OPERATIONAL RIVER-FLOOD FORECASTING BY WIENER AND KALMAN FILTERING

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1 Hydrology Jbr the Water Management of Large Siver Basais (Proceedings of the Vienna Symposium, August 1991). IAHS Publ. no. 201,1991. OPERATIONAL RIVER-FLOOD FORECASTING BY WIENER AND KALMAN FILTERING K. WHJΠFederal Institute of Hydrology, Koblenz, F.R.Germany F. BARTH State Department of Environmental Protection, Karlsruhe, F.R.Germany ABSTRACT Short-term flood forecasts are playing an increasing role for the real-time operation of water resources systems as well as for flood warning systems in the international river Rhine basin. For the extreme flood event of February 1990 the result of the realtime forecasts at the upper Rhine by using Wiener filtering and Kalman filtering are presented. Some operational aspects will be discussed. INTRODUCTION In the Federal Republic of Germany the states are responsible for flood reports and flood forecasts. The states are supported by various federal authorities (e.g. Federal Waterways Administrations) in carrying out the flood reporting service. For the Rhine assistance is given by the Federal Waterways Administrations Southwest in Mainz and West in Munster (see Fig. 1) as well as by the Federal Institute of Hydrology in Koblenz by means of accurately worked out flood report plans. The Federal Waterways Administrations Southwest (from gauge Speyer to gauge Oberwinter) and West (from gauge Bonn up to gauge Emmerich, see Fig. 1) transmit three times a day actual flood forecasts at 16 gauges for the next 24 hours, in arrangement with state authorities to the broadcasting corporations, press and other interested authorities (police, traffic authorities, technical welfare organizations, fire-brigades and so on). These actual forecasts are computed in the Federal Institute of Hydrology using the multichannel Wiener filtering method, whereas the cited Federal Waterways Administrations apply simple empirical methods. It seems indispensable that the forecasts should be prepared with making use of the experience of the hydrologist who, on the basis of his knowledge concerning catchment area and forecasting method, is able to estimate the reliability of his forecast calculations, and who is able to execute corrections without difficulties. Attempts have been made to achieve an automatic adaptation of model parameters to instantaneous flood event processes, e.g. using the Kalman filtering method. This method has been applied in the Neckar catchment (see Figs. 1 and 5) and shall be used for flood forecasts at the Rhine gauges of Maxau, Speyer and Worms in the near future. The most recent extreme flood event in the Rhine basin was in February 391

2 K Wlke&F.Barth Actual forecasts using the Wiener filter are now compared with later on calculations of the Kalman filter for the gauging station of Worms in this paper. Fig. 1. SWITZERLAND Location map of gauging stations in the River Rhine basin (forecast gauges are underlined).

3 393 Flood forecasting by Wiener and Kalman filtering At present time, the results of the Kalman filter application must be considered as preliminary. A direct comparison of both models is not possible therefore. APPLICATION OF THE WIENER FILTER The Wiener filter can be used to analyze the response characteristics of a hydrological system (e.g. catchment area or river reach between two gauges) from the loading of the system (effective precipitation, discharge or stage at the upper gauge) and the result (e.g. direct discharge, measured discharge or stage at the lower gauge) by means of a response function. With a given response function the output time series result from linear convolution of the response function with the hydrograph of the discrete and equidistant loading as input time series. Analogous to the communication theory a filter is understood as a response system which transmits a given discrete input function into a desired but given discrete output function. The response system is completely described by its response function. A multichannel filter exists when more than one input is available. The mathematical method is bases on the theory of stationary continuous time series by Wiener (1949) which has been extended for multichannel discrete input and output time series by Wiggins & Robinson (1965). Early in 1980 the attempt was made to use the Wiener filter for calculation of stage forecasts at the gauge of Koblenz for 6, 12, 18 and 24 hours ahead only on the basis of 6-hourly stage changes of the Rhine, Lahn and Mosel. The flood of February 1980 was operationally forecasted for the first time. The scope of application of the Wiener filter was then extended until actual flood forecasts were calculated for all 16 gauges underlined in Fig. 1. The theory of the model is described by Wilke (1975) and Huthmann & Wilke (1982). Four multichannel filters must be available to compute forecasts at one Rhine gauge for the next 6, 12, 18 and 24 hours. The number of response functions of a multichannel filter corresponds to the number of input gauges. At the time, 64 multichannel filters with more than 500 response functions are used for the 16 mentioned forecasting gauges. The response functions were calculated on the basis of 6-hourly stage and/or discharge changes of historical floods. Each response function consists of 12 coefficients, i.e hourly stage or discharge changes must be on hand for convolution. The stage or discharge changes which have been forecasted by means of four multichannel Wiener filters for the next 6, 12, 18 and 24 hours, are added to the actual known stage or discharge in order to obtain the forecasted values. The forecasting time points are 05.00, 11.00, and hours. At the time, 36 gauges in the Rhine basin are taken into account (see Fig. 1). 32 gauges are equipped with an automatic stage data registration and transmitting system and 4 gauges with an automatic telephone responder. The flood forecasts computed in the Federal Institute of Hydrology are transmitted by telephone to

4 KWlke&F.Barth h a >h # i J»? i <o. ^ "Hi. > V \ " ^--a- H. s /, è S 1 I " «-- m ^.-Y*--*"- -s SP *' 4 ' r '" J ^ ^ 1 i 4 i 4 5, H "" Tf. 5 \ \ "V ' V " a* V K.. W. " w ""'--nr- #- " c -'t--». "*--. «T-*-*' s * -wo - = WORMS/RHEIN -MN - = MANNHE1M/RHEIN -SP - = SPEYER/RHE1N -MX- = MAXAU/RHEIN -RF- = RHEINFELDEN/RHEIN -RC- = ROCKENAU/NECKAR Fig Stage hydrographs at the Rhine gauges of Rheinfelden, Maxau, Speyer, Mannheim, Worms and at the Neckar gauge of Rockenau, February the Federal Waterways Administrations by about 06.00, and hours each day. ACTUAL FORECASTS OF THE FLOOD IN FEBRUARY 1990 AT THE GAUGE OF WORMS The flood in February 1990 was an extreme one in the Rhine basin. The peak

5 395 Flood forecasting by Wiener and Kalman filtering discharges in the upper river Neckar catchment had recurrence intervals of 100 years and more. One of the forecasting gauges is Worms. Using the Wiener filter, 5 input gauges are taken into account, the Rhine gauges Rheinfelden, Maxau, Speyer and Mannheim as well as the Neckar gauge Rockenau (see Fig. 1). Because a station rating curve at gauge Mannheim is lacking, only stages can be used for actual forecasts. The stage hydrographs are shown in Fig. 2. The flood event started at the end of a very low water period in the Rhine basin with stage increases of about 300 cm within 24 hours. At the Neckar gauge Rockenau there has been a maximum increase of 380 cm and a maximum decrease of 460 cm within 24 hours. In order to compare the actual stage forecasts at the gauge of Worms with the discharge forecasts using the Kalman filter, computed later on, the stages are converted to discharges. The operational Wiener filter forecasts are shown in Fig. 3 for the forecasting time intervals of 6, 12, 18 and 24 hours. The corresponding forecasting errors are depicted in Fig. 4. APPLICATION OF THE KALMAN FILTER The state government of Baden-Wurttemberg will establish a new River Flow Forecasting Centre at the State Department of Environmental Protection in Karlsruhe. In the first stage in 1990/91 the forecasting centre should develop the infrastructure and the mathematical models to compute and transmit forecasts for the gauges Maxau, Speyer, Worms at the Rhine and for the gauge Heidelberg at the Neckar. FORECASTS AT THE RIVER NECKAR Up to 1989 only some very rough empirical methods were available for the flood forecasting at the Neckar. Therefore it was necessary to develop a new mathematical forecasting-model. This development was mostly determined by the following three practical aspects: the availability of the model for operational river flow forecasting as soon as possible; the number of gauges which are equipped with an automatic stage data registration and transmitting system (see Fig. 5) the characteristics of the flood regime of the Neckar catchment. A qualitative analysis of the discharge hydrographs of 12 historical flood events showed the complexity of the flood regime in the Neckar catchment. Considering the Neckar flood regime the main problems for forecasting were that the processes of flood wave formation and propagation (12 hours wave travelling time from Plochingen to Heidelberg) are very fast compared to the size

6 K. Wilke&F.Barth 396 6h> D r 2hi 4 18k' X 24h< 4? !f ] i i I i I i Fig hourfy forecasts at the Rhine gauge of Worms using the Wiener filter; actual forecasts from February *-' 1 12h s '18h M «- 1 ' ' ' I ' ' ' ' h Fig. 4. Forecasting errors for 6, 12, 18 and 24 hours using the Wiener filter, February 1990.

7 397 Flood forecasting by Wiener and Kalman filtering km Fig. 5. Location map of the gauging stations in the Neckar catchment (forecast gauges are underlined). of the Neckar catchment ( km 2 ). Besides, the process of flood propagation is not fully determined by the flood hydrographs at the gauging stations because of the influence from small ungauged catchments. With regard to these facts the catchment was divided into several subareas (Fig. 5) where in a first step separate forecasting models were developed. All the

8 K. Wilke&F.Barth 398 submodels are based on an ARMA-relationship (Box & Jenkins, 1976) between the discharges at the upstream gauges and the downstream gauge (Wang, 1987). For the model calibrations 6 historical flood events were used. The model parameters and the order of the ARMA-models were found by recursive least squares (Young, 1984). The lags of the subsystems were determined by the analysis of the discharge hydrographs. For the computation of the forecasts the parameters of the ARMA-models were updated by a Kalman filter (Kalman, 1960). In the statespace formulation the states of the systems are defined as unknown parameters, respectively (Amirthanathan, 1982). The noise statistics which are important for an optimal operation of the filter were found by an adaptive algorithm described by O'Conell (1980). After the submodels showed consistency in the verification with 6 further flood events, the forecasting model for the gauging station Heidelberg was made by combination of these submodels. With that model combination a sufficient forecasting quality at the gauging station Heidelberg could be reached for all 12 historical flood events up to a lead time of 12 hours. The forecast for the extreme flood event in February 1990 is shown in Fig. 6. When computing the various forecasts one could see the advantages of the ARMA-model formulation together with the parameter updating by Kalman filter. The effect of ungauged flows is nearly compensated by the AR-terms (Wood, 1978) and the model is always adapted to the changes in the input information by the parameter updating. In the near future this model will be inserted for the operational river flow forecasting by the River Flow Forecasting Centre of Baden-Wurttemberg. FORECASTS AT THE RIVER RHEM The forecasting model for the gauge of Worms is based on the same model structure as presented for the river Neckar and also the model development was carried out in the same way. As model input the discharges at the stations Maxau and Heidelberg are taken into account (Fig. 1). First computations have shown that by the presented forecasting technique suitable forecast for Worms might only be obtained by using the already existing forecast at the gauge of Heidelberg and the discharges at Maxau as input informations for the model (see Fig. 6). Finally 6 hourly forecasts 24 hours in advance were computed for the 12 historical flood events by using the forecasts 12 hours in advance for the gauging station Heidelberg. The results of the forecasts were sufficient in accuracy for all historical flood events except for the extreme event of February 1990 presented in Fig. 6. The forecasting errors around the flood peaks are not acceptable in view of the operational use of the model, thus the model structure must be checked based on an analysis of these errors. Several further computations with shorter lead-times gave good results (Fig. 7), but with increasing lead-time the automatic parameter updating by the Kalman filter got more and more ineffective, especially for the flood event of February 1990, with its extremely rapid rising and falling levels. The systems behaviour changes significantly within the lead-time of 24 hours and the updated parameters are not representing future systemstates, re-

9 399 Flood forecasting by Wiener and Kalman filtering HEIOELBERG 4 12 H FORECRST HEIDELBERG Q(m?s ') 5600 _ S200 48O0 WORMS l_ < Fig. 6. Discharge hydrographs at the gauges Maxau, Heidelberg, Worms and hourly made forecasts (12 hours ahead) at Heidelberg, using Kalman filter for the flood event of February OBSERVED 5600??...H...FORECAST i^-itj^qrecrst Fig hourly made forecasts at the Rhine gauge of Worms using Kalman filter for the flood event of February 1990.

10 K. Wlke&F.Barth 400 spectively. It is not possible to compensate these effect by giving the Filter more power for the parameter change because then the forecasted hydro-graphs are oscillating around the observed which is not acceptable for real-time-forecasting. The presented results must be thought as preliminary, and further investigations will be made on automatic parameter updating in river flow forecasting, especially in case of such extreme flood events. REFERENCES Amirthanathan, G.E. (1982) Contribution des techniques de filtrage optimal PhD, Université des sciences et techniques du Languedoc. Box, G.E.P.; Jenkins G.M. (1976) Time Series Analysis: Forecasting and Control, Holden-Day Inc., San Francisco. Huthmann, G. & Wilke, K. (1982) Time and frequency domain analyses of hydrological systems./. Hydrol. 55, Kalman, R.E. (1960) A new approach to linear filtering and prediction problems. Trans. ASME, Journal Basic Eng., 82 O'Conell, P.E. (ed.) (1980) Real Time Hydrological Forecasting and control. Proc. 1st in workshop. Institute of Hydrology, July 1977, Wallingford Wang Guang-Te; Yun-Sheng Yu; Wu Kay (1987) Improved flood routing by ARMA modelling and the Kalman Filter Technique. Journal of Hydrology, 93, Wiener, N. (1949) Interpolation, Extrapolation, and Smoothing of Stationary lime Series. MIF Press, Cambridge, Massachusetts. Wiggins, R.A. & Robinson, E.A. (1965) Recursive solution of the multichannel filtering Problem. /. Geophys. Res. 70, Wilke, K. (1975) Principles of hydrological forecasting by multichannel Wiener filtering. In: Mathematical Models in Hydrology and Water Resources System Proc. Bratislava Symp., September 1975), International Association of Hydrological Sciences Publ. no Wood, E.F. (1978) An Application of Kalman Filtering to River Forecasting. In: Proc. Chapmann conference on Application of Kalman filter to hydrology, hydraulics and water resources, Pittsburgh Young, P. (1984) Recursive Estimation and Time-series-analysis. Springer Verlag, Berlin, Heidelberg, New York, Tokyo