Hydrological forecasting - Prévision hydrologiques (Proceedings of the Oxford Symposium, April 1980; Actes du Colloque d'Oxford, avril 1980): IAHS-AISH Publ. no. 129.

Simple self-correcting models for forecasting flows on small basins in real time

R. J. SIMPSON University of Birmingham, UK

T. R. WOOD Severn-Trent Water Authority, Birmingham, UK

M. J. HAMLIN University of Birmingham, UK

Abstract. The authors discuss the advantages of employing simple and self-correcting models t o forecast flows in real time using rainfall data. Two such models which have been designed to tolerate inaccurate data typical of a real time situation, are presented. The first model is based o n regression and the second on unit hydrograph theory. The models have been developed and tes ted on small basins in the Severn-Trent Water Authority area. Results of tests on independent events above the range of model calibration are presented. These are compared with forecasts from a complex model. It is shown that the simple models perform well and that no greater accuracy is achieved by using the complex model.

Modèles simples autocorrigés pour les prévisions de débit en temps réel dans les petits bassins

Résumé. Les auteurs discutent les avantages de l'emploi de modèles simples autocorrigés pour la prévision des écoulements en temps réel, modèles utilisant les données pluviométriques. Deux modèles de ce type, qui ont été conçus pour accepter des données insuffisantes, ce qui correspond bien à la situation réelle, sont présentés. Le premier modèle est basé sur des régressions et le deuxième sur la théorie de l'unité graphique. Les modèles ont été mis au point et essayés pour de petits bassins de la région de la Severn-Trent Water Authority. Des résultats d'essais faits avec des événements indépendants sortant du champ d'étalonnage des modèles sont présentés. Ceux-ci sont comparés à des prévisions obtenues par un modèle complexe. 11 est montré que les modèles simples fonctionnent bien et qu'on n'obtient pas une plus grande précision en utilisant le modèle complexe.

INTRODUCTION

In considering the development of systems for real time forecasting of high flows it may be important to use a model which is not so complex as to exclude the forecaster from any interactive role.

This paper describes two simple models for flood forecasting, one based on regression and the second on the well established principle of the unit hydrograph. Both models are simple in concept and permit an operator to use an element of judgement in considering what action should be taken based on the model output.

The forecasts from these simple models are compared with the output from the more complex basin model HYSIM (Manley, 1974). This comparison, based on the high flow situation, does not include the advantages which may accrue to a Water Authority from the use of a continuously updated model monitoring the whole range of basin outflows.

In the development of the models, split sample testing was used so that output was not tested against events used in the calibration. The test set of data always consisted of events greater in magnitude than the calibration set. Finally, in the forecasting mode the forecast was based only on information known up to the time at which the forecast was made.

THE CASE FOR SIMPLE MODELS

There are few good quality gauging stations where high flows are adequately measured. In addition real time rainfall data are sampled at a limited number of points. Together

433

434 R. J. Simpson et al.

this implies that there are few basins where models can be precisely calibrated or operated. A considerable proportion of the error involved in forecasts might therefore be attributed to poor data rather than to the limitations of a model. As a consequence, it is desirable that a forecasting model should make use of real time flow data and the experience of the forecaster to adjust either the inputs to the model or suitable parameter values. It is therefore a considerable advantage to be able to understand easily the effects of such adjustments on the model output. This is more readily achieved with simpler models and is especially true where models are operated by people who did not construct them.

It is most likely that the greatest benefits of flood forecasting will be realized in extreme floods outside the range of events experienced in a typical record length of 10 or 20 years, provided that the reliability of forecasts is sufficient to ensure public response. A chief requirement of a forecasting model would therefore be that it is robust enough to perform well both within and above its range of calibration. It is possible that a complex model may achieve a good fit in calibration owing to its flexibility rather than its realism and might not be applied confidently outside its range of calibration. Even if this is not true, the use of such a model in extreme events would most likely require an interactive role on the part of the operator together with the model's own self-correction.

Finally, simple models would be cheaper to calibrate and run and may save both run time and core storage costs. The savings could be significant where the models are used frequently.

The arguments in favour of simple models could best be justified if it could be shown that they can achieve a forecasting performance not inferior to a complex model in events above the range of calibration. This is the intention of the following sections of this paper.

SIMPLE SELF-CORRECTING MODELS

The regression model This takes the form of a lumped black box representation of a basin. At each hour (or other short time interval) of an event in real time a forecast of the change in flow from its present monitored value to its value a fixed lead time ahead is determined by a five variable equation. The variables comprise the observed change in flow over a short period immediately prior to the forecast, cumulative rainfall, a rainfall timing index, the current moisture state of the basin and the initial flow. The equation is statistical and the values of exponents of the variables are derived by multiple regression analysis.

The model is not a rainfall—runoff model in the traditional sense since it uses the observed change in flow to incorporate the inertia of the hydrograph. The structure of the model is therefore self-correcting and is not entirely dependent on rainfall data.

The unit hydrograph model The unit hydrograph principle, originally devised by Sherman (1932) is now well established in hydrology. The unit hydrograph can be regarded as a lumped, linear time invariant model of a drainage basin system.

The derivation of the unit hydrograph from recorded events requires effective rainfall and direct runoff to be identified. Effective rainfall was calculated using a technique recommended in the UK Flood Studies Report (Natural Environment Research Council 1975). A constant runoff parameter defines the effective rainfall as a proportion of gross rainfall, adjusted by an index of current basin moisture conditions. Direct runoff was separated from baseflow by a method suggested by Linsleyefa/. (1958).

Simple self-correcting models 435 The 1 mm 1 h unit hydrographs were derived from between three and five events

on each basin by the matrix inversion procedure due to Snyder (1955) and recom­mended by the Natural Environment Research Council (1975). Events with uniform areal rainfall and well-defined beginnings were selected to obtain consistent sets of unit hydrographs. A mean unit hydrograph was obtained by drawing a representative curve, passing through the average peak ordinates and times to peak.

In real time operation a forecast hydrograph is built up by reconvoluting successive effective rainfall inputs. The runoff parameter is estimated from initial moisture conditions using an approximate relationship derived from past events. The runoff parameter is adjusted automatically as telemetered flow data become available. The basis for this self-correction is the matching of observed and modelled volumes of direct runoff during a short period prior to each forecast. This method is intended to achieve a compromise between the stability of using all the observed flows and the flexibility of using only the present flow. The residual error in the modelled flow at the time of forecast can be taken into account over short lead times either by automatic addition or interactively.

PERFORMANCE TESTS OF THE MODELS

The two simple models were calibrated on small basins in the English Midlands. Four basins (the rivers Worfe, Roden, Dove and Avon) were selected to test the unit hydrograph model and two basins (the rivers Worfe and Roden) to test the regression model. The areas of the basins range from 83 to 347 km2 and comprise a variety of geological and topographical characteristics. The basins are mostly unaffected by man-made influences at high flows but one sixth of the Avon basin drains to a reservoir and there is some urban contribution to storm runoff further downstream. Each basin is characterized by very variable storm percentage runoff, ranging from 1

TABLE 1. Performance of the unit hydrograph model

Basin A, T*

Worfe A = 258 T=2S

Roden A =259 r = 2 8

Dove A = 83 T = 8

Avon A = 347 r = 3 3

Max. peak calibration [m'/s]

6.0

10.7

10.2

38.5

Max. rain calibration [mm]

22

31

43

28

Test events

Date

24.9.76 28.9.76 27.1.77 9.2.77

20.12.65 17.10.67 2.7.68 5.8.73 30.12.76

5.8.73 1.1.76 3.1.76 11.2.77

14.5.67 10.7.68 23.2.77

Peak [m3/s]

4.0 5.1 7.3 6.4

18.4 10.2 31.2 19.0 11.9

9.4 12.2 10.2 14.0

39.3 99.2 48.2

Rain [mm]

73 30 19 27

33 41 64 61 23

59 36 35 42

39 82 22

Peak forecast % errorf for lead times of: 1 2 h

+ 13 - 3 - 5 - 1

+ 23 + 13

0 - 2

0

— — — — - 3 8 - 4 7

- 2

4 h

0 - 3 - 3 _ 1

+ 1 - 1 - 2 - 9 - 1

+ 4 + 2

+ 12 - 1 4

- 8 - 3 1

- 8

*A is basin area [km2 ] ; T is unit hydrograph time to peak [h]. fPer cent error calculated as [(observed peak - forecast peak)/observed peak] X 100.

436 R. J. Simpson et at. TABLE 2. Performance of the regression model

Basin

Worfe

Roden

Date

24.9.76 28.9.76 27.1.77 9.2.77

19.12.65 17.10.67 2.7.68

Peak forecast % error for lead time of 6 h

- 5 ^ 8 ~ 3 - 2

+ 1 - 3 - 4

Note. Calibration maxima and other information are as given in Table 1.

to 60 per cent in the case of the Avon. None of the basins is experimental and each contains only one continuously recording raingauge.

The models were tested only on independent events above their range of calibration in terms of rainfall depth, peak flow or both. Real time operating conditions were simulated by using uncorrected rainfall input from only one recording gauge situated within or near the basin and assuming no knowledge of future rainfall.

The results of the tests were generally good, as can be judged from the summary of peak flow forecast accuracies given in Tables 1 and 2. Only the results from the Avon basin give cause for concern, and it may be that this basin is too large (347 km2) for the assumption of uniform rainfall to be always reasonable. The poor forecasts are both from summer storms whilst a good forecast was achieved for the winter event, as shown in Table 1.

Typical long lead time unit hydrograph forecasts from the three smaller basins are shown in Figs 1—3. It can be clearly seen that the quality of successive forecasts of the peak improves on the rising limb, especially in Fig. 2. In this case, the single raingauge

RAIN

MM

FLOW

M VSEC 3

,W

IC0SF0R0I

J1!

OBSERVED

, - ' FORECAST

3 8 T I M OF FORECAST

u 10 20 30 40 50 60

TIME IH0URSI

FIGURE 1. Unit hydrograph model forecasts, River Worfe, 24 September 1976.

Simple self-correcting models 4 3 7

t OBSERVED

' ' FORECAST

4 TIME OF FORECAST

10 20 30 40 50 60 70

TIME (H0URSI

FIGURE 2. Unit hydrogiaph model forecasts, Rivet Roden, 2 July 1968.

RAIN " (MM) 4

2F

or-

1M0NYASH)

12

10

now 8

V13/SEC

6

4

2

,r-1 Ï

;t /

'' / " /

/''6

k' II

'——-A

\ \ \ \ \

8 \ \

^•v

29 ;

\f / '2?

/

6

C \

\ 32 \

OBSERVED

FORECAST

TIME OF FORECAST

15 20 25 30 35 40

TIME HOURS

FIGURE 3. Unit hydrograph model forecasts, River Dove, 1 January 1976.

438 R. J. Simpson et al. 41 fl (COSFORD)

RAW , (MMI 2-

0

FLOW

M3 /SEC

/ OBSERVED

* PREDICTED

10 20 30 40 50 60 70"

TIME IHÛURSI

FIGURE 4. Regression model forecasts, River Worfe, 27 January 1977.

RAIN (MMI 2-i

FLOW

M3/SEC

10-

(SHAWBURYI

J ^ ,

/ OBSERVED

,* PREDICTED

10 20 30 40 50 60 70

TIME (HOURSI

FIGURE 5. Regression model forecasts, River Roden, 17 October 1967.

Simple self-correcting models 439

RAIN IMMI (SHAWBURYI

0 20 3D 40 50 60 70

TIME I HOURS I

FIGURE 6. Regression model forecasts, River Roden, 2 July 1968.

underestimated basin areal rainfall by about 20 per cent, this error being taken up by the self-correction. There is usually a discrepancy between the forecast and observed flows at the time of forecast, which is apparent in all three examples. This residual error is taken into account in issuing short lead time forecasts. Some of the errors apparent in Fig. 3 may be due to the recording raingauge being well outside the basin making real time operation difficult, especially since the basin response is rapid compared with the lead times required.

Typical results of the regression model are shown in Figs 4—6. They are all point forecasts of the flow 6 h ahead, the forecasts being made every 3 h. The results are good although Fig. 4 shows the possible effects of noisy flow data on the forecasts, which could be reduced by smoothing the data in real time.

COMPARISON WITH A COMPLEX MODEL

The unit hydrograph model was compared in terms of performance with a physically-based, distributive, nonlinear conceptual drainage basin model, HYSIM, developed by Manley (1974). HYSIM is an established model which is used for water resource planning by the Severn—Trent Water Authority. The model is distributive since the basin is divided into a number of sub-basins for rainfall—runoff and a number of channel segments for routing by a kinematic method. A detailed description of HYSIM is given by Manley (1974).

440 R. J. Simpson et al. n

ISHAWBURYt

MJ/SEC

\ V / / OBSERVED V - - ^

HYS1M SIMULATION

/ UNIT HYDROGRAPH SIMULATION

10 20 30 40 É 60 70 SO-

TIME (HOURS)

FIGURE 7. Comparison of unit hydrograph and HYSIM simulations, River Roden, 2 July 1968.

151 FLOW

m3/s / OBSERVED / UNIT HYDROGRAPH

HYSIM

1 2 3 4 T I M E (days)

FIGURE 8. Comparison of unit hydrograph and HYSIM simulations, River Roden, 30 December 1976.

The model comparisons were made in two stages: simulating and forecasting with self-correction. HYSIM was calibrated from field measurements and from daily data with parameters adjusted on the basis of the isolated events used to derive the unit hydrograph. Independent test events above the range of calibration were used for the comparisons, which were made on the Roden, Dove and Avon basins. A number of criteria were adopted in the assessment of the test results: accuracy, timing and duration of the peak flow, runoff volume and hydrograph shape. The performance with respect to initial moisture conditions, out of bank events and individual basins was also considered. The results from both stages of testing indicated that the performance of the two models could not be discriminated with respect to any of the

Simple self-correcting models 441 8

M M 6

(MM 4

2

ROW ° M3 /SK

12

10

8

8

4

2

J U_^ iMONYASM

Il f

'<< /

Y *

P

\ \ \ \

^ J1 H r.

,*\\ //\\

=7 / / OBSERVED ^ ^

// HYSIM SIMU.ATI0N

/ UNIT HYDROGRAPH SIMULATION

10 15 20 35 40 45 25 30 TIMEIHOUMI

FIGURE 9. Comparison of unit hydrograph and HYSIM simulations, River Dove, 1 January 1976.

40 FLOW

m3/s

30

20

/ OBSERVED

/ UNIT HYDROGRAPH

/ HYSIM

1 2 3 4 TIME (days)

FIGURE 10. Comparison of unit hydrograph and HYSIM simulations, River Avon, 23 February 1977.

criteria, taking the events as a whole. Examples of the results, shown in Figs 7—12 illustrate this conclusion. Figures 7-10 show that both models give similar results in the simulation mode. The simulations are sometimes very inaccurate (Fig. 10), indicating that self-correction is necessary for real time forecasting. Examples of the performance of HYSIM with self-correction using the flow at the time of forecast are given in Figs 11 and 12. Some more limited comparisons have been made in the forecasting rhode where both the unit hydrograph and HYSIM employed self-correcting techniques. The corresponding unit hydrograph forecasts seem to be somewhat better (Figs 2 and 3).

442 R.J. Simpson et al.

/ OBSERVED

/ FORECAST

7 TIME Of FORECAST

10 20 30 40 50 60 70

TIME (HOURS)

FIGURE 11. HYSIM model forecasts, River Roden, 2 July 1968.

Flow MJ/SEC

5 10 15 20 25 30 35 40

FIGURE 12. HYSIM model forecasts, River Dove, 1 January 1976.

Simple self-correcting models 443 DISCUSSION AND CONCLUSIONS

In judging the accuracy of the results presented it must be remembered that the model calibration suffers from all the inherent errors of hydrological models.

(1) There are errors in the models themselves. These are well stated by Shaw and Austin (1931). 'Every theory of the course of events in nature is necessarily based on some process of simplification of the phenomena and is, to some extent therefore, a fairy tale'.

(2) The errors in assuming that the basin rainfall input is correctly determined from raingauge information and that the river flow is correctly measured by the river stage recorder can never be adequately overcome. These are, at least partially, accounted for by the self-correcting facility of the models.

(3) Timing errors, in both the calibration and test data, cannot be satisfactorily dealt with by the models and will always tend to corrupt the forecasts. Fortunately, in the real time situation, whilst input and output errors may be significant, timing errors will not be. The input information will give, in the correct time sequence, the measured rainfall over the previous time interval and the current value of river stage.

The results presented in this paper show that simple models can be effective in providing timely and accurate forecasts of high flows in real time conditions. Tables 1 and 2 and Figs 1 —6 indicate that a high degree of forecasting accuracy and reliability can be achieved, operating the models above their range of calibration and using limited and error prone data. These results are achievable despite the accepted theoretical criticisms and limitations of such models, which demonstrates that the models are robust and respond well to simple self-correction techniques. The accuracy of the simple models and the evidence, presented in Figs 6-12, that a complex model does not give better results under typical operating conditions, strongly reinforces the case for simple models in real time forecasting.

Acknowledgements. The authors wish to acknowledge the Severn-Trent Water Authority for the financial support of the study and permission to publish this paper; also R. Manley for kindly providing the results from HYSIM.

REFERENCES

Linsley, R. K., Kohler, M. A. and Paulhus, J. L. H. (1958) Hydrology for Engineers: McGraw-Hill, New York, USA.

Manley, R. E. (1974) Catchment models for river management. MSc Thesis, Birmingham University, Birmingham, UK.

Natural Environment Research Council (1975) Flood Studies Report, vol. I, Hydrological Studies, chapter 6: NERC, London, UK.

Shaw, Sir N. and Austin, E. (1931) Manual of Meteorology, vol. I, Meteorology in History. Sherman, L. K. (1932) Streamflow from rainfall by unitgraph method. EngngNews Rec. 108,

501-505. Snyder, W. M. (1955) Hydrograph analysis by the method of least squares. Proc. A mer. Soc. Civ.

Engrs 81, separate no. 793.