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Recent developments in simpleadaptive flow forecasting models inPolandERYK BOBINSKI a & MAŁGORZATA MIERKIEWICZ a

a Institute of Meteorology and Water Management, OperationalHydrology Division , 01673 , Warsaw , Podlesna 61 , PolandPublished online: 21 Dec 2009.

To cite this article: ERYK BOBINSKI & MAŁGORZATA MIERKIEWICZ (1986) Recent developments insimple adaptive flow forecasting models in Poland, Hydrological Sciences Journal, 31:3, 297-320,DOI: 10.1080/02626668609491050

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Hydrological Sciences - Journal - des Sciences Hydrologiques, 31, 3, 9/1986

Recent developments in simple adaptive flow forecasting models in Poland

ERYK BOBINSKI & MALGORZATA MIERKIEWICZ Institute of Meteorology and Water Management, Operational Hydrology Division, 01673 Warsaw, Podlesna 61, Poland

ABSTRACT Real-time flow hydrograph forecasting models have been operational in Poland as a routine service since 1975. A trend in recent years is a shift away from complex conceptual catchment models to simple ones of the storage function type. Another approach has been to forecast the mass flow curve (S-curve) using an ARMA-type model the parameters of which are estimated via a linear Kalman filter algorithm. Examples are shown of the application of the latter approach to models of both the rainfall-runoff category and the channel routing category including estimation of ungauged lateral inflow in the latter models. Previous experience with the application of a linear Kalman filter to flow hydrograph forecasts is discussed. Derivation of the forecast flow hydrograph from the S-curve is done by converting flow volume increments into flow discharges assuming a linear shape for the flow hydrograph within the time step and a trend corresponding to the last observed trend. In the vicinity of the hydrograph peak, the forecast flow values are computed by means of a flood wave equation developed by Strupczewski (1964).

Progrès récents en Pologne dans les modèles simples ajustables de prévision d'écoulement RESUME Les modèles de prévision en temps réel des hydrogrammes d'écoulement ont été opérationnels en Pologne sur une base de routine depuis 1975. Une nette tendance dans les dernières années est la suivante: on tend a délaisser les modèles conceptuels complexes de bassin pour des modèles simples du type à réservoirs. Une autre approche consiste a prévoir la courbe d'écoulement cumulé (courbe en S) en utilisant un modèle de type ARMA, dont les paramètres sont estimés par l'intermédiaire d'un algorithme linéaire de filtre de Kalman. On montre des exemples d'applications de cette dernière approche à la fois à la catégorie des modèles pluie-débit et à la catégorie des modèles de propagation de l'écoulement dans le lit des rivières incorporant l'estimation des apports latéraux non mesurés pour ce dernier type de modèle.

*Paper presented at the Anglo-Polish Workshop held at Jabfonna, Poland, September 1984. (See report in Hydrological Sciences Journal, vol.30, no.l, p.165.)

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298 Eryk Bobinski & Ma/gorzata Mierkiewicz

On analyse les résultats des essais antérieurs de l'application d'un filtre linéaire de Kalman à la prévision de 1'hydrogramme d'écoulement. On déduit 1'hydrogramme d'écoulements prévu de la courbe en S eu transformant l'augmentation de volume d'écoulement par unité de temps en débits en supposant des variations de débit linéaires sur 1'hydrogramme pendant le pas de temps choisi et une tendance correspondant à la dernière tendance observée. Au voisinage de la pointe de 1'hydrogramme les valeurs des prévisions de 1'écoulements sont calculées au moyen de l'équation de l'onde d'écoulement mise au point par Strupczewsfci (1964).

INTRODUCTION

Development of real-time hydrological forecasting based on catchment simulation models started in Poland in 1971 with the application of a rainfall-runoff model of the SSARR type to the flow forecasts of the rivers Sola and Dunajec. The purpose was to improve forecasting for flood protection and reservoir operation. The forecasting systems for these two rivers became fully operational in 1975 and since then underwent several modifications and expansions. A channel routing model for the Lower Vistula River (stage forecasting) was put into operation in 1976 and has been expanded and improved several times since then.

The forecasting systems have been operated as a routine service up to the present. After each year of operation the forecast error statistics have been computed and evaluated. Next, decisions as to further modifications and improvement of the systems have been implemented. After each year this activity is repeated. Considerable experience has been gained and new ideas have emerged (Bobinski et al., 1978, 1980, 1981).

At present, studies are concentrated on the following areas of activity :

(a) the development and implementation of a set of forecasting models simple enough to be run on micro/minicomputers, and, at the same time, as reliable and physically realistic as possible;

(b) a coupling of component models (rainfall-runoff, channel routing, reservoir operation) into one forecast-decision system embracing the sub-basins, river reaches and reservoirs of the Upper Vistula basin; and

(c) the improvement of a forecast by application of various updating techniques. Much of the paper is concerned with the last topic. Discussed first are some recent developments in flow forecast modelling and present views on the subject.

EVOLUTION OF OPERATIONAL RAINFALL-RUNOFF MODELS

Conceptual models mentioned earlier have been operated for the flow forecasting of the Sola and the Dunajec rivers since 1975. They provide flow forecasts 48 h ahead with a time step of 3 h as well as the flow volume forecast with a time step of 12 h within the forecast

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Simple adaptive f low forecasting models 299

period. An example of the model performance for the Dunajec is shown in Fig.l for a 1980 flood event.

Among various input data, the quantitative precipitation forecast, QPF, is input at each forecast run, and consists of two 24 h precipitation totals for the first and second days of a forecast period. The QPF is the main source of the output flow errors. So far, the errors in QPF are of the same range of magnitude as the precipitation actually observed: the heavier the

23..

observed value forecast value observed precipitation, 24-h totals forecast precipitation, 24-h totals date forecast issued

1000

• at

R a

600

400

200

22 23 24 25 26 27 28 29 30 31 1 2 3 * 5 J u l y A u g u s t

1980

Time

Days

Fig. 1 Observed and forecast f low hydrograph and S-curve for the Dunajec River at Nowy Sfcz, 22 Ju ly -8 August 1980. The forecasts are 48-h-ahead ou tpu t by the MONS model. Time steps: A t = 3 h for precipitat ion and f low hydrograph; A t = 12 h for S-curve. Using QPF (see text ) .

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300 Eryk Bobinski & Mafgorzata Mierkiewicz

the rain, the greater the QPF errors. The rainfall totals are usually underestimated at the beginning of a significant rainfall event and overestimated at the time of the rain cessation. The resulting flow hydrograph shows errors of a similar nature.

Besides QPF, the conceptual models used are sensitive to errors in estimation of initial soil moisture conditions of the catchment and to errors in time and space averaging of the rainfall input. In many cases both the soil moisture accounting procedure and the rainfall averaging in these lumped parameter models were found to be inadequate.

To illustrate this statement, Fig.2 shows the forecast for the same event as in Fig.l but computed on the assumption that QPF was equal to the observed 24-h precipitation totals. The forecasts of the rising limb of the hydrograph are better than those in Fig.l, but nevertheless the hydrograph peaks are underestimated as well as the S-curve forecasts. Such a feature for different rainfall-runoff models has been observed by many authors, e.g. by O'Connell & Clarke (1981).

The operation of the above conceptual models has now been transferred from main-frame to microcomputers. At the same time it was noted that the performance of simple forecasting models is no worse than that of sophisticated ones. In addition, the number of modelled catchments and component models mentioned in the Introduction is still increasing. For these reasons, simple flow forecasting models have been chosen for operational purposes.

Two simple storage function models have been adopted, viz. the ISO model (Lambert, 1972) with a logarithmic storage-outflow relation, and the Nash cascade (Nash, 1960), the parameters of which are estimated via the geomorphological instantaneous unit hydrograph (GIUH) theory developed by Rodriguez-Iturbe & Valdes (1979) .

For both models, two different procedures for net rainfall estimation are applied. For the ISO model, precipitation loss due to infiltration is obtained from an exponential recession function of the antecedent precipitation index (API). The function contains two empirical parameters. A great advantage of the ISO model is its simple structure and self-correcting ability. However, without an adequate net rainfall estimation procedure the ISO model performs poorly.

For the Nash-GIUH model, the net rainfall separation is based upon the OS Soil Conservation Service approach (US Department of the Interior, 1965). The latter model and its development is discussed by Zelazinskl (1986).

Both models are being implemented for the subcatchments of the Upper Vistula basin.

Still another set of simple models originates from the endeavour to apply Kalman filter updating techniques (Gelb, 1974) to flow forecasting. This is discussed in the following sections.

UPDATING PROCEDURES WITH APPLICATIONS OF KALMAN FILTER TECHNIQUES

An updating procedure is an indispensable part of a real-time forecasting system. At each forecast run, the output of the basin model is adjusted taking into account new incoming observations and

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Simple adaptive flow forecasting models 301

2 5 -

observed value forecast value observed precip i tat ion, 24 h totals date forecast issued

1000

600

400

200

T=r t/%/

1980

10

20 g

3 0 «

50""

Time Dajra

Fig. 2 Observed and forecast flow hydrograph and S-curve for the Dunajec River at Nowy Sacz, 22 July-8 August 1980. The forecasts are 48-h-ahead output by the MONS model. Time steps: At = 3 h for precipitation and flow hydrograph; At = 12 h for the S-curve. Using observed precipitation (see text).

previous forecast errors. Among various methods of recurrent estimation, the linear Kalman

filter enjoys considerable popularity and has been used for several years in Poland, as well as other methods, for flow forecasting (Mierkiewicz & Szoliosi-Nagy, 1979).

To start with, the performance of the filter procedure was tested on a rainfall-runoff model simple enough for the state-space notation. An ARMA-type model was chosen in which observed flows

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302 Eryk Bobinski & Margorzata Mierkiewicz

preceding the time of the forecast, t0, formed the autoregressive component and observed rainfall amounts formed the moving average component. The state vector of the system consisting of the ARMA-type model parameters was estimated by the filtering algorithm using a random walk approach. A measurement equation produced directly the one-step ahead, t0 + At, forecast. Forecasts for the next time steps, tQ + 2At, t0 + 3At, ..., were obtained by repetition of the one-step forecast, substituting observed elements by predicted ones in the preceding step (river discharges) or forecast independently (rainfall amounts).

The results obtained were unacceptable. Such an approach led to the accumulation of errors at each time step, so that for distant time steps the forecast flows were spread over a very large interval. An example is presented in Fig.3.

200

50

1-step aheaa forecast

observed value 1-step-ahead forecast value

x—x— 3-steps-ahead forecast value

3-steps ahead forecast

3 4 : June 1976

_, î i a e

Fig. 3 One-step ahead and three-steps-ahead f low forecasts f r om the A R M A (3,2) type and Kalman f i l ter model for the Dunajec River at Kowaniec, 31 May-7 June 1976 (from Mierkiewicz & SzôlltSsi-Nagy, 1979).

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Simple adaptive flow forecasting models 303

The introduction of an additional filter for the rainfall, i.e. a simple autoregressive model coupled with the linear Kalman filter, improved the results only slightly. In spite of these failures, useful experience was gained concerning the algorithms estimating the system and measurement noises.

At that time it was hoped that better results could be obtained if a more adequate rainfall-runoff model was applied. The Nash cascade (Nash, 1960) adopted for the state-space notation and expanded by a residuals model was chosen and tested. This improved the one-step-ahead forecast; however, the forecasts for the following time steps remained bad.

Further refinement of the process model seemed inappropriate since it made the space-state notation more difficult. Complex hydrological models to be described in a state-space notation need to undergo such simplifications that lead to an unacceptable deformation of the original model structure. On the other hand, a good performance of the first time step forecast (t0 + At) obtained by the use of an adequate model does not guarantee a good performance for the forecasts for the next time steps (t0 + 2At, t 0 + 3At, . . . ) .

Further attempts to improve operational flow forecasts through the filtering approach have been made. A general conclusion following from these attempts was that the results are good as long as the process trend remains unchanged or changes slowly. When the trend changes sharply, e.g. at the beginning of the flood hydrograph or in the vicinity of the peak, the model fails. Here again, a strong dependence of the model performance on the precipitation input (a disturbance) was demonstrated. The rising of the flow hydrograph in the catchments discussed starts soon after the beginning of a significant rainfall event and a peak indicates the rainfall cessation. In the vicinity of the peak the rainfall input for the next time step forecast becomes equal or near to zero and, as a result, the ARMA-type model output depends mostly on the previously observed flows and follows their rising trend. After passing the hydrograph peak, when a new observation indicates a change of the process trend, the model coupled with a Kalman filter adapts to it, but until that moment the forecast errors remain large.

MASS FLOW CURVE (S-CURVE) FORECASTS

S-curve forecasts derived from rainfall-runoff relations

The next attempt to improve flow forecasting using the Kalman filter technique, keeping in mind previous experience, was to choose a runoff characteristic for which the trend changes slowly. The mass curve (S-curve) was chosen as it represents a monotonie function proceeding with positive increments.

It was assumed that the S-curve operation can be described by a transfer function noise model with an autoregressive moving average structure that is an ARMA-like model. The autoregressive terms represent the S-curve ordinates and the moving average terms represent the rainfall amounts, i.e. disturbance impulses. For the 48-h-ahead S-curve forecast on the Dunajec River the most suitable time step was chosen, after trials, as At = 12 h. The model

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304 Eryk Bobinski & MaKgorzata Wlierkiewicz

parameters are estimated by Kalman filter at each forecast computation.

This approach needs some justification. Dooge (1972), Todini & Bouillot (1975) and other authors have shown that ARMA-type models give an adequate representation of the dynamics of the rainfall-runoff process and can model the short-term behaviour of hydrological systems. Dooge has shown an analogy between an ARMA model and that of the Nash (1960) linear cascade. In our case, the same model structure is applied to the integrated hydrograph or S-curve which has a nonstationary nature (a mean value of the process does not exist). This implies a nonstationary behaviour of the autoregressive terms. On the other hand, it is known that nonstationarity does not complicate the filtering problem (O'Connell, 1980). In the operational adaptive mode, when the S-curve forecast is computed every day, the continuous mass accumulation process in the model is interrupted and only a few time steps backward and forward are considered the same way as is done with the flow forecast. Nevertheless, this implication has been noted and in the following later examples checked against numerical experiments.

To model the S-curve, an ARMA (3,2) type model was chosen, after trials, and the following relations used for computations:

st+l = xiC* + !) st + x 2 ^ t + l) st-l + x3(-t + !) st-2 +

t± TN v 2 4 xi ( t - 2 , t - 4 ) , . n . v 24 _, ( t , t - 2 ) x 4 ( t + 1) E P i + x 5 ( t + 1) E P i +

v ( t + 1) (1 )

where : S t + k is the S-curve ordinate at time (t + k): EiPi

Tj is the 24 h rainfall amount for day T-=; Xj^(.) are parameters of the model; and v(.) is measurement noise (Gaussian) with a mean r and variance matrix R. The time step, At = 12 h.

The measurement vector is:

H(t + 1) = [St,St_1,St_2 ! E iP i< t~ 2' t- 4 ), Z.P.^' 1^] (2)

while the state vector is:

x(t + 1) = [xx(t + 1 ) , ..., x5(t + 1)]T (3)

The measurement equation is:

S(t + 1) = H(t + 1) £ (t + 1) + v(t + 1) (4)

and the state equation is:

x(t + 1) = x(t) + w(t) (5)

where w(.) is the system noise (Gaussian) with mean q and variance matrix Q.

The one-step-ahead forecast is given by:

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Simple adaptive flow forecasting models 305

S ( t + l | t ) = H( t + 1) 3£(t + l [ t ) (6 )

where x_(t + l|t) is an estimate of the ARMA-type model parameters produced by a linear Kalman filter.

A forecast two steps ahead is given by:

S(t + 2Jt) = H(t + 2) x(t + l|t) (7)

H(t + 2) = [St^.St.St.i Z.P.TJ1, Z.P.TJ2] (8)

The next forecasts, for times t + 3 and t +4, are obtained in a similar way, by substituting the forecast values of Sj and P^ instead of observed values for the corresponding time steps.

Examples of the S-curve forecasts computed by the above model are presented in Figs 4(a) and 5(a), for the Dunajec River flood events in 1980 and 1983. In both cases, observed 24-h rainfall amounts are input as QPF.

Another option tried in this study was to insert in equation (1), as the moving average terms, ordinates of the mass curve of precipitation instead of the 24-h rainfall totals. This results in better S-curve forecasts in the vicinity of the flow hydrograph peak and falling limb, and worse S-curve forecasts for the rising limb of the hydrograph (Figs 4 and 5).

To evaluate the adaptive property of the Kalman filter, a comparison was made of the performance of the ARMA-type model of the same structure as in equation (1) but with constant values of all five parameters. The values of xj, ..., X5 were computed as averages of the state vector elements in the filtering algorithm run for the whole six-month measurement sequence (May-October 1980). The forecasts of the S-curve using this model were computed for the first time step by applying equation (6) and for the next time steps - following the same rules as mentioned above - using equation (8). Results for the 1980 flood (the same event as in Fig.4) are shown in Fig.6. There is considerable deviation of the forecasts from observed values in Fig.6. Underestimation on the rising limb of the hydrograph and overestimation on the falling limb are clearly visible. The initial points of the forecasts are quite distant from the observed ones. The inclination of the forecast fragments of the S-curve remains practically unchanged throughout a flood event. This results from the constant ARMA parameters. In simulation mode such a model would produce output deviating significantly from observations. Comparing Figs 6 and 4, one can note quite distinct updating effects of the Kalman filter.

Due to the filtering procedure the values of parameters xl> x2> •••! x5 undergo variations which, for the 1980 flood event, are shown in Fig.7. One can see that, before the first hydrograph peak, the fastest reaction to the rainfall disturbance appears with parameter X5. The greatest variation of the parameters is observed between the first major peak and the last, small peak on the falling limb, which corresponds to the greatest variation of the flow trend. After passing this time interval, the autoregressive parameters, xj, X2, X3, approach stationary values which are close to the average discussed above in the comment on Fig.6. The moving average parameters, x^ and Xg, oscillate around stationary (average)

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308 Eryk Bobinski & Ma/gorzata Mierkiewicz

600

500

3 400

300

200

100

Fig. 6 S-curve forecasts computed by the ARMA (3,2) type model with constant average values of parameters for the flood event shown in Fig. 1.

values, but with smaller amplitudes corresponding to the smaller precipitation impulses.

In Fig.7, all five parameters for the precipitation mass curve option show greater variations than those for the precipitation daily totals. This phenomenon needs more cases to study before arriving at a final conclusion.

From consideration of Figs 6 and 7 one may conclude that the nonstationary behaviour of the cumulative runoff function (the S-curve) does not have much influence on the output of the adaptive forecasting model.

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Simple adaptive flow forecasting models 309

Tiaê time

Time

Days

'< 0 Days August 1980

parameter value for the 24-h

precipi tat ion totals input as MA

terms

parameter value for the precipi tat ion

mass curve ordinates input as MA terms

average A R M A parameter computed

for the s ix-month period

A R M A parameters

Fig. 7 Variation of the ARMA (3,2) type model parameters for two assumed versions of MA terms for the same event as in Figs 4(a) and (b).

Figure 8 shows a plot of the autocorrelation function of the

innovation (forecast errors) of the one-step-ahead forecast sequence

-0 .2

Time lag

autocorrelat ion func t ion

95% confidence l imits (Anderson test)

Fig. 8 Autocorrelation function of one-step-ahead forecasting errors (innovation) over period May-October 1980 for the ARMA (3,2) type, S-curve model coupled with Kalman filter.

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310 Eryk Bobiiiski & MaCgorzata Mierkiewicz

over the period May-October 1980. The 95% confidence limits computed following the Anderson test (Chow, 1964) are also marked. One can see that some small but significant correlation exists, mainly at low lags. This is presumably due to model inadequacy and an incorrect assumption of Gaussian independence for the system noise in the Ealman filter formulation.

Numerical measures In addition to the graphs presented in Figs 4 and 5 to illustrate the model performance, a few numerical measures were computed following WMO (1975):

1. Ratio of relative error to the mean given by:

RE = ^Sl^z^l (9)

2. Ratio of absolute error to the mean given by: v n i i 1 vc " vo

A = -^z—-9± (10) » y0

3. Ratio of standard deviation to the mean given by:

Y = ^

£(yc -

(ii)

where: yc is the computed value; y0 the observed value; and n the number of computed or observed values.

In the above equations the mean observed value, yQ, is:

for the flow hydrograph:

E y y = = 5 (12) yo,Q N Ho v '

for the S-curve:

V s = Sj = 1 <sNsij - sNsi0> <13>

where: Q0 is the mean observed discharge; i the index of a forecast (day); j the index of time step of a forecast (j = 1 , 2, 3, 4); Sj is the mean observed flow volume for the j-th time span; SJJ the value of observed flow volume (an ordinate of an S-curve corresponding to the j-th step of the i-th forecast); S^g the observed ordinate of an S-curve at an initial moment of the i-th forecast, j = 0; and N the number of S-curve forecasts (each consisting of j values) equal to the number of days.

For the evaluation of forecast efficiency, a modified criterion R2 (Nash & Sutcliffe, 1970) was computed:

2n(yc - y 0)2

R2 = 1 S. 2 (14) I A2„

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Simple adaptive flow forecasting models 311

where: Ap is the error of a persistence (inertial) forecast; and

An = y, O, t+L 'o,t (15)

and L is the forecast lead time. Equation (15) means that no change in the flow discharge is

assumed in the persistence forecast over the time span L. For the S-curve it is assumed that the mean discharge remains unchanged over two consecutive time spans, L^_^ = Lj^, and ASj_i = ASj_. In equation (14), the residual sum of squares is related to the initial sum of squares of the persistence forecast instead of that related originally to the prediction based on mean observed discharge. Among these two "naive" forecast options, the persistence forecast gives a narrower scatter and the R2 criterion is more rigorous than that related to the "mean value" forecast.

For the Dunajec River flood events of 1980 and 1983 the values of the listed criteria were computed for the conceptual catchment model and two versions of the S-curve forecast model, one with daily precipitation totals input as moving average terms in equation (1) (version SC DPT), and another with the ordinates of the precipitation mass curve input as MA terms (version SC PMC). The results are shown in Figs 9 and 10. Numerical measures of the two S-curve model versions are also shown in Table 1.

A method of derivation of the flow hydrograph from the S-curve is described in a following section.

From Table 1 and the Figures one can see that, in relation to the conceptual model, the RE criterion is better for both SC model versions for the 1980 flood and worse for that of the 1983 flood. The A criterion values are practically of the same range of values for all models and both flood events. The Y and R2 values are slightly better for the SC DPT model version, or nearly the same for the SC PMC model, compared with the conceptual model for the 1983 flood whereas for the 1980 flood both SC model versions gave slightly worse results.

/ / / / U / V

r-

-/

S? 1

J

/

"\

,

(see legend of Table 1) model SC DPT model SC PMC

Fig. 9 Plots of numerical measures of forecasting model performance for the flood event shown in Figs 4(a) and (b). (a) Numerical measures for flow hydrograph forecasts; and (b) numerical measures for S-curve forecasts.

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312 Eryk Bobiiiski & IVIa/gorzata Mierkiewicz

Jy A /

/ V

r

12 24 36 48

— 0.4

1 i " ~ - - - ~ J6 46 [»») 12 24 36 48 [ h ] 12 24 36 4B [h ] 12 24 36 48 ft]

(see legend of Table 1 )

model MONS

model SC DPT

model SC PMC

Fig, 10 Plots of numerical measures of forecasting model performance for the flood event shown in Figs 5(a) and (h). (aj Numerical measures for flow hydrograph forecasts; (b) numerical measures for S-curve forecasts.

For all models, the R2 values are quite low or even negative for flows over the initial 12 h of forecast lead time. This results from the form of equation (14) involving a persistence forecast which would be presumably better for this period.

The number of events and set of data analysed in this study are certainly insufficient to draw decisive conclusions. For the time being, one can note that the performance of the proposed S-curve model is not worse than that of the conceptual model. The advantages of the S-curve model are its simplicity and the savings in computer time and human effort needed for model calibration and operation.

Table 1 Numerical measures of forecasting performance of models of the rainfall-runoff class for the Dunajec River at Nowy Sacz, flood events of 21 July-7 August 1980 and 17-25 June 1983. Forecast variable: flow volume or increment of the S-curve

Forecast lead t ime Model : (hi MONS

1980 Flood 12 - 0 . 1 0 0 24 - 0 . 1 9 9 36 - 0 . 1 8 9 48 - 0 . 1 9 8

1983 Flood 12 0.025 2 4 - 0 . 0 0 6 36 0.012 48 - 0 . 0 0 3

RE

SCDPT

- 0 . 0 1 1 0.012 0.069 0.064

0.100 0.094 0.148 0.104

SCPMC

- 0 . 0 7 3 - 0 . 0 8 2 - 0 . 0 8 8 - 0 . 0 9 4

0.013 - 0 . 0 6 9 - 0 . 1 1 4 - 0 . 1 6 4

M

Model : MONS

0.105 0.147 0.188 0.199

0.304 0.350 0.360 0.324

easures A

of model performance

Model : SCDPT SCPMC MONS

0.211 0.261 0.218 0.222

0.158 0.231 0.215 0.330

0.187 0.261 0.271 0.278

0.182 0.333 0.380 0.433

0.138 0.217 0.256 0.270

0.488 0.478 0.445 0.394

Y Model :

SCDPT SCPMC MONS

0.327 0.392 0.351 0.349

0.288 0.321 0.286 0.422

0.313 0.420 0.439 0.472

0.294 0.482 0.491 0.539

0.856 0.814 0.815 0.832

< 0 0.443 0.718 0.815

R2

SCDPT SCPMC

0.197 0.393 0.651 0.719

0.395 0.749 0.883 0.786

0.262 0.299 0.455 0.490

0.368 0.433 0.658 0.652

Legend: RE, A , Y , MONS SC DPT

SCPMC

R2 see equations (9), (10), (11), (14) . name of conceputal model . the S-curve forecast model w i t h dai ly prec ip i ta t ion totals input as the moving average terms (see equation (1)). the same as above, but the ordinates of the precepitat ion mass curve input as the moving average terms.

T ime step of the S-curve forecast, A t = 12 h.

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Simple adaptive flow forecasting models 313

The S-curve forecast derived for channel flow routing

The S-curve forecast model described above has been applied also to forecast the inflow volume of the Vistula River to the Wloclawek reservoir. The forecast lead time, L, is 72 h, and the time step, At = 12 h, which means that each forecast consists of six ordinates, 12 h apart, of the mass flow curve.

To forecast an inflow hydrograph to this reservoir, an earlier model based on a nonlinear storage cascade has been operated since 1980. The model performed well for summer flood conditions. However, during low flow and under ice conditions the model performance was poor. To improve the inflow forecast, it was decided to try the model concept described above, equations (1) to (8). In the ARMA (3,1) type model the ordinates of the S-curve served as the AR terms and the 12-h flow volume output from the nonlinear cascade model for the upstream cross section served as the MA term. As a second option for the MA term, the daily flow volume forecast obtained from the crest-stage relationship and the flow rating curve for the same upstream cross section were used. The latter option is available in most cases in the absence of a channel routing model. In either of the two options, the moving average term represents the process disturbance dependent on the upstream river flow development. The ARMA parameters were estimated using the linear Kalman filter algorithm at each forecast operation after incorporating the current input message consisting of the observed reservoir inflow for the last 24-h and the inflow volume forecast as output by the nonlinear cascade channel routing model or obtained from the crest-stage relationship.

The S-curve forecasts computed in this way are shown in Fig.11 where the results of the first MA input version are presented. The results of the second version are practically indistinguishable at the scale of Fig.11. At the beginning of the flow series, 1 March-5 April 1985, shown in Fig.11, an ice cover followed by an ice break-up were observed. Nevertheless, the results look quite good. The numerical measures of the forecast performance are shown in Table 2 in the two versions of the MA input. One can see that criteria RE, A and Y for the S-curve are quite small, within a few per cent of the mean observed flow volume, whereas for the nonlinear cascade the relevant figures are within 20-30%. The criterion Rz, which is computed in relation to a persistence forecast, shows not such impressive values for the suggested model; however the values are better than those for the nonlinear cascade which are negative. The R2 value improves for more distant forecast points which means that the model forecast performs better than the persistence one for these points, which is obvious.

It should be noted that good agreement between forecast and observed inflow volumes has been obtained for the ice conditions period. Here the suggested S-curve forecast model is certainly better than the nonlinear cascade. Forecast 12-h inflow volumes are helpful for the reservoir operation planning three days ahead.

Another example of the proposed model performance is shown in Fig.12 for the Dunajec River 1980 flood at Goïkowice, located 12 km upstream from Nowy Sacz, the cross section for which the examples shown in Figs 1,2, 4 and 5 are presented. In this case, the

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314 Eryk Bobiiîski & Ma/gorzata Mierkiewicz

8000

7500 .

î 7000

5500

5000

4500

M a r o ù 1 9 8 5 A p r i l

Fig. 11 The S-curve forecasts computed via the ARMA (3,1) type model coupled with Kalman filtec The channel routing non-linear cascade model output is entered as MA term. The Vistula River at Wloofawek, flood period: 1 March-5 April 1985.

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Simple adaptive flow forecasting models 315

Table 2 Numerical measures of forecasting performance of models of the channel routing class for the Vistula River at W/odawek, flood event of 1 March-13 April 1985. Forecast variable: flow volume or an increment of the S-curve

Forecast lead t ime (h)

12 24 36 48 60 72

Measures of RE Model : S C N C M

- 0 . 0 2 2 - 0 . 0 2 4 - 0 . 0 3 3 - 0 . 0 3 7 - 0 . 0 4 5 - 0 . 0 4 9

model performance

SCCSR

- 0 . 0 0 8 - 0 . 0 0 1

0.000 0.004 0.004 0.000

A Model : S C N C M

0.058 0.058 0.057 0.062 0.065 0.066

SCCSR

0.059 0.060 0.061 0.063 0.064 0.066

Y Model : S C N C M

0.076 0.074 0.076 0.081 0.084 0.087

SCCSR

0.076 0.074 0.075 0.080 0.083 0.086

R2

Model : S C N C M

0.161 0.203 0.294 0.309 0.354 0.361

SCCSR

0.177 0.219 0.311 0.326 0.374 0.402

RE, A , Y , R J see equations (9), (10), (11), (14). SC NCM the S-curve forecast model w i th the nonlinear cascade mode] outputs entered as moving average term

(see equat ion (1 )). SC CSR the same as above, but the crest-stage relationship used t o obta in the moving average term. T ime step of the S-curve forecast, A t = 12 h.

S-curve forecast model was applied to the estimation of the residual inflow volume representing both the lateral inflow and errors in data and those caused by the channel routing model inadequacy. In this part of the study a channel routing model for a river reach 30 km long was considered. The flow hydrograph from the upstream section was routed using a kinematic wave model. As a result of the channel routing, the residuals between the observed and routed hydrographs at the downstream section, Golkowice, are created. The S-curve of residuals, which can be positive or negative, is forecast using an ARMA (3,1) type model and a Kalman filter in the same way as described earlier (equations (1) to (8)). The observed S-curve ordinates of residuals serve as autoregressive terms and the last observed 24-h precipitation amount serves as the moving average term. The hydrograph of residuals is computed by dividing the one-step volume of residuals obtained from the S-curve by the time step (12 x 3600 s) and then by linear interpolation assuming that the residual hydrograph and the one obtained from the channel routing are parallel within the time step.

The observed and forecast residual flow volumes computed using the approach described are shown at the top of Fig.12.

DERIVATION OF THE FLOW HYDROGRAPH FROM THE S-CURVE

To obtain a flow hydrograph forecast from that of the S-curve the first approach would be the numerical differentiation of the S-curve. This approach, however, is ineffective, probably due to the large time resolution, At = 12 h, of the forecast S-curve and the uncertainty of the flow hydrograph behaviour within this time interval.

The following approach to solve the problem was applied in this study. The flow volume increments over the time step, At, are converted into flow discharges assuming a linear shape of the flow hydrograph within At and a trend corresponding to the last observed time step, i.e. rising or falling. In the vicinity of the

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316 Eryk Bobinski & Ma/gorzata Mierkiewicz

observed value

forecast value

1« 11 12 13 14 15 1 * 17 18 1» JO

A « g a a t 1*80

Bnja

Fig. 12 Observed and forecast flow hydrograph and the 12-h volume of flow residuals for the Dunajec River at Goikowice, 8-20 August 1980. Note: forecast of residuals are computed by the S-curve model, forecasts of the hydrograph by the kinematic wave model; both are added and result is showa

hydrograph peak the forecast flow values are computed using the typical flood wave equation developed by Strupczewski (1964):

:u: e v + b (16)

where : Q-j. is the flow discharge at time t; Q is the peak discharge of the flood hydrograph; t is time to peak; a, b are mean values of parameters estimated from historical data records;

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Simple adaptive flow forecasting models 317

a = 1 b; b = Q0/Qp; QQ is the baseflow preceding the rising limb of the hydrograph at time t0; and m, n are flood hydrograph shape parameters estimated using formulae or graphs developed by Strupczewski (1964).

A typical flood hydrograph for the Dunajec River at Nowy Sacz is shown in Fig.13. The equation for this hydrograph after identification of the parameters in equation (16) is:

Qt = ,{«. 95 -10.0

7.35[1 -1.36

+ 0.05S (17)

To obtain values for Q p and tp needed for computation of Qt by means of equation (16) , one can use additional information available from the precipitation forecast (QPF).

For the Dunajec River and other mountainous tributaries of the Upper Vistula one can assume that the initial time of a significant rainfall event, tQ, corresponds roughly to the rising of the flow hydrograph while the time of the rainfall cessation, tc, corresponds, with negligible lag, to the time of peak flow, Q . The times t 0 and tc have to be provided by the forecaster, the meteorologist on duty an author of the QPF. The forecast of tQ is already done routinely; as to tc, it can be forecast with the accuracy of the time step, At = 12 h.

Within the forecast lead time, tL, two cases are possible:

1. tc = tp = tL Then, assuming a triangular shape of the hydrograph within t^:

2(SL - S0)

*L - to (18)

hydrograph averaged from past floods typical flood hydrograph obtained from equation (171 past flood hydrograph

Fig. 13 The plot of a typical flood hydrograph for the Dunajec at Nowy Sacz. Dimensionless scale.

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318 Eryk Bobinski & Mafgorzata Mierkiewicz

where: S L is the forecast ordinate of the S-curve at the time t^; S 0 is the initial ordinate of the S-curve corresponding to t0 or the beginning of the current forecast; and t^ is the forecast lead time.

2- tc = tp < tL Then:

2(S_ - S,)

where: S c is the forecast S-curve ordinate at the time tc; and Sj is the same as above, but at the time t^, the beginning of the current forecast.

If the forecaster on duty expects that tc > tL, then tc = tL is assumed since no forecast is provided beyond the lead time, tL. Following this approach, the flow hydrograph forecasts were computed for the 1980 and 1983 floods and the results are presented in Figs 4(b) and 5(b). The time t is marked on these Figures. One can see that the flow hydrograph forecasts obtained from the S-curve forecast through the suggested approach appear no worse, in general terms, than those produced by the conceptual model and presented in Fig.2, although the shapes of the forecast hydrograph fragments are different.

In the series of forecasts shown in Figs 4(b) and 5(b) one can note that some, predicting fragments of the rising limb of the hydrograph, are worse than the preceding ones. The model described coupled with the Kalman filter is adapting to the process trajectory through the underestimation-overestimation sequence which is seen in Figs 4(a) and 5(a). From these S-curve forecasts follow the segments of flow hydrograph shown in Figs 4(b) and 5(b). One can note that in Figs 4(a) and 5(a), after a sufficient number of time steps, the S-curve forecasts are approaching the observed S-curve before the hydrograph peak occurs and at the advanced recession.

One of the reasons of the flow forecast instability seen in Figs 4(b) and 5(b) could be an influence of the noise terms in equations (4) and (5) from which follows the forecast equation (6). Standard assumptions of independence and distribution for system noise which are not justified for hydrological systems (O'Connell & Clarke, 1981) can also contribute to the phenomenon discussed.

Keeping in mind the instability of the flow forecast sequence one cannot take a decision based on one single forecast. It is necessary to look at the set of forecasts available up to the present moment and reject forecasts evidently departing from the actual trend. Such an additional filtering procedure needs further development. Still another option for improvement of the suggested model would be to use the net rainfall instead of the total rainfall in equations (1), (2) and (8). This, again, needs to be studied.

CONCLUSIONS

From the experience of recent years, it follows that operational

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Simple adaptive flow forecasting models 319

hydrological forecasting systems should be based on simple and physically realistic models with efficient updating procedures. The performance of simple models in satisfying the conditions discussed is practically as good as that of complex models. Moreover, the simple models are far more economical.

The S-curve forecast seems better suited for updating by linear Kalman filter techniques than the flow forecast. A suggested transfer function model of the ARMA type coupled with a linear Kalman filter for the S-curve forecast gave promising results in application to problems of both the rainfall-runoff class and the channel routing class. In the latter problem the model suggested can be applied also for ungauged lateral inflow estimation. In all applications, the previous ordinates of the S-curve serve as autoregressive terms and precipitation (for rainfall-runoff problems) or upstream inflow forecasts (for channel routing), obtained either from the simple crest-stage relations or existing channel routing models for the upstream river reach, may serve as moving average (disturbance) terms. Due to the adaptive nature of the suggested model, i.e. the Kalman filter algorithm, promising results were obtained in channel routing problems under ice conditions which usually corrupt the flow volume forecast accuracy.

The well-known behaviour of all rainfall-runoff models (underestimation on the rising limb and overestimation on the peak and early recession) are also present in the suggested S-curve forecast model. However the transition from underestimation to overestimation seems not so rapid as in the case of the flow hydrograph forecasting.

Nonstationary behaviour of the cumulative runoff function, following from the examples discussed of simulated forecasts, seems not to have much influence on the forecast accuracy due to the adaptive nature of the model.

Derivation of the flow forecast from that of the S-curve can be done by means of the approach suggested. However, the results so far are not as good as one would like. This problem needs further research.

Another subject for further investigation is the way to input a precipitation in rainfall-runoff S-curve models viz. totals over selected time intervals or ordinates of the precipitation mass curve, and using either observed or net rainfall amounts.

The results obtained from a few examples discussed in the paper seem sufficiently good for a further development of the suggested model for the S-curve forecast.

REFERENCES

Bobinski, E., Piwecki, T. & Zelazinski, J. (1978) Seal-time hydrological forecasting systems in Poland. J. Hydrol. Sci. 5(1).

Bobinski, E., Piwecki, T. & Zelazinski, J. (1980) Real-time hydrological forecasting systems in Poland. In: Real-Time Forecasting/Control of Water Resources Systems (IIASA Proceedings Series, vol.8). Pergamon Press, Oxford.

Bobinski, E., Piwecki, T. & Zelazinski, J. (1981) Efficiency of the real-time hydrological forecasting systems in Poland since their

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implementation in 1975. In: Logistics and Benefits of Using Mathematical Models of Hydrologie and Water Resource Systems (IIASA Proceedings Series, vol.12). Pergamon Press, Oxford.

Chow, V.T. (1964) Handbook of Applied Hydrology, chapter 8. McGraw-Hill, New York.

Dooge, J.C.I. (1972) Mathematic models of hydrologie systems. Proc. International Symposium on Mathematic Modelling Techniques in Water Resources Systems (Ottawa, Canada).

Gelb, A. (1974) Applied Optimal Estimation. MIT Press, Cambridge, Massachusetts, USA.

Lambert, A.O. (1972) Catchment models based on ISO-functions. J. Instn Wat. Engrs 26(8).

Mierkiewicz, M. & Szbllosi-Nagy, A. (1979) Linear Kalman filter for the short-term forecasting of runoff on the Dunajec River. J. Hydrol. Sci. 6(1-2) .

Nash, J.E. (1960) A unit hydrograph study with particular reference to British catchments. Proc. Instn Civ. Engrs 17(5).

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

O'Connell, P.E. (editor) (1980) Real-Time Hydrological Forecasting and Control. Inst, of Hydrology, Wallingford, Oxfordshire, UK.

O'Connell, P.E. & Clarke, R.T. (1981) Adaptive hydrological forecasting - a review. Hydrol. Sci. Bull. 26(2),179-205.

Rodriguez-Iturbe, I. & Valdes, J.B. (1979) The géomorphologie structure of hydrologie response. Wat. Resour. Res. 15(5), 1409-1420.

Strupczewski, W. (1964) Equation of flood crest (in Polish). Reports no.57 of the Hydrological and Meteorological Service, Warsaw, Poland.

Todini, E. & Bouillot, D. (1975) A rainfall-runoff Kalman filter model. In: System Simulation in Water Resources (ed. by G.C.Vansteenkiste). North-Holland Publ. Co., Amsterdam.

US Department of the Interior (1965) Design of Small Dams, appendix A. US Government Printing Office, Washington, DC.

World Meteorological Organization (1975) Intercomparison of Conceptual Models Used in Operational Hydrological Forecasting. WMO no.429, Geneva.

Zelazinski, J. (1986) Application of the geomorphological instantaneous unit hydrograph theory to development of forecasting models in Poland. Hydrol. Sci. J. 31(2), 263-270.

Received 3 June 1985; accepted 24 February 1986.

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