comparative assessment of six automatic optimization techniques for calibration of a conceptual...

This article was downloaded by: [71.196.192.211]On: 12 May 2015, At: 11:22Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Hydrological Sciences JournalPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/thsj20

Comparative assessment of six automaticoptimization techniques for calibration of aconceptual rainfallrunoff modelMONOMOY GOSWAMI a & KIERAN MICHAEL O'CONNOR aa Department of Engineering Hydrology , National University of Ireland , Galway, IrelandE-mail:Published online: 18 Jan 2010.

To cite this article: MONOMOY GOSWAMI & KIERAN MICHAEL O'CONNOR (2007) Comparative assessment of six automaticoptimization techniques for calibration of a conceptual rainfallrunoff model, Hydrological Sciences Journal, 52:3,432-449, DOI: 10.1623/hysj.52.3.432

To link to this article: http://dx.doi.org/10.1623/hysj.52.3.432

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shallnot be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Hydrological SciencesJournaldes Sciences Hydrologiques, 52(3) June 2007 Special issue: Hydroinformatics

Open for discussion until 1 December 2007 Copyright 2007 IAHS Press

432

Comparative assessment of six automatic optimization techniques for calibration of a conceptual rainfallrunoff model MONOMOY GOSWAMI & KIERAN MICHAEL OCONNOR Department of Engineering Hydrology, National University of Ireland, Galway, Ireland [email protected] Abstract In this application-based study, six automated strategies of parameter optimization are used for calibration of the conceptual soil moisture accounting and routing (SMAR) model for rainfallrunoff simulation in two catchments, one small and the other large. The methods used are: the genetic algorithm, particle swarm optimization, Rosenbrocks technique, shuffled complex evolution of the University of Arizona, simplex search, and simulated annealing. A comparative assessment is made using the Nash-Sutcliffe model efficiency index and the mean relative error (MRE) to evaluate the performance of each optimization method. It is found that the degree of variation of the values of the water balance parameters is generally less for the small catchment than for the large one. In the case of both catchments, the probabilistic global population-based optimization method of simulated annealing is considered best in terms of having the least variability of parameter values in successive tests, thereby alleviating the phenomenon of equifinality in parameter optimization, and also in producing the lowest MRE in verification. Key words optimization; SMAR; genetic algorithm; Rosenbrock; simplex; particle swarm optimization; simulated annealing; shuffled complex evolution

Evaluation comparative de six techniques automatiques doptimisation pour le calage dun modle conceptuel pluiedbit Rsum Dans cette tude de mise en application, six stratgies automatiques doptimisation de paramtres sont utilises pour le calage du modle conceptuel soil moisture accounting and routing (SMAR) pour la simulation pluiedbit dans deux bassins versants, lun petit et lautre grand. Les mthodes utilises sont: lalgorithme gntique, loptimisation par essaim de particules, la technique de Rosenbrock, la technique shuffled complex evolution de lUniversit dArizona, la mthode du simplex et celle du recuit simul. Une estimation comparative est mene laide de lindice defficience de modlisation de Nash-Sutcliffe et de lerreur relative moyenne pour valuer la performance de chaque mthode doptimisation. Il apparat que le degr de variation des valeurs des paramtres du bilan hydrologique est gnralement infrieur pour le petit bassin versant que pour le grand. Dans le cas de lensemble des deux bassins, la mthode doptimisation du recuit simul, probabiliste globale base sur la population, apparat tre la meilleure en termes de moindre variabilit des valeurs des paramtres au cours de tests successifs, ce qui attnue le phnomne dquifinalit vis--vis de loptimisation des paramtres et galement de la minimisation de lerreur relative moyenne lors de la vrification. Mots clefs optimisation; SMAR; algorithme gntique; Rosenbrock; simplex; optimisation par essaim de particules; recuit simul; shuffled complex evolution INTRODUCTION Most conceptual models are characterized by a considerably large number of variables and require multiple-parameter optimization involving multi-peak objective function surfaces. Successful application of such models, widely used for simulation of the rainfallrunoff transformation process, invariably depends on the adequacy of the optimization procedures. Numerous efforts have been directed since the 1960s towards efficient determination of the optimum values of the parameters of such models. Good references on earlier work on various methods of optimization can be found in Trn & Zilinskas (1989), Pintr (1995) and Duan (2003). The direct-search methods, also called enumerative methods, use the values of the objective function only (Spendley et al., 1962; Nelder & Mead, 1965; Rosenbrock, 1960; Hooke & Jeeves, 1969), whereas

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15

Comparison of automatic optimization techniques

Copyright 2007 IAHS Press

433

the gradient or calculus-based methods also involve the evaluation of the derivatives of the function (Levenberg, 1944; Marquardt, 1963; Fletcher & Powell, 1963; Fletcher, 1970). However, both categories, some of which are analytically demanding, suffer from the drawbacks of being trapped in local optima around the first relative minimum reached and largely depend on the specified starting values of the parameters. A third category, namely, random search is generally less susceptible to such drawbacks. In this approach, direct-search algorithms are started sequentially from each point of a large sample of points in the parameter space, each aiming to reach the optimum (Price, 1978; Masri et al., 1980; Pronzato et al., 1984). In order to proceed, a pre-defined scheme is followed whereby only the best amongst the number of local optima is retained. It is hoped that this best optimum is the elusive global optimum. The genetic algorithm (Holland, 1975; Goldberg, 1989), simulated annealing (Kirkpatrick et al., 1983; Corana et al., 1987), particle swarm optimization (Kennedy & Eberhart, 1995; Kennedy et al., 2001; Eberhart & Shi, 2004), the shuffled complex evolution of the University of Arizona (Duan et al., 1992, 1993, 1994), adaptive clustering methods (Solomatine, 1999) and some hybrid combinations of these methods (Cheng et al., 2002; Vrugt et al., 2003; Vrugt, 2004; Vugrin, 2005) all fall under this third category of random search techniques. Numerous comparative studies have been made to assess the relative performance of search techniques (e.g. Ibbitt & ODonnell, 1971; Duan et al., 1992; Luce & Cundy, 1994; Gan & Biftu, 1996; Cooper et al., 1997; Kuczera, 1997; Franchini et al., 1998; Freedman et al., 1998; Thyer et al., 1999; Solomatine, 1999; Madsen et al., 2002; Vugrin, 2005). Conclusions of these studies suggest that the global population-evolution-based algorithms are generally more efficient than the multi-start local optimization techniques, which in turn perform better than the pure local (single-start) search methods (Madsen et al., 2002). However, the choice of an appropriate calibration method for a model still remains a relevant research theme. Most studies investigating the efficacy of optimization techniques generally involve finding a parameter set which is likely to produce the optimum value of the objective function. But it is often observed that different sets of parameter values can produce very similar values of the objective function, even for the same model structure, and that each of these objective function values may well be deemed acceptable within the limit of accuracy involved. This leads to the well-documented phenomenon of equifinality, ambiguity, non-uniqueness, ill-posedness and identifiability (Beven, 2006) of the parameters, and results in uncertainty associated with the flow forecasts. Some streamflow analyses ideally require uniqueness of parameter values, such as those for relating the parameters of a conceptual rainfallrunoff model to physical catchment descriptors and for continuous streamflow estimation in ungauged catchments using regional analysis of the rainfallrunoff transformations. In this study, the effectiveness of a selection of six optimization methods applied for parameter estimation of the conceptual soil moisture accounting and routing (SMAR) model is discussed in the context of low variability of the parameter values over a large number of tests and acceptable model performance. The genetic algorithm, particle swarm optimization, Rosenbrocks technique, shuffled complex evolution of the University of Arizona, simplex search and simulated annealing are considered for this purpose and the best overall method is identified. A brief outline of these methods and some of their hydrological applications are given below.

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15

Monomoy Goswami & Kieran Michael OConnor


434

BRIEF DESCRIPTION OF THE OPTIMIZATION METHODS Genetic algorithm (GA) Genetic algorithms (Holland, 1975; Goldberg, 1989) are probabilistic global search algorithms based upon the mechanics of natural selection and natural genetics. Genetic algorithms can be numerically implemented either in binary coding or in real coding. In this study, binary coding is used. A good description of the genetic algorithm can be found in Duan (2003). The search steps of the genetic algorithm start by randomly generating a population (N points) in the feasible parameter space which are ranked according to the value of the objective function at those N points. Two parents are selected from the population and offspring are generated randomly using genetic operators (e.g. cross-over and mutation). The two worst points in the population are replaced by the two newly-generated offspring. The process of selection of parents and their replacement by offspring is repeated until a convergence criterion is satisfied. Genetic algorithms have been widely used in many disciplines, including hydrology. Wang (1991) showed that the genetic algorithm is a robust and efficient method for the calibration of hydrological conceptual models. A good account of reference to its recent applications in water resources and hydrology can be found in Anctil et al. (2006). Particle swarm optimization (PSO) Like the genetic algorithm, this search method (Kennedy & Eberhart, 1995) also starts with a population of potential solutions called a swarm. The members in the swarm, called particles, undergo change (i.e. evolve and learn) over time relying on their own experience and that of other particles in the swarm. See Clerc & Kennedy (2002) for a description of this method. In this method, each particle is considered to have (a) a current position, (b) a memory of the direction it followed in reaching that position, (c) a memory of its own best previous position, and (d) a memory of the best previous position of any other particle in the swarm. In order to change its position from the current position, each particle may move in either (i) the same direction that it came from, (ii) the direction of its best previous position, or (iii) the direction of the best previous position of any other particle in the swarm. The algorithm considers the actual direction of change of the particle as a weighted combination of all these three possibilities. The optimization stops when either all particles arrive at the same position, or the specified maximum number of iterations is reached. Although PSO has been widely used in neuro-computing, environmental science and many other fields, its application in hydrology is less common. Two recent appli-cations in hydrology are the parameter estimation of the Sacramento soil moisture accounting model (Gill et al., 2006) and in the training algorithm for an artificial neural network (ANN) in stage prediction of a river in Hong Kong (Chau, 2006). Rosenbrocks technique (RNB) In this direct local-search method for optimization (Rosenbrock, 1960), the first para-meter, which is made active for optimization, is given the greatest weight. The search

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



435

is initiated by using the original coordinate system for the first iteration, the axes of the system being the parameters to be optimized. The axes are searched in turn, the strategy of direction of the search vector being success, success, , failure, , success, where a success refers to a lower value of the objective function than the value at the start of the iteration and the subsequent success steps, whereas a failure corresponds to a value higher than that obtained at the preceding success step. With each monotonically decreasing value of the objective function, the length of the search vector is increased. At failure, the direction of the search vector is reversed, and the length of the search vector is reduced with each search step that fails, until a success is encountered. At this point the original parameter value is replaced by the new value at which it is held, and the procedure is repeated with the same strategy for the next parameter axis. The iteration is complete when all axes are searched, a new minimum is attained, or the initial parameter setting is retained. After this first iteration step, the method provides for rotation of the coordinate systems by aligning the axis of the primary parameter with the vector joining the origin and the latest minimum. The new axes are searched in turn, and the iterations continue until a stopping criterion is fulfilled. With modifications made for use with hydrological models, the method was found to be robust by Ibbitt & ODonnell (1971) who conducted a study comparing nine different methods for fitting hydrological models. Rosenbrocks technique was used by Kachroo et al. (1992) for illustrating the application of linear rainfallrunoff models in 14 catchments and by Liang et al. (1992) for demonstrating the application of two linear flow routing methods on three rivers in China. Shuffled complex evolution University of Arizona (SCE-UA) The shuffled complex evolution method developed at the University of Arizona by Duan et al. (1992, 1993) is another probabilistic global search method which is intended to combine the strength of the simplex search with the concepts of controlled random search, competitive evolution, and shuffling of complexes or communities. The algorithm begins by randomly selecting s points in the parameter space such that s = p m where p is the number of complexes each having m points. After evaluating the function at each point, the points are sorted in order of the increasing value of the objective function and are stored in an array that is partitioned into m complexes. Each complex is processed independently through an evolutionary system called the competitive complex evolution (CCE) strategy (Duan et al., 1992) by randomly forming a simplex containing n + 1 vertices, n being the number of parameters to be optimized. New points replace points with the largest values of the objective function using a single iteration of the local-search simplex method (see the next subsection) and the generation of random points within the feasible parameter space. The updated points are returned to the complex where new n + 1 points are randomly selected to form a new simplex, and the procedure is repeated a specified number of times. After processing through the CCE, the elements of the complexes are shuffled by replacing the complexes in the original array, sorting the points in that array in order of the increasing value of the objective function, and partitioning the array into m complexes as before. The procedure is repeated until a stopping criterion, e.g. the specified number of function evaluations, is satisfied.

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



436

Since the introduction of the SCE-UA method by Duan et al. (1992) to locate the global optimum of a conceptual rainfallrunoff model, a large number of researchers in hydrology have employed it for parameter optimization. Applications include those by Cooper et al. (2007), Thyer et al. (1999), Kuczera (1997), Gan & Biftu (1996) and Sorooshian et al. (1993) for parameter identification of conceptual rainfallrunoff models, by Abdulla et al. (1999) for estimation of the ARNO model baseflow parameters using daily streamflow data, by Freedman et al. (1998) for identification of erosion parameters for a process-based model of catchment runoff and sediment yield, and by Luce & Cundy (1994) for finding the parameter values of a process-based model for studying runoff from forest roads, etc. Simplex search (SIM) The simplex search is a direct local-search method for parameter optimization. Originally suggested by Spendley et al. (1962) and subsequently improved by Nelder & Mead (1965), it involves a search of the parameter space using a geometric figure called a simplex having the number of vertices which is greater by one than the number of parameters. A simplex in n-dimensional space thus has a set of n + 1 vertices. The method starts with function evaluation at all vertices of an initial simplex. The vertex having the highest error is replaced by a new vertex having lower error to form a new simplex. To determine the location of this new vertex, that having the highest error is reflected through the centroid of the remaining vertices. If the function evaluated at that new vertex fails to reduce the error, then another new vertex is generated by contraction towards the centroid. If neither of these methods succeeds in finding a vertex with lower error, then the entire simplex is contracted towards the vertex having the lowest error. The search is terminated when a stopping criterion is fulfilled. The stopping criterion may be a tolerance limit on the vector distance moved in a step, or a tolerance limit on the objective function value. The simplex search method has been widely used for parameter optimization in hydrology for: training a three-layer feed-forward ANN to model stage level and streamflow (e.g. Filho & dos Santos, 2006); for unbiased parameter estimation of the Neyman-Scott model for rainfall simulation (Favre et al., 2004); for the derivation of unit hydrographs of the quick and slow response runoffs using a conceptual model (Yue & Hashino, 2000); and to compare the relative performance of a number of optimization algorithms for parameter identification of hydrological models (e.g. Abdulla et al., 1999; Freedman et al., 1998; Gan & Biftu, 1996; Luce & Cundy, 1994; Duan et al., 1992). Simulated annealing (SA) Simulated annealing (Kirkpatrick et al., 1983) is a probabilistic global search method. For a description of this method, see Corana et al. (1987). According to this method, for a given temperature parameter, candidate points are generated around a point in the parameter hyperspace by applying a cycle of random moves each of which is along one of the n coordinate directions, n being the number of parameters to be optimized. The new coordinate values are uniformly distributed in intervals centred around the

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



437

corresponding coordinate of the current point. Half the size of these intervals along each coordinate is stored in the step vector. A candidate point is accepted or rejected according to the Metropolis criterion (Metropolis et al., 1953). The cycles are repeated until the total number of cycles exceeds a user-specified limit. The function is evaluated at every changed location of the points, and the optimum point reached thus far is recorded. After the user-specified number of cycles is performed with a given step vector, the step vector is adjusted according to the value of a user-specified step-varying criterion for each direction of movement. The cycles of movement are repeated for each of the user-specified number of step-vector adjustments and the optimum point reached thus far is noted. After these cycles of movement, conceptually resulting in a thermal equilibrium corresponding to the starting temperature, the temperature is reduced by a fraction, and a new sequence of moves is made starting from the optimum point reached thus far until thermal equilibrium is reached again. Iteration ceases when a stopping-criterion, such as a tolerance limit of the difference in successive values of the objective function, or the maximum number of successive temperature reductions, etc. is fulfilled. Simulated annealing has been used for optimal selection of the number and location of rainfall gauges for rainfall estimation (Pardo-Igzquiza, 1998), and simulated annealing combined with the simplex method was used for the calibration of a semi-distributed model for conjunctive simulation of surface and groundwater flows (Rozos et al., 2004), for optimization of a conceptual rainfallrunoff model (Thyer et al., 1999; Sumner et al., 1997), and for the estimation of baseflow parameters of the ARNO model (Abdulla et al., 1999). THE RAINFALLRUNOFF MODEL The soil moisture accounting and routing (SMAR) model is a nine-parameter lumped quasi-physical conceptual rainfallevaporationrunoff model, which has been developed from the Layers conceptual rainfallrunoff model of OConnell et al. (1970). In this study, the SMARG form of the SMAR model (Kachroo, 1992; Tan & OConnor, 1996; Shamseldin et al., 1997; Shamseldin & OConnor, 1999; Fazala et al., 2005) with Liangs groundwater modification (Liang, 1992) is used. In this model, the input variables, i.e. rainfall and evaporation, are transformed into simulated discharge through a series of steps which, in a very simplified manner, mimic the dominant physical processes (excluding snowmelt) in the rainfallrunoff transforma-tion. The schematic diagram of the SMAR model is given in Fig. 1. The nonlinear water balance (or soil moisture accounting) component of this model preserves the balance between the rainfall, the evaporation, the generated runoff and the changes in the layers of soil moisture storage. Five parameters, namely, Z (moisture holding capacity of soil layers), T (evapotranspiration conversion factor), H (fast response separation factor), Y (infiltration excess separation term) and C (factor for soil moisture depletion by evapotranspiration), control the overall operation of the water budget component of the model. The generated surface and groundwater runoff components are routed through linear time-invariant storage elements. At each time step, the outputs of the two routing elements combine to produce the simulated discharge. In this study, the classic Gamma (Nash IUH) is used for routing the

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



438

Rainfall in excess of infiltration capacity [Y]

Soil moisture storage [Z]

Evaporation [C]

Total estimated discharge

Evaporation E RainfallR

Layer 1Layer 2

Layer (Z/25)

oo

(1 H)X Y r2 = (1 H)X Y if (1 H)X > Y

Generated surface runoff

TE

r1= H X

r3

rg rs

Parameter symbols are shown within square brackets

Groundwatercomponent

Direct runoff[H]

Conversion to potential rate

[T]

Excess rainfallX = R - TE

Moisture in excess of soil capacity

[G]

Linear routing component [N], [NK]

A linearreservoir

[Kg]

Fig. 1 Structure of the nine parameter SMAR conceptual model (Liang, 1992).

generated surface runoff and a single discrete linear reservoir is used for routing the generated groundwater runoff. Four parameters control the operation of the routing component: G (saturation excess separation factor), N (shape factor of the Nash cascade model for surface water routing), NK (lag of the Nash cascade model) and Kg (linear reservoir constant for groundwater routing). THE TEST CATCHMENTS Two test catchments, namely the Brosna in central Ireland and the Baihe in north-eastern China, are used in the study. Figures 2 and 3 show the stream layout of these two catchments, their location in the respective countries and the distribution of raingauge stations. The Brosna catchment (area 1 207 km2) is predominantly flat with some relief formed by glacial deposits and two lakes of about 26 km2 at the upper end. Daily rainfall data for six years (19962001) are lumped by averaging the data from six raingauge stations while, the daily evaporation data are obtained from one centrally located meteorological station. Concurrent daily discharge data are obtained from the Ferbane gauging station shown in Fig. 2. Rainfall is nearly uniformly distributed throughout the year and usually of low intensity. The average daily rainfall, evapora-

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



439

Fig. 2 Location of the Brosna River catchment in Ireland and the Brosna catchment showing the distribution of 6 raingauge stations in it.

Fig. 3 Location of the Baihe River catchment in Shanxi Province, China, and the Baihe catchment showing the distribution of 16 raingauge stations.

tion and discharge for the Brosna catchment are 2.6, 1.6 and 1.1 mm (15.2 m3 s-1) respectively, the corresponding maximum values being 58.2, 5.1 and 6.5 mm (91.5 m3 s-1) and minimum being 0.0, 0.1 and 0.02 mm (0.3 m3 s-1), respectively. The Baihe catchment has an area of 61 789 km2. For this catchment, eight years of data (19721979) from 16 raingauge stations and six meteorological stations are used to obtain lumped daily average rainfall and evaporation. Concurrent daily discharge data at the Baihe station (Fig. 3) are also used. The catchment is mountainous and semi-dry, located in a typical monsoon climate region with rainfall mainly during the summer and autumn. Normally, evaporation is greater than rainfall from November to April and in August, whereas rainfall is less in the other months. The average daily rainfall, evaporation and discharge for the Baihe catchment are 2.6, 2.8 and 1.0 mm (696 m3 s-1), respectively, the corresponding maximum values being 80.0, 12.8 and 28.2 mm (20 200 m3 s-1) and minimum being 0.0, 0.0 and 0.1 mm (60.5 m3 s-1), respectively. METHODOLOGY Pre-processed daily rainfall, evaporation and discharge data from two test catchments are used in this study adopting the split-record calibration and evaluation procedure. Accordingly the SMAR model is calibrated with two-thirds of the data (four out of six years) in the case of the Brosna catchment and with three-quarters of the data (six out of eight years) in the case of the Baihe catchment, the remaining data being used for verification. In four sets of tests, the SMAR model is run 25, 50, 100 and 150 times with each of the six optimization methods using the data of each catchment. The

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



440

parameter values and the corresponding efficiencies in each set of tests for each of the two test catchments are noted. The ordinary least-squares objective function, i.e. minimization of the global mean square error (MSE) is considered for optimization. For every run with the local search methods, the same set of starting parameter values is considered, whereas for the global search methods, the numbers of starting para-meter values or points, as required by the different algorithms, are produced from within the hyperspace defined by the parameter bounds by randomly moving away from the starting parameter set. Although some of the nine parameters of the SMAR model could be fixed, resulting in fewer parameters to be optimized and thereby reducing the time of optimization, it is decided to allow all nine parameters to vary in this study in order to see how each parameter behaves in successive tests using the different optimization methods. The parameter bounds of all parameters are kept the same in all tests for both catchments. The relative efficiency of the model is assessed by the widely-used Nash-Sutcliffe R2 efficiency index (Nash & Sutcliffe, 1970) and the mean relative error (Elshorbagy et al., 2000). For the purpose of this study, the MRE is defined as the average of the absolute values of the errors between the estimated and the observed flows expressed as fractions of the corresponding observed flows. Although the SMAR model in each test with each optimization method is calibrated for obtaining the best performance over the calibration period, the relative efficiencies of the different optimization methods are determined on the basis of their corres-ponding performances in verification and on the degree of variability of the parameter values in successive tests. RESULTS The mean and the coefficients of variation (CV) of R2 and MRE in calibration and verification by each optimization method, for the four sets of 25, 50, 100 and 150 tests, are presented for the Brosna and the Baihe catchments in Figs 4 and 5 respectively. It is observed from these two figures that, except for the PSO and the RNB methods, the other four optimization methods generally produce similar values of mean and CV for the different sets of tests, the variations of the mean and the CVs of the efficiency indices being marginal for the sets of 50, 100 and 150 tests. From the results of efficiency indices for the Brosna catchment it was observed that, in a few of the tests in each set, in which the initial parameter values produced very low or negative values of the R2 efficiency index, the PSO method failed to optimize, retaining the initial set of parameter values. This can be seen in the plots of the R2 efficiency index for the set of 50 tests in Fig. 6, for both calibration and verification which provides representative plots showing the variation of the calibrated values of the parameters T, H, Y, N and NK of the SMAR model and the R2 efficiency index in both calibration and verification. The plots of the other parameters and the MRE efficiency index show similar variation but for brevity are not included here. In the case of the Baihe catchment, the corresponding plots are given in Fig. 7. For this catchment, it is seen that the RNB method fails to optimize in those few cases in which the initial parameter values produced very low or negative values of the R2 efficiency index, with the PSO method producing sub-optimal efficiency in such cases.

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



441

60

70

80

90

100

25 50 75 100

125

150M

ean

R2 (%

) in

calib

0.00.10.20.30.40.50.6

25 50 75 100

125

150

CV o

f R2

in c

alib

0.30.40.50.60.70.80.91.01.1

25 50 75 100

125

150M

ean

MRE

in c

alib

0.0

0.1

0.2

0.3

0.4

25 50 75 100

125

150C

V o

f MRE

in c

alib

60

70

80

90

100

25 50 75 100

125

150

No. of optimisat ion runs

Mea

n R2

(%) i

n ve

rif

0.00.10.20.30.40.50.6

25 50 75 100

125

150


CV o

f R2

in v

erif

0.30.40.50.60.70.80.91.01.1

25 50 75 100

125

150

No. of optimisation runsM

ean

MRE

in v

erif

0.0

0.1

0.2

0.3

0.4

25 50 75 100

125

150

No. of optimisation runs

CV o

f M

RE in

ver

if

0.5

0.6

0.7

0.8

GA SA SCE-UA PSO RNB SIM Fig. 4 Mean and coefficient of variation of R2 and MRE in calibration and verification for different sets of tests by each optimization technique (Brosna catchment).

60

70

80

90

100

25 50 75 100

125

150M

ean

R2 (%

) in

calib

0.00.10.20.30.40.5

25 50 75 100

125

150

CV o

f R2 i

n ca

lib

0.50.50.60.60.70.70.8

25 50 75 100

125

150M

ean

MRE

in c

alib

0.00.10.20.30.40.5

25 50 75 100

125

150C

V o

f MRE

in c

alib

60

70

80

90

100

25 50 75 100

125

150


Mea

n R2

(%) i

n ve

rif

0.00.10.20.30.4

0.5

25 50 75 100

125

150


CV o

f R2

in v

erif

0.50.50.60.60.70.70.8

25 50 75 100

125

150

No. of optimisation runs

Mea

n M

RE in

ver

if

0.00.10.20.30.40.5

25 50 75 100

125

150

No. of opt imisation runs

CV o

f M

RE in

ver

if

0.5

0.6

0.7

0.8

GA SA SCE-UA PSO RNB SIM Fig. 5 Mean and coefficient of variation of R2 and MRE in calibration and verification for different sets of tests by each optimization technique (Baihe catchment).

With the PSO and the RNB methods, failure to optimize, when it occurred, resul-ted in large variations in the mean and CV values of the efficiency indices as compared with the corresponding values for the other four methods. For the same reason, the mean and particularly the CV values for these two methods also varied from one set of tests to the other. Thus, when the initial parameter values were such as to produce poor model performance, both the PSO and the RNB methods failed to find the global optimum in the case of both test catchments. The marginal variations of the mean and the CV of the R2 and MRE indices for the sets of 50, 100 and 150 tests by the GA, the SA, the SCE-UA and the SIM methods demonstrate that results from 50 tests can be considered as being adequate for investigating the relative variations of the parameter

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



442

0.50.60.70.80.9

1

0 5 10 15 20 25 30 35 40 45 50

T (B

rosn

a)

00.20.40.60.8

1

0 5 10 15 20 25 30 35 40 45 50

H (B

rosn

a)

020406080

100

0 5 10 15 20 25 30 35 40 45 50

Y in

mm

(Bro

sna)

02468

10

0 5 10 15 20 25 30 35 40 45 50

N (B

rosn

a)

02468

10

0 5 10 15 20 25 30 35 40 45 50

NK

in d

ay-1

(Bro

sna)

02040

6080

100

0 5 10 15 20 25 30 35 40 45 50

R2 (%

) in

calib

ratio

n (B

rosn

a)

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40 45 50Test no.

R2 (%

) in

verif

icat

ion

(Bro

sna)

0

0.4

0.5

0.6

0.7

GA SA SCE-UA PSO RNB SIM Fig. 6 Representative plots showing the variation of the SMAR parameters T, H, Y, N, and NK and the R2 efficiency in calibration and verification in successive tests by each optimization technique (Brosna catchment).

values of the SMAR model in successive tests. On the basis of this observation, and also from the consideration of presenting relatively less complex graphical displays, the set of 50 tests only was chosen for drawing the main conclusions presented in this study. For the Brosna catchment, as discussed in the previous paragraph, and as can be seen from Fig. 6, the R2 values in calibration and verification for the PSO method are

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



443

0.50.60.70.80.9

1

0 5 10 15 20 25 30 35 40 45 50

T (B

aihe

)

00.20.40.60.8

1

0 5 10 15 20 25 30 35 40 45 50

H (B

aihe

)

020406080

100

0 5 10 15 20 25 30 35 40 45 50

Y in

mm

(Bai

he)

02468

10

0 5 10 15 20 25 30 35 40 45 50

N (B

aihe

)

02468

10

0 5 10 15 20 25 30 35 40 45 50NK

in d

ay-1 (B

aihe

)

020406080

100

0 5 10 15 20 25 30 35 40 45 50R2 (

%) i

n ca

libra

tion

(Bai

he)

020406080

100

0 5 10 15 20 25 30 35 40 45 50Test no.

R2 (%

) in

verif

icat

ion

(Bai

he)

0

0.4

0.5

0.6

0.7

0.8

GA SA SCE-UA PSO RNB SIM Fig. 7 Representative plots showing the variation of the SMAR parameters T, H, Y, N, and NK and the R2 efficiency in calibration and verification in successive tests by each optimization technique (Baihe catchment).

observed to deviate significantly in a few cases from their values in the other tests. The CE-UA also produced R2 values in calibration and verification which are somewhat lower than the corresponding values produced by the other methods. The local-search SIM and RNB methods are found to produce higher efficiency values which are comparable to the results produced by the global GA and SA methods. In order to differentiate further in deciding on the suitability of the four remaining methods (GA, SA, SIM and RNB), the graphs showing the variation of the parameter

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



444

Table 1 Coefficient of variation (CV) of the parameters of the SMAR model and the efficiency indices in 50 optimization runs of the program by each optimization method.

CV of SMAR parameters CV of R2 CV of MRE Optim. method T H Y Z C G N NK Kg Calib. Verif. Calib. Verif.Brosna catchment: GA 0.03 0.26 0.19 0.07 0.10 0.32 0.26 0.30 0.50 0.01 0.03 0.06 0.25 PSO 0.06 0.24 0.13 0.05 0.07 0.50 0.60 0.45 0.69 0.25 0.31 0.26 0.32 RNB 0.02 0.58 0.27 0.09 0.06 0.20 0.52 0.56 1.06 0.02 0.02 0.04 0.23 SA 0.00 0.13 0.02 0.00 0.00 0.21 0.25 0.56 0.02 0.00 0.02 0.03 0.23 SCE-UA 0.06 0.61 0.29 0.23 0.17 0.23 0.59 0.38 0.53 0.05 0.09 0.21 0.36 SIM 0.03 0.24 0.29 0.07 0.07 0.35 0.51 0.44 0.78 0.01 0.02 0.06 0.28 Baihe catchment: GA 0.02 0.27 0.46 0.28 0.11 0.11 0.13 0.03 0.07 0.00 0.06 0.05 0.05 PSO 0.06 0.20 0.15 0.25 0.15 0.16 0.09 0.02 0.10 0.07 0.21 0.11 0.09 RNB 0.15 0.43 0.44 0.46 0.28 0.24 0.43 0.55 0.42 0.33 0.42 0.21 0.15 SA 0.04 0.32 0.09 0.46 0.08 0.08 0.03 0.00 0.01 0.00 0.06 0.05 0.03 SCE-UA 0.13 0.34 0.33 0.36 0.16 0.18 0.32 0.07 0.38 0.01 0.07 0.09 0.12 SIM 0.03 0.14 0.49 0.22 0.28 0.07 0.07 0.01 0.28 0.00 0.04 0.04 0.07 values and Table 1 giving the CV values are studied. It may be observed that for the SA, the variations of the parameter values in successive tests are small in comparison to the other methods and the values of the CVs of the parameters are generally the lowest. This indicates that while all four methods produce similar efficiency values for parameter estimation in the case of the Brosna catchment, the SA is most likely to produce the sets of parameter values which do not differ from test to test. Table 2 provides the values of the parameters and the efficiency indices for the test corres-ponding to the maximum R2 value in verification for the Brosna catchment. Figure 8(a) shows scatter plots of the errors as ratios of observed flows and the observed flows as fractions of the peak flow (91.5 m3 s-1) produced by the SA method for that test. It is observed from Table 2 that, although the R2 value in calibration by the SA method is the highest, the R2 value in verification by this method is the second lowest, the lowest being that by the SCE-UA. The fact that the R2 value in calibration is very high compared with its relatively low value in verification is investigated with reference to the flow data. It is observed that very high flow values including the highest and the second highest occurred in the calibration period whereas the magnitudes of the peak flows in the verification period are relatively low, as indicated also in Fig. 8(a). As the length of data for calibration is relatively short (only four years of daily data) and the global MSE-based objective function is optimized, the model, while producing a good fit of the high flow values in calibration, fails to adequately simulate the lower flow peaks in verification. Hence, in order to draw further conclusions, the additional MRE statistic is used. It can be observed that the MRE value in verification by the SA method is the second lowest, the lowest being that by the SCE-UA. As indicated in Table 3, the SA method proved to be computationally less efficient than the other methods in respect of the total number of function evaluations. However, from consideration of least variability of the parameter values in successive tests combined with a low MRE in verification, the SA method can be considered suitable for para-

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



445

Table 2 Values of parameters of the SMAR model and the efficiency indices corresponding to the run producing best R2 efficiency value in verification amongst 50 optimization runs of the program by each optimization method.

Optim. Test Parameter R2 (%) MRE Method no. T H Y Z C G N NK Kg Calib. Verif. Calib. Verif.Brosna catchment: GA 12 0.93 0.21 60.91 119.1 0.95 0.40 1.06 7.87 156.84 82.76 82.88 0.41 0.89 PSO 2 0.97 0.28 55.12 124.99 0.87 0.38 1.09 7.62 182.74 83.12 83.46 0.37 0.84 RNB 44 1.00 0.76 100.00 122.33 0.78 0.89 1.00 10.00 28.68 79.47 83.03 0.42 0.63 SA 32 0.93 0.20 53.28 125.00 0.87 0.93 3.44 2.57 24.48 84.86 77.44 0.39 0.55 SCE-UA 19 0.97 0.19 98.00 117.74 0.90 0.95 8.15 3.84 24.16 77.33 76.55 0.42 0.53 SIM 16 0.93 0.28 66.51 125.00 0.87 0.36 1.00 8.87 157.54 83.49 82.41 0.39 0.86 Baihe catchment: GA 4 0.63 0.29 91.47 61.85 0.50 0.60 4.98 2.57 177.63 84.87 79.96 0.51 0.68 PSO 28 0.69 0.38 24.93 48.04 0.56 1.00 4.24 2.65 199.27 75.26 84.28 0.59 0.71 RNB 31 0.82 0.43 46.24 25.48 0.50 0.75 4.87 2.56 140.15 84.20 81.43 0.50 0.60 SA 38 0.66 0.38 30.28 59.84 0.50 0.64 4.49 2.61 200.00 85.05 74.03 0.48 0.63 SCE-UA 33 0.70 0.34 65.34 31.50 0.63 0.80 8.18 2.37 143.98 82.90 82.14 0.59 0.75 SIM 50 0.71 0.47 99.93 46.53 0.50 0.74 4.70 2.58 185.38 84.88 78.45 0.47 0.60

Optimisation by: SACalibration period

0.0

0.2

0.4

0.6

0.8

1.0

-2 0 2 4 6 8 10Error/Obs. flow

Optimisat ion by: SAVerification period

0.0

0.2

0.4

0.6

0.8

1.0


Optimisation by: SACalibration period

0.0

0.2

0.4

0.6

0.8

1.0


Optimisat ion by: SAVerification period

0.0

0.2

0.4

0.6

0.8

1.0


Fig. 8 Scatter plots of errors as ratios of observed flows and observed flows as fractions of the peak flow corresponding to the highest R2 index produced by the SA method: (a) Brosna catchment; and (b) Baihe catchment.

meter estimation of the SMAR model for the Brosna catchment. Figure 8(a) shows that, in calibration, while the errors corresponding to the medium and low flow values are generally well distributed about the vertical line of zero error, the high flow values are generally underestimated. In verification, the pattern of errors indicates general overestimation of the flow values. These observations explain why the MRE values in calibration are lower than those in verification.

(a)

(b)

Obs

.flow

/Pea

k flo

w

Obs

.flow

/Pea

k flo

w

Obs

.flow

/Pea

k flo

w

Obs

.flow

/Pea

k flo

w

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



446

Table 3 Number of function evaluations in the 50th test and the relative time taken for 50 optimization runs by each optimization method.

No. of function evaluations Total time taken in 50 tests (h:min:s) Optim. method Brosna Baihe Brosna Baihe GA 5000 5000 0:04:48 0:05:11 PSO 5001 5001 2:22:9 2:28:19 RNB 961 1333 0:02:27 0:01:53 SA 22501 26101 0:26:35 0:27:30 SCE-UA 5092 5092 1:08:47 1:07:30 SIM 394 700 0:00:41 0:00:41 For the Baihe catchment, it can be seen from Fig. 7 and Table 1 that the efficiency indices for the PSO, RNB and SCE-UA methods vary considerably from test to test. In contrast, the GA, SA and SIM methods generally produce similar values of these indices in the different tests. The coefficients of variation of five out of the nine SMAR parameters, namely Y, C, N, NK and Kg, and the MRE in verification are seen to be lowest for the SA method. The performance of the simplex method is also high because it produces the lowest coefficients of variation of the parameters H, Z and G, and of the R2 index in verification. From Table 2, which provides the values of the parameters and the efficiency indices for the test producing the highest R2 efficiency in verification, it is seen that, for the Baihe catchment, the SA produces the highest R2 efficiency in calibration but the least R2 efficiency in verification. Scatter plots of the errors as ratios of observed flows and the observed flows as fractions of the peak flow (20 200 m3 s-1) produced by the SA method for that test are shown in Fig. 8(b). The reason for the apparent discrepancy in the R2 values is similar to that described in the previous para-graph for the Brosna catchment, i.e. the highest flow peaks are in the longer calibration period. As for the Brosna, the MRE in verification for the Baihe is also relatively low. The SIM method produces a little higher value of R2 and a little lower value of the MRE in verification, in comparison to the SA method. From consideration of the number of function evaluations and the time taken for optimization indicated in Table 3, the SIM method would appear to outperform the SA method. However, from consideration of the least variability of the greatest number of parameters, the global-search SA method can also be considered suitable for parameter estimation of the SMAR model for the Baihe catchment. The pattern of errors for this catchment, shown in Fig. 8(b), and the observation concerning the lower value of MRE in calibration are similar to those for the Brosna catchment discussed in the preceding paragraph. Thus, for both catchments, the values of the parameters optimized by the SA method differ least from test to test. This implies that equifinality in the parameter space can be expected to be less pronounced when the SA method is used for optimization of the SMAR model for the two test catchments. In effect, the values of the parameters obtained by application of the SA method can be reliably considered as being representative for these catchments for applications which ideally require uniqueness of parameter values, as when relating the parameters of the SMAR model to physical catchment descriptors, and in regional studies involving the SMAR model for continuous streamflow estimation. A comparison of the CV values obtained by the SA method for the Brosna and the Baihe catchments, as given in Table 1, shows that the CVs of the five water balance

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



447

parameters of the SMAR model, namely, T, H, Y, Z and C are larger in the case of the Baihe catchment. Generally such larger variations for the Baihe catchment are also observed for the other optimization methods. This can be explained by the larger size of the Baihe catchment (about 51 times larger than the Brosna) which experiences larger variability in rainfall and produces highly variable flow magnitudes. Such variability of input data results in a more poorly-behaved parameter space as reflected in the CV values of the parameters. The wider variability in the values of the water balance parameters for this large catchment also reflects the inadequacy of the lumped rainfall approach in conceptual rainfallrunoff modelling in this case. CONCLUSION The behaviour of the optimization methods in producing optimum parameter values does not differ when 50, 100 and 150 tests are considered. From consideration of the least variability in parameter values in successive tests and of the least value of the MRE in verification, the probabilistic global simulated annealing (SA) search method can be considered best for optimization of the parameter values of the SMAR model in the case of the two test catchments. This also implies that for the test cases, the equifinality in the parameter space can be expected to be less pronounced when the SA method is used. The variability in the values of the water balance parameters of the SMAR model in successive optimization tests is influenced by the size of the catchment. For the larger Baihe catchment, which is characterized by hydrometeoro-logical and physiographical heterogeneity and the consequent variability in the input data series, most of the optimization methods produce high variation of parameter values in successive tests, but for the SA method such variation is found to be relatively the lowest. As confirmed in this study, for flow simulation with a given model, different optimization methods perform differently for different catchments. Hence, it is recommended that a series of repetitive tests with different starting values of the parameters by each optimization method be undertaken for deciding on the most appropriate method of optimization. Acknowledgement The authors are grateful to the two anonymous reviewers whose detailed comments and constructive suggestions contributed greatly to the improve-ment and readability of this paper. REFERENCES Abdulla, F. A., Lettenmaier, D. P. & Liang, X. (1999) Estimation of the ARNO model baseflow parameters using daily

streamflow data. J. Hydrol. 222, 3754. Anctil, F., Lauzon, H., Andrassian, V., Oudin, L. & Perrin, C. (2006) Improvement of rainfallrunoff forecasts through

mean areal rainfall optimization. J. Hydrol. 328, 717725. Beven, K. (2006) A manifesto for the equifinality thesis. J. Hydrol. 320, 1836. Chau, K. W. (2006) Particle swarm optimization training algorithm for ANNs in stage prediction of Shing Mun River.

J. Hydrol. 329, 363 367. Cheng, C. T., Ou, C. P. & Chau, K. W. (2002) Combining a fuzzy optimal model with a genetic algorithm to solve multi-

objective rainfallrunoff model calibration. J. Hydrol. 268(1/4), 7286. Clerc, M. & Kennedy, J. (2002) The particle swarmexplosion, stability, and convergence in a multidimensional complex

space. IEEE Trans. Evolutionary Comput. 6 (1), 5873.

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



448

Cooper, V. A., Nguyen, V.-T.-V. & Nicell, J. A. (2007) Calibration of conceptual rainfallrunoff models using global optimization methods with hydrologic process-based parameter constraints. J. Hydrol. 334(3/4), 455466.

Cooper, V. A., Nguyen, V. T. V. & Nicell, J. A. (1997) Evaluation of global optimization methods for conceptual rainfallrunoff model calibration. Water Sci. Tech. 36(5), 5360.

Corana, A., Marchesi, M., Martini, C. & Ridella, S. (1987) Minimizing multimodal functions of continuous variables with the simulated annealing algorithm. ACM Trans. Math. Software 13(3), 262280.

Duan, Q. (2003) Global optimization for watershed model calibration. In: Calibration of Watershed Models (ed. by Q. Duan, H. V. Gupta, S. Sorooshian, A. N. Rousseau & R. Turcotte), 89104. Water Science and Application 6, Am. Geophys. Union, Washington DC, USA.

Duan, Q. Y., Gupta, V. K. & Sorooshian, S. (1993) Shuffled complex evolution approach for effective and efficient global minimization. J. Optimiz. Theory Appl. 76(3), 501521.

Duan, Q. Y., Sorooshian, S. & Gupta, V. K. (1992) Effective and efficient global optimization for conceptual rainfallrunoff models. Water Resour. Res. 28(4), 10151031.

Duan, Q. Y., Sorooshian, S. & Gupta, V. K. (1994) Optimal use of the SCEUA global optimization method for calibrating watershed models. J. Hydrol. 158, 265284.

Eberhart, R. C. & Shi, Y. (2004) Guest editorial on special issue on particle swarm optimization. IEEE Trans. Evolutionary Comp. 8(3), 201203.

Elshorbagy, A., Simonovic, S. P. & Panu, U. S. (2000) Performance evaluation of Artificial Neural Networks for runoff prediction. J. Hydrol. Engng 5(4), 424427.

Favre, A. C., Musy, A. & Morgenthaler, S. (2004) Unbiased parameter estimation of the Neyman-Scott model for rainfall simulation with related confidence interval. J. Hydrol. 286, 168178.

Fazala, M. A., Imaizumia, M., Ishidaa, S., Kawachib, T. & Tsuchihara, T. (2005) Estimating groundwater recharge using the SMAR conceptual model calibrated by genetic algorithm. J. Hydrol. 303, 5678.

Filho, A. J. P. & dos Santos, C. C. (2006) Modelling a densely urbanized watershed with an artificial neural network, weather radar and telemetric data. J. Hydrol. 317, 3148.

Fletcher, R. (1970) A new approach to variable metric algorithms. Comput. J. 13, 317322. Fletcher, R. & Powell, M. (1963) A rapidly convergent descent method for minimization. Comput. J. 6, 163168. Franchini, M., Galeati, G. & Berra, S. (1998) Global optimization techniques for the calibration of conceptual rainfall

runoff models. Hydrol. Sci. J. 43(3), 443458. Freedman, V. L., Lopes, V. L. & Hernandez, M. (1998) Parameter identifiability for catchment-scale erosion modelling: a

comparison of optimization algorithms. J. Hydrol. 207, 8397. Gan, T. Y. & Biftu, G. F. (1996) Automatic calibration of conceptual rainfallrunoff models: optimization algorithms,

catchment conditions, and model structure. Water Resour. Res. 32(12), 35133524. Gill, M. K., Kaheil, Y. H., Khalil, A., McKee, M. & Bastidas, L. (2006) Multiobjective particle swarm optimization for

parameter estimation in hydrology. Water Resour. Res. 42(7), W07417, doi:10.1029/2005WR004528. Goldberg, D. E. (1989) Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Boston, USA. Holland, J. H. (1975) Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor, USA. Hooke, R. & Jeeves, T. A. (1969) Direct search solutions of numerical and statistical problems. J. Ass. Comp. Mach. 8(2),

212229. Ibbitt, R. P. & ODonnell, T. (1971) Fitting methods for conceptual catchment models. J. Hydraul. Div., Proc. ASCE

97(9), 13311342. Kachroo, R. K. (1992) River flow forecasting, Part 5: Applications of a conceptual model. J. Hydrol. 133, 141178. Kachroo, R. K., Sea, C. H., Warsi, M. S., Jemenez, H. & Saxena, R. P. (1992) River flow forecasting, Part 3, Applications

of linear techniques in modelling rainfallrunoff transformations. J. Hydrol. 133(1/2), 4197. Kennedy, J. & Eberhart, R. C. (1995) Particle swarm optimization. In: Proc. IEEE Int. Conf. on Neural Networks (ICNN,

1995), 4, 19421948. The University of Western Australia, Perth, Australia. Kennedy, J., Eberhart, R. C. & Shi, Y. (2001) Swarm Intelligence. Morgan Kaufmann, San Francisco, USA. Kirkpatrick, S., Gelatt, C. D., Jr & Vecchi, M. P. (1983) Optimization by simulated annealing. Science 220, 671680.

doi:10.1126/science.220.4598.671. Kuczera, G. (1997) Efficient subspace probabilistic parameter optimization for catchment models. Water Resour. Res.

33(1), 177185. Levenberg, K. (1944) A method for the solution of certain problems in least squares. Quart. Appl. Math. 2, 164168. Liang, G. C. (1992) A note on the revised SMAR model. Workshop Memorandum. National Univ. of Ireland, Galway

(unpublished). Liang, G. C., Kachroo, R. K., Kang, W. & Yu, X. Z. (1992) River flow forecasting, Part 4, Applications of linear

modelling techniques for flow routing on large catchments. J. Hydrol. 133(1/2), 99140. Luce, C. & Cundy. T. W. (1994) Parameter identification for a runoff model for forest roads. Water Resour. Res. 30(4).

10571069. Madsen, H., Wilson, G. & Ammentorp, H. C. (2002) Comparison of different automated strategies for calibration of

rainfallrunoff models. J. Hydrol. 261 (1/2), 4859. Marquardt, D. W. (1963) An algorithm for least squares estimation of nonlinear parameters. SIAM J. Ind. and Appl. Math.

11, 431441. Masri, S. F., Bekey, G. A. & Safford, F. B. (1980) A global optimization algorithm using adaptive random search. Appl.

Math. Comput. 7, 353375. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., Teller, E. (1953) Equation of state calculations by fast

computing machines. J. Chem. Phys. 21, 10871090.

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15



449

Nash, J. E. & Sutcliffe, J. V. (1970) River flow forecasting through conceptual models, Part 1, A discussion of principles. J. Hydrol. 10, 282290.

Nelder, J. A. & Mead, R. (1965) A simplex method for function minimization. Computer J. 7, 308313. Pardo-Igzquiza, E. (1998) Optimal selection of number and location of rainfall gauges for rainfall estimation using

geostatistics and simulated annealing. J. Hydrol. 210, 206220. Pintr, J. D. (1995) Global Optimization in Action. Kluwer, Amsterdam, The Netherlands. Price, W. L. (1978) A controlled random search procedure for global optimization. In: Towards Global Optimization 2

(ed. by C. W. L. Dixon & G. P. Szeg), 7184. North-Holland, Amsterdam, The Netherlands. Pronzato, L., Walter, E., Venot, A. & Lebruchec, J. F. (1984) A general purpose global optimizer: implementation and

applications. Math. Comput. in Simulations 26, 412422. Rosenbrock, H. H. (1960) An automatic method for finding the greatest or least value of a function. Comput. J. 7, 175

184. Rozos, E., Efstratiadis, A., Nalbantis, I. & Koutsoyiannis, D. (2004) Calibration of a semi-distributed model for

conjunctive simulation of surface and groundwater flows. Hydrol. Sci. J. 49(5), 819842. Shamseldin, A. Y. & OConnor, K. M. (1999) A real-time combination method for the outputs of different rainfallrunoff

models. Hydrol. Sci. J. 44(6), 895912. Shamseldin, A. Y., OConnor, K. M. & Liang, G. C. (1997) Methods for combining the outputs of different rainfallrunoff

models. J. Hydrol. 197, 203229. Solomatine, D. P. (1999) Two strategies of adaptive cluster covering with descent and their comparison to other

algorithms. J. Global Optimization 14(1), 5578. Sorooshian, S., Duan, Q. & Gupta, V. K. (1993) Calibration of rainfallrunoff models: Application of global optimization

to the Sacramento soil moisture accounting model. Water Resour. Res. 29(4), 11851194. Spendley, W., Hext, G. R. & Himsworth, F. R. (1962) Sequential application of simplex designs in optimization and

evolutionary design. Technometrics 4, 441461. Sumner, N. R., Flemming, P. M. & Bates, B. C. (1997) Calibration of a modified SFB model for twenty-five Australian

catchments using simulated annealing. J. Hydrol. 197(1/4), 166188. Takagi, T. & Sugeno, M. (1985) Fuzzy identification of systems and its application to modelling and control. IEEE Trans.

Syst. Man. Cybern. 15, 116132. Tan, B. Q. & OConnor, K. M. (1996) Application of an empirical infiltration equation in the SMAR conceptual model.

J. Hydrol. 185, 275295. Thyer, M., Kuczera, G. & Bates, B. C. (1999) Probabilistic optimization for conceptual rainfallrunoff models: a

comparison of shuffled complex evolution and simulated annealing algorithms. Water Resour. Res. 35(3), 767773. Trn, A. A. & Zilinskas, A. (1989) Global Optimization. Lecture Notes in Computer Science no. 350, Springer-Verlag,

Berlin, Germany. Vrugt, J. A. (2004) Towards improved treatment of parameter uncertainty in hydrologic modeling. PhD Thesis, University

of Amsterdam, The Netherlands. Vrugt, J. A., Gupta, H. V., Bouten, W. & Sorooshian, S. (2003) A shuffled complex evolution metropolis algorithm for

estimating posterior distribution of watershed model parameters. In: Calibration of Watershed Models (ed. by Q. Duan, H. V. Gupta, S. Sorooshian, A. N. Rousseau & R. Turcotte), 105112. Water Science and Application 6, Am. Geophys. Union, Washington DC, USA.

Vugrin, K. E. (2005) On the effects of noise on parameter identification optimization problems. PhD Dissertation, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA.

Wang, Q. J. (1991) The Genetic Algorithm and its application to calibrating conceptual rainfallrunoff models. Water Resour. Res. 27(9), 24672471.

Yue, S. & Hashino, M. (2000) Unit hydrographs to model quick and slow runoff components of streamflow. J. Hydrol. 227, 195206.

Received 3 October 2006; accepted 26 March 2007

Dow

nloa

ded

by [7

1.196

.192.2

11] a

t 11:2

2 12 M

ay 20

15

comparative assessment of six automatic optimization techniques for calibration of a conceptual...

Documents