Macroscale Modelling of the Hydrosphere (Proceedings of the Yokohama Symposium, July 1993). I A H S P u b l . n o . 214,1993.

Assessing uncertainties in hydrological response to climate at large scale

A. J. JAKEMAN, T. H. CHEN, D. A. POST Centre for Resource and Environmental Studies, Institute of Advanced Studies, Australian National University, Canberra, ACT 2601, Australia G. M . HOINBERGER Department of Environmental Sciences, University of Virginia, Charlottesville, Virginia 22903, USA

I. G. MTTLEWOOD & P. G. WHITEHEAD Institute of Hydrology, Wallingford, Oxfordshire OX10 8BB, UK

Abstract The construction of useful models for assessing hydrological response requires parameter estimation from observational data, predominantly precipitation, temperature and discharge time series. A lumped parameter model is applied to two basins with around three decades of such observations to elucidate the uncertainties associated with the simulation of discharge, and hence evaporative losses, at basin scale. Model performance is assessed over a range of historical conditions. This allows prescribed changes in air temperature and basin rainfall to be translated into effects on évapotranspiration and streamflow. The study provides both an indication of the level of uncertainties to be expected and a methodology for assessing response in other basins. Requirements for extending the work to continental scale are data-based and do not depend on major advances in scientific knowledge.

WmODUCIION

Appropriate models are required to assist with the assessment of macroscale hydrological impacts arising from fluctuations in climatic forcing variables and/or spatially extensive changes in land use. Hydrological models for separating the water balance, including inference of the évapotranspiration component, are also required to assist in studying the global redistribution of solar energy and especially for inclusion in global climate models. Rind et al. (1992) point out that the results from hydrological impact models depend crucially on their representation of the hydrological cycle, and argue that uncertainties in both climate and impact models limit confidence in current assessments.

Detailed physically-based models for these purposes would be ideal; it would be convenient if spatially and temporally distributed inputs of rainfall, radiation and other climate variables could be used to drive models which, given spatial descriptions of vegetation, soils, terrain and other important physical catchment descriptors, yield as output the dynamic water (and energy) fluxes to the land and back to the atmosphere. But this requires a thorough understanding of key biological-hydrological interactions of the soil-vegetation-atmosphere system. The dynamics of the hydrological processes in this interactive system are only poorly understood (e.g. IGBP, 1991). Even at the hillslope scale there still exist many problems associated with the use of spatially-distributed models for predicting runoff and évapotranspiration losses from precipitation. These include the estimation and uncertainty problems which accompany

38 A. J. Jakeman et al.

over-parameterization, resulting largely from lack of observations of internal mass fluxes, as well as heavy computational demands.

It is important to build up the level of physical detail in hydrological models slowly by including only relevant mechanisms identifiable from, or at least consistent with, observational data. Following initial work by Jakeman et al. (1990) on small catchments, Jakeman & Hornberger (1993) have illustrated that, after adjustment for antecedent conditions, only a quick and slow flow response can usually be identified from time series data of precipitation and runoff, and that each flow response can be characterised by just two parameters. Their findings are consistent with the work of Loague & Freeze (1985), Hornberger et al. (1985), Beven (1989) and Jakeman et al. (1990) who note or show the ability of small parameterizations in rainfall-runoff models to reproduce most of the information in a hydrological record. The study of Jakeman & Hornberger (1993) covers catchments from 490 m2 to 90 km2 in a wide range of humid hydroclimatological regimes and with very different physical characteristics. In most cases, daily data on rainfall, air temperature and streamflow were used to extract the streamflow component responses. The quick and slow flow responses identified, and consequently the associated évapotranspiration losses, represent spatially integrated quantities over the catchment to the stream discharge measurement location.

Inclusion in distributed models of information on physical catchment descriptors (PCDs), in addition to climatic and runoff data, does not improve the identifiability problem (Jakeman & Hornberger, 1993) because PCDs are used mainly to define the spatial distribution of hydrological units within the catchment; each unit is selected to have generally similar hydrologie behaviour on the basis of its vegetation, soils and terrain. Parameters related to travel times and volumes of throughput still need to be estimated for each hydrological unit from observed rainfall and runoff data. Therefore, where more than a few hydrological units are required for each set of monitored rainfall-runoff data, distributed parameter rainfall-runoff models are not presently capable of being used with good predictive accuracy to determine the dynamic water balance in unmonitored subareas of the catchment.

To be useful for climate modelling and climate change impact studies, a hydrological model must be capable of predicting the evaporative losses and runoff components over large basins when they are subjected to climatic inputs at least as variable as that in available hydrological and meteorological records. Hence the parameters of the model must be capable of low variance estimation from a reasonable length of time series data. Also, the parameters should not be dependent on the climate sequence in the estimation record, as was found by Gan & Burges (1990) for estimating parameters of the Sacramento model from simulated data.

The issues of model predictiveness and parameter estimation for variable climatic conditions are investigated in this paper by examining rainfall, air temperature and runoff data from two large basins, one in the UK and the other in the US, using the lumped watershed model described in the next section. With the time series available in each case spanning several decades, it is possible to investigate both model estimation performance on numerous subperiods of the series, and simulation performance of the variously estimated models on independent periods. Uncertainties in any such modelling application will depend upon several factors: coverage of contributing areas within the basin by the raingauge network; temporal nature of the

Assessing uncertainties in hydrological response to climate 39

incident rainfall; properties of stream gauge errors; length of data available for model construction; underlying nature of the catchment response characteristics; effect in the records analysed of changes in the basin resulting from vegetation and land use changes, abstractions and regulation. The two basins analysed here give an indication of the level of performance that can be expected from hydrological models in the presence of such uncertainties. The variability of the model parameters estimated are also a useful guide to the lower bounds that can be expected with the use of more highly parameterized models.

UNIT HYDROGRAPH MODEL

The unit hydrograph model similar to the one described in Jakeman & Homberger (1993) i s used here to examine the basic issues outlined in the introduction. This model is adopted because of its ability to represent the basic information in climatic (rainfall, temperature) and streamflow data with a minimal number of parameters. It has seven parameters, four in a non-linear module to model rainfall to rainfall excess and three in a linear module of rainfall excess to streamflow. In the non-linear module, a catchment wetness index, sk, is calculated at each time step k by a weighting of the rainfall time series rk, the weights decaying exponentially backwards in time from step k, viz.

h = ch + Q--T»)h-x ( 1 )

The parameter TW is the time constant, or inversely the rate, at which the catchment wetness declines in the absence of rainfall. Hence a large value of TW gives more weight to the effect of antecedent rainfall on catchment wetness than a smaller one. The rainfall excess or effective rainfall is computed using

uk = rksk (2)

The value of the parameter c in (1) is set such that the volume of rainfall excess is equal to the total streamflow volume over the estimation period. It is the increase in catchment wetness index per unit rainfall in the absence of any decrease due to évapotranspiration. To account for fluctuations in évapotranspiration, a function of temperature and catchment wetness index at the previous time step can be used to modulate the rate at which the catchment dries out. Then rw in (1) is replaced with the function

frwC*) = rwexp(20/-y) exp(-jp^_1) (3)

where tk is the temperature in °C at time step k. In this way, TW is inversely related to the rate at which catchment wetness declines at 20 °C as the catchment wetness index approaches zero. The parameter / i s a temperature modulation factor, which determines how rw(tk) changes with temperature for a constant sk. The value of p determines the nonlinear effect of catchment wetness on actual évapotranspiration. When p = 0 évapotranspiration can be considered to occur at a potential rate, and when p > 0 it occurs a t a reduced rate according to the value of sk.

The linear module of the model converts excess rainfall uk at time step k to streamflow qk. It has quick and slow flow components, where the quick and slow streamflow outputs are parameterized as

40 A. J. Jakeman et al.

4 = -aqXk-i+Pquk ( 4 )

4 = -aX-i + ̂ uk (5)

qk = xg+xt + £k (6)

The term %k represents the addition of all data and model errors. The parameter a (or as) describes the rate of decay, or equivalently the time

constant r (rs), of the quick (slow) flow hydrograph following a unit input of rainfall excess; T = -A/ln(-a?), TS = -A/ln(-as). The parameter Pq (0S) defines the peak of the quick (slow) component of the unit hydrograph. The volume vq (vj of water passing through the quick (slow) component is a function of aq and 0 (as and I3S); vq = /3q/(7 + aq), vs = (8/fl + a J where A is the time sampling interval. The quantities c, f, p, TW, rq TS, v vs are dynamic response characteristics (DRCs) of the catchment.

BASM STUDIES

The unit hydrograph model was applied to two basins. One is the French Broad River at Blantyre, a 767 km2 catchment in western North Carolina. Wood et al. (1992) applied a bucket model and a variable infiltration capacity (VIC) model to this basin. The other is the Teifi at Glan Teifi, an 894 km2 catchment in Wales. A preliminary analysis of this basin is undertaken by Littlewood & Jakeman (1993) who also demonstrated the applicability of a reduced version of the model described above to two other large basins in the UK, the Exe at Thorverton (601 km2) and the Thames at Kingston (9948 km2).

The Time Series Date

The periods of daily rainfall, streamflow and temperature records available for analysis were 1953-1988 for the French Broad River and 1961-1989 for the Teifi. Rainfall and temperature for the former are from a single meteorological station near Blantyre. Rainfall for the latter is an average for 8 to 22 raingauges in and around the basin, depending upon the period analysed. Mean monthly temperatures representative of a 40 km by 40 km area were used for the Teifi basin. For any month, daily temperatures were estimated by setting all daily values to the mean monthly value.

Periods of Analysis and Their Hydroclimatology

Periods for model parameter estimation were selected to begin and end on low flows of similar magnitude just before a rainfall event. In this way, initial and final catchment storages are relatively small and change in storage over the period can be neglected (an assumption implicit in the calculation of c in (1) as described above). The length of periods analysed was two to three years to balance problems of variance and bias. Periods of one year generally yield model parameters with too high variance while

Assessing uncertainties in hydrological response to climate 41

periods greater than a few years are more likely to encompass changes in the system, such as vegetation cover, or changes in measurement variability, such as drift or shift in the stage-discharge rating curve.

For the French Broad River, models were estimated initially for overlapping periods of three water years (low flows on which to commence the analysis could generally be found close to the beginning of a water year), the starting point for each period advancing a year at a time. Periods with poorer model fits (coefficient of determination D, which is fraction of variance of streamflow explained, less than 63 per cent) were deleted from further consideration, leaving 20 out of 31 original three year periods for analysis. Periods with low D values were considered to be more frequently associated with events where the single raingauge was unrepresentative of basin precipitation. A subset of 10 periods was formed comprising those which gave estimated models with the highest D values. Percentage runoff over these periods varied from 70 to 95 (Table 1). For the Teifi, some short periods of apparently unrepresentative data were identified by running a model estimated from a period of good quality records in simulation mode on the whole data set. Periods where the simulated flow differed substantially from the recorded flow were omitted from further analysis. Nine non-overlapping periods, each of approximately two years duration (ranging from 706-751 days), were selected for closer study. These cover the period

Table 1 Summary of French Broad River and Teifi hydroclimatology in data periods analysed.

French Broad River at Blantyre

Period

Start Da te

1

Aug 1954

2

Oct 1956

3

Nov 1961

4

Sept 1963

5

Oct 1965

6

Oct 1970

7

Oct 1971

8

Sep 1972

9

Sep 1980

10

Sep 1981

Rain (mm/day) 3.28 3.85 3.84 4.06 3.47 4.37 4.54 4.43 3.39 4.02

Flow (mm/day) 2.31 3.17 3.13 3.48 3.10 3.81 4.11 4.21 2.81 3.53

Temp1 C°C) 13.2 12.9 12.7 12.3 12.1 12.9 13.5 13.6 13.7 13.8

% Runoff 70.4 82.4 81.5 85.8 89.3 87.3 90.5 94.9 82.9 87.8

Teifi at Glan Teifi

Period

Start D a t e

1

Aug 1961

2

Sep 1964

3

Apr 1967

4

Sep 1971

5

Apr 1974

6

Oct 1977

7

Feb 1981

8

Sep 1984

9

Oct 1986

Rain (mm/day) 3.54 3.78 3.99 3.39 3.37 3.79 4.27 4.01 4.33

Flow (mm/day) 2.56 3.01 2.98 2.42 2.37 2.81 3.31 2.79 3.27

Temp2(°C) 7.0 8.3 8.2 8.5 8.5 8.0 8.5 7.7 8.0

% Runoff 72.4 79.7 74.7 71.3 70.4 74.1 75.7 69.4 75.4

'Average daily 2 Average monthly

42 A. J. Jakeman et al.

from 1961-1989 reasonably uniformly. Percentage runoff for these periods range from 69 to 80 (Table 1).

Table 1 also shows other indicators of the climate in the 9 different periods of analysis for Teifi and the subset of 10 for French Broad River. The essential differences between the basins are higher temperatures and larger variations in rainfall, flow and percentage runoff for French Broad River.

RESULTS

Estimation and Simulation Performance on Independent Data Sets

For the 10 French Broad River models with the highest coefficients of determination, and for the 9 Teifi models, estimation and simulation performance on independent periods was investigated. Along with 10(9) values of D from the estimated models, another 90(72) were obtained by simulating each model on the remaining 9(8) independent data sets. In general the D values on the estimation period were quite high, the mean of the D values over the 9(8) simulation periods were quite close to that for the estimation period, and the standard deviations were small (Table 2). Visually, the fits of the models to the measured data reflected the high D value, simulated flows matching observed quite well (e.g. Figs 1 and 2). Mismatches occur principally due to gross error in the representativeness of the rainfall recorded (January 1964 in Figure 1) or when snow is treated as rainfall (February 1982 in Fig. 2).

Only model 5 for the French Broad River and model 1 for the Teifi appear substantially inferior to other models in explaining the variance of observed streamflow over other periods (Table 2). The first period for French Broad River indicates the verification (simulation) period used by Wood et al. (1992) whose application of the VIC model in the French Broad River basin yielded a significant improvement in performance over a bucket model. They noted that the standard deviation of simulated streamflow from the bucket model was 3.86 times the standard deviation of the observed streamflow. For a VIC model with just four parameters this factor reduced to 0.82. The runoff simulated using the VIC model also compared better qualitatively

Table 2 Coefficient of determination value D for estimation periods and mean and variance of D over simulation periods.

French Broad River Model

D mean D st. dev. D

Teifi Model

1

.78

.70

.05

1

2

.77

.70

.04

2

3

.73

.70

.04

3

4

.71

.70

.04

4

5

.73

.66

.06

5

6

.78

.70

.05

6

7

.74

.69

.05

7

8

.72

.71

.04

8

9

.73

.70

.04

9

10

.72

.70

.04

D .84 .86 .84 .85 .85 .88 .86 .83 .90 mean£> .75 .80 .82 .82 .81 .80 .83 .82 .80 st. dev. D .05 .03 .02 .02 .03 .02 .03 .03 .05

Assessing uncertainties in hydrological response to climate

30 i

43

| 20

o H— E m 2 10 (0

Observed Modelled

' -USJUJ i i i i i i > i ) i i i i i i i i i ) i i

Nov March Jul Nov March Jul i i i J i t i i i i i i i

Nov March Jul

1962 1963 1964

Fig. 1 Observed and modelled daily streamflow (mm/day) over the three year period from November 1961 for the French Broad River at Blantyre, North Carolina.

30-1

E 20-

o 4— E ta £ 10H CO

Pe ïibruary May Aug Nov February May Aug Nov February

1981 1982 1983

Fig. 2 Observed and modelled daily streamflow (mm/day) over the two year period from February 1981 for the Teifi at Glan Teifi, Wales.

with the observed than did the bucket model. The latter had characteristics more similar to daily precipitation than runoff, with the simulation of flow between rain events generally zero. The fraction of the variance of streamflow, D, explained by the VIC model on their verification period was 0.54, whereas for the bucket model the variance of model residuals is larger than that of the observed streamflow. The average D value for the nine models (estimated from each of the remaining periods) on the corresponding period is 0.74 with a standard deviation of 0.03.

A measure of model bias over different periods can assist with the detection of obvious changes in data measurement and human or vegetation influences in a catchment. It must also be tolerably low enough in a model for it to be used

44 A. J. Jakeman et al.

successfully to predict any absolute changes in streamflow for any climatic forcing. The average percentage error in modelled streamflow (compared to observed) for each model on each of the 10(9) periods can be used to assess bias (Figs 3(a) and (b)).

For the French Broad River, model 5 can be seen to perform poorly over other periods; in addition all other models show a large bias in fitting period 5. This is the case to a lesser extent with model 8. On this basis, it was judged that periods 5 and 8 should not be used in assessing model performance; only estimated models 5 and 8 perform well in those periods. For the Teifi there is an overall, though not monotonie, change in bias of all models between periods 1-5 and 6-9.

20

H—H Model 1 M Model 2 •»-"• Model 3 # - * Model 4

" Model 5

(a)

*-•*. Model 6 "Model 7

-«H« Model 8 >• Model 9

AS- Model 10

4 5 6 7 Period Number

4 5 6 7 Period Number

Fig. 3 Results of simulation of estimated models over different periods: (a) average difference between daily simulated and observed streamflow expressed as

a percentage of observed streamflow — French Broad River; (b) average difference between daily simulated and observed streamflow expressed as

a percentage of observed streamflow — Teifi; (c) reduction in simulated streamflow for a 1°C rise in all mean rise in all mean

temperatures — Teifi; (d) reduction in simulated streamflow for a 1°C rise in all mean temperatures —

French Broad River.

Dependence of Model Parameters on Estimation Period

There were 20 three year periods for the French Broad River which yielded adequate performance as previously specified. The dependency of the dynamic response

Assessing uncertainties in hydrological response to climate 45

characteristics of the 20 estimated models on climate sequence in the relevant estimation record was investigated by plotting DRCs estimated in each period against corresponding percentage runoff (Fig. 4). Standard regression tests for any systematic variation were not applied as formal analysis requires robust techniques which can treat outliers. However, the scatter plots are sufficiently encouraging to indicate there is no general trend. Fig. 4 also indicates the variability in parameter estimates over different estimation periods. Because of an insufficient number of periods (9) available to investigate independency, similar plots are not given here for the Teifi.

T (days)

30

2 0 -

10-

60 ' 70 90 Too

0.14

0.12-

0.10

0.08

0.06

0.04-

0.02-

0.00 60 7*0 ' 8b

0.0020-

0.0015'

0.0010-

0.0005-

90 0.0000-

60 W 100

x q (days)

6 0 7 0 90 % Runoff

140-

120-

100-

8 0 -

6 0 -

4 0 -

2 0 -

0 60

T , (days)

70 ' 8b ' 9b % Runoff

0.60-

0.50-

0.40-

0.30-

0.20-60 70 ' 8b ' 9'0 ' 10

% Runoff

Fig. 4 Estimated dynamic response characteristics against percentage runoff for each of the 20 French Broad River periods.

Sensitivity to Temperature Rise

The effects of rises in temperature on discharge were investigated for both basins. The temperatures tk at each time step were all increased firstly by 1°C. All 10(9) estimated models for the French Broad River (Teifi) historical periods were used to simulate streamflow using the 10(9) historical rainfall series and the (increased) temperature series as revised inputs. This yielded 10(9) streamflows series for each revised set of 10(9) historical inputs. This exercise was repeated for increases of 2, 3 and 4°C. For each basin, the simulated decrease in average streamflow over any period was almost linear per unit temperature rise, there being a slightly smaller decrease the higher the temperature.

46 A. J. Jakeman et al.

The reduction in average streamflow for a 1°C rise in temperature is around 2 to 6% for the Teifi and around 4 to 7% for the French Broad River (Figs 3(c) and (d)). If there has been no substantial changes in hydrological response or its measurement in the Teifi basin over the period of record, then each 1°C rise in temperature can be expected to reduce average streamflows by between 2 and 6%. The grouping in Fig. 3(c) indicates the possibility that sensitivity of Teifi streamflow to changes in mean temperature is dependent on non-climatic factors which have not been considered here. For example, preliminary analysis not reported here indicates a relationship between the number of raingauges used to calculate average areal rainfall in different periods for the Teifi and the four model groupings in Fig. 3(c). Any analysis which could tie the biases of the different estimated models to specific factors would allow narrowing of the range of uncertainties.

CONCLUSIONS

Two basins were selected to perform extensive model parameter estimation and simulation exercises using a conceptually simple, lumped, unit hydrograph model. The results show that it is possible to construct a hydrological model to predict daily runoff, and hence evaporative losses, in a large catchment from a few years of precipitation, temperature and discharge time series. In general, the estimated models derived here have good properties in terms of predictiveness on independent, climatically different periods and reasonable consistency of sensitivity to changes in temperature and precipitation across different estimation periods. Because such a model can be estimated on two to three years of observations, the methodology has the potential to detect systematic changes in the nature of a catchment's response or in measurement of observational time series data spanning successive model estimation periods.

When these types of systematic changes are not present in rainfall and streamflow data it is possible, in the manner described, to assess quite accurately the sensitivity of streamflow to changes in air temperature, e.g. as prescribed in climate change scenarios. For the two catchments investigated a 1°C rise in temperature resulted in a decrease in average streamflow of 4 to 7% for the French Broad River and 2 to 6% for the Teifi basin. In a similar way the sensitivity of streamflow to prescribed changes in rainfall, or both rainfall and temperature together, could be assessed.

Key hydrological features of the model are: prediction of both quickflow and slowflow response, the latter being more important the more substantial the component of slow subsurface flow in a catchment; description of the state of catchment wetness at a given time step in terms of its value at the previous time step and incoming rainfall; rate of catchment drying in the absence of rainfall is related to a potential rate of évapotranspiration by a temperature function; potential évapotranspiration is reduced to actual évapotranspiration through dependency on the wetness state of the catchment. Furthermore, once a model is estimated for a basin the computation of runoff and évapotranspiration losses is simple and straightforward, of the same order as required for a bucket model.

Previous work by Jakeman et al. (1990) and Jakeman & Hornberger (1993) indicate strongly that, wherever reasonable quality rainfall, streamflow and temperature data are available, the model is transferable across a wide range of hydroclimatological regimes and catchment sizes. Work by Littlewood & Jakeman (1993) and the study

Assessing uncertainties in hydrological response to climate 47

reported here provide encouragement for the model's applicability to large basins. In principle, the model could be applied at continental scale. It would have a coarse horizontal resolution consisting of a continent's major basins, in each of which daily runoff, and hence losses, could be calculated.

The present limitations are data rather than knowledge-based. Data required are basin precipitation, temperature and discharge. In basins subject to substantial river abstraction and regulation, a time series of naturalised flow at the basin outlet must be constructed. In basins subject to substantial snowfall, a submodule to convert precipitation to rainfall excess must be developed. Such a device must be temperature sensitive and of compatible conceptual simplicity to the existing non-linear module in the model.

Acknowledgement The authors are grateful to Sue Kelo for processing the manuscript.

REFERENCES

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dynamics of small hypothetical catchments. 2. Hydrologie responses for normal and extreme rainfall. Water Resour. Res. 26(7), 1605-1619.

Hornberger, G. M., Beven, K. J., Cosby, B. J. &Sappington,D. E. (1985) Shenandoah watershed study: calibration of a topography-based, variable contributing area hydrological model to a small forested catchment. Water Retour. Res. 25, 2177-2185.

IGBP (1991) Plant-water interactions in large-scale hydrological modelling. Global Change Report No. 17, Report of a workshop in Vadstena, Sweden, 5-8 June 1990. International Geosphere-Biosphere Programme, 44 pp.

Jakeman , A. J. & Hornberger, G. M. (1993) How much complexity is warranted in a rainfall-runoff model? Water Retour. Res. In press.

Jakeman, A. J., Littlewood, I. G. & Whitehead, P. G. (1990) Computation of the instantaneous unit hydrograph and identifiable component flows with application to two small upland catchments. J. Hydrol. 117, 275-300.

Littlewood, I. G. & Jakeman, A. J. (1993) Characterisation of quick and slow streamflow components by unit hydrographs for single- and multi-basin studies. In: M. Robinson (ed.) Methods of Hydrologie Basin Comparison, Institute of Hydrology Report No. 120, 94-105.

Loague, K. M. & Freeze, R. A. (1985) A comparison of rainfall-runoff modelling techniques on small upland catchments. Water Resour. Res. 21, 229-248.

Rind, D . , Rosenzweig, C. & Goldberg, R. (1992) Modelling the hydrological cycle in assessments of climate change. Nature 358, 119-122.

Wood, E , F., Lettenmaier, D. P. & Zartarian, V. G. (1992) A land-surface hydrology parameterisation with subgrid variability for general circulation models. / . Geophys. Res. 97(D3), 2717-2728.