Ju
l1-A
ug
11-A
ug
21-A
ug
31-A
ug
(e) daily event calibration data (first event)
(b) 2-daily calibration data
(c) weekly calibration data
CP
(gm
-3)
CP
(gm
-3)
CP
(gm
-3)
Figure 6.4 Calibration data and “true” model result for 1998. Only one of five realisations of data error is shown.
As a baseline for studying the value of data, error-free in-river data are generated using
the phosphorus transport model together with assumed true parameter values, listed in
Table 6.1. These parameter values are the mid-points of ranges of values perceived to be
feasible (also see Table 6.1) so that the generated data represent a feasible, although
hypothetical and initially idealised system response. The data which are used for
calibration are those generated for the last river reach (river kilometre 40) between the 1st
July and the 31st of August, initially at a sampling frequency of 1 day (Figure 6.4(a)).
This period is chosen because it contains the significant rainfall events of both the studied
years. The calibration is repeated using the same data with increased sampling intervals
of 2 days, which still captures the storm events (Figure 6.4(b)), and 7 days, which
includes only one data point during each of the events (Figure 6.4(c)). Two subsets of the
daily calibration data set are also created (Figures 6.4(d) and 6.4(e)). The first of these
contains only the data within the first of the 1998 storm events, and the second contains
those data within all three of the significant 1998 events.
137
Following a series of model calibrations (described below) using these error-free data,
noise is introduced to the calibration data;
( )ipipip CCC 25.0,Norm' = i = 1, Nres (6.6)
where iP is the ith sample of error-free data, 'iP is the ith sample of noisy data,
( )σµ ,Norm signifies a random sample from a normal distribution with mean µ and
standard deviation σ , and Nres is the number of data points available (listed in Table 6.2).
Thus, the introduced error is Gaussian, unbiased, not autocorrelated, and has a time-
variable standard deviation proportional to the magnitude of the associated error-free data
point. Figures 6.4(a) to 6.4(e) include samples of the noisy time-series. Noise is also
introduced also to the 62 points of daily data which describe the load from upstream (i.e.
Cpu), and to that from the distributed source (Cpd), according to Equations 6.7(a) and
6.7(b).
( )ipuipuipu CCC 25.0,N' = i = 1,62 (6.7a)
( )ipdipdipd CCC 25.0,N' = i = 1,62 (6.7b)
This is to examine the effect on the calibration of possible error in these loads (which are
themselves the output of a simulation model). In looking at effects of randomly generated
noise, it is expected that results will depend upon the particular realisation of noise used.
To gauge this dependency, five alternative realisations are used, for each investigated
scenario of calibration data frequency.
To define the predictive task of the calibrated models, the upstream boundary condition is
changed to simulated data from July and August 1999, providing a significantly different
time-series of flow and phosphorus at Beikouqian, and a new set of idealised data are
generated using the true model parameters. The resulting true model result at 40km
downstream becomes the data to which the calibrated models are tested in forecasting
mode.
138
6.4 The calibration, prediction and performance evaluation
procedures
The parameters which, throughout the investigation, are considered uncertain and to be
calibrated are kd, ks, τcr, Hs, vp, and n. The initial condition of the sediment phosphorus
concentration Sp0 in each reach is also considered as an uncertain parameter, as only
sparse measurements are available. The ranges considered feasible prior to calibration are
given in Table 6.1, and during calibration these ranges are considered to be uniform and
independent distributions. Latin hypercube sampling (LHS) is used as the calibration
procedure. 5000 sample sets of parameters are taken in each calibration, and a simulation
is performed using each sampled set. The degree to which each of the corresponding
model results matches the calibration data is measured using the following sum-of-
squared-residuals objective function,
( )∑=
−=resNi
ikppk CCOF,1
2
, (6.8)
where OFk is the kth of 5000 sampled objective function values, ( )ikpp CC,
− is the ith of
the Nres residuals between the true phosphorus concentration pC and the modelled
concentration Cp ( pC is replaced by 'pC in the case of noisy data) obtained from the kth
sampled parameter set, and Nres is the number of data points in that particular data set (see
Table 6.2). For example, a high objective function value implies that the model result
gives a poor fit to the data, and the optimum parameter set is that which minimises the
objective function value. The main limitation of LHS as an optimisation procedure is that
the possible combinations of parameter values are sampled relatively sparsely. This
means that successive calibrations to one data set give alternative realisations of the
optimum parameter set, whereby the LHS procedure introduces its own source of error
into model results. The significance of this error compared with those arising from data
limitations is analysed as part of the discussion below.
The experimental procedure is summarised in Figure 6.5. For each calibration data set, a
series of ten independent calibrations are done, thereby identifying ten alternative
'optimum' parameter sets and corresponding time-series for July to August, 1998. These
parameter sets are then applied to modelling the 1999 response. The variation in the ten
139
optimum modelled time-series is an estimate of result uncertainty, and visual comparison
indicates the degree to which this estimated uncertainty explains the idealised data.
Daily
2-daily
Weekly
All events
1st event
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
Monitoringprogramme
Datarealisation
Calibration(10 x 5000 runs)
Prediction(10 x 1 run)
Performanceevaluation
1. Standard error2. Failure rate3. Bias4. Time-series
comprison
Daily
2-daily
Weekly
All events
1st event
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
Monitoringprogramme
Datarealisation
Calibration(10 x 5000 runs)
Prediction(10 x 1 run)
Performanceevaluation
1. Standard error2. Failure rate3. Bias4. Time-series
comprison
Figure 6.5 Experimental procedure. Five data realisations shown for the case of noisy calibration data. Only one data realisation is used for the case of error-free data.
At this stage, the model’s original task should be recalled – to estimate monthly
phosphorus loads to the Dahuofang reservoir. While confidence in the identified model is
related to comparisons of true and modelled time-series, it is not adequate to evaluate
proposed monitoring schemes (or our scenarios of data and model errors) solely on that
basis. In particular, the modelling purpose does not necessarily require that the low flow
phosphorus concentrations are simulated accurately, but that the monthly integration of
flow and concentration over time is adequate. Therefore, an analysis of errors in modelled
monthly exports also needs to be done. The modelled export is defined as,
∑ ⋅= pp CQE 4.86 (6.9)
where, in this context, Q (m3s-1) and Cp (gm-3) are daily modelled flow and phosphorus at
the downstream boundary (river kilometre 40), Ep (kg) is the modelled monthly
phosphorus export, and 86.4 effects the necessary unit conversions. Three measures of the
error in Ep are applied. Firstly, the standard error of Ep, for each calibration data scenario
separately, is calculated as the standard deviation of Ep for the corresponding set of ten
‘optimum’ modelled time-series. This is expressed as a percentage of the mean modelled
Ep, and represents the uncertainty in Ep due to difficulty of identifying a single optimum
140
model from the data. Secondly (again for each individual calibration data scenario), the
difference between the mean Ep and the known true value is expressed as a percentage of
the latter, representing the actual bias in Ep. The third measure of error is the percentage
of realisations of Ep which fail to fall within a specified tolerance of the true value (from
here on, such realisations are referred to simply as ‘failed results’). For the current study,
this tolerance is specified as %25± , which is considered necessary to drive a useful
model of the Dahuofang Reservoir. This proportion of realisations that lead to failure of
the performance criteria may be considered a measure of the probability of obtaining an
unhelpful outcome, should we choose to employ a single ‘optimum’ model.
6.5 Results, discussion and supplementary experiments
Under the various experimental conditions of data and model structure errors, Tables 6.2
to 6.4 list the three alternative evaluations of error in monthly exports, as defined above.
In Tables 6.3 and 6.4, where data error is involved, the range of results over the five data
error realisations is given. The results in parenthesis are those obtained using a genetic
algorithm for calibration (see below).
Figure 6.6 shows the ten optimum time-series of Cp identified using the trial sets of error-
free calibration data shown in Figures 6.4(a) to 6.4(d). Figures 6.6(a), 6.6(c), 6.6(e) and
6.6(g) show the 1998 results (i.e. performance under calibration conditions) and Figures
6.6(b), 6.6(d), 6.6(f) and 6.6(h) show the 1999 results (i.e. performance in predictive
mode). The respective wet season hydrographs are included in Figures 6.6(i) and 6.6(j).
Regarding Figure 6.6(a), the limitations of the LHS calibration procedure have caused
some uncertainty in the optimum model result (note that the model is known to be
capable of perfectly fitting the true data). While this uncertainty is quite minimal under
calibration conditions, Figure 6.6(b) indicates that there is significant divergence of
solutions in predictive mode. This indicates that the information retrieved during
calibration has lost relevance under the changed boundary conditions.
While it might be supposed that an improved search procedure will reduce the prediction
error, the degree to which this is true depends also on the significance of equifinality of
parameter sets under calibration conditions. That is, even if a more efficient global search
algorithm is employed, a unique optimum parameter set, which allows a unique forecast
of the future, still may not be found. To investigate the significance of this effect, the
experiment using the error-free data was repeated, this time using a genetic algorithm that
141
is part of the WaterRAT tool (McIntyre and Zeng 2002). This algorithm evolves a sample
of parameter sets until their objective function values (from Equation 6.8) converge to
within a specified tolerance of the sample optimum (a sample of 100 and a tolerance of
0.1% were used in this case). The average number of simulations required as part of this
procedure was approximately 8100.
2-Jul
12-Ju
l
22-Ju
l
1-Aug
11-A
ug
21-A
ug
31-A
ug0
100
200
300
400
500
2-Jul
12-Ju
l
22-Ju
l
01-A
ug
11-A
ug
21-A
ug
31-A
ug
4(i) 1998 flow data 4(j) 1999 flow data
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
(a) 1998 result; daily calibration data
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
(c) 1998 result; 2-daily calibration data
(e) 1998 result; weekly calibration data
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
(g) 1998 result; event daily calibration data
(b) 1999 result; daily calibration data
(d) 1999 result; 2-daily calibration data
(f) 1999 result; weekly calibration data
(h) 1999 result; event daily calibration data
Phos
phor
us c
once
ntra
tion
(mg/
l0Ph
osph
orus
con
cent
ratio
n (m
g/l0
Phos
phor
us c
once
ntra
tion
(mg/
l0Ph
osph
orus
con
cent
ratio
n (m
g/l0
Flow
(m
3 /s)
2-Jul
12-Ju
l
22-Ju
l
1-Aug
11-A
ug
21-A
ug
31-A
ug0
100
200
300
400
500
2-Jul
12-Ju
l
22-Ju
l
01-A
ug
11-A
ug
21-A
ug
31-A
ug
4(i) 1998 flow data 4(j) 1999 flow data
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
(a) 1998 result; daily calibration data
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
(c) 1998 result; 2-daily calibration data
(e) 1998 result; weekly calibration data
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
(g) 1998 result; event daily calibration data
(b) 1999 result; daily calibration data
(d) 1999 result; 2-daily calibration data
(f) 1999 result; weekly calibration data
(h) 1999 result; event daily calibration data
Phos
phor
us c
once
ntra
tion
(mg/
l0Ph
osph
orus
con
cent
ratio
n (m
g/l0
Phos
phor
us c
once
ntra
tion
(mg/
l0Ph
osph
orus
con
cent
ratio
n (m
g/l0
Flow
(m
3 /s)
Figure 6.6 The optimum model results retrieved from the calibrations using an error-free model and error-free data at different frequencies
142
Table 6.2 Model performance with no data or structural errors. The results in parenthesis are those obtained using a genetic algorithm for calibration.
Year Calibration data frequency Nres
Standard error (%)
Bias (%) No. failed (%)
1998 Daily 62 4 (3) 2 (1) 0 (0)
2-daily 31 8 (5) 4 (3) 0 (0)
Weekly 9 17 (16) 11 (6) 30 (25)
Daily event (all events) 11 6 (4) 1 (6) 0 (0)
Daily event (first event) 4 5 (6) 9 (6) 0 (0)
1999 Daily 62 15 (4) 10 (13) 20 (15)
2-daily 31 13 (5) 21 (18) 35 (20)
Weekly 9 14 (7) 28 (22) 45 (40)
Daily event (all events) 11 17 (4) 16 (16) 25 (20)
Daily event (first event) 4 19 (14) 23 (28) 30 (40) Table 6.3 Model performance with data error and no structural error
Year Calibration data frequency Nres
Standard error (%)
Bias (%)
No. failed (%)
1998 Daily 62 4 - 8 2 - 3 0
2-daily 31 5 - 7 1 - 5 0
Weekly 9 10 - 15 18 - 23 0 - 40
Daily event (all events) 11 5 - 9 3 - 5 0
Daily event (first event) 4 6 - 8 5 - 8 0 - 10
1999 Daily 62 9 - 20 13 - 27 20 - 45
2-daily 31 17 - 19 10 - 35 20 - 50
Weekly 9 12 - 21 27 - 56 40 - 55
Daily event (all events) 11 17 - 21 9 - 24 25 - 40
Daily event (first event) 4 15 - 27 22 - 45 25 - 70
Table 6.4 Model performance with data error and structural error
Year Calibration data frequency Nres
Standard error (%)
Bias (%)
No. failed (%)
1998 Daily 62 5 - 9 5 - 9 0 – 5
2-daily 31 6 - 8 5 - 8 0
Weekly 9 6 - 7 5 - 7 0 – 15
Daily event (all events) 11 7 - 11 4 - 9 0 – 10
Daily event (first event) 4 7 - 11 5 - 8 0 – 10
1999 Daily 62 12 - 13 17 - 170 40 – 55
2-daily 31 10 - 19 26 - 157 20 - 50
Weekly 9 9 - 12 13 - 188 30 – 55
Daily event (all events) 11 9 - 38 10 - 81 30 – 55
Daily event (first event) 4 8 - 19 12 - 103 15 - 50
143
The result was that, although the genetic algorithm led to the average standard error of the
1998 monthly export using daily data reducing from 4% to 3% and the bias from 2% to
1% (see Table 6.2), the bias in the predicted 1999 export increased from 10% to 13%.
Similar observations can be made for the other calibration data frequencies, with the
genetic algorithm generally improving performance under calibration conditions, but
providing no significant improvement in the forecasts of export. This indicates that using
a more powerful search algorithm during calibration will reduce result uncertainty under
idealised calibration conditions but, due to the equifinality caused by over-
parameterisation, will not necessarily improve the quality of model forecasts.
The step-down to 2-daily data, from Table 6.2, leads to a consistent deterioration in
performance from that achieved with daily data. Reducing the calibration data to weekly
further worsens the reliability of results (Figures 6.6(e) and 6.6(f), and Table 6.2), with a
45% (40% using the genetic algorithm) rate of failed results in predictive mode, and an
average 28% (22%) bias in the monthly export compared to the standard error of 14%
(7%). (The increase in bias is because the July P fluxes are more consistently under-
estimated when the less frequent calibration data are employed. Although not obvious in
Figure 6.6, the error in concentration is compounded when multiplied by high flows). The
estimated standard error is a poor measure of the reliability of the model, as it is often
significantly lower than the actual bias, and it may be concluded that, even in these
idealised conditions, a more robust estimator of model uncertainty is needed.
The use of daily event calibration data (11 samples covering all 3 main events of the 1998
wet season; and 4 samples covering only the first event) leads to better reliability than
achieved using weekly sampling. For the 4-sample option this improvement is not
significant, for example reducing average bias from 28% to 23% using LHS, and
increasing it from 22% to 28% using the genetic algorithm. While there are potential
economies to be made in pursuing event-based monitoring, small-sample results are
likely to be particularly sensitive to the introduction of model structural errors (Chatfield
1995), as evaluated below.
The use of error-free data has allowed us to gauge the effects of parameter equifinality
and search algorithm limitations. It provides an indication of the limit to which improved
data quality, and model structural quality, can improve export forecasts at various
calibration data frequencies. The results exemplify the fundamental reliability problems
in predictive modelling, due to equifinality and the biases introduced by restricting
144
calibration data to a frequency below that of the model response time (daily, in this case).
The results in Table 6.2 show that biases, which we would wish to assume much smaller
than the estimated standard errors, are likely to become larger as calibration data become
typically sparse.
The introduction of data error (both in loads Cpu and Cpd, and in calibration data pC )
provides a more realistic basis upon which to review the benefit of increased sampling
frequency. For this experiment, the 1998 calibration and loading data sets have five
realisations of Gaussian error imposed, all described by Equations 6.6 and 6.7. The 1999
loading data are left error-free, allowing us to concentrate on examining errors introduced
at calibration stage. Table 6.3 shows the range of performances obtained over the average
over the five realisations of data noise. These ranges are wide, with, for example, the
number of prediction failures associated with the one-event data ranging from 5 to 14,
implying that the predictive performance depends largely on the particular realisation of
data noise. Noting that, in general practice, only one realisation of calibration data is
available, and estimating the noise distribution is a fundamental part of the calibration
problem (see Chapter 1).
So far, in reviewing the relative effects of the calibration algorithm limitations,
equifinality of parameter sets, data errors and data sparseness, the important issue of
model structure error has been deliberately set aside. However, it is clear that the model
proposed in Equations 6.1 to 6.6 is a great simplification of the real Hun River system,
and that it has a number of significant structural faults that will in practice further
complicate the quest for prediction reliability. To review the implications of the inevitable
structural error on the value of calibration data, a manufactured structural error is
imposed upon the model. It is supposed that evaluating the performance of this erroneous
model will be indicative of more general structural problems, which may affect the
planning of the monitoring programme. The newly contrived model is based on the
resuspension model described by Blom and Aalderink (1998), in which the resuspension
rate is proportional to velocity-over-threshold (rather than the shear stress relationship
used in Equation 6.2),
( ) uu CvSuukR crpppcrssp >−−= for (6.10a)
for uu CvR crppsp ≤−= (6.10b)
145
where ucr is the water velocity at which resuspension is initiated, and the other terms are
already defined. The model is calibrated using the noisy data that were previously
generated by the error-free model structure. Again, ten LHS calibrations are done for each
of the five realisations of data noise.
The calibrated (1998) and predictive (1999) results for the daily, 2-daily, weekly and first
event data are shown in Figure 6.7, and Table 6.4 summarises the model performances.
The model appears to have had general success in simulating the 1998 resuspension
events, but inspection of the event recessions reveals consistent misrepresentation as
evidence of the structural fault. Nevertheless, Table 6.4 shows that under calibration
conditions the model performs no less impressively than its error-free alternative. On the
other hand, the 1999 prediction results indicate deteriorated performance, with, for
example, percentage bias of up to 170% using the daily data set. Differences in the
number of failed results are not so significant, because the large percentage bias is caused
by a small number of very biased results (again it is worth emphasising that in a
conventional, deterministic calibration, there would be no indication that such a result
was so misrepresentative). Another important result in Table 6.4 is that using event-based
calibration data has led to the generally less biased results. While not conclusive, this
indicates that the value of additional information in the calibration data (which was
evident in Table 6.2) is compromised by the presence of model error; moreover, it
indicates that the additional information contained in the low-flow conditions has become
an impediment, at least in some realisations. In the presence of structural error, the
objective function has become a poorer evaluator of the sampled model, identifying
optimum parameter sets which compensate for the structural fault under calibration
conditions. The preferred data set (and the preferred objective function) are those which
limit the adverse nature of this compensatory effect, with respect to the predictive task of
the model. For example, this study implies that neglecting the low flow data may improve
estimates of monthly export. Generally speaking, the utility of data cannot be judged
without consideration of the complexity of the modelling problem - the interactions
between the model structure, data reliability and objective function. More specifically, the
calibration data should be restricted to those which are more relevant to the intended
application of the model.
As a final experiment, sediment data were generated to explore the benefits which might
arise from their inclusion in the calibration. Again, nominal noise is introduced into the
data using Equation 6.6, and the calibration objective function is modified to Equation
6.11.
146
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0 .01
0 .02
0 .03
0 .04
0 .05
0 .06
0 .07
0 .08
0
0 .01
0 .02
0 .03
0 .04
0 .05
0 .06
0 .07
0 .08
0
0 .01
0 .02
0 .03
0 .04
0 .05
0 .06
0 .07
0 .08
(a) 1998 result; daily calibration data (b) 1999 result; daily calibration data
(c) 1998 result; 2-daily calibration data (d) 1999 result; 2-daily calibration data
(e) 1998 result; weekly calibration data (f) 1999 result; weekly calibration data
(g) 1998 result; event daily calibration data (h) 1999 result; event daily calibration data
2-Jul
12-Ju
l
22-Ju
l1-A
ug
11-A
ug
21-A
ug
31-A
ug2-J
ul12
-Jul
22-Ju
l
01-A
ug
11-A
ug
21-A
ug
31-A
ug
Phos
phor
us c
once
ntra
tion
(mg/
l0Ph
osph
orus
con
cent
ratio
n (m
g/l0
Phos
phor
us c
once
ntra
tion
(mg/
l0Ph
osph
orus
con
cent
ratio
n (m
g/l0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0 .01
0 .02
0 .03
0 .04
0 .05
0 .06
0 .07
0 .08
0
0 .01
0 .02
0 .03
0 .04
0 .05
0 .06
0 .07
0 .08
0
0 .01
0 .02
0 .03
0 .04
0 .05
0 .06
0 .07
0 .08
(a) 1998 result; daily calibration data (b) 1999 result; daily calibration data
(c) 1998 result; 2-daily calibration data (d) 1999 result; 2-daily calibration data
(e) 1998 result; weekly calibration data (f) 1999 result; weekly calibration data
(g) 1998 result; event daily calibration data (h) 1999 result; event daily calibration data
2-Jul
12-Ju
l
22-Ju
l1-A
ug
11-A
ug
21-A
ug
31-A
ug2-J
ul12
-Jul
22-Ju
l
01-A
ug
11-A
ug
21-A
ug
31-A
ug
Phos
phor
us c
once
ntra
tion
(mg/
l0Ph
osph
orus
con
cent
ratio
n (m
g/l0
Phos
phor
us c
once
ntra
tion
(mg/
l0Ph
osph
orus
con
cent
ratio
n (m
g/l0
Figure 6.7 The optimum model results retrieved from the calibrations using a model with structural fault and data with Gaussian error at different frequencies
( ) ( )kNi Nj
jppippkres res
SSCCOF
−−= ∑ ∑
= =,1 ',1
22 (6.11)
where pS is the reference sediment data, Nres’ is the number of points of sediment data
included (= Nres for all experiments), and the other variables are as previously defined.
The new calibration was performed using the same frequencies of Cp and Sp data as in the
previous calibration, with both the true and erroneous model structures, using the
experimental procedure outlined in Figure 6.5. The ‘true’ Sp result along with the error-
147
free data and one realisation of erroneous data are illustrated in Figure 6.8, and Tables 6.5
- 6.7 summarise the performance in predicting monthly export (see the end of section 6.4
for definitions of performance measures). Comparing Tables 6.5 and 6.6 with Tables 6.2
and 6.3 indicates that the additional information within the Sp data is generally valuable,
in terms of reducing bias and failed results, where there is no structural error. With the
introduction of structural error (compare Tables 6.4 and 6.7), while the extreme instances
of bias in Table 6.4 have been avoided, the number of failed results is generally slightly
higher. Thus the value of the extra data seems to be more questionable when model
structural error is present but not addressed prior to calibration. Another issue with using
sediment data would be the practical difficulty of achieving representative Sp
measurements, especially considering the conceptual nature of the modelled sediment
phosphorus store.
It is notable from the results in Tables 6.3, 6.4, 6.6 and 6.7 that, when data and structural
error are allowed to affect the calibration process, the intuitive expectation that model
performance is proportional to calibration data frequency (and prior perceptions of
information content) is not at all clear. Far more important controls on model
performance, as measured by percentage bias, are data noise and model structural error.
This leads to the suggestion that resources should be initially directed at collecting short
periods of high quality data, and evaluating and developing this data in a quest to resolve
the structural error. Specifically, in the Hun River study, quality data from the first event
would be recommended. However, it is naive to suggest that a single event’s data may be
adequate for structural identification, especially given the inevitable structural change that
arises in water quality systems (Van Straten 1998; also see Section 2.2.3 of this
dissertation). There is also the practical limitations of sampling and measurement
precision – the size of the residuals in Figure 6.7 indicates that visually identifying the
structural fault would be problematic, should there be even a small amount of error.
Furthermore, the error in the phosphorus loading data seems bound to contribute
significantly to the identification problem, irrespective of the quality of the in-river
calibration data. Improving the quality of the loading data is an ominous task, as it is a
function of rainfall and diffuse loads (among other inputs) from the large and sparsely
gauged upper Hun River catchment (Suttamanutwong 2001). Also, while the 1999 load
data were assumed to be precise in the experiments described above (to allow focus on
the value of calibration data), their errors would directly and significantly add to
prediction errors. Commitment of resources to in-river sampling should not, therefore, be
undertaken without a broader review of data requirements, especially with regard to
identification and modelling of diffuse sources.
148
Table 6.5 Performance with no data or structural errors (with sediment data)
Year Calibration data frequency Nres
Standard error (%)
Bias (%)
No. failed (%)
1998 Daily 62 8 3 0
2-daily 31 9 8 5
Weekly 9 9 2 0
Daily event (all events) 11 5 7 0
Daily event (first event) 4 7 3 0
1999 Daily 62 12 4 10
2-daily 31 14 6 10
Weekly 9 12 2 5
Daily event (all events) 11 13 2 15
Daily event (first event) 4 14 9 20
Table 6.6 Performance with data error and no structural error (with sediment data)
Year Calibration data frequency Nres
Standard error (%)
Bias (%)
No. failed (%)
1998 Daily 62 9 4 - 10 0 - 5
2-daily 31 7 - 11 2 - 6 0
Weekly 9 9 - 12 2 - 5 0 - 5
Daily event (all events) 11 8 - 11 4 - 10 0 - 10
Daily event (first event) 4 6 - 11 2 - 13 0 - 5
1999 Daily 62 1 - 19 2 - 14 0 - 15
2-daily 31 9 - 11 2 - 4 0
Weekly 9 12 - 17 2 - 5 5 - 10
Daily event (all events) 11 9 - 33 2 - 23 5 -35
Daily event (first event) 4 9 - 14 3 - 25 0 - 40
Table 6.7 Performance with data error and structural error (with sediment data)
Year Calibration data frequency Nres
Standard error (%)
Bias (%)
No. failed (%)
1998 Daily 62 6 - 9 17 - 18 5 - 10
2-daily 31 5 - 6 19 - 21 15 - 25
Weekly 9 4 - 6 16 - 20 5 - 20
Daily event (all events) 11 8 - 13 13 - 16 10 - 15
Daily event (first event) 4 4 - 8 13 - 22 5 - 45
1999 Daily 62 13 - 25 24 - 26 30 - 50
2-daily 31 6 - 9 8 - 34 33 - 50
Weekly 9 4 - 7 32 - 34 45 - 50
Daily event (all events) 11 29 - 39 24 - 35 50 - 60
Daily event (first event) 4 13 - 31 25 - 98 65 - 100
149
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.50
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
S P(g
m-2
)S P
(gm
-2)
S P(g
m-2
) Error-free calibration sediment dataSample of calibration sediment data with error
“True” model result
2-Jul
12-Ju
l
22-Ju
l
1-Aug
11-A
ug
21-A
ug
31-A
ug
2-Jul
12-Ju
l
22-Ju
l
1-Aug
11-A
ug
21-A
ug
31-A
ug
(a) daily calibration data (d) daily event calibration data (all events)
(e) daily event calibration data (first event)
(b) 2-daily calibration data
(c) weekly calibration data
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.50
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.50
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
S P(g
m-2
)S P
(gm
-2)
S P(g
m-2
) Error-free calibration sediment dataSample of calibration sediment data with error
“True” model result
2-Jul
12-Ju
l
22-Ju
l
1-Aug
11-A
ug
21-A
ug
31-A
ug
2-Jul
12-Ju
l
22-Ju
l
1-Aug
11-A
ug
21-A
ug
31-A
ug
(a) daily calibration data (d) daily event calibration data (all events)
(e) daily event calibration data (first event)
(b) 2-daily calibration data
(c) weekly calibration data
Figure 6.8 Sediment calibration data and “true” model result for 1998. Only one of five realisations of data error is shown.
The results have clearly shown that the various sources of error frequently cause the
calculated standard error of monthly exports of P to be lower than the actual (and in
practice unknown) bias of the result. The modelled uncertainty has, in this study, merely
represented the sampling error in the calibration procedure along with the equifinality of
parameter sets, and is not an adequate indicator of total model uncertainty. Even if the
error in the calibration data could be estimated, and some suitable allowance made
(Chapters 2, 7), this does not allow for the important influence of the unknown but
inevitable structural error. Intelligent judgement of the scope for model and data errors is
necessary, together with a less restricted estimator of model parameter uncertainty.
Established approaches to doing so are Bayesian parameter estimation (e.g. Reichart and
Omlin 1998), set-theoretic methods (Van Straten and Keesman 1991), the use of a
number of candidate model structures (Chatfield 1995), and an integrated framework of
these called Generalised Likelihood Uncertainty Estimation (GLUE; Beven and Binley
150
1992). While these methods cannot solve the uncertainty estimation problem (i.e. cannot
reliably estimate the scope for biases in model predictions), they provide a more
reasonable alternative to objective methods which inevitably over-simplify the
complexity and extent of the issue. Where permitted by constraints of available modelling
resources, established (e.g. Beck 1983) and novel (e.g. Wagener et al. 2002b) system
identification techniques should be considered as means of reducing model structural
error.
6.6 Summary
A conceptual model of river phosphorus mobilsation and transport has been proposed,
and applied to the upper Hun River in Liaoning, China over the wet seasons of 1998 and
1999. The aim was to investigate the benefits associated with alternative programmes of
in-river calibration data collection, in terms of the achievable reliability of model
predictions.
Alternative programmes of in-river data collection to support model calibration were
evaluated. Using synthesised, but reasonable scenarios of data error and model structural
error, the model performance was tested in terms of the reliability and precision of its
predictions of monthly phosphorus exports. The experiments show that, when data and
structural errors are present, decreasing the total number of samples in a 2-month period
from 62 to 4 led to only marginal deterioration in performance. This is partly because the
information content of the data varied markedly within the 2-month period, with low flow
data being information-poor, or in cases detrimental. Model predictive performance was
strongly deteriorated by the investigated scenarios of data noise and model structural
error, despite reasonable performance under calibration conditions.
Failure to identify a unique optimum set of parameters caused significant variability
amongst calibration results, even using an error-free model and data set. The limitations
of the Latin Hypercube Sampling calibration procedure in finding a unique global
optimum parameter set were examined by comparisons with results of a genetic
algorithm. While the latter method reduced uncertainty under calibration conditions (i.e.
improved convergence of the calibration objective function), it led to no significant and
consistent improvements in model predictions, (i.e. no improvements in convergence of
the calibrated model). While the alternative realisations of optimum parameter sets were
carried forward to give an estimate of standard error in the modelled phosphorus exports,
151
this estimate was, in many cases, under-representative of the actual error. More liberal,
robust estimators of uncertainty are needed, which aim to limit under-representation of
the scope for prediction error. The pollution load errors are thought to be another limiting
control on model performance, and it is reasonable to suggest that improved nutrient load
models should take priority over the in-river transport model. Further study of the
sensitivity of nutrient export to a wider range of inputs (to include, for example, rainfall
and fertiliser application) is needed.
In summary, the experiments indicate a high risk that in-river data collection will fail to
support the sought modelling capability, given the wider modelling problems that are
perceived to be present. Therefore it is recommended that a minimalist approach to
sampling is initially taken, which concentrates on the first major storm event where data
information content is considered to be relatively high and robust to structural error.
However, recommendations cannot be restricted to the in-river sampling regime, but
should encompass the broader modelling strategy, expectations, and general data
priorities.
152
7. Risk-based modelling of surface water quality: a case study of the
Charles River, Massachusetts
A model of chlorophyll-a, dissolved oxygen and nutrients is presented and applied to the
Charles River, Massachusetts within a framework of Monte Carlo simulation. The model
parameters are conditioned using data from eight sampling stations along a 40km stretch
of the Charles River, during a (supposed) steady-state period in the summer of 1996, and
the conditioned model is evaluated using data from later in the same year. Regional multi-
objective sensitivity analysis is used to identify the parameters and pollution sources most
affecting the various model outputs under the conditions observed during that summer.
The effects of Monte Carlo sampling error are included in this analysis, and the
observations which have least contributed to model conditioning are indicated. It is
shown that the sensitivity analysis can be used to speculate about the factors responsible
for undesirable levels of eutrophication, and to speculate about the risk of failure of
nutrient reduction interventions at a number of strategic control sections. The analysis
indicates that phosphorus stripping at the CRPCD wastewater treatment plant on the
Charles River would be a high-risk intervention, especially for controlling eutrophication
at the control sections further downstream. However, as the risk reflects the perceived
scope for model error, it can only be recommended that more resources are invested in
data collection and model evaluation. Furthermore, as the risk is based solely on water
quality criteria, rather than broader environmental and economic objectives, the results
need to be supported by detailed and extensive knowledge of the Charles River problem.
153
7.1 Introduction
7.1.1 Motivation
The reasons for and significance of uncertainty in predictions of water quality are widely
recognised and documented elsewhere (e.g. Hornberger and Spear 1980, Beck 1983,
Beck 1987, Reckhow 1994, Reichart and Omlin 1996, Van Straten 1998, Adams and
Reckhow 2001, McIntyre et al. 2001). The inevitability of significant uncertainty, and the
need to account for it in water quality management, has been recognised in the
development of some decision-support tools, for example QUAL2E-UNCAS (Brown and
Barnwell 1987), DESERT (Ivanov et al. 1996), SIMCAT (UK Environment Agency
2001a) and WaterRAT (McIntyre and Zeng 2002). An alternative modelling philosophy
is to aim to reduce uncertainty to an insignificant level through refinement of scale and
process representation (Young et al. 1996, Beck 1999). While so refined models have
proven valuable in a number of applications (see the review of Ambrose et al. 1996),
there are four reasons why this modelling philosophy seems to be of restricted value in
practice. 1) Field data are not usually adequate to identify the boundary conditions and
parameter values of such models (Beck 1997). 2) In any case, more data do not
necessarily lead to better models or system understanding (Chatfield 1995, Beck 1999).
3) Human resource constraints often preclude intricate, data-intensive, modelling
exercises (Reckhow 1994, van Straten 1998). 4) Results of complex models are more
easily mis-interpreted, while not necessarily being any more reliable (e.g. Gardner et al.
1980, Van der Perk 1997). Therefore, there remains a need to promote uncertainty
estimation, and to continue to develop models and analytical tools which permit such
analysis, and which reflect the resource constraints of users.
This chapter will exhibit and review the utility of the WaterRAT tool for water quality
management, using the Charles River, Massachusetts as a case study.
7.1.2 Scope and objectives
The case study is approached in a way which addresses the difficulty of applying a water
quality model to decision support on the basis of limited supporting data and modelling
resources. This is done in the context of five tasks which are set for the study.
• To condition the model using the water quality data observed at various control
sections of the Charles River on the 20th August 1996.
154
• To evaluate the conditioned model with respect to its success in representing the
water quality observed on the 8th October of 1996.
• To identify the principal factors affecting water quality on 20th August 1996.
• To support the appraisal of options for reducing eutrophication in the Charles
River, specifically limiting chlorophyll-a to less than 10mgm-3 at a number of
control sections. Proposed water quality interventions will be evaluated on the
basis of associated risk of failure to achieve the specified target.
• To identify ways to reduce the element of risk which stems from model
prediction uncertainty.
Arguably, confronting these tasks rigorously would require careful consideration of
different modelling tools, and selection of one or more approaches. Furthermore, a critical
review of data reliability including evaluation of sources of sampling and measurement
errors would normally be recommended, followed by iterative model structure
adjustments and parameter calibrations. However, within the scope of this chapter, none
of these practices are adopted. Instead, this investigation starts from the premise that
modelling resources are limited so that only one model structure can be used, and that the
readily available data must be interpreted without researching quality control issues.
Additionally, the view is taken that the in-river data are too sparse to be usefully analysed
using traditional maximum likelihood techniques. Instead, this investigation is founded
on the methodologies of Regional Sensitivity Analysis (RSA; Hornberger and Spear
1980) and Generalised Likelihood Uncertainty Evaluation (GLUE; Beven and Binley
1992), whereby qualitatively derived constraints are used to supplement the information
in the sparse data set. In adopting these approaches, we set out to address the question “If
human resources and observed data are limited, as they typically are, what degree of
support can be given to strategic management of water quality?”
The study does not take account of a large number of factors presently affecting policies
for management of the Charles River, nor of many observations outwith the 1996 study.
The study is primarily a demonstration of methods, and all results should be seen in this
context.
7.1.3 The case study
Water quality problems associated with the Charles River in previous decades were
industrial pollution and combined sewer overflows which led, among other unwelcome
effects, to nutrient enrichment and eutrophication. Storm-water interceptions and other
interventions in the 1990s have greatly improved the overall ecology and amenity value
155
of the river, although they have failed to control eutrophication satisfactorarily. Further
measures are currently being implemented by installing phosphorus stripping facilities at
a number of wastewater treatment plants (CRWA 2000). However, such control of point
sources will not necessarily solve the problem, as phosphorus from non-point sources
may enter the river directly or via tributaries. Furthermore the phytoplankton may be
resilient to low phosphorus concentrations. There is therefore a need for decision support
tools which can estimate the residual eutrophication given various options for point and
non-point interventions.
This study looks at the 40km length of the Upper Charles River, between the Populatic
Pond in Medway County and the Cochrane Dam in Dover County, on two days in the
summer to autumn of 1996 (the 20th August and the 8th October), when data were
collected at nine sections along the river (CDM 1997), shown in Figure 1.5. The
determinands measured include,
• chlorophyll-a (Ca),
• dissolved oxygen (Cox),
• 5-day biochemical oxygen demand (Ccf),
• organic phosphorus (Cps) and orthophosphates (Cpo),
• organic nitrogen (Cns), nitrates (Cni) and ammonium (Cna).
• flow (Q), water depth (Hw) and water temperature (Tw),
These measurements were made three times on each day to estimate a daily mean and an
expected diurnal range. In addition, the major point sources to the river (two wastewater
discharges and seven tributaries) were monitored, and daily pollution and hydraulic loads
were estimated. An additional pollution and hydraulic load was assumed to be evenly
distributed along the studied length of river, based on measurements at a number of minor
inlets to the river. The three dates were chosen from periods when the river was
considered to be near steady-state, and a steady-state assumption is maintained in this
exercise.
7.2 Model Structure and Methods
7.2.1 Specification of the model structure
We begin with the premise that a model structure which is adequate for the tasks can be
adopted, and all the uncertainty in the model structure can be represented by parameter
uncertainty. While not analytically ideal, this premise is consistent with the constraints of
156
time and resources which are normal in practice, and the adequacy of the model structure
will be reviewed and discussed as part of the model evaluation. The philosophy of
parsimonious modelling has been rejected, as one important task is to explore the risk
stemming from model components which are not identifiable during conditioning, but are
relevant to future scenarios. Importantly, it should be noted that the selected structure,
summarised below, is not intended to represent our full prior knowledge of phytoplankton
dynamics and nutrient cycling due to the unmanageable complexity this would entail
without expected improvements in predictive performance, but is a simplification based
on our prior experience of what the principal components of the system are likely to be.
Notwithstanding the simplifications, the model is of a similar complexity to the
commonly used QUAL2E model (Brown and Barnwell 1987).
The selected model structure includes all the monitored determinands previously listed.
The in-river nutrient and oxygen cycling processes are represented by the set of
differential equations presented below, and the interactions of the water quality
determinands are summarised in Figure 7.1. The descriptions below are purposefully brief
- a more in-depth description and discussion of the formulations are given in McIntyre
and Zeng (2002) and Chapra (1997).
Phytoplankton
)()( aaadaagaa CCCkCk
dtdC
∆+Φ+−= (7.1)
where t is time (in units of s); Ca is phytoplankton concentration measured as
Chlorophyll-a (gm-3); ∆(Ca) represents the advective and dispersive transport of
phytoplankton to and from adjacent control volumes of water; Φ(Ca) represents the
external load of phytoplankton; kga (s-1) is the net photosynthesis rate of phytoplankton
(includes effect of phytoplankton respiration); and kda (s-1) is the death rate of
phytoplankton. kga is a function of water temperature Tw (oC), light availability I (Jm-2 s-1),
nutrient availability and the maximum net photosynthesis rate at Tw = 20oC, kga20.
Similarly, kda is a function of water temperature and the maximum death rate at Tw =
20oC, kda20;
20,)(),,()( gawTponinaNIga kTfCCCfIfk = (7.2)
20,)( daTda kTfk = (7.3)
157
where the functions fI, fN and fT are respectively based on the Steele model of light
limitation, the Michaelis-Menten equation of nutrient limitation (where fN is defined by
the minimum of nitrogen and phosphorus limitation), and the Arrhenius formula of
temperature effect (see Chapra 1997: p40, p607, p610). The coefficient θ which defines
the relationship between reaction rates and temperature (Equation 3.13) is, for this study,
assumed to have a common value over all the model components.
Ca
Cox
Ccs Cns
CONTROL VOLUMEOF WATER
Ccf
Cni
Cni and Ccf lostCcf lost
Cns lostCcs lost
ATMOSPHERE
Hydrolysis
Hydrolysis
Resp-iration
SedimentationSedimentation
SEDIMENT
Denitri-fication
Sediment oxygen demand
Nitrifi-cation
Carbon fixed
Photo-synthesis
Phytoplankton death
Cpo
Cps
Cps lost
FixationPhytoplankton death
Hydrolysis Phytoplankton death
Fixation
Fixation
Aeration
Cna
Cox lost/gained
Ca
Cox
Ccs Cns
CONTROL VOLUMEOF WATER
Ccf
Cni
Cni and Ccf lostCcf lost
Cns lostCcs lost
ATMOSPHERE
Hydrolysis
Hydrolysis
Resp-iration
SedimentationSedimentation
SEDIMENT
Denitri-fication
Sediment oxygen demand
Nitrifi-cation
Carbon fixed
Photo-synthesis
Phytoplankton death
Cpo
Cps
Cps lost
FixationPhytoplankton death
Hydrolysis Phytoplankton death
Fixation
Fixation
Aeration
Cna
Cox lost/gained
Figure 7.1 Schematic of conceptual processes and state interactions
Dissolved oxygen
Dissolved oxygen Cox (gm-3) is gained or lost through exchange with the atmosphere
which occurs in direct proportion to the oxygen deficit (Cos - Cox) at a rate kra (ms-1),
where Cos is the dissolved oxygen concentration in equilibrium with the atmosphere
calculated as a function of water temperature (Equation 1.15). Cox is gained due to
photosynthesis, at a rate of one unit of oxygen to roa units of chlorophyll-a produced; lost
due to bacterial respiration at a rate koc (s-1); lost due to nitrification at the rate of
nitrification kon multiplied by the oxygen demand of a unit mass of ammonium nitrogen
ron; and finally dissolved oxygen is lost due to sediment oxygen demand SOD (gm-2s-1)
which is a spatially varying model parameter.
158
( ) )()( oxoxw
naononcfocagaoaoxosw
raerox CCH
SODCrkCkCkrCCH
kkdt
dC∆+Φ+−−−+−=
(7.4)
The aeration rate kra (gm-2s-1) is calculated through relationships with water velocity, Hw
and Tw (see Chapra 1997: 377), and the effect of a weir or sluice on the aeration rate is
modelled using an additional empirical relationship (see Chapra 1997: 380). Error in
these aeration formulae is allowed for by assuming that any deviation from the
relationship increases linearly with kra at a rate given by parameter ker.
Non-living organic carbon
Non-living organic carbon (i.e. excluding that in phytoplankton) is modelled using two
conceptual fractions, all modelled in units of equivalent oxygen demand. The first is 5-
day biochemical oxygen demand Ccf representing the fast-decaying dissolved organic
carbon, and the second is Ccs representing the slow-decaying particulate detritus, which
slowly hydrolyses into Ccf.
)()( cfcfnionpdncfoccshccf CCCrkCkCk
dtdC
∆+Φ+−−= (7.5)
)()( cscsaoadacsw
cscshc
cs CCCrkCHv
Ckdt
dC∆+Φ++−−= (7.6)
where khc (s-1) is the hydrolysis rate of Ccs which is dependent on Tw (as already described
for kda); koc (s-1) is the decay rate of Ccf which is similarly dependent on Tw and is also
limited by Cox using the previously mentioned Michaelis-Menten formulation; vcs (ms-1) is
the sedimentation rate of Ccs; kdn (s-1) is the rate of denitrification of nitrate Cni, a process
which consumes ronp units of Ccf for each unit of Cni (see Chapra 1997: 478).
Nitrogen
Nitrogen is included in the model as non-living organic nitrogen Cns, ammonium (plus
ammonia) Cna, and nitrate Cni. Nitrite is omitted under the assumption that the conversion
of ammonium to nitrite is the rate-limiting process (Chapra 1997: 422). Nitrogen is
allowed to be lost by sedimentation of Cns at a rate vn and by denitrification of Cni at a rate
kdn (s-1) which is a function of Tw as previously described (while denitrification generally
occurs only in anoxic waters, the kdn term represents the effect of denitrification processes
159
in the sediments – see Whitehead and Toms 1993). Similarly, the rate of nitrification of
Cna to Cni is dependent on Tw, and is also limited by Cox using the previously mentioned
Michaelis-Menten formulation. Both ammonium and nitrate are assimilated at a rate kga in
proportion to the nitrogen-chlorophyll-a ratio rna in the phytoplankton. The phytoplankton
consume Cni and Cna in a proportion (which is defined by coefficient kna) to the relative
availability of these two nutrients.
)()( nananshnnaonananagana CCCkCkCkrk
dtdC
∆+Φ++−−= (7.7)
)()()1( nininidnnaonananagani CCCkCkCkrk
dtdC
∆+Φ+−+−−= (7.8)
)()( nsnsnsw
nsnshnanada
ns CCCHv
CkCrkdt
dC∆+Φ+−−= . (7.9)
Phosphorus
Phosphorus is represented by non-living organic phosphorus Cps and inorganic
phosphorus Cpo.
)()( popopshpapagapo CCCkCrk
dtdC
∆+Φ++−= (7.10)
)()( pspspsw
pspshpapada
ps CCCHv
CkCrkdt
dC∆+Φ+−−= (7.11)
where rpa is the ratio of phosphorus to chlorophyll-a and vps is the effective sedimentation
rate of organic phosphorus.
The pollution transport model (through which the ∆ terms above are calculated) is a one-
dimensional control volume solution of the advection-dispersion equation (see Chapra
1997: 215), i.e. a different application of Equation 6.4. The control volumes are defined
by a series of 44 cells (average length of 910m) which make up the full length of the
river. Adjacent cells with similar hydro-geometric properties are grouped together into
reaches, giving the 12 reaches illustrated in Figure 7.2. The flow routing model is a non-
160
linear store whereby the flow out of each cell Q (m3s-1) is proportional to a power q2 of
the average depth in that cell H (m), and a constant of proportionality q1 (m3-b s-1),
2
1q
wHqQ = (7.12)
The rate of change of water volume V (m3) in each cell is simply the balance of flow from
the upstream cell Qup, flow out of the cell Q, and external sources Φ(Q),
)(QQQdtdV
up Φ+−= (7.13)
Hw is a function of V and the geometric properties of the cell. Each cell is conceptualised
as having a symmetrical trapezoidal cross-section, so these properties are the cell length,
the base width and the side-slope. The dispersion between cells is calculated from an
empirical relationship with water velocity (Chapra 1997: p245). The water temperature is
prescribed on the basis of observations.
2897
6759
1223
0
1502
0
1759
5
2019
6
2371
0
2649
9
2950
3
3210
5
3566
1
3926
6
CRPCDWWTW
MillRiver
Head-water
StopRiver
Medfield WWTW
BogastowBrook
WabanBrook
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
1 2 3 4 5 6 8 9 10 11 127
TroutBrook
SewallBrook
CochraneDam
0
IndianBrook
2897
6759
1223
0
1502
0
1759
5
2019
6
2371
0
2649
9
2950
3
3210
5
3566
1
3926
6
CRPCDWWTW
MillRiver
Head-water
StopRiver
Medfield WWTW
BogastowBrook
WabanBrook
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 441 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
1 2 3 4 5 6 8 9 10 11 127
TroutBrook
SewallBrook
CochraneDam
0
IndianBrook
Figure 7.2 River reaches, spatial grid and point sources
The set of ordinary differential equations described above is integrated using the Fehlberg
scheme described in Chapter 5. The integration was performed during the 12 days leading
up to the observed days, on the assumption that the daily average pollution loads and
other boundary conditions were constant during this period. 12 days was found to be
more than sufficient to allow the water quality to reach steady-state.
7.2.2 Specification of prior parameters and uncertainty in observed data
The model includes 24 biochemical parameters which are all considered to be uncertain.
The advective and dispersive terms (∆) in Equations 7.1 to 7.11 are assumed not to carry
uncertainty. The dispersion is time-variable as a function of velocity (Chapra 1997:
p245), and the q1 parameter is calibrated independently of water quality, on a reach-by-
161
reach basis, with q2 fixed at a value of 1. The ranges of possible values of the biochemical
parameters, prior to model conditioning, are taken from various reviews (Brown and
Barnwell 1987, Bowie et al. 1987, Thomann and Mueller 1982, Chapra 1997). All prior
parameter ranges together with references are listed in Table 7.1. All values within these
ranges are taken to be independent and equally likely prior to conditioning.
Table 7.1 Prior parameter ranges
Model
component Parameter Unit Min. Max. [a] Ref [b]
Slow carbon khc20 s-1 0.001 0.1 Bo vsc m.s-1 0.05 1 Br, C Fast carbon koc20 s-1 0.1 3 T kochs mgO.L-1 0.1 1 Bo Oxygen ker % -100 100 C kwe 0.0325 1.26 C SOD g.m-2 s-1 0.2 1 C Nitrogen khn20 s-1 0.001 0.4 Br, Bo vsn m.s-1 0.05 1 (as vsc) kon20 s-1 0.1 1 Br ron gO.gN-1 4.57 4.57 C konhs mgO.L-1 0.1 1 (as kochs) kdn20 s-1 0 0.1 Bo ronp gO.gN-1 2.86 2.86 C Phosphorus khp20 s-1 0.01 0.7 Br vsp m s-1 0.05 1 (as vsc) Phytoplankton kga20 s-1 1 2.5 Br, Bo kgahsn mgN.L-1 0.01 0.03 Br, Bo kna mgN.L-1 0 1 Br, Bo kgahsp mgP.L-1 0.001 0.05 Br, Bo kgahsl W.m-2 3.8 24 Br, Bo kda20 s-1 0.003 2 Bo rna gN.gChla-1 3.6 18 C roa gO.gChla-1 55 280 C rpa gP.gChla-1 0.5 2.5 C Common θ 1.024 1.15 Br, Bo
Notes: [a] If Max. = Min. then the parameter is taken as known with certainty. [b] Br = Brown and Barnwell (1987); Bo = Bowie et al. (1987); T = Thomann and Mueller (1982); C = Chapra (1997). Parenthesised entries indicate that the range has been assumed the same as that of another parameter.
As well as the high number of parameters, the sources of pollution are also not known
precisely. To make justified inferences about the parameters, this additional source of
uncertainty must be allowed for during the model conditioning. As is often the case, the
available data for sources of pollution are daily means with no supporting quality control
162
information to indicate the scope for procedural errors. Therefore we initially assume that
all sources are described as uniform distributions with ranges ±30% around the given
mean value. If there is evidence in the data to suggest that this assumption is important
and unreasonable, then this will become evident during model evaluation and sensitivity
analysis.
The in-river water quality data consists of three measurements per determinant per
monitoring section (see Figure 1.5) on both of the monitored days – one of the three
measurements was taken early in the morning, one at mid-day and one in early evening.
Therefore, the variability observed within each set of three will consist of errors (e.g.
sampling errors) and diurnal variations. For this study, it is appropriate to use the mean of
the three measurements to condition the model, as the model is assumed steady-state at a
daily time scale, and is driven by mean daily inputs. The uncertainty in the estimation of
the daily mean water quality is difficult to estimate due to the limited statistical
significance of such small sets of data, and due to unknown procedural biases. The
problem of describing data error structures using typically available data sets is a major
obstacle to objectivity in water quality model uncertainty analysis, and some degree of
simplification and subjectivity is required. For the purpose of this investigation, the
simplifying assumption is made that the uncertainty bounds on the observed mean water
quality are represented by the maximum and minimum daily values - all values between
these bounds are perceived to be equally likely. Exceptions to this rule are when the
concentrations are below the detection limits, in which case the daily mean is taken as
anywhere between zero and the detection limit, which occurs frequently for Cna.
Therefore, as well as having a prior estimation of the uncertainty in all the model’s input
variables (24 parameters plus 7 types of pollution for each of 9 point sources plus the
headwater), we have an estimation of the uncertainty in the data (7 determinands at each
of 8 downstream monitoring sections). These data are central to the first of our modelling
tasks – to condition the model using the observations made on the 20th August 1996.
Clearly, there is very large number of factors to be taken into account in confronting this
task.
The following calibration samples realisations of both the model inputs and the
observations of river water quality, as well as the model parameters. The aim is to
integrate the effects of input and output observation uncertainty into the calibrated
parameter distributions, and to test the relative sensitivity of the objective function to the
perceived uncertainty in the observations. The logic behind the method is that the
163
objective function is regarded as a function with two uncertain inputs; 1) the output of the
model (itself a function of uncertain parameters and observed pollution load data) and 2)
the observations of water quality. It is then reasonable, within the Monte Carlo analysis,
to sample realisations of both the model output and the observations of water quality for
the calculation of the objective function. It follows that the posterior joint parameter
distribution will be derived by integrating across all sampled values of the observed
pollution load and river water quality data sets. Similarly, posterior joint distributions of
observed pollution load, and of observed river water quality can be derived. ‘Factor’ is
used through the remainder of this chapter to describe all three types of uncertain input to
the objective function.
This approach is in contrast to many previous applications which do not explicitly
represent scope for error in the observed calibration data. Instead, it is usually considered
that the objective function implicitly integrates the data error into the posterior parameter
response surface. This is valid using likelihood functions that are designed explicitly
recognising a data error distribution (this is supported by the experimental and theoretical
comparisons in section 2.4.3), but the usefulness is not clear when subjective measures of
performance are being employed. It is proposed that this new approach will improve
robustness of the estimated parameter uncertainty, and will indicate to what extent the
perceived error in the data is controlling the value of the calibration exercise. The primary
limitation is the simplistic representation of prior distributions of data error, although
arguably this is no more an issue than the assumption of uniform independent priors for
the parameters.
7.2.3 Multi-objective model conditioning
The initial objectives of the model are to adequately replicate the observed concentrations
of the eight determinands observed on the 20th August, 1996. The degrees to which the
eight objectives are met are measured by eight objective functions (OFs), one for each
determinant;
OF1 ≡ replicate the observations of Ca
OF2 " Cox
OF3 " Ccf
OF4 " Cns
OF5 " Cni
OF6 " Cna
164
OF7 " Cps
OF8 " Cpo
(7.14a-i)
To condition the model to meet each of these objectives individually, the values of the
OFs are calculated for each of a number of randomly sampled sets of factors α within a
Monte Carlo procedure. This involves running a simulation of the Charles River for each
of the sampled sets and calculating the corresponding OF values for all 8 determinands.
For the kth of Nsam sampled sets of factors αk and the mth determinant, OFk,m is defined
as the inverse of the sum of the squared residuals for that determinant,
( )1
2,,,
−
−= ∑
nnmkmk CCOF k=1,Nsam; m=1,8; n=2,9 (7.15)
where ( ) nmkCC ,,− is the residual between the observed value of the mth determinant
C and the corresponding model result C at the nth of the nine monitoring sections (the
headwater section, n = 1, is not included) obtained from the kth sampled set of factors.
Therefore, if the model result closely replicates the data then OFk,m will be high and it will
be nearly zero if the result is far from the data. Nsam sets of factor values and
corresponding OF values are obtained. These OF values are multiplied by the prior
probability Lp of α and normalised so that they sum to unity,
[ ][ ]∑ ⋅
⋅=
k,
,, OF
OF
mkp
mkpmk L
LL k=1,Nsam; m=1,8 (7.16)
Then, all Lk,m can be regarded as values of probability mass from a conditioned (posterior)
joint distribution of factors. In this case Lp is constant at 1/Nsam for all k and m, and its
terms in Equation 7.16 cancel each other. However, the analysis will later incorporate
Bayesian updating of a model (i.e. incorporating new data or knowledge into an existing
model using probability theory) whereby the old posterior becomes the new prior (L →
Lp), and in the general case Lp would become an essential term in Equation 7.16.
The result of Equation 7.16 defines the conditioned distribution of model parameters,
representing the uncertainty in the model. It should be noted that this uncertainty has been
subjectively defined, because Equation 7.15 is not a likelihood function which is based on
165
statistical evidence or probability theory. Also, in this study there are eight independently
conditioned models, associated with the eight determinands, which may or may not be
complementary. That is to say, although the concept of the model is founded on the
speculated interactions between these eight determinands, our model identification
criteria neglect this. Evaluation of the uncertainty in model predictions which this may
introduce can be approached using multi-objective processing, and this is discussed later.
7.2.4 Graphical model evaluation
The justification of the model as a tool for sensitivity and scenario analysis depends on
the model structure being a good conceptual representation of the actual water quality
processes under all conditions of interest. To some extent, the structure is known to be
reasonable a priori because the formulations (Equations 7.1 to 7.13) are based on
common knowledge of the principal processes affecting water quality. On the other hand,
some processes known to have the potential to affect water quality are not represented.
For example, sediment-water nutrient interactions are not represented explicitly, but
might become important following nutrient load reduction, and therefore their omission
casts doubt on the model’s reliability for appraising intervention scenarios. Ideally, the
model structure would be evaluated under a wide range of conditions to identify the
importance of these and other “missing “ structural components. Additional discussion of
approaches to model structure evaluation is given in Wagener et al. (2001).
The traditional criterion for evaluating a model structure is its ability to achieve a
satisfactory distribution of model residuals (i.e. the residuals between the model result
and the corresponding observed data point) (see Kuczera 1983, Beck 1987). This
approach is rejected here because the observations in this study (and many other
distributed water quality studies) are not of sufficient quality and quantity to allow any
useful statistical inference about the residuals. It is more useful to interpret observations
subjectively alongside the prior knowledge contained in the model result. Arguably, in
cases of unknown data reliability, it may be equally justifiable to invalidate the data on
the basis of an a priori hypothesis of the model structure as vice-versa (see the discussion
of Chatfield (1995) on data-mining).
For this study, the model is evaluated using the Charles River data from the 20th August
and the 8th October 1996. The spatial variations in water quality on these two dates are
stochastically modelled using the conditioned values of probability mass, with each
determinant m modelled using its own OFm (see Equations 7.14 to 7.16). For the 20th
August, the variations modelled along the river have been conditioned on the same day’s
166
observed data and therefore, excepting serious structural error, would be expected to
explain the data relatively well. For the later day, an additional Monte Carlo analysis is
needed which incorporates the changed boundary conditions - pollution loads, hydraulic
loads, water temperature and light intensity. Graphically comparing the modelled spatial
variations with the in-river data indicates, on a subjective basis, the suitability of the
model structure and the suitability of the definition of uncertainty given by Equations
7.15 and 7.16. That is, this evaluation seeks to answer the question “are the observations
sufficiently described by the modelled confidence limits?” (see Section 7.3.1 for results).
7.2.5 Regional sensitivity analysis
Regional Sensitivity Analysis (RSA) identifies the factors which have significant
probability of influencing the degree of achievement of each of the objectives. It is an
essentially probabilistic sensitivity analysis, as opposed to traditional derivative-based
methods. This means that the measure of regional sensitivity assigned to each factor is not
only dictated by the sensitivity of the model outcome to a unit perturbation in that factor,
but by the relative responses due to all the factors, and due to the relative uncertainties in
all the factors.
The employed RSA procedure is founded on the behavioural analysis described by
Hornberger and Spear (1980) and Spear and Hornberger (1981), together with the
Generalised Likelihood Uncertainty Estimation (GLUE) procedure of Beven and Binley
(1992). The methods of sensitivity analysis are employed in multi-objective mode, using
a similar approach to that described by Bastidas et al. (1999) and Meixner et al. (1999).
The procedure may be summarised as follows. The marginal frequency distribution of
each of the factors for all eight objectives is derived. The marginal distribution function
Pm(γ), indicates the probability that objective m will be met across the range of factor γ,
given the uncertainty in all the other factors. Pm(γ) is derived by totalling the values of L
(for objective m) within each of a number of equally sized bins of γ, which is illustrated
schematically in Figures 7.3(a) and (b). The difference between the cumulative marginal
distribution and the factor’s prior distribution (which is shown as uniform in Figure 7.3)
is summarised by the Kolmogorov-Smirnoff statistic, KSm(γ) (see Ang and Tang 1975,
pp. 277-280). The significance of this statistic is illustrated in Figure 7.3(c).
167
P(γ)
unconditionedconditioned
KS statistic = KSi(γ)
Like
lihoo
d Figure 7.3(a)Scatter-plot ofOFi over rangeof factor γ
Figure 7.3(b)Marginalprobabilitydistributionof factor γ
Figure 7.3(c)Conditioned andunconditionedcumulativedistributionsof factor γ
unconditionedconditioned
γ
γ
γ
Cum
ulat
ive
P(γ)
P(γ)
unconditionedconditioned
KS statistic = KSi(γ)
Like
lihoo
d Figure 7.3(a)Scatter-plot ofOFi over rangeof factor γ
Figure 7.3(b)Marginalprobabilitydistributionof factor γ
Figure 7.3(c)Conditioned andunconditionedcumulativedistributionsof factor γ
unconditionedconditioned
γ
γ
γ
Cum
ulat
ive
P(γ)
Figure 7.3 Derivation of K-S statistic
Figure 7.3(a) draws attention to the fact that different posterior probabilities are found for
any one value of γ - the probability is not solely a function of γ, but also of the values of
all the other interacting factors. This is why joint and marginal distributions must be
reported rather than uni-variate distributions. High correlation between factors will tend
to diminish their regional sensitivity, and so the results of the RSA should be interpreted
in conjunction with the factor covariance matrix.
Note from Equation 7.15 that different realisations of the in-river data are being used as
part of the sensitivity analysis. This is done to improve robustness of the sensitivity
analysis to data uncertainty, and to indicate the importance of the data uncertainty
compared to that of the a priori parameter uncertainty. For example, referring again to
Equation 7.15, if the OF is shown to be more sensitive to the C terms than the C terms
then it may be argued that the model cannot usefully be conditioned given the quality of
the available data (i.e. the task of replicating the observations is too badly defined). The
penalty for sampling the data errors is that there are more interacting factors so that less
information is retrieved about the effect on the posterior marginals of high order factor
interactions. This is an issue of sampling adequacy, which is discussed later.
For this sensitivity analysis, 10000 samples are taken from the prior ranges using Latin
hyper-cube sampling (McKay et al. 1979). Latin hyper-cube sampling ensures optimum
coverage of the individual ranges, and with 10000 samples gives relatively
168
comprehensive representation of two-, three-, and four-factor interactions, but sampling
of higher level interactions is sparse. In theory, this means that if the overall significance
of a factor is dependent on the simultaneous value of more than three other factors, then
its evaluation will not be reliable. In practice, however, it tends to be only the lower
interactions that affect the results of a sensitivity analysis (Henderson-Sellers and
Henderson-Sellers 1993). Therefore, 10000 samples are assumed to be sufficient. Due to
the semi-random nature of Latin hyper-cube sampling, values of the KS statistic beneath
an arbitrary level will not be significant. While significance levels can easily be
calculated as a function of the number of samples for fully random uni-variate
experiments (see Ang and Tang 1975, pp. 278), this is not valid in the context of Latin
hyper-cube sampling. Also, the KS statistic refers to the difference between the marginal
posterior and prior distributions, but the sampling is from the multivariate prior
distribution. While this does not preclude the KS test, it makes analytical derivation of the
significance level very difficult. To address this issue of identification of meaningful
significance levels for the KS statistic, a number of control factors – factors which are
known not to have any significance – are included. Significant factors can then be
identified as those whose KS statistics clearly above those of the control factors.
Although the KS statistic is a potentially insightful summary of model sensitivity, it can
diminish the importance of local effects, especially at extreme values. Where significant
factors are reported by the KS statistic, or where significant factors are expected but not
reported, the sensitivity associated with that factor can be visualised by the scatter-plot of
the probability mass or the marginal distribution (Figure 7.3). This isolates important
local sensitivities not identifiable from summary statistics.
The ultimate task of the Charles River model is to identify pollution management
scenarios which lead to an acceptable risk of failing to limit phytoplankton concentrations
(Ca) to below 10mgm-3 of chlorophyll-a. It is not safe to assume that the model
components which are important under this new objective are the same as those identified
for the objective of replicating the 1996 data, and another sensitivity analysis is
warranted. The same algorithm is used, but in this case it is known precisely what the
objectives are (<10mgm-3 at each monitoring section), and so there is no need to treat
aC as a randomly sampled variable (although there may be practical cases where
regulatory objectives are not so exact, with the qualitative objectives set by the European
Community’s Water Framework Directive (CEC 2000) being a prime example).
169
The previously conditioned parameter sets and associated probabilities α1,k, L1,k : k=1,
Nsam are employed to define the prior distribution for this second sensitivity analysis.
This is based on the premise that model parameters represent the principal physical
components of the system and will not change under future pollution load scenarios (as
opposed to the pollution loads, for which the preferred changes are under investigation).
The probabilities are translated to the prior probabilities for this second sensitivity
analysis,
kpk LL →,1 k=1,Nsam (7.17)
The sampled pollution loads within α1,k, L1,k : k=1, Nsam, which previously represented
the perceived range of errors in the historic pollution load data, are overwritten by
random samples from feasible ranges of future pollution reductions. These ranges are
defined as from zero up to the average pollution loads measured in the summer-autumn of
1996 (i.e. from 100% reduction to no substantial change). This provides the basis for
investigating phytoplankton responses to pollution interventions.
Nine objectives are defined – the constraint Ca < 10mgm-3 at each of nine control sections
(Sections A to I in Figure 1.5). The constraints are defined by upper and lower constraints
lCa and uCa (in this case, 0 and 10mgm-3 respectively), and are imposed on the model
results using Boolean objective functions,
otherwise 0OF for 1OF
,
a,
=
≤≤=
mk
uaalmk CCC k=1,Nsam; m=1,9 (7.18)
Equation 7.16 is then applied and Pm(γ) and KSm(γ) are derived as before (although this
time the prior marginals of the parameters are not uniform), and factors which will dictate
our ability to achieve the chlorophyll-a management objectives are speculated.
7.2.6 Risk-based appraisal of intervention strategies
Although the results of the second sensitivity analysis will indicate the pollution sources
most affecting our ability to achieve the objective of eutrophication reduction, a useful
question to ask is “on the evidence of the model (and given all the uncertainties involved)
what is the probability that a specified intervention will produce the desired effect?”. This
question can be answered by further processing of the results of the RSA. Following
application of Equations 7.18 then 7.16, and derivation of the factor’s marginal
170
distribution as in Figure 7.3, Pm(γ) is the probability of factor value γ given that Ca <
10mgm-3 at the mth control section. To derive the probability that Ca < 10mgm-3 given a
value of γ is a simple application of Bayes Theorem,
( ) ( ) ( )( )γ
γγ
p
mamma L
CPPCP
10.10
<=< m=1,9 (7.19)
where Lp(γ) is the prior probability of γ and ( )10<maCP is the overall probability of
success at control section m,
( ) ∑=<k
mkkpma LCP ,OF10 k=1,Nsam; m=1,9 (7.20)
( )γ10<maCP is not conditional on the value of any factor other than γ, and will reflect
the risk that the uncertainty in the other factors will cause non-achievement of the target
across the range of γ. For example, the risk of not achieving the target water quality at
each control section can be plotted against the degree of a proposed pollution load
intervention. This uni-variate report of risk could be extended into a bi-variate plot,
although more realisations may be required to identify a usefully smooth risk surface. For
this study, each potentially important pollution load intervention is analysed individually.
The other pollution loads, as well as the model parameters, are kept as conditioned by the
August 1996 observations.
7.3 Results and discussion
7.3.1 Preliminary model evaluation
The modelled spatial variation and 90% confidence limits of all 8 determinands for the
20th August are shown in Figures 7.4(a) to 7.4(h), together with the data of the observed
water quality and their error bounds. It is seen that the model, with isolated exceptions, is
successfully representing spatial variations in water quality on that date.
171
(h) Spatial variation in inorganic phosphorus Cpo 20/8/96
0
0.02
0.04
0.06
0.08
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cpo
(mgP
/L)
(b) Spatial variation in disolved oxygen Cox 20/8/96
Distance downstream of headwater (km)
0
4
8
12
0 7.5 15 22.5 30 37.5
Cox
(mgO
/L)
(a) Spatial variation in phytoplankton Ca 20/8/96
Distance downstream of headwater (km)
0.00
0.02
0.04
0.06
0 7.5 15 22.5 30 37.5
Ca
(mgC
hl-a
/L)
(d) Spatial variation in organic nitrogen Cns 20/8/96
0
0.4
0.8
1.2
1.6
2
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Ns
(mgN
/L)
(f) Spatial variation in ammonia Cna 20/8/96
0
0.05
0.1
0.15
0.2
0.25
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cna
(mgN
/L)
(e) Spatial variation in nitrate Cni 20/8/96
0.5
1.0
1.5
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cni
(mgN
/L)
(g) Spatial variation in organic phosphorus Cps 20/8/96
0
0.05
0.1
0.15
0.2
0.25
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cps
(mgP
/L)
(c) Spatial variation in biochemical oxygen demand Ccf 20/8/96
0
1
2
3
4
5
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Ccf
(mgO
/L)
(h) Spatial variation in inorganic phosphorus Cpo 20/8/96
0
0.02
0.04
0.06
0.08
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cpo
(mgP
/L)
(b) Spatial variation in disolved oxygen Cox 20/8/96
Distance downstream of headwater (km)
0
4
8
12
0 7.5 15 22.5 30 37.5
Cox
(mgO
/L)
(a) Spatial variation in phytoplankton Ca 20/8/96
Distance downstream of headwater (km)
0.00
0.02
0.04
0.06
0 7.5 15 22.5 30 37.5
Ca
(mgC
hl-a
/L)
(d) Spatial variation in organic nitrogen Cns 20/8/96
0
0.4
0.8
1.2
1.6
2
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Ns
(mgN
/L)
(f) Spatial variation in ammonia Cna 20/8/96
0
0.05
0.1
0.15
0.2
0.25
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cna
(mgN
/L)
(e) Spatial variation in nitrate Cni 20/8/96
0.5
1.0
1.5
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cni
(mgN
/L)
(g) Spatial variation in organic phosphorus Cps 20/8/96
0
0.05
0.1
0.15
0.2
0.25
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cps
(mgP
/L)
(c) Spatial variation in biochemical oxygen demand Ccf 20/8/96
0
1
2
3
4
5
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Ccf
(mgO
/L)
*
(h) Spatial variation in inorganic phosphorus Cpo 20/8/96
0
0.02
0.04
0.06
0.08
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cpo
(mgP
/L)
(b) Spatial variation in disolved oxygen Cox 20/8/96
Distance downstream of headwater (km)
0
4
8
12
0 7.5 15 22.5 30 37.5
Cox
(mgO
/L)
(a) Spatial variation in phytoplankton Ca 20/8/96
Distance downstream of headwater (km)
0.00
0.02
0.04
0.06
0 7.5 15 22.5 30 37.5
Ca
(mgC
hl-a
/L)
(d) Spatial variation in organic nitrogen Cns 20/8/96
0
0.4
0.8
1.2
1.6
2
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Ns
(mgN
/L)
(f) Spatial variation in ammonia Cna 20/8/96
0
0.05
0.1
0.15
0.2
0.25
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cna
(mgN
/L)
(e) Spatial variation in nitrate Cni 20/8/96
0.5
1.0
1.5
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cni
(mgN
/L)
(g) Spatial variation in organic phosphorus Cps 20/8/96
0
0.05
0.1
0.15
0.2
0.25
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cps
(mgP
/L)
(c) Spatial variation in biochemical oxygen demand Ccf 20/8/96
0
1
2
3
4
5
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Ccf
(mgO
/L)
(h) Spatial variation in inorganic phosphorus Cpo 20/8/96
0
0.02
0.04
0.06
0.08
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cpo
(mgP
/L)
(b) Spatial variation in disolved oxygen Cox 20/8/96
Distance downstream of headwater (km)
0
4
8
12
0 7.5 15 22.5 30 37.5
Cox
(mgO
/L)
(a) Spatial variation in phytoplankton Ca 20/8/96
Distance downstream of headwater (km)
0.00
0.02
0.04
0.06
0 7.5 15 22.5 30 37.5
Ca
(mgC
hl-a
/L)
(d) Spatial variation in organic nitrogen Cns 20/8/96
0
0.4
0.8
1.2
1.6
2
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Ns
(mgN
/L)
(f) Spatial variation in ammonia Cna 20/8/96
0
0.05
0.1
0.15
0.2
0.25
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cna
(mgN
/L)
(e) Spatial variation in nitrate Cni 20/8/96
0.5
1.0
1.5
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cni
(mgN
/L)
(g) Spatial variation in organic phosphorus Cps 20/8/96
0
0.05
0.1
0.15
0.2
0.25
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cps
(mgP
/L)
(c) Spatial variation in biochemical oxygen demand Ccf 20/8/96
0
1
2
3
4
5
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Ccf
(mgO
/L)
*
Figure 7.4 Modelled and observed spatial variations in water quality 20th August 1996 (mean value and 90% confidence limits shown). *error bounds on Cna data signify that all observations were below the detection limit of 0.1mgN/L.
The estimated confidence limits for the phytoplankton concentration Ca are interesting, as
they are not constrained by the observations as much as might be expected relative to the
other determinands. In particular, the lower confidence limit at the downstream reach
diverges from the observed data. It is speculated that this is due to the high order nature
of the phytoplankton model (evident in Equations 7.1 to 7.3), together with the non-
discriminating nature of the phytoplankton OF defined by Equation 7.15. For example, a
high numerical order would mean that the Ca result could be a good replication of the
observed data until the downstream stretch when it could swing rapidly to a poor
172
replication. As the OF defined by Equation 7.15 is aggregated over all of the monitored
sections (except the headwater), such a result would be given significant probability of
occurrence. Hence the lower confidence limit in Figure 7.4(a) is not reflecting the
observed data at monitoring section (I). This is a case where, arguably, the discrimination
between alternative sets of factors is not high enough, and Equation 7.15 should be re-
designed so that the model uncertainty is reduced, and the confidence limits are narrower.
On the other hand, more discrimination would mean that the estimation of uncertainty is
less robust to structural error, data bias, and inadequate sampling of the prior ranges of
factors (the value of Nsam). That is, narrower confidence limits are less likely to explain
the effects of these sources of error. For example, as one point of data on Figure 7.4(g)
and another on 7.4(h) are not explained by the combined estimates of model and data
error, it may be argued that the confidence limits on the Cps and Cpo models are not wide
enough. Essentially, this issue needs to be addressed using the judgement of the modeller
– to achieve a balance between robustness (presumed higher model uncertainty), and
precision (presumed lower uncertainty) which is appropriate to,
• the modeller’s judgement of data precision and reliability,
• the modeller’s judgement of model structure validity,
• the answers the modeller requires from the model, and
• the available modelling resources.
As the observed data in Figure 7.4 were used to condition the model, this result does not
demonstrate the model’s predictive capability. To attempt to do so, the conditioned model
is applied to prediction of the water quality on the 8th October, and the results are shown
in Figures 7.5(a) to 7.5(h). The confidence limits are generally wider than in Figure 7.4
indicating that the principal processes affecting water quality have changed from August
to October, so that the result is more dependant on the poorly identified model
parameters. This is clear for Ca, for which the upper confidence limit diverges greatly
from the observed data. Clearly, we are not able to accurately predict chlorophyll-a
concentrations under the October conditions, given the information retrieved from the
conditioning. Notwithstanding the lack of precision, we are able to predict with
confidence that the Ca values on the 8th October are less than the target of 10mgm-3.
173
(g) Spatial variation in organic phosphorus Cps 8/10/96
0
0.1
0.2
0.3
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cps
(mgP
/l)
(h) Spatial variation in inorganic phosphorus Cpo 8/10/96
0
0.04
0.08
0.12
0.16
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cpo
(mgP
/l)
(f) Spatial variation in ammonia Cna 8/10/96
0
0.1
0.2
0.3
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cna
(mgN
/l)
(d) Spatial variation in organic nitrogen Cns 8/10/96
0
0.25
0.5
0.75
1
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cns
(mgN
/l)
(a) Spatial variation in phytoplankton Ca 8/10/96
0.000
0.004
0.008
0.012
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Ca
(mgC
hl-a
/L)
(b) Spatial variation in dissolved oxygen Cox 8/10/96
0
4
8
12
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cox
(mgO
/l)
(c) Spatial variation in biochemical oxygen demand Ccf 8/10/96
Distance downstream of headwater (km)
0
1
2
3
4
5
0 7.5 15 22.5 30 37.5
Ccf
(mgO
/l)
(e) Spatial variation in nitrate Cni 8/10/96
0.0
0.4
0.8
1.2
1.6
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cni
(mgN
/l)
(g) Spatial variation in organic phosphorus Cps 8/10/96
0
0.1
0.2
0.3
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cps
(mgP
/l)
(h) Spatial variation in inorganic phosphorus Cpo 8/10/96
0
0.04
0.08
0.12
0.16
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cpo
(mgP
/l)
(f) Spatial variation in ammonia Cna 8/10/96
0
0.1
0.2
0.3
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cna
(mgN
/l)
(d) Spatial variation in organic nitrogen Cns 8/10/96
0
0.25
0.5
0.75
1
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cns
(mgN
/l)
(a) Spatial variation in phytoplankton Ca 8/10/96
0.000
0.004
0.008
0.012
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Ca
(mgC
hl-a
/L)
(b) Spatial variation in dissolved oxygen Cox 8/10/96
0
4
8
12
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cox
(mgO
/l)
(c) Spatial variation in biochemical oxygen demand Ccf 8/10/96
Distance downstream of headwater (km)
0
1
2
3
4
5
0 7.5 15 22.5 30 37.5
Ccf
(mgO
/l)
(e) Spatial variation in nitrate Cni 8/10/96
0.0
0.4
0.8
1.2
1.6
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cni
(mgN
/l)
*
(g) Spatial variation in organic phosphorus Cps 8/10/96
0
0.1
0.2
0.3
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cps
(mgP
/l)
(h) Spatial variation in inorganic phosphorus Cpo 8/10/96
0
0.04
0.08
0.12
0.16
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cpo
(mgP
/l)
(f) Spatial variation in ammonia Cna 8/10/96
0
0.1
0.2
0.3
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cna
(mgN
/l)
(d) Spatial variation in organic nitrogen Cns 8/10/96
0
0.25
0.5
0.75
1
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cns
(mgN
/l)
(a) Spatial variation in phytoplankton Ca 8/10/96
0.000
0.004
0.008
0.012
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Ca
(mgC
hl-a
/L)
(b) Spatial variation in dissolved oxygen Cox 8/10/96
0
4
8
12
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cox
(mgO
/l)
(c) Spatial variation in biochemical oxygen demand Ccf 8/10/96
Distance downstream of headwater (km)
0
1
2
3
4
5
0 7.5 15 22.5 30 37.5
Ccf
(mgO
/l)
(e) Spatial variation in nitrate Cni 8/10/96
0.0
0.4
0.8
1.2
1.6
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cni
(mgN
/l)
(g) Spatial variation in organic phosphorus Cps 8/10/96
0
0.1
0.2
0.3
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cps
(mgP
/l)
(h) Spatial variation in inorganic phosphorus Cpo 8/10/96
0
0.04
0.08
0.12
0.16
0 7.5 15 22.5 30 37.5
Distance downstream of headwater (km)
Cpo
(mgP
/l)
(f) Spatial variation in ammonia Cna 8/10/96
0
0.1
0.2
0.3
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cna
(mgN
/l)
(d) Spatial variation in organic nitrogen Cns 8/10/96
0
0.25
0.5
0.75
1
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cns
(mgN
/l)
(a) Spatial variation in phytoplankton Ca 8/10/96
0.000
0.004
0.008
0.012
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Ca
(mgC
hl-a
/L)
(b) Spatial variation in dissolved oxygen Cox 8/10/96
0
4
8
12
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cox
(mgO
/l)
(c) Spatial variation in biochemical oxygen demand Ccf 8/10/96
Distance downstream of headwater (km)
0
1
2
3
4
5
0 7.5 15 22.5 30 37.5
Ccf
(mgO
/l)
(e) Spatial variation in nitrate Cni 8/10/96
0.0
0.4
0.8
1.2
1.6
0 7.5 15 22.5 30 37.5Distance downstream of headwater (km)
Cni
(mgN
/l)
*
Figure 7.5 Modelled and observed spatial variations in water quality 10th October 1996 (mean value and 90% confidence limits shown). *error bounds on Cna data signify that all observations were below the detection limit of 0.1mgN/L.
The purpose of Figures 7.4 and 7.5 is not to evaluate the model structure alone, but to
jointly evaluate the structure, the GLUE likelihood measure defined by Equation 7.15 and
the data. The inseparability of these three facets of model evaluation, which is endemic in
modelling water quality using sparse and/or unreliable data, must be approached using
subjectively orientated visualisation.
The approach was taken that the model should be conditioned individually for each of the
eight determinands, so that there were eight joint distributions of model parameters. This
174
is justified because it maximises the information retrieved about the sensitivities of the
determinands, as individual entities, to all the factors. However, arguably such an
approach fails to rigorously estimate the uncertainty in the model because, for every
determinant, the information contained in the data of the other determinands is neglected.
In fact, if the model had been required to explain the variations in all the determinands
simultaneously as a function of one joint distribution of factors, the confidence limits
would have inevitably been significantly wider. A related observation was made with
respect to the Ca spatial variation in Figure 7.4. In that case, the model was expected to
explain the variations in Ca at the upstream sections simultaneous to explaining those
downstream, leading to confidence limits which were not intuitive from the observations.
This prompts a discussion of how to use multiple objectives to robustly represent model
uncertainty, and to diagnose why and in what respects the model is failing to achieve all
its objectives simultaneously. Such discussion is not pursued here, but the reader is
referred, for in-depth discussion, to Gupta et al. (1998) and Wagener et al. (2001). For
current purposes, it is proposed that the conditioned model is sufficiently explaining the
spatial variation and error in the observed data, and that, in the context of limited
resources and high uncertainty, the model conditioned by the chlorophyll-a observations
may usefully be applied to the remaining tasks of this study.
7.3.2 Sensitivity analysis (1996 conditions)
Rather than tabulating the values of the KS statistic, the model sensitivities are reported
graphically, in Figure 7.6. Within this figure, there are six graphs which report the model
sensitivities measured as described in Section 7.2.5 using the Cna, Cni, Ca, Cpo, Cox and Ccf
data (only the Cns and Cps results are not illustrated). The common x-axis of these graphs
is the series of factors, comprising of model parameters, point loads and observed in-river
data. On the y-axes, the value of the KS statistic corresponding to each of these factors is
plotted, and these points are joined to give a trajectory, the significantly high peaks of
which indicate significant factors. The significance level, which is shown on Figure 7.6 as
the horizontal dashed line, is defined by the maximum KS statistic identified for the
control factors (factors to which the specific OFs are known to be independent). For
clarity, only the evidently significant factors are labelled.
In Figure 7.6, the mean observed water quality for determinant m at the nth control section
(from Figure 1.5) is signified by dn,m. For example d2,na signifies the mean observation of
ammoniacal nitrogen at the second control section, i.e. downstream of Mill River whereas
d8,a signifies the mean observation of chlorophyll-a at the section upstream of the Trout
Brook confluence. Similarly, the point load of determinant m at the nth point source (from
175
Figure 1.5) is signified by wn,m. The special case of n = 0 signifies the point load from the
headwater.
Factors
Parameters Pollutionsources
Data error
0.00
0.25
KS
stat
istic
OF6 (Ammonia, Cna)
d2,na
d3,na
d4,na d5,na
d6,na d7,nad8,na
kon20
w0,ns
w1,na
khn20
w4,nakda20
kgahsn
0.00
0.25
KS
stat
istic
OF8 Orthophosphate, Cpo
kga20
kda20
khp20:vsp
w1,po
w3,pow4,po d8,po
0.00
0.50K
S st
atis
tic
OF1 (Phytoplankton, Ca)kda20
kga20
roa:rpa:vsn:kdn20:w0,a d8,a
0.00
0.50
KS
stat
istic
OF2 (Dissolved oxygen, Cox)kga20
kda20
roa: koc20: khc20: ker
kon,20 w1,nad2,ox d3,ox
d4,ox d5,ox d6,ox d7,ox
d8,ox
0.00
0.50
OF3 (Biochemical oxygen demand, Ccf)
vsn: kdn20: khp20, w0,cf
kga,20, kda20: roa:, rpa: khc20: koc20
w3,po d8,cf
KS
stat
istic
0.00
0.50
KS
stat
istic
OF5 (Nitrate, Cni)
kon20
kdn20
w1,Ni
d2,ni
Factors
Parameters Pollutionsources
Data error
0.00
0.25
KS
stat
istic
OF6 (Ammonia, Cna)
d2,na
d3,na
d4,na d5,na
d6,na d7,nad8,na
kon20
w0,ns
w1,na
khn20
w4,nakda20
kgahsn
0.00
0.25
KS
stat
istic
OF8 Orthophosphate, Cpo
kga20
kda20
khp20:vsp
w1,po
w3,pow4,po d8,po
0.00
0.50K
S st
atis
tic
OF1 (Phytoplankton, Ca)kda20
kga20
roa:rpa:vsn:kdn20:w0,a d8,a
0.00
0.50
KS
stat
istic
OF2 (Dissolved oxygen, Cox)kga20
kda20
roa: koc20: khc20: ker
kon,20 w1,nad2,ox d3,ox
d4,ox d5,ox d6,ox d7,ox
d8,ox
0.00
0.50
OF3 (Biochemical oxygen demand, Ccf)
vsn: kdn20: khp20, w0,cf
kga,20, kda20: roa:, rpa: khc20: koc20
w3,po d8,cf
KS
stat
istic
0.00
0.50
KS
stat
istic
OF5 (Nitrate, Cni)
kon20
kdn20
w1,Ni
d2,ni
Figure 7.6 Value of the Kolmogorov-Smirnoff statistic across the model factors for objectives of replicating August 1996 observed data. The common x-axis of these six graphs is the series of factors, comprising of model parameters, point loads and observed in-river data. The significance level of the KS statistic is shown as the horizontal dashed line. For clarity, only the evidently significant factors are labelled.
176
It is seen that the Cna objective has been affected largely by the data uncertainty (perhaps
predictable because ammonium was below the detection limit for all but one of the
monitored sections). Lack of more precise measurements of Cna has restricted the
information that can be retrieved about lesser factors such as the model parameters.
Nevertheless, four parameters (khn20, kon20, kda20, and kgahsn) are identified as significantly
affecting the value of the Cna objective function. Considering the role of these four
parameters in the model concept (Figure 7.1), all three arrows leading to or from the Cna
box may be considered “active” components of the model. There is no evidence,
therefore, upon which to reduce the complexity of the Cna representation. Returning to
Figure 7.6, three point sources (organic nitrogen from the headwater, ammoniacal
nitrogen from the CRPCD wastewater treatment plant and ammoniacal nitrogen from the
Medfield wastewater treatment plant) are clearly most affecting the modelled Cna
pollution under the conditions of the 20th August, 1996.
The objective of replicating the Cox data is suggested by Figure 7.6 to be dominated by
parameters (kga20, kda20, roa, koc20, khc20, kon20 and ker). Thereby, with the exception of the
sediment oxygen demand component, all the model components affecting Cox are implied
as “active” and there is no justification contained in Figure 7.6 for removing them from
the model. That Cox should be largely affected by phytoplankton dynamics is somewhat
predictable (at least for a water quality expert) from general knowledge of the issues
affecting the Charles river (e.g. CRWA 2000). However, that the uncertainty in the
phytoplankton’s chemical composition (parameter roa) may be more important to our
success in modelling Cox (and hence the broader ecological status of the river) than, for
example, improved knowledge of the organic pollution loads, is an example of the insight
which the sensitivity analysis can offer.
The result for Ccf indicates the dependency of Ccf on nitrogen and phosphorus
concentrations, which can only be caused by the role of Ca in the model of the carbon,
nitrogen and phosphorus cycles. Why then is the Ccf significantly influenced by the 3rd
point source of phosphates (w3,po) but Ca has not been implied as so? This is because there
is some information contained in the Ccf data which has improved the chance of w3,po
standing out as an individual factor, and apparently less such information in the Ca data.
This demonstrates that no inferences should be made about individual factors without
taking into account all the available evidence, and that the most informative data are not
always where they might be expected.
177
If the current objectives were limited to the replication of the nitrate data, Figure 7.6
indicates that barely two parameters would be justified, and only one point source
(CRPCD wastewater treatment plant) would be significant in effect. The sensitivity of the
Cni objective to d2,ni, the Cni data at monitoring section 2 (shortly downstream of Mill
River), is because this point of data is the most uncertain of all the Cni data. This is
evident from Figure 7.4. The significance of the point source from the CRPCD
wastewater treatment plant (w1,ni) is predictable because this is such an intense source of
nitrate (six times the average from the other point sources). It should be noted that a
perceived uncertainty of ±30% rather than a statistically identified distribution was
applied for this source, and the value of the KS statistic depends on this assumption.
However, inspection of the scatter plot of posterior probability against w1,ni, Figure 7.7,
implies significant responses even for example in the ±5% bracket.
0.0001
0.0002
0.0003
0.0004
0.0005
-30% -20% -10% 0% 10% 20% 30%Point source 1 (CRPCD WWTW Nitrate % deviation from measured average)
Like
lihoo
d
0.0001
0.0002
0.0003
0.0004
0.0005
-30% -20% -10% 0% 10% 20% 30%Point source 1 (CRPCD WWTW Nitrate % deviation from measured average)
Like
lihoo
d
Figure 7.7 Scatter-plot of posterior probability for the error in nitrate load from CRPCD
Finally, from Figure 7.6, it is seen that the data uncertainty at the eighth analysed
monitoring section (section I in Figure 1.5) is repeatedly implied to significantly affect
the ability of the model to meet the objectives. In other words, the “goalposts “ are being
moved too much, through sampling of C in Equation 7.15. This suggests that the
uncertainty in the data at that monitoring section is limiting the information retrievable
about the other factors, and that this section is a priority for more data collection and/or
more precise measurement techniques. This suggestion may equally well be applied to the
data shortcomings previously identified specifically for the Cna and Cni models. Thus, it is
proposed that application of regional sensitivity analysis in the described manner has
potential value in designing and updating field monitoring progammes, and in the
objective management of sources of data error. Clearly, significant investment would not
be justified solely on the basis of a preliminary analysis, like that presented here. Instead,
178
for example, the analysis could be repeated to incorporate hypothesised reductions in
sampling and measurement error, to evaluate associated reductions in model uncertainty
and improvements in decision-making.
7.3.3 Sensitivity analysis (eutrophication reduction)
Figure 7.8 shows the superimposed trajectories of the KS statistic for each of the
eutrophication reduction objectives (keeping chlorophyll-a concentrations below 3mgm10 − of chlorophyll-a at each of the nine control sections, A-I). The overwhelming
implication of the analysis is that the concentration of chlorophyll-a in the headwater w0,a
is mainly responsible for the occurrences of failing to achieve the objective. For the
sections further down the river, the growth and death rates of phytoplankton become
more significant, as the phytoplankton have had more time to grow, or to die.
FactorsParameters Pollution sources
kga20
kda20
khp20
khn20
vsn
r pa
w0,a
w1,pok gah
sik g
ahsp
0
1
KS
stat
istic
vspkdn20 w3,po w4,po
FactorsParameters Pollution sources
kga20
kda20
khp20
khn20
vsn
r pa
w0,a
w1,pok gah
sik g
ahsp
0
1
KS
stat
istic
vspkdn20 w3,po w4,po
Figure 7.8 Plot of Kolmogorov-Smirnoff statistic across the model factors for objectives of reducing eutrophication at each of nine control sections
Various other phytoplankton parameters are implied to be relevant, so that the uncertainty
in the phytoplankton properties is a source of risk which may lead to nutrient reduction
interventions being ineffective. For example, this suggests that it is important to know the
dominant species of phytoplankton (e.g. blue-greens, diatoms, etc) in identifying low-risk
interventions for eutrophication management, as species are known to have individual
maximum growth rates, carbon to chlorophyll-a ratios, etc. Both the phytoplankton
phosphorus and light half-saturation constants (kgahsp and kgahsl) are included in the
significant parameters, implying that there is insufficient evidence to predict whether the
system would be phosphorus or light limited, although there is evidence that it would not
be nitrogen limited. The evidence for phosphorus limitation is corroborated by the
observation that three point sources of phosphorus have significant KS values - at the
179
CRPCD wastewater treatment plant, at the Stop River and at the Medfield treatment
plant.
7.3.4 Appraisal of intervention strategies
Before appraising options for reduction of summer eutrophication, it is worth noting from
Figure 7.4(a) that a “do nothing “ strategy has little chance of working, assuming that
1996 conditions are typical. It is seen that all the observed chlorophyll-a concentrations
are well above the target chlorophyll-a concentration of 10mgm-3, and the lower 90%
percentile of the model result just dips below the target at the downstream end of the
studied reach (although, as previously argued, this percentile is an outcome of the
limitations of the GLUE likelihood measure, and may be overly optimistic).
Firstly, the risk associated with reducing the phosphate load from the CRPCD treatment
plant was investigated. Even with 100% reduction in this load, results indicated
insignificant probability of achieving the target at any of the control sections. Further to
this, the effect of reducing the total phosphorus load from the CRPCD treatment plant
was investigated (the organic fraction of the total phosphorus load was lumped into the
inorganic fraction). Again, this was ineffectual. The results are strong evidence that
phosphorus stripping at the CRPCD treatment plant alone is unlikely to adequately
control eutrophication. The residual phosphorus in the river is estimated as adequate to
sustain undesirable phytoplankton growth under almost all feasible summer conditions.
Additional or alternative measures are required. These results also emphasise the lack of
detail given by the sensitivity analysis results in Figure 7.8. Figure 7.8 indicates the
factors for which the successful factor values are significantly different from the
unsuccessful values, not the level of success.
0.0
0.2
0.4
0.6
0.8
1.0
-100% -75% -50% -25% 0%Reduction
Prob
abili
ty o
f ach
ievi
ng o
bjec
tive
(B)(A)
(C)
(D)
(E)
(F)
(I)
(H)
(G)0.0
0.2
0.4
0.6
0.8
1.0
-100% -75% -50% -25% 0%Reduction
Prob
abili
ty o
f ach
ievi
ng o
bjec
tive
(B)(A)
(C)
(D)
(E)
(F)
(I)
(H)
(G)
(B)(A)
(C)
(D)
(E)
(F)
(I)
(H)
(G)
Figure 7.9 - Risk associated with reduction in chlorophyll-a in headwater.
180
(A) = probability of achieving objective at control section (A); (B) = probability of
achieving objective at control section (B); etc.
The third scenario involved leaving all phosphorus loads at their 1996 levels, and
investigating the effect of reducing the concentration of chlorophyll-a in the headwater.
The results are presented in Figure 7.9 for the sections (A) to (I) indicated on Figure 1.5.
These suggest, for example, that reducing the headwater Ca by 60% will guarantee to
satisfactorily reduce Ca at control section (A) (although this is a trivial result, as section
(A) is the headwater); has a 80% chance of success at control section (B); 30% chance at
(C); 5% chance at (D); and no chance at any of the further downstream sections.
Alternatively, 100% reduction in the headwater Ca is more or less guaranteed to be
effectual upstream of South Natick Dam, and even has a 30% chance of success at the
Cochrane dam.
For the fourth scenario, the concentration of Ca in the headwater was fixed at 50% of the
average value observed in the summer of 1996, and the risk associated with reducing the
CRPCD total phosphorus concentration was investigated. The risk associated with
reducing the CRPCD total phosphorus load under these new conditions is plotted in
Figure 7.10. Firstly, it is noted that the Ca is almost bound to be below the target at
section (A) which is consistent with the result in Figure 7.9. In general, Figure 7.10 shows
that even after reducing Ca in the headwater, phosphorus stripping at the CRPCD plant is,
by itself, a high-risk intervention. For example, 95% reduction in phosphorus loading is
suggested as having a 40% chance of adequately reducing chlorophyll-a at section (D), a
25% chance at section (E) and no chance of success further downstream, following the
influences of Medfield WWTP, Sewall Brook and Bogastow Brook. However, this
intervention is likely to be effective at sections (B) and (C). Clearly, the specific objective
of the planners is crucial in this case.
This demonstration has highlighted the considerable degree of uncertainty, or risk,
attached to any water quality intervention. Importantly, the derived risk should not be
interpreted as the expected frequency of failure. This would imply that the model
parameters behave randomly as described by their conditioned joint distribution when, in
fact, we simply do not know what their statistical properties or time-variance should be.
Although predicted fluctuations in the water quality can be allowed for in the measure of
risk (for example by using a dynamic simulation or, as done here, by treating diurnal
variations as random effects), a large part of the risk stems from the low reliability of
181
model predictions. Therefore, interventions which can be modelled with relative precision
- which are not affected by highly uncertain components of the model - will be identified
as preferable. For example, the reduction in chlorophyll-a concentration which would be
achieved by doubling the flow in the Charles River would be identified as a low risk
intervention, as it does not rely on highly uncertain biochemical properties. However,
given that much of the risk comes from the model’s limitations, it is sensible and
probably economical to explore methods of improving the reliability of the model, rather
than opt for a low-risk intervention.
-100% -75% -50% -25% 0%Reduction
(B)
(A)
(C)(D)(E)
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty o
f ach
ievi
ng o
bjec
tive
-100% -75% -50% -25% 0%Reduction
(B)
(A)
(C)(D)(E)
(B)
(A)
(C)(D)(E)
0.0
0.2
0.4
0.6
0.8
1.0
Prob
abili
ty o
f ach
ievi
ng o
bjec
tive
Figure 7.10 Risk associated with reduction in phosphorus load from CRPCD. (A) = probability of achieving objective at control section (A); (B) = probability of achieving objective at control section (B); etc.
A primary motivation for risk-based modelling is that management decisions should not
be restricted to those which prioritise water quality interventions. A judicious decision
may recognise that a sufficiently low-risk intervention is unattainable, and instead call for
model improvements, review of water quality targets or further data collection. Further
data collection has been identified as a priority for this investigation of the Charles River,
due to the restrictions which the employed data (and their assumed errors) imposed upon
model reliability. By implication, the reliability of any decisions about preferred
interventions is degraded, and this is represented in the high risk levels reported in
Figures 7.9 and 7.10. A useful extension to this investigation, therefore, would be to
evaluate how the attainable quality of decision (reduction in risk) would respond to
proposed investments in data collection. This could simply involve repeating the above
conditioning, sensitivity and risk analyses with respecified data errors – in particular,
Figure 7.8 indicates that the value of reducing errors associated with the headwater
182
chlorophyll-a may significantly improve the reliability of the decisions which need to be
made.
7.4 Summary
A Monte Carlo-based framework of sensitivity analysis and risk evaluation has the
capacity to support risk-based management of surface water quality through,
• calculating the response of model outputs to uncertain or stochastic model inputs
(boundary conditions, pollution loads and model parameters), allowing the
distributions of model outputs to be reported at all points within the model’s
space and time domains,
• calculating the response required of model inputs to meet constraints imposed
upon the model output, allowing the posterior distributions of model inputs to be
reported. The imposed constraints are either defined by observed water quality
(with the purpose of conditioning the model, and exploring the sensitivities of the
system), or by prescribed water quality targets (with the purpose of appraising
intervention scenarios, and identifying realistic targets).
This has been demonstrated in this chapter using the upper Charles River, Massachusetts
as a case study. A model structure has been selected based on a prior hypothesis of the
principal water quality processes. The model has been conditioned using observations of
water quality observed on 26th August 1996 at a number of control sections along the
river. The model was conditioned with respect to each of eight objectives, one for each of
the eight key water quality determinands, and was evaluated through visualisation of the
spatial variation of modelled water quality and observations (Figures 7.4 and 7.5). The
superposition of confidence limits (estimated uncertainty in the model) and error bars
(estimated uncertainty in the observed data) allowed informed judgement upon the
relative reliabilities of the model structure and the measured data.
In pursuit of rigorous evaluation of risk, all uncertain factors potentially affecting the
achievement of water quality objectives should be included in the sensitivity analyses.
This may include model parameters, pollution and hydraulic sources, boundary
conditions, and perhaps the water quality objectives themselves. It was shown (Figure
7.6) that treating the water quality objectives as random variables allows the relative
183
importance of data error to be indicated. This feature also has potential for integrating the
effects of uncertain regulations into the appraisal of scenarios (e.g. which scenario is
safest given qualitative or otherwise uncertain water quality criteria?), and in offsetting
different targets against the cost of conforming (i.e. what are viable targets?).
Figures 7.6 and 7.8 are examples of how the factors significantly affecting water quality
are identified using the Kolmogorov-Smirnov statistic to summarise the results of a
regional sensitivity analysis. These results should be seen as indicators of the factors most
likely to be affecting the respective objectives given the knowledge embodied in the
model structure and parameter ranges. It is a useful approach to screening the model for
unexpected sensitivities, and for identifying factors to be taken forward to more detailed
analysis. For example, Figure 7.8 clearly identified that reducing the headwater Ca and the
Cpo load from the CRPCD treatment plant were priorities for further investigation –
because there is indication that the former might be an effective way forward, and that the
latter might be less so. Further investigation resulted in Figures 7.9 and 7.10 which
confirmed these suspicions in terms of risk of failure associated with various magnitudes
of intervention.
The overriding and unavoidable limitation to the methods employed here, at least in
applications where data are typically sparse and imprecise, is that results are partly
subjective. This is because model evaluation must be based on judgement of the relative
unbiasedness of the model structure and the observed data. Tools such as WaterRAT
(McIntyre and Zeng 2002) are needed to support the modeller in these judgements, and
allow justifiable use of results in a decision support role, within the practical constraints
of data and modelling resources.
184
8. Conclusions
The path that this dissertation has followed so far may be summarised as - establishment
of motivation (Chapter 1); review of previous models and methodologies (Chapters 2 and
3); description of the developed tool (Chapter 4); insight into numerical issues
encountered in the development (Chapter 5); a priori exploration of model identification
issues (Chapter 6); and application to a management problem (Chapter 7). Each chapter,
especially the latter three, has raised issues relating to the Thesis which need further
discussion, and Chapter 8 picks up these issues, expands upon them and integrates them
into a set of conclusions. Additional discussions of the practical obstructions encountered
in Hun River modelling study, and of current trends in the field of water quality
modelling in the UK are also included.
185
8.1 Summary
The principal achievements of this Thesis have been;
• A statement of the increasing motivation for risk-based modelling of water
quality and continuing relevance of model uncertainty analysis.
• Instructive inter-comparison between uncertainty estimation methods.
• Provision of software for model uncertainty analysis and risk-based decision-
support.
• Identification and exploration of numerical issues that control formal risk
evaluation, with limited guidance on how to deal with these issues. This included
the relevance of numerical approximations, field experiment design, model
structural errors, boundary and initial condition errors, and calibration data
information content.
• A framework for establishing links between model uncertainty, data quality and
decision-making risk.
• An agenda for further research and development needed to cope with persisting
and emerging problems (this chapter).
Thus, this dissertation may be regarded as a valuable resource for any modeller charged
with estimating, reporting and reducing uncertainty in model predictions, and for anybody
wishing to rigorously evaluate sources of risk to water quality status and in associated
decisions.
8.2 Review of GLUE
Arguably GLUE is now the most used and best recognised approach to the evaluation of
model uncertainty in the field of hydrological modelling. This may be because of the
relatively simple concepts and theory upon which it is based compared, for example, to
Monte Carlo Markov Chains and recursive parameter estimation methods. It is also due
partly to the relative ease with which GLUE can be implemented using plain Monte Carlo
sampling, and extended to regional sensitivity analysis and Bayesian analysis as
presented in Chapter 7.
The GLUE methodology is a framework within which Bayesian methods are applied to
parameter and model output uncertainty estimation, and regional sensitivity analysis. It
may also be seen as an extension of the set-based method introduced by Hornberger and
186
Spear (1980) and Spear and Hornberger (1980), introducing Bayesian or fuzzy
discrimination between behavioural models and promoting scope for Bayesian updating
of models. The originality of GLUE, therefore, is not in providing a new analytical
method, but by providing an analytical methodology that, rather than promoting specific
measures of uncertainty, requires modellers to define uncertainty according to their
perceptions of the model and data errors (as introduced in Chapter 1), and the proposed
use of the model. Thus, the role of the modeller’s judgement is recognised, and his
subjective input is objectively (i.e. numerically) expressed so that it is open to audit
(Beven and Freer 2001). This is consistent with the view expressed, in the wider context
of Bayesian modelling, by West and Harrison (1997) – that likelihood functions have a
subjective foundation for which modellers should be accountable.
It may be said that the research reported within this dissertation, although limited in a
variety of regards, is one of the more extensive explorations of application of GLUE to
water quality modelling. This included the introductory demonstrations of Chapter 2, the
integration into WaterRAT in Chapter 4, identification of the need for a GLUE-type
approach in Chapter 5, the estimation of confidence limits for river water temperature in
Chapter 6 and the sensitivity analysis and extension to risk analysis in Chapter 7.
The primary difficulty with using GLUE is justifying the measure of likelihood. Beck
(1987) points out that set-based method of Hornberger and Spear (1980), which he calls
the HSY method, may be preferred for this reason, simply because the origin of the
estimated model uncertainty can be relatively intuitive. Early in this dissertation, it was
suggested that we are only warranted in investing resources in uncertainty estimation to
the degree that it could affect management decisions. Given the innumerable contributing
political, management, and economic factors affecting decisions, whether discrimination
within the behavioural models is influential in practice, and worth the extra complexity
and degree of subjective input that GLUE may introduce, is open to argument and study.
On the other hand, it may be argued that it is intuitive to give more weight to the
behavioural models that appear to better fit the data, and that not doing so is neglecting
relevant information.
8.3 Prior and posterior knowledge
Following from the arguments about the value of GLUE, the question must be raised “in
what cases is it a waste of time attempting to condition the model?”, irrespective of the
187
conditioning algorithm used. For example, this can easily be argued when the information
content in the data is too poor and/or the uncertainty in the model boundary and initial
conditions is too high to allow that information to be extracted. A considerable amount of
this research has focussed on model conditioning – yet in the motivating case study of the
Hun River, reducing uncertainty through conditioning proved to be futile (see Section
8.9). Even with the relatively reliable data used in the Charles River study, the effect of
model conditioning was limited by the perceived scope for error in the loading and in-
river data. An interesting extension to Chapter 7 would be to investigate the effects on the
report to the decision-maker (e.g. Figures 7.9 and 7.10) that the conditioning stage can
initiate, given the data limitations. While model conditioning is arguably important in
data-rich studies (although, even in such cases, the effect on ultimate decisions appears
not to have been demonstrated in the literature) this study has raised the question of its
importance in studies with severe or typical data-limitations.
Notwithstanding the popularity of QUAL2E-UNCAS and SIMCAT, one of the general
challenges for proponents of the Monte Carlo method remains achieving its widespread
application in ‘mainstream’ water quality modelling. If the data are sufficient, model
conditioning means reduced risk in decision-making, but is a stage that inevitably adds to
the perceived complexity of the Monte Carlo analysis, and draws on resources of the
investigators. Arguably, therefore, model conditioning is a secondary challenge to
achieving recognition and formal incorporation of a priori uncertainty.
Given that in some studies, due to data limitations and/or the philosophy or resources of
the modeller, the uncertainty analysis will be limited to propagation of a priori model
uncertainty, what degree of model complexity is justified? Should we be reverting to the
most basic a priori knowledge of mass balance (in the Hun River study even this proved
difficult – see Section 8.9), or should we be maximising the prior knowledge contained in
the model? One argument is that, if a process is known to be potentially significant, and
the prior knowledge and computational resources are available, then there is no reason
not to include it; while the opposing view is that the inclusion of processes that cannot be
identified from the data means unwarranted assumptions and complexity potentially
leading to unjustified modelling effort and ill-founded conclusions, and increases the
danger of ‘over-fitting’ noisy data. Reducing model complexity to suit the data makes the
model more specific to the conditions of calibration, and so the reliability of extrapolation
is more questionable, while including freedom which is redundant under conditions of
calibration generally leads to equifinality and higher parameter uncertainty. While, then,
it can be argued that a combination of a complex model and a Monte Carlo uncertainty
188
analysis which thoroughly explores the a priori parameter space is the preferred solution,
the human and computational effort required to do so successfully is unlikely to be
available. In particular, if the sampling error associated with the Monte Carlo method,
which is inevitably more significant when applied to models with a large number of
uncertain parameters, becomes the prevalent limitation to the justification of results, the
object of the uncertainty analysis is defeated, and the modeller must take the
responsibility of avoiding this.
The reliability of model conditioning and/or uncertainty propagation using Monte Carlo
analysis depends upon the validity of the model structure, its numerical implementation,
the representation of data uncertainty, the adequacy of the number of Monte Carlo
samples of model inputs, and the likelihood measure that was used to condition the
model. The modeller is faced with a difficult and important task in balancing the effort
directed at researching and resolving each of these issues, and decisions about the model
structure (either towards making it more complex or more simple) should consider the
relative importance of the other contributing factors, as well as resource constraints and
the actual answers that are sought from the model output. This dissertation has proposed a
framework and associated modelling tool through which this task can be approached.
8.4 Monte Carlo sampling
The comparisons of GLUE with MCMCs and Monte Carlo sampling of data provided in
Chapter 2 has shown that each are different approaches to exploring a response surface,
and assuming that the theoretical definition of this response surface is consistent, the
methods can be expected to produce consistent results (the inconsistencies being due to
the mathematical approximations used in the OF definition, and the sampling efficiency
of the exploration). This result provides a basis for further, case-study comparisons of the
efficacy of the alternative methods, but the wider objectives of this research meant that
this was not pursued. While Kuczera and Parent (1998) found that MCMC is significantly
more efficient than importance sampling at analysis of extreme values, this apparently
contradicts the observation in Chapter 2 that GLUE using uniform prior distributions is
better suited to analysis of extremes. However, these investigators assumed normal and
uniform posterior distributions rather than the posterior distributions obtained from
GLUE, and their conclusions seem somewhat misleading. Clearly, some additional
189
research into the issue of extreme value analysis using GLUE and other Monte Carlo
methods is warranted.
While in Chapters 2 and 5 stratified sampling was used, Chapters 6 and 7 go on to use
Latin hypercube sampling (LHS) (McKay et al. 1979 review the differences). Yu et al.
(2001) extol the value of the LHS method for evaluating parameter uncertainty in rainfall-
runoff modelling, noting that it reduced the required number of samples by 90%.
However, as Press et al. (1988) note, LHS can only be expected to be advantageous when
the sampling is so sparse that plain random sampling may not even identify univariate
effects, and it cannot provide an advantage in identifying factor interactions (see also
Section 8.5). In particular, LHS is most valuable where the uncertainty is predominantly
arising from univariate effects of some of the factors, but of which factors is unknown a
priori. Therefore, it is speculated that that the conclusion of Yu et al. (2001) arose
because of the limited significance of interactions in their particular study, and it should
not be seen as a general result. Furthermore, it may be argued that the use of LHS in
Chapters 6 and 7 of this dissertation, rather than plain sampling, is unlikely to have added
considerable significance to results.
It is important to note that, while a number of studies have been done comparing
sampling methods and response surface exploration algorithms (e.g. Kuczera and Parent
1998, Thyer et al. 1999, Portielje et al. 2000, Melching and Bauwens 2001, Yu et al.
2001), these are in the context of relatively simple water quality and rainfall-runoff
models, with considerably fewer parameters than the thermodynamic models and water
quality models employed in Chapters 5 and 7. Furthermore, these other studies have not
been done in the context of possibly large data and model structure errors, and perhaps
therefore emphasise the overall importance of sampling efficiency more than is
appropriate in water quality modelling using typical data. The exploration of the
multivariate response surface of parameters of the Chapter 5 and Chapter 7 models is
inevitably very sparse. As raised in Chapter 7, the relevant question is whether the higher
order interactions that are not sampled extensively are significant to the modelling task;
and (returning us to the discussion above), if higher interactions are not important, then
does the model need to be so complex? Far-reaching audits of decisions and associated
costs and models are required to answer this.
190
8.5 Review of WaterRAT
Further to the discussion in 8.3, a priority development of WaterRAT would be to make
the selection of the GLUE likelihood measure easier, or at least to make the significance
of the measure more transparent to the user. At present, there is a menu which gives a
variety of options for the likelihood measure, and the user must refer to the manual to
view the definitions (see McIntyre and Zeng 2002). In addition, behavioural zones can be
specified by entering limits on spreadsheets, and the determinands, and the time and
space domain to be included in the objective function is defined on another menu. To
simplify the GLUE likelihood specification, various alternatives could be removed,
reducing the GLUE to its parent HSY format. The Excel interface provides an especially
useful means of ‘drawing’ behavioural zones. On the other hand, for those potential users
who insist on the discrimination allowed by GLUE, an improvement would be to require
the user to write the GLUE likelihood definition themselves, rather than offer a
prescribed choice of functions. This is more in line with the philosophy of GLUE, forcing
the modeller to be aware of and responsible for that definition.
Statistical analysis of residuals is not presently included in WaterRAT, and some
functions for looking at this (e.g. residual autocorrelation, heteroscedacity, outliers and
non-normality) would allow some assessment of the presence of model structure error
and data bias (see section 2.2.3). Should these errors be small, analysis of residuals would
also allow data error variance to be added to model result variance obtained on the basis
of likelihood functions (see section 2.2.6).
The version of WaterRAT delivered to the European Commission did not include the
MCMC algorithm, as its value remains to be tested. Furthermore, the genetic algorithm is
at a relatively undeveloped stage, unable to make any justifiable estimation of parameter
uncertainty. It would seem particularly valuable to develop a multiple-objective capability
of this algorithm, to avoid the especially computationally demanding task of identifying
Pareto optimal parameters using random sampling and the Pareto filter, as currently used
in WaterRAT (refer to McIntyre and Zeng 2002).
Another component of WaterRAT which was not included in the delivered version is
factorial sensitivity analysis (Henderson-Sellers and Henderson-Sellers 1993) which
allows for two-factor interactions between model inputs. A two-factor factorial analysis
may be regarded as the antithesis of Latin hypercube sampling – in the factorial analysis
the prior marginal distributions of individual factors are represented by just two points so
191
that two-factor interactions can be explored rigorously, while Latin hypercube sampling
relegates exploration of interactions in favour of a much more thorough sampling of the
univariate form. For model screening, two-factor factorial analysis may be preferred to
Monte Carlo-based methods of analysis, as it can be used to concentrate on the
sensitivities associated with extreme values (Kleijnen 1997) (although it still cannot
identify non-linearities in the response surface). Use of factorial sensitivity analysis of
factors affecting oil pollution in the Hun River was undertaken by Sherif (2000) using
WaterRAT.
The limited choice of model structures within WaterRAT could limit the applicability of
the tool. For example, the lack of a hydrodynamic model together with the need for direct
estimation of shear friction to drive the sediment resuspension model, meant that the
quasi-steady friction formula model was employed in Chapter 6, where a fully dynamic
model would have been helpful, at least to prove or disprove the sufficiency of the quasi-
steady approximation. Furthermore, the capability of WaterRAT to model sediment-water
interactions is not fully realised due to the absence of a choice of sorption models, for
example for modelling toxins, and absence of a model of redox status that may control
phosphorus release. In these regards, and with regard to conceptualisation of other
perceived important processes, there is much to be said for the framework adopted for
DESERT (Ivanov et al. 1996) whereby the user can specify his own model structure,
giving ultimate flexibility. Of course, this would require advanced modelling expertise
and added time needed to apply the tool, and certainly would be inappropriate for the
TOPLEM application of WaterRAT where, ultimately, minimal end user expertise is
available.
8.6 Review of Chapter 5: numerical issues
The numerical study presented in Chapter 5 was born out of necessity, when it became
clear that Monte Carlo simulation involving solution of systems of space-time partial
differential equations was not viable without careful attention to solution schemes and
tolerances. Not only was the time required to maintain numerical stability using the
originally devised fixed grid integration scheme too large, but the differences in
numerical precision from one Monte Carlo realisation to the next were a considerable and
unwanted complication in the overall analysis. The solution of the river
thermodynamic/ice growth model incorporated significant variations in time-step
192
requirements within and between realisations, and this was also observed in the water
quality studies of Chapters 6 and 7, although to a lesser extent.
Chapter 5 had a number of outcomes worthy of discussion. The river ice model was an
interesting stage of development – the work involved was justified by the perceived
dominant effect of the ice on the pollution transport and fate during the critical winter dry
season on the Hun River. Unfortunately, this was another aspect that could not be
developed within this project due to data limitations. More relevant to the Thesis, the
chapter draws attention to the benefits to be gained from reconciling numerical tolerances
with the overall task and achievable precision of the model. There is clearly more work to
be done in this regard, as the chapter took only a cursory look at what tolerance was
actually appropriate – the study did not attempt to identify the relative benefits of
increasing the number of samples, and reducing numerical precision. In the wider context,
given the inevitable limitations caused by errors in the model structure, calibration and
boundary condition data, and given the subjective nature of the GLUE likelihood, how
may samples at what numerical precision are really justified? It may be argued that the
constantly increasing power of desktop computers is removing numerical restrictions, and
makes such questions less relevant. However, we are clearly very far from achieving
satisfactorily fast Monte Carlo simulation on personal computers, and far from knowing
what is satisfactory given a certain model and its various uncertainties. In addition,
various demands for the additional computer power are evolving (such as GIS interfaces),
which are in competition with improved application of Monte Carlo methods.
The numerical investigation also raised the issue of design of the spatial grid, as, while
the truncation errors in the time domain were automatically controlled, the spatial errors
were managed manually. Not only should the modeller think of what scale he wishes to
be modelling on (e.g. at the reach scale, or at the sub-reach scale, referring to Figure 5.7),
but ideally either he would know a priori what spatial grid-size is needed to achieve
adequate precision at this scale, or the grid would adapt in interaction with that in the time
domain. While the spatial truncation errors may be lumped into the overall error and the
effects included in parameter estimates and their uncertainty, this is not ideal as it
constitutes another loss of rigour and reliability in extrapolation to changed modelling
tasks. The automatic generation of spatial grids is a field which has not been touched
upon in this work, and seems to be relatively untouched in water quality modelling.
193
8.7 Review of Chapter 6: prior identification of data needs and
assessment of model capability.
Much of the review in Chapter 2 was based on rainfall-runoff modelling which typically
uses daily input-output data, and in which it is generally assumed that the input data are
error free (notable exceptions are Xu and Vandewiele 1994, Shah et al. 1996). Water
quality data are considerably more costly to collect than streamflow data, and may be
considerably less accurate both in terms of sampling errors and measurement errors.
Adding to the problem is the general desire, whether well-founded or not, to use models
that are mathematically over-parameterised. Chapter 6 recognised this fundamental
problem and set out to explore what we might expect to achieve out of a data-limited
water quality modelling study using a priori modelling experiments. At the same time,
that chapter covered one of the practical problems encountered in allocating TOPLEM
resources to monitoring.
Chapter 6 is limited in scope. It is hypothetical, based on synthetic data and untested
models, and so only gives an indication of the extent of difficulties that can arise from
data limitations. It also assumes that some important data (namely the upstream flow and
phosphorus concentrations, and the diffuse sources) were available at the desired
frequency, and while effects of random error in these inputs were explored, serious biases
were not. However, the investigation exposed the degree of difficulty arising from model
and data limitations in a way which can only be done on a synthetic basis. In particular,
the significance of a (relatively minor) structural fault on both the preferred monitoring
programme and the achievable reliability of forecasts is revealing. In ‘real’ studies, the
effects of various sources of error (e.g. data error and model structural error) cannot be
separated, and therefore little insight can be gained into controls on model uncertainty
and reliability.
The large number of sources of error in real modelling studies have effects on model
reliability that are inextricable from each other, and therefore leave us no option but to
use devices such as GLUE that agglomerate errors in a manner that is not easy to justify
either to an objective-minded mathematician or to a ‘layman’ decision-maker, and will
always leave doubt about the usefulness of forecast error estimates. While semi-synthetic
experimental modelling will never eradicate errors or remove the need to subjectively
estimate uncertainties in forecasts, it can at least be used to indicate the likely significance
of individual error sources, as in Chapter 6. Extending the investigation to evaluating the
194
ability of GLUE and multiple-objective optimisation to improve the robustness of the
error forecasts, is a matter for further research.
Following the conclusions of Chapter 6, together with recognition of its limited scope, it
is arguable that, in general, extensive numerical studies are warranted before embarking
on costly monitoring programmes data to support predictive modelling. These should be
both of the type in Chapter 6, whereby speculative modelling is used to enhance the
efficiency of data collection, and of the recursive modelling-monitoring type, where
model and monitoring programme developments feed off each other (Somlyody 1995,
Van Straten 1998).
8.8 Review of Chapter 7: a framework for model conditioning, sensitivity and risk analysis
Firstly, it is noted that, for the purpose of Chapter 7, the arguments for parsimonious
modelling have been rejected. That study aimed to relate model sensitivities to reasonably
meaningful model parameters, and reducing the physical meaning by lumping processes
into a parsimonious form was considered contrary to this aim. Also, parameters to which
the algae output showed no sensitivity under calibration conditions were indicated as
important when investigating future scenarios.
Using multiple objectives for the definition of parameter uncertainty would be an
interesting development, which seemingly has not yet been done in water quality
applications, apart from the simple demonstrations given by McIntyre et al. (2001). While
Chapter 7 measured sensitivity simultaneously with respect to multiple objectives, it did
not go so far as to define parameter and model output uncertainty this way, in the manner
of Yapo et al. (1998) for example. Clearly, following the discussion in Chapter 2 (see also
the discussions of Beven (2000b) and, Kavetski et al. (2002) in the rainfall-runoff
modelling context), the Pareto definition of uncertainty is quite different from that which
might be identified using GLUE, and there seems to be some interesting scope for
extending the genericism of GLUE to encompass multi-objective considerations (e.g. the
fuzzy union of two equally likely response surfaces).
Multi-objective conditioning has recently been extended by Wagener et al. (2002a, b) to a
dynamic conditioning, which they call DYNIA. Using DYNIA, the objective function is
defined by the residuals within a window (of a predefined size) which moves over a
195
whole time-series of data. Therefore, instead of the distinct multiple objectives used for
example in Chapter 7, there is a ‘continuum’ of objectives over the time domain. This
allows the relative importance of each model parameter and its optimum value to be
assessed over the time domain, thus identifying the system response modes that are most
relevant to each parameter, and indicating structural weaknesses from time-variations in
optimum parameter values. In principle, DYNIA could be applied to evaluation of water
quality models either over the time-domain or, in the same manner, over the space
domain. However, it would be expected that, applied to studies such as that in Chapter 7
where data are quite sparse and potentially imprecise, the results of DYNIA would be
very sensitive to data errors. Like other ‘dynamic’ methods of parameter estimation, such
as Kalman filtering (Beck 1983), DYNIA is suited to relatively intense and good quality
data sets.
The study in Chapter 7 may be regarded as innovative in that all the factors affecting the
objective function value are considered as uncertain and treated the same way in the
Monte Carlo analysis. This is an advance on the common approach of estimating
parameter uncertainty (e.g. Whitehead and Hornberger 1984, Freer et al. 1996, McIntyre
et al. 2001, Wagener et al. 2002b) with no explicit recognition of data uncertainty, or
boundary and initial condition uncertainty. For some objectives, the relative importance
of in-river data error was highlighted, whilst for others it was the significance of
uncertainty in pollution load estimates, indicating the impediments to successful model
conditioning. Chapter 7 also tried to present the application of GLUE from its Bayesian
foundation, and thereby to develop the regional sensitivity analysis results into estimates
of probability of failing prescribed objectives (where the objective would be some water
quality target or targets). It was argued that the risks of failure of environmental
management arise largely from the assumptions and approximations employed at the
modelling stage, and so risks can largely be managed by investment in the monitoring and
modelling process, rather than in pollution control. For this to be pursued requires that all
significant assumptions and approximations may be represented as uncertainties. It was
emphasised in Chapter 7 that these may include the uncertainty in the objective itself.
This raises opportunities for exploring questions such as “is it the hydro-chemical model
and its input uncertainty that leads to risk in making decisions about aquatic ecosystem
management (i.e. the basis for this dissertation), or is it our inability to link the chemistry
to the ecosystem?”; what relative efforts should be put into improving the hydro-chemical
model versus the chemico-ecological model?
196
Chapter 7 also introduced the concept that evaluation of uncertainty is not restricted to
identifying and reducing uncertainties in the model, but, in cases of sparse and unreliable
data, may be used to identify data of dubious quality. The visualisation methods used in
the Charles River study (Figures 7.4 and 7.5) aim to assess data using simulation
modelling and GLUE-based confidence limits. While, of course, any conclusions about
data reliability must be made in the context of the limitations of the model and its
uncertainty estimation, the same is equally true for well-established methods of data
quality control using time-series methods (e.g. Tsakalias and Koutsoyiannis 1999).
The subjective input to practical modelling studies is evident in the work of Chapter 7.
Not only is the structure judged to represent the main water quality processes, but the
prior ranges of parameter values are taken from the literature, and the prior uncertainty in
point pollution sources is based on perceived possible error in the daily load estimates.
While it might be argued that the variability in point sources can be measured, in practice
generally only the inputs to sewage treatment works are monitored, and measuring the
discharges as part of a modelling exercise is costly. In practice, daily discharges are often
approximated by assumed reductions to the input, or by assuming direct proportion to the
served population (e.g. Daldorph et al. 2000). The estimation and consideration of
pollution load uncertainties, therefore, is needed in order to qualify any results, for
example by expressing uncertainty as confidence limits or as risk associated with a
decision. The subjective nature of this and all other subjective aspects of the uncertainty
analysis (the GLUE likelihood measure has already been highlighted) must be
emphasised as so by the modeller, and laid open to scrutiny and review.
When estimating scope for error in data, the modeller needs to have some knowledge of
how the data were collected. Quality control is essential, and ideally the modeller would
observe, and have some expertise in, the monitoring and measurement processes.
Unfortunately, in practice, modelling, monitoring and measuring tend to be separate skills
and disciplines (Somlyody 1995). Fundamental communication problems (as encountered
in the Hun River study) and tendency of data to be freely available without the supporting
quality control documentation (as with the Charles River data) mean that the modeller
may have insufficient insight into the significance of the data that is being used.
Modellers have no chance of communicating the significance of model results to the
decision-maker if they themselves lack basic appreciation of how the results originated.
This basic problem has limited this dissertation to an exploration of issues and
methodologies rather than case-specific conclusions about water quality and/or suitable
models.
197
8.9 The Hun River case study
The Hun River case study was supposed to provide a basis for developing and testing the
modelling framework and associated methodology, and it largely failed in that respect.
Discussion of why it failed allows broader understanding of uncertainty, and allows
comment upon the worth of modelling and uncertainty evaluation in difficult practical
circumstances, which might be expected in developing country studies.
The summarised outcomes of the river and lake water quality modelling work packages
of the TOPLEM project are numbered below (from McIntyre 2002), and each is followed
by additional discussion.
1) Within the limitations of the Hun River modelling study, there is minimal
evidence that the COD, ammonia and nitrate pollution levels in the Hun River
are significantly affected by in-river processes. There is somewhat stronger
evidence that oil and phenol concentrations are significantly affected by in-river
mass losses6. This leads to the opinion that very simple models are appropriate
for immediate strategic decision-support tasks.
The hypothesis that the water quality was sensitive to in-river processes was tested using
models of lesser complexity (see McIntyre 2002) than that used in the Charles River
study, together with the objective function, regional sensitivity analysis and the KS
significance test as described for the Chapter 7 study. For each factor, this test poses the
question “given the overall variation in the model output (i.e. the objective function
value) due to all the input uncertainties and their interactions, and given the sampling
error that is part of the Monte Carlo method, is there significant evidence that this factor
is affecting the model output”. That the answer, for COD, nitrate and ammonia, was
negative indicates that, given the limitations of the modelling method and the perceived
data uncertainties, the inclusion of water quality processes in the model was arguably
superfluous. The message regarding these pollutants is simple; that the managers of the
Hun River quality need not be accounting for in-river processes in current pollution
management strategy, at least not before better models can be identified. Furthermore, a
clear message to the Hun River modellers is that, with currently available knowledge and
data-bases, and for currently relevant modelling tasks, including in-river processes in the 6 Chemical oxygen demand (COD), ammonia, nitrate, phenol and oil were the five Hun River pollutants investigated.
198
models of COD, nitrate and ammonia is a waste of human and computer resources
(except as a demonstration of this same point).
2) As the key sources of pollution are known prior to any modelling and no other
key factors affecting water quality can be identified from the available evidence,
arguably computer models have a limited role at this stage in management of the
Hun River water quality. However, their role of providing graphic reports of
spatially and temporally varying water quality with which to support decisions is
undoubtedly of value, and for identifying key information which needs to be
collected in order to advance modelling/management strategy.
The value of water quality models for studying rivers badly polluted by known discharges
lies largely in reporting expected improvements following interventions, rather than their
ability to represent environmental processes. In particular, decision-makers and other
stake-holders may have little conception of causes and effects of spatial and temporal
water quality variations (even though they may appear obvious to a modeller), which can
be effectively reported using model interfaces. For example, Montgomery Watson
(2001b) have employed simple models with GIS interfaces for effective communication
of the Hun River (and wider Liaoning water pollution) situation. Similarly graphic reports
of uncertainty would provide a higher level of information for decision-making purposes.
3) As there is no databased evidence of seasonal variation in the pollution loads, the
loads were regarded as stationary inputs to the model, represented by the raw
data collected from the Hun River tributary sewers and rivers. The very high
variability in each pollution load was represented in the model as a stochastic
process, and was the major cause of uncertainty in model results. For this
reason, the model parameters could not be successfully conditioned by in-river
observed water quality. Therefore, a modelling priority is to reduce uncertainty
through coupling the river models with the TOPLEM load estimation model,
which was not achieved within this (TOPLEM) project.
As the in-river water quality observations allowed conditioning of the pollution loads but
not the model parameters, it can be said that the conditioning failed, and all risk
evaluations were done effectively using the a priori model. As in Chapter 6, the value of
surface water quality models and hence the risk of poor decisions was shown to be
controlled by lack of knowledge of model boundary conditions especially pollution loads,
together with the uncertainty in the in-river water quality data. Effective management of
199
modelling projects must involve suitable allocation of resources to modelling and
monitoring of boundary conditions, and effective communication between modelling
parties. A modelling issue that follows on from (3) is the justification of Monte Carlo
simulation - if the predominant future uncertainty is in the pollution load, then
propagation of uncertainty through the water quality model is a first order task, for which
the first order second moment method (see Chapter 2) may be more suitable.
4) All results presented in this report (i.e. McIntyre 2002) are conditional on the
precision of the flow data, and the unconditioned hydraulic model. These
components were not modelled as uncertain inputs due to the difficulty in
interpreting available data and quality control information. Had they been input
to the model with nominal uncertainty, this would have led to the simple (but
valid and important) conclusion that flow is a principal factor affecting water
quality. Therefore, useful practical modelling of the Hun River quality must be
preceded by (at least) daily, well-documented flow gauging at all key sections
and/or improved communication of existing data.
A flaw of the TOPLEM modelling project was that the political/institutional/cultural
obstacles to delivery of key data (especially flow and stage data), and to delivery of
supporting quality control documentation, were neither fully foreseen nor overcome. This
aspect of modelling cannot be completely neglected, as it seems that modellers, and the
reliability of their models, will always rely on the efficiency of other parties in delivery of
adequate data and supporting documentation. The success of the modelling (including
uncertainty estimation and minimisation) was ultimately controlled by the project
management issues, more than by the limitations of modelling methodology.
Management and communication are elements of uncertainty analysis which have been
almost completely outwith the scope of this dissertation, but their general importance
should be emphasised.
Following (4), it is noted that the severe pollution problem in the Hun River, as in many
developing country situations, is as much due to water shortage rather than to lack of
wastewater treatment. Therefore, some estimation of the uncertainty in flow data, leading
to the flow data being shown by the sensitivity analysis to be amongst the most
significant factors, would be advisable.
200
8.10 A look to the future
This dissertation has been non-committal about the relative merits of parsimonious and
more complex models, and has argued that the model design should depend upon the case
rather than any predetermined mindset. Increased numbers of uncertain inputs introduce
extra difficulty in conducting and interpreting sensitivity analyses, model conditioning
and uncertainty propagation; on the other hand, parsimony leaves extra doubt about the
reliability of the model structure. While some modellers call for identification of simple
cause-effect relationships (e.g. Beck 1997) others justify increased parameterisation and
numbers of state variables (Shanahan et al. 2001). While some have identified the
principal components of interacting parameters in order to reduce the dimensionality of
the parameter space (Kuczera 1990), others have the view that dimensionality should not
be reduced but handled using Monte Carlo analysis (Reichart and Omlin 1996). There is
no conflict; simply differing needs and resources (e.g. data, human and computer
resources). The challenge is to allow modellers, through provision of suitable modelling
toolkits and guidance, to recognise the implications of their needs and resource
limitations.
It is argued that there is nothing wrong with over-parameterised models per say, so long
as 1) at calibration, the extra degree of freedom is not used to explain what are, for all
intents and purposes, data errors; 2) increased model elaboracy is not, without evidence,
taken to imply increased prediction accuracy; and 3) the additional demands on human,
computer and field resources do not draw attention from more constructive, relevant
tasks. On the first point, a priori constraints on ranges of parameters and a priori
specification of perceived data error bounds, will avoid the extreme consequences of
over-fitting data. On the second, while the more complex model may be inaccurate in a
deterministic sense, it is likely to be a better tool for exploring the set of possible futures,
and a more powerful decision-making tool if parameter uncertainties could be
comprehensively managed. If they could (there is no evidence that they are), it would be
difficult to find a purely mathematical argument to support a simple conceptual model
over a more physically based model. The third point of reservation, relating to complex
models being an unjustified use of resources, is the heart of the matter. Is a complex
model, with or without adequate supporting data, with or without adequate uncertainty
analysis, likely to be cost effective in improving quality of decisions? While there are
(very few) reports of model post-audits in terms of fitting data, there are (apparently)
none that audit the cost-effectiveness of the modelling exercise in terms of quality of
decisions.
201
Returning to the objective of providing frameworks for uncertainty analysis, it seems that
a valuable development in water quality modelling would be toolkits that provide
flexibility in selecting model structures. This would encourage exploration and
recognition of structural uncertainty. WaterRAT provides some capacity to test
alternative structures and modelling scales; DESERT (Ivanov et al. 1996) seems to
provide more choice of model structure, with a loss of ease of use. Ease of use is a
paramount consideration, and recent software advances (e.g. SIMULINK, Mathworks
2002) may provide a platform from which multiple structures can easily be devised and
tested. It is easy to say that toolkits could automatically combine results from alternative
models into an ‘ensemble’ estimation of uncertainty, but in practice this idea may be
premature, considering that few modellers yet even consider parameter uncertainty.
Apart from uncertainty analysis, current developments in water quality modelling include
models which represent runoff-groundwater-river-lake interactions (e.g. Daldorph et al.
2000), integrated urban sewerage–river water quality models (e.g. Lau 2002), and new
extensions of hydro-chemical models to include hydro-ecology (e.g. Wade et al. 2002). A
more general development is towards distributed or semi-distributed catchment-wide
models, often using geographical information system (GIS) databases and interfaces (e.g.
Crosetto et al. 2000). The computational demands of the GIS itself, and the increased
spatial dimension of the problem, pose additional problems for the implementation of
uncertainty analysis. Apparently, there has been almost no research into how the benefits
of increased spatial representation can be reconciled with the high and unknown
uncertainty in results. Cerati (2002) demonstrates that, using established Monte Carlo
methods and current desktop hardware, rigorous sensitivity analysis of such models is
difficult, and maybe impossible.
Surface water quality modelling has traditionally been the field of civil engineers (Chapra
1997). Increasing relevance of groundwater, ecology and GIS has meant that the
collaboration of hydrologists, ecologists and geographers is now essential. Also, now
more than ever before, the skills of mathematicians are needed to develop efficient and
reliable algorithms for model solving, uncertainty analysis and optimisation. As was
stated near the outset of this dissertation, the value of Monte Carlo methods and
importance of uncertainty analysis should not be diminished by the perceived inefficiency
of the numerical implementations. The efforts directed at numerical efficiency within
WaterRAT, in particular the work reported in Chapter 5, have merely scratched the
surface of the numerical modelling issues. Furthermore, the full potential of the MCMC
202
and the genetic algorithm in application to uncertainty estimation can best be attained
with in-depth understanding of the mathematics. Recent initiatives (NERC 2003) point
the way forward for improved cross-collaboration between mathematicians and
environmental modellers.
In the context of systems identification of wastewater treatment models, Beck (1999)
observes that better technology for data collection does not necessarily permit improved
understanding of system processes, as key responses will always remain unobserved. In
the context of prior perceptions of environmental models, Beven (2002c) notes that the
key system characteristics may be “essentially unknowable”. While the contexts of these
two references are different - the latter is referring to identification of inputs to a priori
models, while the former refers to measurement of the output state variables – both imply
that, despite research and emerging technology, a complete set of information with which
to develop an accurate model will never be available. There are two pragmatic responses
to this concern. Firstly, for decision-support purposes, process understanding at the
fundamental scale is not essential if justified uncertainty analysis is undertaken, and
neither is a precise model. Secondly, the future of environmental modelling relies on
improved monitoring technology to facilitate affordable, justifiable uncertainty analysis.
At least in developing country applications where continuous monitoring is not
practicable, data quantity will remain low and modelling methods must strive to account
for the associated uncertainty. It is the opinion of the author of this Thesis that the future
of uncertainty analysis in modelling data poor environments, as far as uncertainty
estimation is concerned, lies in the well-established set-based method introduced to
environmental modelling by Hornberger and Spear (1980), which was later applied to
uncertainty forecasting by Van Straten and Keesman (1991). This method provides a
basis for incorporating alternative structural assumptions, as well as parameter, and initial
and boundary condition uncertainties. It provides a transparent method, which uses a
minimum number of clearly defined assumptions, for conditioning, sensitivity testing and
uncertainty propagation. The use of more elaborate methods of response surface analysis
(such as might be applied using data sampling, MCMCs or GLUE as in Chapter 2)
involves extra assumptions that are likely to obstruct the acceptance of the result rather
than to fortify it, due to their subjective and sometimes quite complex basis. The primary
objective is to achieve recognition of parameter and structural uncertainty, their effects on
decisions and the reasons for them; the best way to achieve such recognition is to keep
the analysis as simple as possible.
203
In developed country applications, where water quality constituents and hydrodynamics
might be effectively monitored on a time scale of minutes (at least in the case of laterally
mixed rivers) to give a truly dynamic picture of the hydro-chemistry, there may be scope
in the future for exploring the model-data error structure. Using time-varying
identification of optimum model parameter values (e.g. Beck 1983, Wagener et al. 2002b)
allows combinations of structural and data error to be hypothesised and potentially
resolved, or represented as uncertainty. Of course, the success of such research depends
on adequate boundary condition identification, which may be problematic in models of
diffuse-sourced runoff. In the cases of surface water where two- and three-dimensional
variations in water quality exist, the same methods could be aspired to, but achieving
adequate data of the system is obviously much more difficult. Looking at the wider
picture, specifically the growing relevance of ecological modelling, there will inevitably
be strong competition for monitoring and modelling resources. Again, the question to ask
is “where is the risk of not achieving our water quality management objectives coming
from?” and that is where resources should be directed. Furthermore, there is going to be a
need for significant trade-offs between protection of aquatic ecology, and the resulting
social and economic costs. A future modelling problem then becomes not only identifying
the risks of failing different criteria, but establishing acceptable compromises between the
risks.
The overall impression gained from the development of this Thesis is that, at the moment,
further research into improving river water quality models (and perhaps environmental
models in general) for decision-support is not warranted. Real improvements in the
practical value of models lie in the willingness of modellers to seriously promote and
confront the problem of uncertainty in research and tool development; and for decision-
makers to accept and use the outcomes, and integrate uncertainty in results into risk
assessments. The complexity of commonly used models far surpasses the complexity of
thought given to using the results properly. This view might be extended to say that
further development of modelling methods (possibly including development of tools for
uncertainty analysis) is not warranted by the current inclination of decision-makers to
properly use them. After all, a minority of modellers have been promoting the need for
such analysis and tools since at least 1970 (see Beck 1987) without proportional uptake of
the advice. However, there are signs of changing views on how models should be used
(e.g. Cipra 2000; Beven 2000c), with a number of important policy documents and codes
of practice now referring to integration of model uncertainty into risk management
(DETR et al. 2001; UK Environment Agency 2002). Additionally, at least in Europe, the
stakes are currently being raised by the Water Framework Directive - the potential cost of
204
sufficient water quality management is high; the probability of unsuccessful investment is
high; worst case scenarios cannot be considered; the necessary data to reliably inform
decisions will not be available.
205
References Adams, B. and Reckhow, K.H. 2001. An examination of the scientific basis for
mechanisms and parameters in water quality models. Unpublished paper. Available
from second author at www2.ncsu.edu/ncsu/CIL/WRRI/AdamsReckhow.pdf
Ambrose, R.B., Wool, T.A. and Martin, J.L. 1993. The water quality analysis simulation
program WASP5 Version 5.10 Part A: Model Documentation. Environmental
Research Laboratory, Office of Research and Development, US EPA, Athens,
Georgia, USA.
Ambrose, R.B., Barnwell, T.O., McCutcheon, S.C. and William, J.R. 1996 Computer
models for water quality analysis. In Water Resources Handbook, Mays, L.W. (Ed),
McGraw-Hill, pp14.1-14.28.
Ang, A.H.S. and Tang, W.H. 1975. Probability Concepts in Engineering Planning and
Design, Vol.1- Basic principles, John Wiley, pp 56, 222, 232-248..
Ang, A.H.S. and Tang, W.H. 1984. Probability Concepts in Engineering Planning and
Design, Vol.2- Decision, risk and reliability, John Wiley, pp 274.
Ashton, G. (Ed.). 1986. River and Lake Ice Engineering. Book Crafters, Inc. MI. USA.
Ashton, G. 1979. Modelling of ice in rivers. In Modelling of Rivers, Shen, H.W. (Ed.),
John Wiley.
Bastidas, L.A., Gupta, H.V., Sorooshian, S., Shuttleworth, W.J. and Yang, Z.L. 1999.
Sensitivity analysis of a land surface scheme using multicriteria methods. Journal of
Geophysical Research 104(D16), 19481-19490.
Beasley, D., Bull, D.R. and Martin, R. 1993. An overview of genetic algorithms: Part 1,
Fundamentals. University computing 15(2), 58-69.
Beck, M.B. 1983. Uncertainty, system identification, and the prediction of water quality.
In Uncertainty and forecasting of water quality, Beck, M.B. and Van Straten, G.
(Eds.), Springer-Verlag, pp3-68.
Beck, M.B. 1987. Uncertainty in water quality models. Water Resources Research 23(8),
1393.
Beck, M.B. 1997. Applying systems analysis in managing the water environment:
Towards a new agenda. Water Science and Technology 36(5), 1-17.
Beck, M. B. 1999. Coping with ever larger problems, models, and data bases. Water
Science and Technology 39(4), 1-11.
Beer, T. and Young, P.C. 1983. Longitudinal dispersion in natural streams. ASCE Journal
of Environmental Engineering 109(50), 1049-1067.
206
Bencala, K.E. and Walters, R.A. 1983. Simulation of solute transport in a mountain pool
and riffle stream: a transient storage model. Water Resources Research 19(3), 732-
738.
Berezin, I.S. and Zdidkov, N.P. 1965. Computing Methods, Permagon Press.
Berthouex, P.M. and Brown, L.C. 1994. Statistics for environmental engineers, CRC
Press.
Beven, K.J. and Binley, A.M. 1992. The future of distributed models; model calibration
and predictive uncertainty. Hydrological Processes 6, 279-298.
Beven, K.J. 1993. Prophecy, reality and uncertainty in distributed hydrological
modelling. Advances in Water Resources 16(1), 41-51.
Beven, K.J. 1998. Generalised Likelihood Uncertainty Estimation (GLUE) User Manual.
Lancaster University, Lancashire, UK.
Beven, K.J. 2000a. On modelling uncertainty, risk and decision making. Hydrological
Processes 14, 2605-2606.
Beven, K.J. 2000b. Rainfall-runoff modelling: The Primer, John Wiley.
Beven, K. J., 2002c. Towards a coherent philosophy for environmental modelling,
Proceedings of the Royal Society of London (A) 458, 2465-2484.
Beven, K.J. and Freer, J. 2001. Equifinality, data assimilation, and uncertainty estimation
in mechanistic modelling of complex environmental systems using the GLUE
methodology. Journal of Hydrology 249(1-40), 11-29.
Beven, K.J., Freer, J., Hankin, B. and Schulz, K. 2001. The use of generalised likelihood
measures in high order models of environmental systems. In Non-linear and non-
stationary signal processing, Fitzgerald, W.S., Smith, R.C., Walden, A.T., Young,
P.C. (Eds.), Cambridge University Press.
Bicknell, B.R., Imhoff, J.C., Kittle, J.L., Donigan, A.s. and Johanson, R.C. 1997.
Hydrologic simulation program – Fortran. Users manual for version 11. EPA/600/R-
97/080.
Bierman, V.J. and Dolan, D.M. 1986. Modeling of phytoplankton in Saginaw Bay: II.
Post-audit phase. ASCE Journal of Environmental Engineering 112(2), 415-429.
Binley, A.M., Beven, K.J., Calver, A. and Watts, L.G. 1991. Changing responses in
hydrology: assessing the uncertainty in physically based model predictions. Water
Resources Research 27, 1253-1261.
Blom, G. and Aalderink, H.R. 1998. Calibration of three resuspension/sedimentation
models. Water Science and Technology 37(3), 41–49.
Boers, P.C.M. Van Raaphorst, W. and Van der Molen, D. T. 1998. Phosphorus retention
in sediments. Water Science and Technology 37(3), 31–39.
207
Bowie, G.L., Williams, B.M., Porcella, D.B., Campbell, C.L., Pagenkopf, J.R., Rupp,
G.L., Johnson, K.M., Chen, P.W.H. and Gherini, S.A. 1985. Rates, Constants and
Kinetics formulations in Surface Water Quality (2nd Edition). Report no. EPA/600/3-
85/040, US EPA, Athens, Georgia, USA.
Boyle, J.D. and Scott, J.A. 1984. The role of benthic films in an East Devon River. Water
Research 18, 1089-1099.
Box, G.E.P. and Jenkins, G.E. 1970. Time series analysis forecasting and control,
Holden-Day, pp 208.
Bras, R.L. 1990. Hydrology, Addison-Wesley.
Brooks, S. 1998. Markov chain Monte Carlo method and its application. Journal of the
Royal Statistical Society, Series D 47(1), 69-100.
Brown, L.C. 2002 Modeling uncertainty – QUAL2E-UNCAS A case study. Water
Environment Federation. Unpublished. www.wef.org/pdffiles/TMDL/Brown.pdf.
Brown, L.C. and Barnwell, T.O. 1987. The enhanced stream water quality models
QUAL2E and QUAL2E-UNCAS: Documentation and user manual. Report no.
EPA/600/3-87/007, US EPA, Athens, Georgia, USA.
Burns, L.A. 2000. Exposure analysis modeling system: User’s guide and system
documentation. National exposure research laboratory, EPA/600/R-00/081. US EPA.
Butcher, J.C. 1987. The numerical analysis of ordinary differential equations, John
Wiley.
Camacho, L.A. 2000. A hierarchical modelling framework for solute transport in rivers
under unsteady flow conditions. PhD Dissertation. Imperial College of Science,
Technology and Medicine.
Camacho, L.A. and Lees, M.J. 1999. Multilinear discrete lag-cascade model for channel
routing. Journal of Hydrology 226(1-2), 30-47
Carver, M.B. 1976. The choice of algorithms in automated method of lines solution of
partial differential equations. In Numerical methods for differential systems, L.
Lapidus and W.E. Schiesser (Eds.), Academic Press.
Cash, J.R. and Karp, A.H. 1990. A variable order Runge-Kutta method for initial value
problems with rapidly varying right-hand sides. ACM Transactions on Mathematical
Software 16, 201-222.
CDM. 1997. Charles River Pollution Control District/Town of Holliston, Massachusetts:
Upper Charles River Wasteload Allocation Study, Camp, Dresser and McKee Inc.,
Cambridge, MA, USA.
CEC. 2000. Directive 2000/60/EC of the European Parliament and of the Council of 23rd
October 2000 establishing a framework for Community action in the field of water
policy. Council of the European Community, Brussels.
208
Cerati, M. 2002. Phosphorus export at the catchment scale – analysis and mapping of
sensitivities using GIS. MSc Project Dissertation. Imperial College London,
September 2002.
Chapra, S.C. 1991. A toxicant loading concept for organic contaminants in lakes. ASCE
Journal of Environmental Engineering 117(5), pp656-677.
Chapra, S.C. 1997. Surface Water-Quality Modeling, McGraw-Hill.
Chapra, S.C. 1999. Organic carbon and surface water quality modelling. Progress in
Environmental Science 1(1), 49-70.
Chapra, S.C. and Canale, R.P. 1998. Numerical Methods for Engineers (3rd Edition),
McGraw-Hill, 710-718.
Chapra, S.C. and Runkel, R.L. 1998. Modelling impact of storage zones on stream
dissolved oxygen. ASCE Journal of Environmental Engineering 125(5), 415-419.
Chatfield, C. 1995. Model uncertainty, data mining and statisitcal inference. Journal of
the Royal Statisitcal Society (A) 158(3), 419-466.
Chen, G.H., Leong, I.M., Liu, J. and Huang, J.C. 1999. Study of oxygen uptake by tidal
river sediment. Water Research 33(13), 2905-2912.
Chen, J. and Wheater, H. 1999. Identification and uncertainty analysis of soil water
retention models using lysimeter data. Water Resources Research 35(8), 2401-2414.
Chow, V.T. 1959. Open Channel Hydraulics, McGraw-Hill.
Christian, J.T. and Baecher, G.B. 1999. Point-estimate method as numerical quadrature.
Journal of Geotechnical And Geoenvironmental Engineering 125(9), 779-786.
Cipra, B. 2000. Revealing uncertainties in computer models. Science 287, 960-961.
Cochran, W.G. 1977. Sampling techniques (3rd edition). John Wiley, pp 72.
Cole, T.M. and Wells, S.A. 2000. CE-QUAL-W2: A two-dimensional, laterally-averaged
hydrodynamic and water quality model, Version 3.0. User manual. U.S. Army Corps
of Engineers. Instruction Report EL-00-1.
Connolly, J.P. and Coffin, R.B. 1995. Model of carbon cycling in planktonic food webs.
ASCE Journal of Environmental Engineering 121(10), 682-690.
Crosetto, M., Tarantola, S. and Saltelli, A. 2000. Sensitivity and uncertainty analysis in
spatial modelling using GIS. Agriculture, Ecosystems and Environment 81, 71-79.
CRWA, 2000. Charles River Watershed Association Annual Report 2000. (Available
from www.crwa.org).
Daldorph, P.W.G., Lees, M.J., Wheater, H.S. and Chapra, S. 2000. Integrated Lake and
Catchment Phosphorus Model: A Eutrophication Management Tool: I Model Theory.
Journal of The Chartered Institution of Water and Environmental Management 15(3),
174-181.
209
DAFF. 2000. National Water Quality Management Strategy, 9, Rural Land Uses and
Water Quality. Australian Department of Agriculture Fisheries and Forestry,
Canberra, Australia.
DEFRA. 2002. The Government’s strategic review of diffuse water pollution from
agriculture in England: Paper 1: Agriculture and water: a diffuse pollution review.
Chapter 4 - Current status of waters in England and Wales, DEFRA June 2002.
DETR, Environment Agency and Institute for Environment and Health. 2000. Guidelines
for environmental risk assessment and management. HMSO, London.
De Marchi, C., Ivanov, P., Jolma, A., Masliev, I., Smith, M.G. and Somlyody, L. 1999.
Innovative tools for water quality management and policy analysis DESERT and
STREAMPLAN. Water Science and Technology 40(10), 103-110.
DHI, 2000. MIKE 11: A modelling system for rivers and channels: User guide. DHI
Software.
Di Toro, D.M. and Fitzpatrick, J.J. 1993. Chesapeake Bay Sediment Flux Model. U.S.
Army Corps of engineers, Waterways Experiment Station, Technical Report EL-93-2.
Draper, D. 1995 Assessment and propagation of model uncertainty. Journal of the Royal
Statistical Society, Series B 57(1), 45-97.
Duan, Q., Gupta, V.K. and Sorooshian, S. 1993. A shuffled complex evolution approach
for effective and efficient global minimization. Journal of Optimization Theory and
Applications 76(3), 501-521.
Engelund, F. and Fredsoe, J. 1976. A sediment transport model for straight alluvial
channels. Nordic Hydrology 7, 239-307.
Enggrob, H. 1997. Danubian Lowland - Ground Water Model: Part 3 - Sediment
Transport Modelling. DHI user conference proceedings 1997.
Evans, E.C., McGregor, G.R. and Petts, G.E. 1998. River energy budgets with special
reference to river bed processes. Hydrological Processes 12(4) 575-595.
Fonseca, C.M. and Fleming, P.J. 1995. An overview of evolutionary algorithms in multi-
objective optimization. Evolutionary computing 3(1), 1-16.
Franks, S. and Beven, K.J. 1997. Bayesian estimation of uncertainty in land-surface-
atmosphere flux predictions, Journal of Geophysical Research 102(D20), 23991-
23999.
Franks, S.W., Beven, K.J. and Gash, J.H.C. 1999. Multi-objective conditioning of a
simple SVAT model. Hydrology And Earth System Sciences 3(4), 477- 489.
Freer, J., Beven, K. and Ambroise, B. 1996. Bayesian estimation of uncertainty in runoff
prediction and the value of data: an application of the GLUE approach. Water
Resources Research 32(7), 2161.
210
Gardner, R.H., O’Neill, R.V., Mankin, J.B. and Kumar, D. 1980. Comparative error
analysis of six predator prey models. Ecology 6(2), 323-332.
Gear, C.W. 1971. Numerical initial value problems in ordinary differential equations,
Prentice-Hall.
Goldberg, D.E. 1989. Genetic Algorithms in Search, Optimization and Machine
Learning, Addison-Wesley, Reading, MA.
Greenberg, A.E., Clesceri, L.S. and Eaton, A.D. (Eds.) 1992. Standard methods for the
examination of water and wastewater (18th Edition), APHA-AWWA-WEF.
Gulliver J.S., Wilhelms S.C. and Parkhill K.L. 1998. Predictive capabilities in oxygen
transfer at hydraulic structures. ASCE Journal of Hydraulic Engineering 124(7), 664-
671.
Gunduz, O., Soyupak, S. and Yurteri, C. 1998. Development of water quality
management strategies for the proposed Isikli reservoir. Water Science and
Technology 37(2), 369-376.
Gupta, H.V., Sorooshian, S. and Yapo, P.O. 1998. Toward improved calibration of
hydrologic models: Multiple and noncommensurable measures of information. Water
Resources Research 34(4), 751-763.
Gustafsson, K. 1993. Control of error and convergence in ODE solvers. PhD Thesis,
Lund Institute of Technology, Sweden.
Hankin, B.G. and Beven, K.J. 1998. Modelling dispersion in complex open channel flow;
2. Fuzzy calibration. Stochastic Hydrology and Hydraulics 12, 397-411.
Harr. M.E. 1989. Probabilistic estimates for multi-variate analyses. Applied Mathematical
Modeling 13(5), 313-318.
Harremoes, P. 1982. Immediate and delayed oxygen reaction in rivers. Water Research
16, 1093-1098.
Hartigan, J.P., Friedman, J.A. and Southerland, E. 1983. Post-audit of lake model used for
NPS management. ASCE Journal of Environmental Engineering 109(6), 1354-1370.
Havnø, K., Madsen, M.N. and Dørge, J. 1995. MIKE 11 - A Generalized River Modelling
Package. In Computer Models of Watershed Hydrology, V.P. Singh (Ed.), Water
Resources Publications, pp733-782.
Henderson-Sellers, B. and Henderson-Sellers, A. 1993. Factorial techniques for testing
environmental model sensitivity. In Modelling Changes in Environmental Systems,
Jakeman, A.J., Beck, M.B. and McAleer, M.J. (Eds.), John Wiley, pp. 59-76.
Holland, J.H. 1975. Adaption in Natural and Artificial Systems, Ann Arbor.
Hondzo, M. and Stefan, H.G. 1994. River-bed heat conduction prediction. Water
Resources Research 30(5), 1503-1513.
211
Hong, H.P. 1998. An efficient point estimate method for probabilistic analysis. Reliability
Engineering and System Safety 59(3), 261-267.
Hornberger, G.M. and Spear, R.C. 1980. Eutrophication in Peel Inlet, 1. Problem-
defining behaviour and a mathematical model for the phosphorus scenario. Water
Research 14, 29-42.
House, W.A. and Denison, F.H. 1998. Phosphorus Dynamics in a Lowland River. Water
Research 32(6), 1819-1830.
Howarth, R.W., Schneider, R. and Swaney, D. 1996. Metabolism and organic carbon
fluxes in the tidal freshwater Hudson River. Estuaries 19(4), 848-865.
Hulme, M., Jenkins, G.J., Lu, X., Turnpenny, J.R., Mitchell, T.D., Jones, R.G., Lowe, J.,
Murphy, J.M., Hassell, D., Boorman, P., McDonald, R. and Hill, S. 2002. Climate
change scenarios for the United Kingdom: The UKCIP02 Scientific Report. Tyndall
Centre for Climate Change Research, School of Environmental Sciences, University
of East Anglia.
Ivanov, P., Masliev, I., Kularathna, M., De Marchi, C. and Somlyódy, L. 1996. DESERT
User Manual. International Institute for Applied Systems Analysis, Laxenburg,
Austria / Institute for Water and Environmental Problems, Barnaul, Russia.
Jarvie, H.P., Withers, P.J.A. and Neal, C., 2002. Review of robust measurement of
phosphorus in river water; sampling, storage, fractionation and sensitivity. Hydrology
and Earth System Sciences 6(1), 113-132.
Jorgensen, S.E., Kamp-Nielsen, L., Christensen, T., Windolf-Nielsen, J. and Westergaard,
B. 1986. Validation of a prognosis based upon a eutrophication model. Ecological
Modelling 32, 165-182.
Kavetski, D., Franks, S.W. and Kuczera, G. 2002. Confronting uncertainty in
environmental modelling: the case for a systems perpective. Advances in calibration
of watershed models.
Keith, L.H. 1990. Environmental Sampling: A Summary. Environmental Science and
Technology 24(5), 610-617.
Kitanidis, P.K. and Bras, R.L. 1980. Real-time forecasting with a conceptual hydrologic
model: 1. Analysis of uncertainty. Water Resources Research. 16(6), 1025-1033.
Kleijnen, J.P.C. 1997. Sensitivity analysis and related analyses: a review of some
statistical techniques. Journal of Statistical Computation and Simulation 57(1-4),
111-142.
Kronvang, B. Laubel, A. and Grant, R. 1997. Suspended sediment and particulate
phosphorus transport and delivery pathways in an arable catchment, Gelbæk Stream,
Denmark. Hydrological Processes 11(6), 627-642.
212
Kuczera, G., 1983. Improved parameter inference in catchment models 1. Evaluating
parameter uncertainty. Water Resources Research 19(5), 1151-1162.
Kuczera, G. 1990. Assessing hydrologic model nonlinearity using response surface plots.
Journal of Hydrology 118(1-4), 143-161.
Kuczera, G. and Parent, E. 1998. Monte Carlo assessment of parameter uncertainty in
conceptual catchment models. Journal of Hydrology 211(1), 69-85.
Kusuda, T., Futawatari, T. and Oishi, K. 1994. Simulation of nitrification and
denitrification processes in a tidal river. Water Science and Technology 30(2), 43-52.
Lai, C.H. 2002. Use of genetic algorithms in calibration of water quality models. MSc
Project Dissertation. Imperial College London, September 2002.
Lal, A.M. and Shen, H.W. 1993. Mathematical model for river ice processes. Journal of
hydraulic Engineering 119(11),1231.
Lau, K.T. 2002. Optimising storage within the integrated urban wastewater system. PhD
Thesis, Imperial College London, March 2002.
Lees, M.L., Camacho, L.A. and Chapra, S.C. 2000. On the relationship of transient
storage and aggregated dead zone models of longitudinal transport in streams. Water
Resource Research 36(1), 213-224.
Li, S.Y. and Chen, G.H. 1994. Modelling the organic removal and oxygen consumption
by biofilms in an open channel flow. Water Science and Technology 30(2), 53-61.
Lick, W. 1982. Entrainment, transport and deposition of the fine grained sediments in
lakes. Hydrobiologia 91, 31-40.
Ma, R. 2001.An option appraisal for Hun River pollution control using a stochastic water
quality simulation model. MSc Project Dissertation. Imperial College London,
September 2001.
MacKay, M.D., Beckman, R.J. and Conover, W.J. 1979. A comparison of three methods
for selecting values of input variables in the analysis of output from a computer code.
Technometrics 21(2), 239-245.
Mailhot, A., Gaume, E. and Villeneuve, J.P. 1997. Uncertainty analysis of calibrated
parameter values of an urban storm water quality model using Metropolis Monte
Carlo algorithm. Water Science and Technology 36(5), 141-148.
Mathworks 2000. SIMULINK: Dynamic system simulation, The Mathworks Inc. Product
description available from www.mathworks.com/products/simulink.
Maunula, M. 1992. Finnish experience with river ice in China. Proceedings of the 9th
International Northern Research Basins Symposium/Workshop, NHRI Symposium
No. 10, Canada 1992.
213
McIntyre, N. 1998. A eutrophication model incorporating sediment-water interactions
and macrophyte-phytoplankton competition with the Venice Lagoon as a case study.
MSc. dissertation. Imperial College London, September 1998.
McIntyre, N. 2000. Estimation and propagation of parametric uncertainty in
environmental models (first draft). Unpublished.
McIntyre, N. Lees, M.J. and Wheater, H.S. 2001. A review and demonstration of methods
of uncertainty analysis in numerical environmental modelling. Proceedings of the 8th
Europia International Conference - Advances in Design Sciences and Technology,
Delft, The Netherlands, 183-196.
McIntyre, N. and Zeng, S. 2002. River and lake water quality models – Final Report:
WaterRAT User Manual. A report to the European Commission for the 4th
Framework Project “TOPLEM”. Report no. PL972722 D4.3(10). Department of Civil
and Environmental Engineering, Imperial College, London, UK.
McIntyre, N. 2002a. River and lake water quality models – Final Report: Hun River case
study. A report to the European Commission for the 4th Framework Project
“TOPLEM”. Report no. PL972722 D4.3(7). Department of Civil and Environmental
Engineering, Imperial College, London, UK.
McIntyre, N., Jackson, B., Wheater, H.S. and Chapra, S. 2003. Numerical efficiency in
Monte Carlo simulations - a case study of a river thermodynamic model. ASCE
Journal of Environmental Engineering (Accepted).
McNeil, J., Taylor, C. and Lick, W. 1996. Measurements of erosion of undisturbed
bottom sediments with depth. ASCE Journal of Hydraulic Engineering 122(6), 316-
324.
Meixner, T., Gupta, H.V., Bastidas, L.A. and Bales, R.C. 1999. Sensitivity analysis using
mass flux and concentration. Hydrological Processes 13, 2233-2244.
Melching, C. 1995. Reliability estimation. In Computer Models of Watershed Hydrology,
Singh, V.P. (Ed.), Water Resource Publications.
Melching, C.S. and Bauwens, W. 2001. Uncertainty in coupled nonpoint source and
stream water-quality models. ASCE Journal of Water Resources Planning and
Management 127(6), 403-419.
Metcalf and Eddy Inc. 1991. Wastewater engineering – Treatment disposal and reuse 3rd
Edition. McGraw-Hill.
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, F. 1953.
Equation of state calculations by fast computing machines. Journal of Chemical
Physics 21, 1087.
Montgomery Watson. 2001a. Liaoning Integrated Environmental Programme. Water
quality modelling report Annex 4. Unpublished Report.
214
Montgomery Watson. 2001b. Liaoning Integrated Environmental Programme. Lioa River
Basin Plan: Executive Summary. Montgomery Watson, Shenyang, March 2001.
Mulligan, A.E. and Brown, L.C. 1998. Genetic algorithms for calibrating water quality
models. ASCE Journal of Environmental Engineering 124(3), 202-211.
Nakato, T. 1990. Tests of selected sediment-transport formulas. ASCE Journal of
Hydraulic Engineering 116(3), 362-379.
Nash, J.E. and Sutcliffe, J.V. 1970. River flow forecasting through conceptual models
Part 1 – A discussion of principles. Journal of Hydrology, 121, 217-238.
Neitsch, S.L., Arnold, J.G., Kiniry, J.R., Williams, J.R. and King, K.W. 2002. Soil and
water assessment tool theoretical documentation. GSWRL Report 02-01. Grassland,
Soil and Water Research Laboratory, Temple, Texas.
NERC. 2003. Environmental mathematics and statistics. Thematic Programme. NERC
(www.nerc.ac.uk/funding/interdisciplinary)
Orlob, G. T. 1992. Water-quality modeling for decision making. Journal of Water
Resources Planning and Management 118, 3, 295-307
Park, S. and Lee, Y.S. 2002. A water quality modeling study of the Nakdong River,
Korea. Ecological Modelling 152(1), 65-75.
Portielje, R., Hvitved-Jacobsen, T. and Schaarup-Jensen, K. 2000. Risk analysis using
stochastic reliability methods applied to two cases of deterministic water quality
models. Water Research 34(1), 153-170.
PRC SEPA. 1999. Environmental quality standard for surface water. Regulation GHZB1-
1999. State Environmental Protection Administration, Beijing, People’s Republic of
China.
Press, W.H., Teukolsky, S.A., Flannery, B.P. and Vetterling, W.T. 1988. Numerical
Recipes in C: The Art of Scientific Computing. Cambridge University Press, pp308.
Protopapas, A.L. and Bras, R.L. 1990. Uncertainty propagation with numerical models
for flow and solute transport in the unsaturated zone. Water Resources Research
26(10), 2463-2474.
Qian, S.S. 1997. An illustration of model structure identification. Journal of American
Water Resources Association 33(4), 811-824.
Qinghua University. 2001. Pollution load estimation methodology: The estimation of
pollution load of the river sources of Dahuofang Reservoir. A report to the European
Commission for the 4th Framework Project “TOPLEM”. Report no. PL972722
D3.3(10). Qinghua University, Department of Environmental Science, Beijing,
China.
Ranjie, H. and Huimin, L. 1987. Modelling of BOD-DO Dynamics in an ice-covered
river in northern China. Water Resources 21(3), 247-251.
215
Rauch, W., Henze, M., Koncsos, L., Reichert, P., Shanahan, P., Somlyody, L. and
Vanrolleghem, P. 1998. River water quality modelling: I. State of the art. Water
Science and Technology 38(11), 237-244.
Reckhow, K.H. 1994. Water quality simulation modeling and uncertainty analysis for risk
assessment and decision making. Ecological Modelling 72(1-2), 1-20.
Reckhow, K.H. and Chapra, S.C. 1983a. Engineering approaches for lake management.
Vol.1: Data analysis and empirical modeling. Butterworth.
Reckhow, K.H. and Chapra, S.C. 1983b. Engineering approaches for lake management.
Vol.2: Mechanistic modeling. Butterworth.
Reichart, P. and Omlin, M. 1996. On the usefulness of over-parameterised ecological
models. Ecological Modelling 95, 289-299.
Robson, A.J. and Neal, C. 1997. Regional water quality of the river Tweed. Science Of
The Total Environment 194, 173-192
Romanowicz, R., Beven, K.J. and Tawn, A. 1994. Evaluation of predictive uncertainty in
non-linear hydrological models using a Bayesian approach. In Barnett, V. &
Turkman, K.F. (eds), Statistics for the environment 2. John Wiley.
Rosenblueth, E. 1981. Two-point estimates in probabilities. Applied Mathematical
Modelling 5, 329-335.
Runkel, R.L. 1998. One-dimensional transport with inflow and storage (OTIS): A solute
transport model for streams and rivers. Water Resources Investigation Report 98-
4018, USGS
Rubinstein, R.Y. 1981. Simulation and the Monte Carlo Method, Wiley, pp 240.
Rutherford, J.C. 1994. River Mixing, Wiley.
Rutenbar, R.A. 1989. Simulated annealing algorithms: an overview. IEEE Circuits and
Devices Magazine Jan, 1989, 19-26.
Sawyer, C.N., McCarty, P.L. and Parkin, G.F. 1994. Chemistry for environmental
engineering; 4th edition, McGraw-Hill.
Shah, S.M.S., O’Connell, P.E. and Hosking, J.R.M. 1996. Modelling the effects of spatial
variability on catchment responses. 2. Experiments with distributed and lumped
models. Journal of Hydrology 175, 89-111.
Shanahan, P., Henze, M., Koncsos, L., Rauch, W., Reichert, P., Somlyody, L. and
Vanrolleghem, P. 1998. River water quality modelling: II. Problems of the art. Water
Science and Technology 38(11), 245-252.
Shanahan, P., Borchardt, D., Henze, M., Rauch, W., Reichert, P., Somlybody, L. and
Vanrolleghem, P. 2001. River water quality model no. 1 (RWQM1): I. Modelling
approach. Water Science and Technology 43(5), 1-9.
Shao, J. and Tu, D. 1995. The jackknife and bootstrap. Springer-Verlag, pp2.
216
Shao, J. 1996. Bootstrap model selection. Journal of the American Statistical Association
91(434), 655-665.
Shen, H.T. and Chaing, L.A. 1984. Simulation of growth and decay of river ice cover.
ASCE Journal of Hydraulic Division 110(7), 958-971.
Shen, H.T. and Yapa, P.D. Oil slick transport in rivers. ASCE Journal of Hydraulic
Engineering, 114(5), 529-543.
Shepherd, B., Harper, D. and Millington, A. 1999. Modelling catchment-scale nutrient
transport to watercourses in the UK. Hydrobiologia 396, 227-237.
Sherif, E. 2000. An oil model for polluted rivers with the Hun River, China as a case
study. MSc Project Dissertation. Imperial College London, September 2000.
Sincock, A. M., Lees., M. J. 2000. Extension of the QUASAR river water quality model
to unsteady flow conditions, Journal of The Chartered Institution of Water and
Environmental Management 16(1), 12-17.
Somlyody, L. 1995. Water quality management: can we improve integration to face
future problems. Water Science and Technology 31(8), 249-259.
Somlyody, L. 1997. Use of optimization models in river basin water quality planning.
Water Science and Technology 36(5), 209-218.
Somlyody, L., Henze, M., Koncsos, L., Rauch, W., Reichert, P., Shanahan, P. and
Vanrolleghem, P. 1998. River water quality modelling: III. Future of the art. Water
Science and Technology 38(11), 253-260.
Sorooshian, S. and Dracup, J.A. 1980. Stochastic parameter estimation procedures for
hydrologic rainfall-runoff models: correlated and heteroscedastic error cases. Water
Resources Research 16(2), 430-442.
Sorooshian, S. and Gupta, V.K. 1995. Model calibration. In Computer Models of
Watershed Hydrology, Singh, V.P. (Ed.), Water resources Publications.
Spear, R. C. and Hornberger, G. M., 1980. Eutrophication in peel inlet - 2. Identification
of critical uncertainties via generalized sensitivity analysis. Water Research 14(1),
43-49.
Streeter, H.W. and Phelps, E.B. 1925. A Study of the Pollution and Natural Purification
of the Ohio River, III, Factors Concerned in the Phenomena of Oxidation and
Reareation. U.S. Pub. Health Serv., Pub. Health Bulletin, 146, pp.75.
Suttamanutwong, W. 2001. A comparison of empirical nitrogen runoff models in
application to an agricultural catchment in China. MSc Project Dissertation, Imperial
College London, September 2002.
Tate, C.M., Broshears, R.E. and McKnight, D.M. 1995. Phosphate dynamics in an acidic
mountain stream: interactions involving algal uptake, sorption by iron oxide and
photoreduction. Limnology and Oceanography 40(5), 938.
217
Taylor, G.I. 1954. The dispersion of matter in turbulent flow through a pipe. Proceedings
of the Royal society of London, Series A 223, 446-468.
Tellinghuisen, J.A. 2000. Monte Carlo study of precision, bias, inconsistency, and non-
Gaussian distributions in non-linear least squares. Journal of Physical Chemistry
104(12), 2834- 2844.
Thomann, R.V. and Mueller, J.A. 1987. Principles of surface water quality modeling and
control. Addison-Wesley.
Thomann, R.V. 1989. Bioaccumulation model of organic chemical distribution in aquatic
food chains. Environmental Science and Technology 23, 699-707.
Thomann, R.V. 1998. The future "golden age" of predictive models for surface water
quality and ecosystem management. ASCE Journal of Environmental Engineering
124(2), 94-103.
Thyer, M., Kuczera, G., Bates, B.C. and Bryson C. 1999. Probabilistic optimization for
conceptual rainfall-runoff models: A comparison of the shuffled complex evolution
and simulated annealing algorithms. Water Resources Research 35(3), 767-773.
Tsakalias, G. and Koutsoyiannis, D. 1999. A comprehensive system for the exploration
and analysis of hydrological data. Water Resource Management 13, 269-302.
Tung, Y.K. 1996. Uncertainty and reliability analysis. In Water Resources Handbook,
Mays, L.W. (Ed.), McGraw-Hill.
Tye, R., Jepsen, R. and Lick, W. 1996. Effects of colloids, flocculation, particle size, and
organic matter on the adsorption of hexachlorobenzene to sediments. Environmental
Toxicology and Chemistry 15(5), 643-651.
UK Environment Agency. 1998. The State of the Environment of England and Wales:
Fresh Waters. The Stationery Office, London.
UK Environment Agency. 2001a. SIMCAT 7.6: A Guide and Reference for Users. UK
Environment Agency, Bristol, UK.
UK Environment Agency. 2001b. Water resources for the future – A summary of the
strategy for England and Wales. UK Environment Agency, Bristol, UK.
UK Environment Agency. 2002. The Water Framework Directive: Guiding Principles on
the Technical Requirements, Chapter 7: Review of the impacts of human activity. UK
Environment Agency, Bristol, UK.
US EPA. 1998. Water quality criteria and standards plan – priorities for the future. US
EPA 822-R-98-003.
US EPA. 1999. BASINS (Better Assessment Science Integrating Point and Nonpoint
Sources) Version 2 User's Manual. Report no. EPA#: 823/B-98-006, US EPA,
Athens, Georgia, USA.
218
Valdes, J.B., Rodriguez-Huras, I. and Vicens, G.J. 1980. Choosing among hydrologic
regression models 2. Extensions to the standard model. Water Resources Research
16(3), 507-516.
Van der Perk, M. 1997. Effect of model structure on the accuracy and uncertainty of
results from water quality models. Hydrological Processes 11(3), 227-239.
Van Straten, G. 1983. Maximum likelihood estimation for phytoplankton models. In
Uncertainty and forecasting of water quality, Beck, M.B. and Van Straten, G (Eds.),
Springer-Verlag.
Van Straten, G. and Keesman, K.J. 1991. Uncertainty propagation and speculation on
projective forecasts of environmental change: a lake eutrophication example. Journal
of Forecasting 10, 163-190.
Van Straten, G. 1998. Models for water quality management: the problem of structural
change. Water Science and Technology 37(3), 103–111
Vanrolleghem P., Borchardt D., Henze M., Rauch W., Reichert P., Shanahan P. and
Somlyody L. 2001. River water quality model no. 1 (RWQM1): III. Biochemical
submodel selection. Water Science and Technology 43(5), 31-40.
Vrugt, J.A., Gupta, H.V., Bouten, W. and Sorooshian, S. 2003a. A shuffled complex
evolution Metropolis algorithm for optimization and uncertainty assessment of
hydrologic model parameters. Water Resources Research 39(8), SWC1:1-1:16.
Vrugt, J.A., Gupta, H.V., Bastidas, L., Bouten, W. and Sorooshian, S. 2003b. Effective
and efficient algorithm for multi-objective optimisation of hydrologic models. Water
Resources Research 39(8), SWC5:1-5:16.
Wade, A.J., Hornberger, G.M., Whitehead, P.G. and Flynn, N. 2001. On modeling the
mechanisms that control in-stream phosphorus, macrophyte and epiphyte dynamics:
An assessment of a new model using general sensitivity analysis. Water Resources
Reasearch 37(11), 2777-2792.
Wagener T., Boyle D.P., Lees M.J., Wheater H.S., Gupta H.V. and Sorooshian S. 2001. A
framework for the development and application of hydrological models. Hydrological
and Earth System Sciences 5(1), 13-26.
Wagener, T., Camacho, L.A., Lees, M.J. and Wheater, H.S. 2002a. Dynamic parameter
identifiability analysis of a solute transport model. Journal of Hydroinformatics 4(3),
199-212.
Wagener, T., McIntyre, N., Lees, M.J., Wheater, H.S. and Gupta, H.V. 2002b. Reducing
uncertainty in conceptual rainfall-runoff modelling: Dynamic identifiability analysis.
Hydrological Processes 17(2), 455-476.
219
Wagner, B.J. and Harvey, J.W. 1997. Experimental design for estimating parameters of
rate-limited mass transfer: analysis of stream tracer studies. Water Resources
Research 33(7), 1731-1741.
Wallingford Software. 2002. ISIS Technical Review. Wallingford Software Ltd, UK.
(Available from www.wallingfordsoftware.com/products/PDF/ISIS_tech_review.pdf).
Warwick, J.J., Cockrum, D. and McKay, A. 1999. Modelling the impact of subsurface
nutrient flux on water quality in the lower Truckee river, Nevada. Journal of
American Water Resources Association 35(4), 837-851.
Watanabe, M., Harleman, D.R.F. and Vasiliev, O.F. 1983. Two- and three-dimensional
mathematical models for lakes and reservoirs. In Mathematical modelling of water
quality, Orlob, G.T. (Ed.), Wiley.
Weber, E.J. 1996. Iron-mediated reductive transformations: Investigation of reaction
mechanism. Environmental Science and Technology 30(2), 716-719.
Weglarczyk, S. 1998. Interdependence and applicability of some statistical quality
measures for hydrological models. Journal of Hydrology 206(1-2), 98-103.
West, M. and Harrison, J. 1997. Bayesian forecasting and dynamic models (2nd edition),
Springer-Verlag, New York.
Wheater, H.S., Bishop, K.H. and Beck, M.B. 1986. The identification of conceptual
hydrological models for surface water acidification. Hydrological Processes 1, 89-
109.
Whitehead, P.G. and Hornberger, G.M. 1984. Modelling algal behaviour in the River
Thames. Water Research 18(18), 945-953.
Whitehead, P.G. and Toms, I.P. 1993. Dynamic modelling of nitrate in reservoirs and
lakes. Water Research 27(8), 1377-1384.
Whitehead, P.G., Williams, R.J. and Lewis, D.R. 1997a. Quality simulation along river
systems (QUASAR): model theory and development. The Science of the Total
Environment 194-195, 447-456.
Whitehead, P.G., Howard, A. and Arulmani, C. 1997b. Modelling algal growth and
transport in rivers: a comparison of time-series analysis, dynamic mass balance and
neural network techniques. Hydrobiologia 349, 39-46.
Whitehead, P.G., Wilson, E.J. and Butterfield, D. 1998. A semi-distributed Integrated
Nitrogen model for multiple source assessment in Catchments (INCA): Part I - model
structure and process equations. The Science of the Total Environment 210(1-6), 547-
558.
WHO. 1996. Guidelines for drinking-water quality : health criteria and other supporting
information 2nd Edition. World Health Organization.
220
Wierman, M.J. 1996. Assessing fuzzy sets and the potential of possibility theory.
Information Sciences 88(1-4), 247-261.
Xianxin, C. and Yongjiu, B. 1991. Water pollution and control measures in Liaoning
Province. Waterlines 10(1), 21-23.
Xu, C.Y. and Vandewiele, G.L. 1994. Sensitivity of monthly rainfall-runoff models to
input errors and data length. Hydrological Sciences 39(2), 157-176.
Yang C.T. and Molinas, A. 1982. Sediment transport and unit stream power function.
ASCE Journal of the Hydraulics Division 108(6), 774-793.
Yapa, P.D. and Shen, H.T. 1995. Modelling river oil spills: a review. Journal of
Hydraulic Research 32(5), 765-782.
Yapo, P.O., Gupta, H.V. and Sorooshian, S. 1998. Multi-objective global optimization for
hydrological models. Journal of Hydrology 204(1-4), 83-97.
Yeh, K., Yang, J.C. and Tung, Y.K. 1997. Regionalization of unit hydrograph
parameters; 2. Uncertainty analysis. Stochastic Hydrology and Hydraulics 11, 173-
192.
Young, P.C. and Wallis, S.G. 1993. Solute transport and dispersion in channels. In
Channel Network Hydrology, Beven, K.J. and Kirkby, M.J. (Eds.), John Wiley,
pp129-174.
Young, P., Parkinson, S. and Lees, M. 1996. Simplicity out of complexity in
environmental modelling: Occam's razor revisited. Journal of Applied Statistics
23(2), 165-210.
Yu, P-S., Yang, T-C.and Chen, S-J. 2001. Comparison of uncertainty analysis methods
for a distributed rainfall–runoff model, Journal of Hydrology 244(1-2), 43-59.
Zadeh, L.A. 1978. Fuzzy sets as a basis for the theory of possibility. Fuzzy Sets and
Systems 1, 3-28.
Zeng, W. and Beck, M.B. 2001. Development and evaluation of a mathematical model
for the study of sediment-related water quality issues. Water Science and Technology
43(7), 47-54.
221
Notation al Atmospheric transmission of long-wave as Atmospheric transmission of short-wave Ac Cross-section area of water Ai Coverage of ice as fraction of Aw As Surface area of sediment Aw Surface area of water b Bowen’s coefficient B Stefan Boltzman constant c Cloud cover C Concentration of arbitrary pollutant C’ Concentration of arbitrary pollutant in transient storage zone Cag Concentration of phytoplankton Cc Concentration of biodegradable carbon Ccf Concentration of 5-day BOD Ccs Concentration of slow reacting carbon Cna Concentration of ammonium plus ammonia Cni Concentration of nitrate plus nitrite Cns Concentration of organic nitrogen Cos Saturated concentration of dissolved oxygen Cox Concentration of dissolved oxygen Cp Concentration of total phosphorus Cpd Concentration of total phosphorus in distributed load Cpo Concentration of inorganic phosphorus Cps Concentration of organic phosphorus Cpu Concentration of total phosphorus at upstream boundary Csa Salinity Css Concentration of suspended solids di Density of ice dw Density of water D Diffusivity (units length2/time) D’ Diffusion coefficient (units length3/time) Dx’ Diffusion to transient storage zone (units length/time) eair Vapour pressure of air eairs Vapour pressure of air at saturation EP Monthly export of phosphorus fb Bulk heat exchange due to flow and dispersion fc Convective heat exchange fe Heat exchange due to evaporation fi Light limitation factor fiw Ice-water heat exchange fl Long-wave exchange fp Heat input due to precipitation fs Short-wave heat exchange fsw Sediment-water heat exchange F() Mass flux due to transport processes
222
Fsp Diffusive flux of phosphorus from sediment g Gravitational constant Hi Thickness of ice Hs Sediment depth Hw Water depth I Light intensity J Heat m Coefficient of emissivity kai Air-ice heat exchange coefficient kAice Empirical coefficient relating Aice to Hi kd Diffusion rate between sediment and water (units length/time) kda (kda20) Phytoplankton death rate (max. Rate at 20oc) kdn (kdn20) Inorganic nitrogen denitrification (max. Rate at 20oc) ker Error in aeration formula kga (kga20) Phytoplankton growth rate (max. Rate at 20oc) kgahsl Phytoplankton light half-saturation constant kgahsn Phytoplankton nitrogen half-saturation constant kgahsp Phytoplankton phosphorus half-saturation constant khc (khc20) Slow carbon hydrolysis rate (max. Rate at 20oc) khn (khn20) Organic nitrogen hydrolysis rate (max. Death rate at 20oc) khp (khp20) Organic phosphorus hydrolysis rate (max. Rate at 20oc) ki Conductivity of ice kiw Heat transfer coefficient (ice-water) kna Ammonium preference coefficient koc (koc20) Fast carbon oxidation rate (max. Rate at 20oc) kochs Fast carbon oxygen half-saturation kon (kon20) Inorganic nitrogen nitrification rate (max. Rate at 20oc) konhs Inorganic nitrogen oxygen half-saturation kra Reaeration rate ks Scour rate ksw Heat transfer coefficient (sediment-water) kwe Weir aeration coefficient kW Convective heat transfer coefficient K Defined constant KM1 Constant used in metropolis algorithm KM2 Constant used in metropolis algorithm KR Root constant in GLUE likelihood KS Kolmogorov-Smirnov statistic li Latent heat of melting of ice lw Latent heat of evaporation L Posterior likelihood Lp Prior likelihood M Notional model result n Mannings coefficient N Defined integer Ndat Number of samples of data sets Npar Number of model parameters Nres Number of residuals Nsam Number of parameter samples
223
Nvar Number of model responses O Notional observation OF Objective p Model order P Probability P’ Random number q1 Linear flow residence time parameter q2 Non-linear flow residence time parameter Q Flow Qev Evaporation rate Ql Flow loss Qs Flow source Qu Flow at upstream boundary Qup Flow from upstream cell r Hydraulic radius rcn Carbon demand of denitrification rna Phytoplankton ratio Nitrogen:Chl-a rN Nutrient limitation factor roa Oxygen:Chl-a ration ron Oxygen demand of nitrification rpa Phosphorus:Chl-a ratio rT Temperature limitation factor Ri Reflectance of ice Rsp Resuspension of P from sediment Rw Reflectance of water s Short wave radiation reaching outside of atmosphere sw Specific heat capacity of water Sp Concentration of phosphorus in sediment SOD Sediment oxygen demand t Time T Residence time Ta Ambient air temperature Tp Water temperature of pollution source Ts Temperature of deep sediment Tw Water temperature
wT Measured water temperature Tw’ and Tw’’ Intermediate evaluations of Tw u Water velocity ucr Critical water velocity vp Total phosphorus settling velocity vsc Slow carbon settling velocity vsn Organic nitrogen settling velocity vsp Organic phosphorus settling velocity vss Suspended solids settling velocity V Volume of water in control volume W Wind speed x Distance downstream, or arbitrary parameter X Vector of arbitrary model outputs Y Vector of arbitrary model inputs
224
Z Degree-days below freezing ∆() Derivative matrix Ω Problem dependant error constant α A sample set of factors β Arbitrary dependant variable δ2 Variance of model result around optimum model result ε Model residual η Step adaption safety constant γ Arbitrary factor λ Approximated local truncation error µ Mean value σ2 Variance of observed data around model result σm
2 Variance of observed data around optimum model result θ Arrhenius coefficient for all reactions ξ Specified tolerance Φ Pollution or hydraulic load τcr Critical shear stress at bed τ Shear stress at bed ζ Local truncation error