flash flood forecasting model for the ayalon stream,...

Hydrohgical Sciences -Journal- des Sciences Hydrologiques,40,5, October 1995 5 9 9

Flash flood forecasting model for the Ayalon stream, Israel

VLADIMIR KHAVICH & ARIE BEN-ZVI Israel Hydrological Service, PO Box 6381, Jerusalem, Israel

Abstract A composite model for real time forecasting of flash floods in the Ayalon stream in central Israel has been constructed. The model is composed of four kinds of sub-models: an autoregressive model for discharges at upstream stations on the two major tributaries; a travel-time model for the flow from these stations to the downstream station located on the main stem of the stream; a time-area concentration curve for sub-watershed drainage between the upstream and downstream stations; and a recession curve for the downstream station. The model incorporates an adaptive mechanism for continuous correction of forecast errors. This mechanism is calibrated during an initial period of operation, and is subsequently operated throughout a flow event. The model issues simultaneous forecasts for seven lead times ranging from 0.5 to 3.5 h. This provides a proper input for a flood warning system which is required for safe operation of a major highway running along the banks of a torrent stream in the metropolitan area of Tel-Aviv.

Modèle de prévision des crues éclair du fleuve Ayalon Résumé Un modèle de prévision en temps réel des crues éclair a été établi pour le fleuve Ayalon se situant au centre d'Israël. Ce modèle est composé de quatre sous-modèles: un modèle autorégressif pour les débits en amont des stations de jaugeage sur les deux principaux affluents; un modèle de temps de parcours pour l'écoulement des stations de jaugeage se situant en aval du cours d'eau principal; une courbe de concentration temps-surface pour la surface du bassin versant comprise entre les stations de jaugeage de l'amont et celles de l'aval; et une courbe de tarissement pour la station en aval du cours d'eau. Le modèle comprend une procédure de correction en continu des erreurs de prévision. Cette procédure est calée durant la période initiale de mise en marche et est ensuite appliquée durant un épisode de crue. Le modèle effectue simultanément des prévisions pour 7 horizons de temps allant de 0.5 à 3.5 heures. Ceci procure une importante contribution au système d'annonce de crue nécessaire à la l'utilisation en toute sécurité de l'autoroute longeant les berges du fleuve qui traverse Tel-Aviv.

INTRODUCTION

The Ayalon is a torrent stream, the lower reach of which runs through the Tel-Aviv metropolitan area (Fig. 1). A major highway has recently been constructed along the banks of this reach with a 30-year recurrence interval of flooding. Safe operation of this highway requires the use of a sensitive flood warning system with an alert control and evacuation organization. This paper

Open for discussion until 1 April 1996

600 V. Khavich & A. Ben-Zvi

Fig. 1 Map of study area.

describes the hydrological model upon which the system could be based. A telemetric network, presently under design, would use this model for flood warnings.

The Ayalon stream drains an area of 815 km2 lying in the central region of Israel. The basin length is about 40 km, its width is about 20 km, and the main direction of flow is northwestwards (see Fig. 1). The watershed is composed of three morphological units: mountainous, hilly and plain. The lower lying portion of the plain is densely populated while the higher lying portion is intensively cultivated. An earth dam constructed on the major tributary stream in the hilly region has reduced the effective area of the catchment to 665 km2. The vulnerable sector of the stream, where the major highway is constructed, is the northbound lowest reach, which ends at the confluence with the Yarkon. The design discharge of this sector is only 300 m3 s"1 whilst the highest recorded discharge, upstream from that reach, is 377 m3 s"1. A new hydrometric station (station E) was constructed in 1993 by the Israel Hydrological Service at the upstream end of this reach. The few data available for this station are insufficient for supporting any hydrological forecast model. The most appropriate station for forecasting discharges in the lower reach is located at Bet-Dagan-Yehud Road (station B). A forecast of dis-

Flash flood forecasting model 601

charges at station B can be obtained from information on discharges passing through upstream stations. The appropriate stations are located on the Ayalon stream at Lod (station L) and on the Natuf stream near El-Al Junction (station N). The catchment areas of these three stations are, respectively, 526 km2, 136 km2 (excluding the catchment area upstream from the dam), and 251 km2. Evidently, the catchment area of station B, below those of stations L and N, (area A) is 139 km2. The station best representing rainfall intensities over area A is located at Bet-Dagan and is operated by the Meteorological Service. The flow at station B is composed of the flow at stations L and N, and of the contribution from area A.

BASIC MODEL

The discharge at station B is composed, with due lag times, of the discharges at stations L and N, and of the discharge generated by area A. Owing to variations in rainfall distribution, the relative magnitudes of the three components vary between runoff events. This variation is evident from the data on seven high flow events presented in Table 1. Therefore, a forecast model which considers contributions from the upstream tributaries should refer to the sum of those components.

Table 1 Selected high events at the Ayalon stream

Day

09.12.63 22.03.69 17.01.74 21.01.74 04.02.92 25.02.92 16.12.92

2peak

B

230 263 194 152 258 165 339

(rn3 s"1)

L

48 16 42 20 130 114 99

N

152 230 126 75 135 60 112

Rain (mm)

57.2 17.5 78.9 52.8 73.0

126.0 186.0

API (mm)

50.9 47.2

150.2 92.6 55.7 78.6

127.2

B, L and N refer to the stations located as shown on Fig. 1. Rain is the depth of the precipitation event at Bet-Dagan. API is computed for data from this station only.

A simple summation of the components does not take into consideration the storage processes in the main channels between the stations. Consequently, such summation would result in a too steep hydrograph for station B, i.e. forecast peaks would appear higher and earlier, and forecast recessions lower than measured ones. In order to overcome this problem, a damping term is added to the summation. This addition breaks the seeming obedience of the summation to the mass conservation law.


The basic model is thus formulated as:

QB(t) = QL(t -1) + Q^t - g + £ flj.P(f - 0 +/(ÔS(0)) 1

(1)

where Q is the discharge; subscripts B, L, and N refer to the appropriate stations; t is the forecast time; tl and tn are the lag times of the flow at station B behind the flows at stations L and N, respectively; i is the time unit; a is the area of a sector of area A; P is the depth of effective precipitation; mdf(QB(0)) is a function of the discharges at station B prior to the preparation time of the forecast. Values of P for times later than the preparation time are assumed to be zero. Values of the functions and parameters appearing on the right-hand side of equation (1) are obtained from analyses of recorded data.

The recession function_/(<2B(0)) is determined from records of high flow events through a method developed by Nejichovski (1974). Discharges lagging by one unit of time are plotted against each other, as shown in Fig. 2. A lower enveloping curve is drawn on the graph. This curve relates appropriate values of <2B(0 to given values of QB(t — 1). For any arbitrarily selected value of QB(t - 1), a related value of QB(t) can be obtained through Fig. 2. A value of QB(t + 1 ) , which is related to QB(t), can then also be obtained through Fig. 2. Upon repetitions of this procedure, values of QB(t + 2), QB(t + 3), ... , etc. are obtained. This process is terminated when the discharge vanishes. The length of

. / . / . 'J"

• /°

J-200 300 400

Earlier discharge Q{t - 1) (m3 s'1)

Fig. 2 Recession relationship.


the series obtained provides the recession time T(Q) elapsing from the occurrence of the selected discharge QB(t - 1) till the end of the series. An analysis of the discharge values in the series yields the time related relationships:

QB(t)/QB(0) = exp{-6.866 * (tlT(Q))1*5} (2)

T(Q) = 2.566 <2B(0)1/3 (3)

where the discharge is measured in m3 s'1 and the time in hours. The lag times tt and tn are determined from measured velocities at the

relevant stations. Stage-discharge and stage- velocity relationships are routinely prepared by the Israel Hydrological Service for each station. Upon elimination of the stage, a discharge-velocity relationship is obtained. The relationships for the three stations under consideration are displayed in Fig. 3.

The travel time of a given discharge along a stream reach is assumed as being equal to the length of the reach divided by the mean of the velocities for this discharge at the upstream and downstream stations. Subsequently, a discharge-lag function is prepared for each reach. Such functions for the present case are:

tt = 9200/(0.785 +0.0296«2L-0.000136(2L2) for Q < 107

(4) tt = 3850 for Q > 107

t„ = 8200/(0.507+ 0.0268g»,-0.000126O2) for Q < 107 (5)

tn = 4250 for Q > 107

where lengths are measured in metres, discharges in m3 sA and time in seconds. These relationships indicate that discharges higher than 100 m3 s"1

travel between the stations in less than 1.2 h. The time-area concentration curve is normally prepared by means of

elaborate travel time computations along the many flow paths of a watershed. It can be shown that these are non-unique functions of the watershed (Ben-Zvi, 1974). Field observations at hydrometric stations, such as those displayed in Fig. 3, as well as the theoretical non-linear relationship between velocity and resistance to flow, indicate that for high discharges the velocity does not increase any further with the discharge. Therefore, for small watersheds, where spatial variations in rainfall are relatively small, and in cases of high discharges, the time-area concentration curve can be accepted as an adequate model for describing the response to temporal rainfall variations. In view of observed terminal velocities at the stations studied, and at other stations located in the Coastal Plain of Israel, which vary between 2 and 3 m s"1, a velocity of 2ms"1 was selected for construction of the time-area concentration curve. For reasons of convenience, the unit time for computations was selected as 0.5 h.

Application of the time-area concentration curve requires the use of


•J STATION N

50 100 150 Q (m3/s)

Discharge

50 100 150 2 0 0 250 Q ( m 3 / s )

Discharge Fig. 3 Discharge-velocity relationships.

effective rainfall rather than the recorded data. These differ from each other by the rainfall losses or abstractions. The rate of abstraction is high during the initial stages of rainfall, but declines thereafter. The initial rate depends upon antecedent moisture conditions. A well-known and simple technique for estimation of these conditions is via the API model (e.g. Linsley et al., 1982).


This model yielded satisfactory results in previous studies of rainfall-runoff relationships in the Coastal Plain of Israel (Ben-Zvi, unpublished reports) and takes the form:

API(f) = P*(t)+ k API(f - 1) (6)

where API(f) is the antecedent precipitation index for day t (mm), P*(t) is the recorded precipitation depth on day t (mm), and k < 1 is a parameter. For practical purposes it is sufficient to apply equation (6) with data beginning 30 days prior to the commencement of a high runoff event. A value of 0.7, which is consistent with previously applied values, was selected here for the parameter k.

Rates of abstractions were determined from empirical comparisons of high rainfall and runoff events in which area A was the major contributor to the discharge at station B. The initial rate varies with the API(O) according to the regressed relationship:

G0 = 2.532-0.014 API(O) (7)

Subsequent abstraction rates decline according to the geometrical series formula:

G(i) = rG(i - 1) (8)

in which G(f) is the abstraction rate at time unit i, and r < 1 is a parameter. For time units of 0.5 h, the value of r was empirically found to be 0.9.

The discharges forecast through the foregoing ensemble of sub-models were compared with ones recorded during high flow events. A summary of the forecast errors for the 0.5-hour lead time is presented in Table 2. For the individual events, 50% of the forecast errors lie within ±5 to ±15% of the observed discharge values, and 75% of the errors lie within 10 to 20% of the observed values. The results indicate two shortcomings of the basic model: differences in lead times of the various components, and relatively large forecasting errors. These have been dealt with by means of the improvements described below.

Table 2 Error distribution for the basic model

Error Event number rate (%)

<1 1-5 5-10 10-15 15-20 20-25 >25

Total

1 No.

4 9 13 14 4 1 4

49

%

8.2 18.3 26.5 28.6 8.2 2.0 8.2

2 No.

4 7 4 3 3 0 6

27

%

14.8 25.9 14.8 11.1 11.1 0.0 22.2

3 No.

6 21 12 3 0 0 0

42

%

14.3 50.0 28.6 7.1 0.0 0.0 0.0

4 No.

7 11 18 0 0 0 0

36

% 19.4 30.6 50.0 0.0 0.0 0.0 0.0

5 No.

5 13 10 8 2 3 9

50

%

10.0 26.0 20.0 16.0 4.0 6.0 18.0

6 No.

6 22 23 14 7 6 9

87

%

6.9 25.3 26.4 16.1 8.0 6.9 10.3

7 No.

3 24 11 8 1 3 0

50

%

6.0 48.0 22.0 16.0 2.0 6.0 0.0

Note: lead time is 0.5 h.


MODEL IMPROVEMENTS

The lead times of components contributed via stations L and N are about 1 h, while that contributed via area A varies from 0.5 to 3.5 h. An extension of the former is achieved through inspection of hydrographs of stations L and N. The discharges show a well-defined serial correlation. The autocorrelation coefficients for different lag times are presented in Figs 4 and 5.

16.12.92

3 <

1.0

0.5

9.12.63

0.5

17.1.74

1 2 3 4

Time lag (hj 2 3 4

Time lag (h! 2 3 4

Time lag (h)

Fig. 4 Discharge correlation: station L.

17-1-79

2 3 4

Time lag (h)

Fig. 5 Discharge correlation: station N.

0 1 2 3 4

Time lag (h) 2 3 4

Time lag (h)


Discharges lagging by 0.5 and 1.0 h behind each other were found to be well-correlated. Taking advantage of these correlations, autoregressive models have been constructed for discharges at stations L and N. Their formulae with parameter values obtained through regressions for two high-flow events are:

QL(i) = U66QL(i - 1) - 0J69QL(i - 2) (9)

Q^O = 1.6590*0- - 1) - 0 . 6 5 9 2 ^ - 2) (10)

The squared correlation coefficients, found for equations (9) and (10) are, respectively, 0.982 and 0.993, while the standard errors of estimate are 0.98 m3 s"1 and 3.68 m3 s'1.

The foregoing equations were obtained through a multiple regression, yet their form indicates a gradual variation in the rate of change of the discharges. This conclusion is apparent by means of the following variation of equations (9) and (10):

QLii) ~ QLd - 1) = 0.77[gL(i - 1) - QL(i - 2)]

QNH) ~ QN(i ~ 1) = O.ôôf^z - 1) - QJi - 2)]

(11)

(12)

The goodness of fit for equations (9) and (10) with records of events different from those utilized for calibration is demonstrated in Figs 6 and 7.

Fig. 6 Fit of forecast discharges: station L.

70 80 90 100 110 Observed discharge (m3 s~1)


I20r

110-

tu 100

90

8 0

TO

50

4 0

3 0

10

16.12.92

10 20 30 40 5 0 60 TO 80 90 100 110 120 Observed discharge (m3 s"1)

Fig. 7 Fit of forecast discharges: station N.

The numerical magnitudes of these fits are presented in Tables 3 to 5. Ninety percent of the forecast errors for the 0.5-hour lead time attain relative values lower than 15% of the observed discharges. Forecasts of further discharges are of lower accuracy, but are still valuable.

Table 3 Error distribution according to lead time for station L

Error rate (%)

< 1 1-5 5-10 10-15 15-20 20-25 >25

Lead time (h)

1.0 No.

6 25 10 11 10 6 15

% 7.2

30.1 12.0 13.3 12.0 7.2

18.1

1.5 No.

5 9 11 14 5 8

31

% 6.0

10.8 13.3 16.9 6.0 9.6

37.3

2.0 No.

2 3 9 8 9 8

44

% 2.4 3.6

10.8 9.6

10.8 9.6

53.0

2.5 No.

0 2 7 5 6 12 51

% 0.0 2.4 8.4 6.0 7.2

14.5 61.4

3.0 No.

0 3 3 5 2 8

62

%

0.0 3.6 3.6 6.0 2.4 9.6

74.7

Note: data are for the 16.12.92 event.


Table 4 Error distribution according to lead time for station N

Error rate Lead time (h) (%)

<1 1-5 5-10 10-15 15-20 20-25 >25

1.0 No.

3 9 12 10 3 3 5

% 6.7

20.0 26.7 22.2 6.7 6.7

11.1

1.5 No.

2 6 5 7 11 1

13

% 4.4

13.3 11.1 15.5 24.4 2.2

28.9

2.0 No.

1 3 6 2 5 11 17

% 2.2 6.7

13.3 4.4

11.1 24.4 37.8

2.5 No.

2 1 5 4 1 5

27

% 4.4 2.2

11.1 8.9 2.2

11.1 60.0

3.0 No.

0 3 2 3 5 3

29

% 0.0 6.7 4.4 6.7

11.1 6.7

64.4


Table 5 Forecast efficiency for the upstream stations

Station

L N

Data points

88 50

Mean Q (m3 s"1)

40.6 63.2

StdO (m3 s'1)

20 30

S E 0 (m3 s"1)

3.67 4.04

SE/Std

0.183 0.135


Forecast errors are traditionally attributed to model structure, data and parameter accuracy, and sampling variations. Sampling variations in the present case stem mainly from the high temporal and spatial variations in rainfall intensity and duration which affect the relative contributions of the different sub-watersheds (see Table 1). The errors can largely be overcome by the incorporation of an adaptive mechanism into the model (e.g. WMO, 1992). This mechanism continuously updates parameter values with respect to forecast errors accumulating for the event under consideration.

The best-known techniques for the introduction of an adaptive mechanism are those of Box & Jenkins (1970), Kalman Filtering (e.g. Cooper & Wood, 1982), and Neural Network (e.g. Kang et al., 1993). A variation of the Box & Jenkins (1970) mechanism was selected for the present work. The extended model has the formula:

QB(t) = QL(t - tx) + Q^t - tn) + £ aj\t - i) +/(<2B(0)) + Z{t) (13) l

where Z(t) is the forecast error. The forecast errors are considered to follow an autoregressive process:

Z(t) = Y,w(i)Z(t-i)+e(t) (14) i = i

where n is a limit on the computation, w(i) is a weighting parameter, and e(f) is a random shock variable assumed to follow a normal process.


A simple and quickly applied procedure for a continuous updating of the parameter values of equation (14) is:

m

l[z(0l minimum (15)

where m is the number of forecast errors for the event considered known at time t.

The shock variable is forecast as:

e*(t) m

i=l It

(16)

where e*(t) is a forecast value of e(t). A short time after t, the measured values of QB(t), QL(t), Q^t), and P(t)

are known and Z(t) and e(t) can be computed by:

Z(t) = QB*{t)-QB(t) (17)

(18) e{t) = Z*(t)-Z(t)

in which Q%(t) is the discharge forecast by equation (1), and Z*(t) is the error forecast by equations (14) and (16).

Based upon equation (15), and a fast trial and error technique, new values are obtained for the weighting parameters w(i). These values range in the present case between 0.2 and 1.6, with a decreasing trend as i increases. The value of the limiting parameter, n, is selected through analyses of past events. The value selected for the present case is n = 5. These values are substituted into equations (14) to (16), yielding values for e*(t + 1) and Z*(t + 1). These allow a new application of equation (13) for the forecasting of future discharges.

08:00 10:30 13:00 Time (h)

Fig. 8 Forecast stages: station B.

15:30


The adaptive mechanism is initially applied in a training mode when the shock variable is not forecast and by letting n = t. Subsequently, when t is longer than a given duration (2.5 h in the present case), these restrictions are relaxed. An example for the use of this mechanism is illustrated in Fig. 8.

RESULTS

The above model has been applied to seven high flow events for the Ayalon stream presented in Table 1. For each application, non-adaptive parameter values were obtained from data of events other than the one under application. Seven updated forecasts were issued every 0.5 h, for lead times ranging from 0.5 to 3.5 h. An example is displayed in Fig. 8. The training mode terminates there at 08:30. Forecasts issued at times 09:00, 10:00, 11:00 and 12:00 are explicitly presented. All other forecast points are of the last updated value, i.e. for the 0.5-hour lead time.

The long lead time forecasts were found to yield significant underestimation errors, but with time and reducing lead time, the forecasts improved. The distribution of forecast errors with respect to lead time is presented in Table 6. For the event presented 80% of the last updated forecasts (i.e. for the 0.5-hour lead time) are smaller than 10% of their relevant discharge values. Results for the other events yielded similar statistics. The error distribution for the 0.5-hour forecasts and an analysis of forecast efficiency are presented, respectively, in Tables 7 and 8. On average, 72% of the forecast errors lie within ±5% of the values of the observed discharges, and 93% lie within +20% of the observed discharges.

The ratio of the standard error of estimate to the standard deviation of the observed discharges is considerably reduced by the adaptive mechanism. Values of this ratio for applications of the basic model on the seven events studied range from 0.150 to 0.319, whereas those for the entire model range from 0.063 to 0.164. The efficiency of the adaptive mechanism is illustrated in Fig. 9.

Table 6 Error distribution according to lead time for station B

Error Lead time (h) rate (%)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 No. % No. % No. % No. % No. % No. % No. %

< 1 1-5 5-10 10-15 15-20 20-25 >25

8 25 7 2 4 2 2

16.0 50.0 14.0 4.0 8.0 4.0 4.0

4 19 10 3 2 2 10

8.0 38.0 20.0

6.0 4.0 4.0

20.0

1 14 9 6 6 3 11

2.0 28.0 18.0 12.0 12.0 6.0

22.0

1 9 10 4 7 6 13

2.0 18.0 20.0

8.0 14.0 12.0 26.0

3 4 7 10 4 7 15

6.0 8.0

14.0 20.0

8.0 14.0 30.0

1 5 5 7 9 4 19

2.0 10.0 10.0 14.0 18.0 8.0

38.0

0 7 6 4 6 4

23

0.0 14.0 12.0 8.0

12.0 8.0

46.0

Note: data are for the 4 February 1992 event.


Table 7 Error distribution by events for station B

Error rate (%)

<1 1-5 5-10 10-15 15-20 20-25 >25

Total

Event number

1 No.

8 19 11 5 1 2 3

49

% 16.3 38.8 22.4 10.2 2.0 4.1 6.1

2 No.

2 14 5 0 1 0 5

27

% 7.4

51.9 18.5 0.0 3.7 0.0

18.5

3 No.

14 28 0 0 0 0 0

42

% 33.3 66.7 0.0 0.0 0.0 0.0 0.0

4 No.

9 23 2 2 0 0 0

36

% 25.0 63.9

5.6 5.5 0.0 0.0 0.0

5 No.

8 20 7 2 4 2 2

50

% 16.0 50.0 14.0 4.0 8.0 4.0 4.0

6 No.

16 50 15 1 0 3 2

87

% 18.4 57.5 17.2

1.1 0.0 3.4 2.3

7 No.

20 22 5 3 0 0 0

50

% 40.0 44.0 10.0 6.0 0.0 0.0 0.0


Table 8 Forecast efficiency

Item (m3 s"1)

Mean Q Std

Basic model: SE SE/Std

Entire model: SE SE/Std

Event number

1

93.4 54.4

11.8 0.216

8.9 0.163

2

114.9 83.3

26.6 0.319

13.6 0.164

3

112.2 44.7

6.7 0.150

3.0 0.066

4

89.4 31.2

5.2 0.168

3.1 0.098

5

127.9 73.3

14.6 0.199

7.3 0.100

6

70.4 43.8

7.6 0.174

3.5 0.080

7

216.1 78.1

12.0 0.154

5.0 0.063


250 250

200

150

100

-50

0 -rn—r~!—i—i—t—i i i—!—t~i—r~i—i—\—i—i—i—i—Î—i—r-i—i—i—i—i—i—i—i—i—r~i—i—i—i—i—i—i—r0

07:00 10:00 13:00 16:00 19:00 22:00 01:00 04:00 07:00 Time (h)

entire model —•— observed basic model

Fig. 9 Forecast results station B.


In view of the error statistics and comparisons of forecast and recorded hydrographs, it appears that the model functions properly. A reasonably accurate flood warning can thus be issued every 0.5 h, with improving accuracy. The final forecast is issued 0.5 h in advance. This allows sufficient time for the distribution of a flood alarm to all concerned offices and to drivers on the endangered highway. Further improvements in the forecast may be expected when sufficient records are available for station E, and quantitative rainfall forecasts become available.

SUMMARY AND CONCLUSIONS

A flood forecasting model has been developed for a stream reach which endangers a busy highway. The model is composed of four kinds of sub-models and an adaptive mechanism. Of the four sub-models, one autoregresses discharges at stations which are located on upstream tributaries, one routes discharges from these stations to the target station, one convolutes contributions from rain falling over the area between the stations, and one estimates flow recession at the target station. The adaptive mechanism compensates for variations in rainfall distribution as well as for errors in model formulation and calibration. It autoregresses forecast errors, minimizes them, and yields correction terms for the forecast.

The model issues simultaneous forecasts for seven lead times, ranging from 0.5 to 3.5 h, which are updated every 0.5 h. Forecasts for the longer lead times involve relatively large errors of underestimation, while those for the shorter lead times are generally accurate. Results obtained for seven high flow events appear to provide accurate and dependable forecasts, particularly for short lead times. Such forecasts enable the issuing of alarms through a flood warning system to be established for the lower reaches of the Ayalon stream.

REFERENCES

Ben-Zvi, A. (1974) The velocity assumption behind linear invariant watershed response models. In: Mathematical Models in Hydrology (Proc. Warsaw Symp., Poland, July 1971), 758-761. IAHS Publ. no. 101.

Box, G. E. P. & Jenkins, G. M. (1970) Time Series Analysis, Forecasting and Control. Holden-Day, San Francisco, California, USA.

Cooper, D. M. & Wood, E. F. (1982) Parameter estimation of multiple input-output time-series models: application to rainfall-runoff processes. Wat. Resour. Res. 18, 1352-1364.

Kang, K. W., Park, C. Y. & Kim, J. H. (1993) Neural network and its application to rainfall-runoff forecasting. Korean J. Hydrosci. 4, 1-9.

Linsley, R. K., Kohler, M. A. & Paulhus, J. L. H. (1982) Hydrology for Engineers, 3rd ed. McGraw-Hill, New York, NY, USA.

Nejichovski, R. A. (1974) Channel network in a basin and formation of runoff. Hydrometeorol. Inst., Leningrad, USSR (in Russian).

WMO (1992) Simulated real time intercomparison of hydrological models. WMO Report no. 779. World Meteorol. Organization, Geneva, Switzerland.

Received 16 September 1994; accepted 19 March 1995

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