Vízügyi Közlemények, 1985 (67. évfolyam)
2. füzet - Mekis Éva-Szöllősi-Nagy András: Determinisztikus, sztochasztikus és egyesített determinisztikus-sztochasztikus rekurzív hidrológiai előrejelző modellek összehasonlító vizsgálata
240 Mekis E. és Szöllösi-Nagy A. Comparative analysis of recursive deterministic, stochastic and joint deterministic-stochastic real-time hydrological forecasting models É. MEKIS (Miss), Meteorologist, Mathematician and Dr. A. SZÖLLÖSI-NAGY, Civil Engineer The objective of the paper is to compare different recursive models suitable to forecast river flow. The advantage of recursive models is that they can be applied for microcomputers and that the updating of forecasts is relatively straightforward. The following models are compared using the same data base. - Purely deterministic discrete linear cascade model (DLCM); - Purely stochastic autoregressive moving avarage (ARMAX) time series model; and - Joint deterministic (DLCM)/stochastic (ARMA) model. Equestions (1) and (2) are the basic recursive formulas of DLCM (Szöllösi-Nagy 1982) starting off with (6) as initial condition. (The experiences of operational use are reported by Bartha-SzöllősiNagy-Harkányi, 1983). The test data set for comparison is the daily discharge time series of the Dunaföldvár section of River Danube, 1982. The input time series is that of the Budapest section, River Danube. Harkányi's (1982) direct search technique is used to optimize model parameters, yielding /i = 2 and K= 0,32 [day]. Figure I shows the measured and one-day-ahead DLMC discharge forecasts for Dunaföldvár. There is a persistency of errors as indicated by Fig. 2. DLCM gives biased forecasts, (£ = — 111.3 cums) and also negative error is followed by negative error, positive errors are followed by positive error. Indeed, as Fig. 3. shows there is a strong serial correlation in the error sequence (r (1) = 0.74), indicating that there are some information not utilized by DLCM. As il is known, ffr optimal forecasts the error sequence should be a Gaussian white noise sequence. The state space formulation (12) of the ARMAX model (9) enables the recursive estimatior of the random walk parameters, Eq. (13). Figure 4 displays the measured and N= 8 dimensior ARMAX forecasted (by linear Kaiman filter) one-day-ahead discharge time series. Forecasts ars very good as indicated by the autocorrelation function (Fig. 6) of the error sequence (Fig. 5), whicl is essentially the ACF of a white noise sequence with r (1)— -0.03. Relatively large errors durinj the first 50 steps are due to the recursive learning property of the algorithm, as shown by the changi of the trace of the estimation error covariance matrix (Fig. 7) that is stabilized after the 50 t h step After this the matrix changes according to the fine tuning only. Although the ARMAX mode resulted white noise residuals the multi-step extension yielded forecast divergency. The model is no able to consider the physics of runoff explicitlely and parameter changes cannot be explained oi a physical basis. I he correlated error sequence (24) of the DLCM can be described by an ARMA model. Th' state space formulation (26) of the ARMA model, the DLCM and the error model can be pu together in a joint deterministic-stochastic model (see Eqs (34)—(39)). The recursive conditiona forecasting and updating of the extended state variables is performed by the linear Kalman-filtei see Eqs (40)—(44). The conditional output forecast is achieved by the linear projection of the a prioi state vector. Forecasts are gradually improved and converge to the real values by the error feedback The correlated DLCM error sequence (Fig. 2) can be modelled by an ARMA (1.1) model wit a, =0.74. Figure 10 shows the results of the joint model (cf. with Fig. 1). It can be seen that the mod< filtered out all information contained in the error. Figure 11 displays the innovation sequence с the joint model with the ACF shown in Fig. 12. Lag-one correlation coefficient, r (1) = 0.08, i much smaller than (1). The joint model reduced DLCM bias to — 5.69 cums while the standar deviation of the forecast decreased from 110 to 78. The joint model really used all availabl information. Table 1 summarize the error statistic of the DLCM, ARMAX and DLCM-ARMA mode together with the coefficient of effectioness, Eq. (47). DLCM yielded less efficient forecasts than th ARMA model, while the joint DLCM-ARMA model significantly increased the effectivenes Table 2 contains river flow and model statistics, again indicating that the joint model is the be: approximation. Numerical investigations lead to the conclusion that the joint deterministic-stochastic mod is the most efficient forecasting model of all the three recursive techniques compared.
