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APPLICATION OF DIFFUSION PROCESSES TO RUNOFF ESTIMATION
By Richard Labib,1 Mario Lefebvre,2 Joseph Ribeiro,3 Jean Rousselle,4 and Hau Ta Trung5
ABSTRACT: In order to forecast drainage basin runoff, mathematical models involving diffusion processes aretested against hydrological data obtained from the hydrographic basin of the Saguenay-Lac-St-Jean, located innortheastern Quebec. For one-day-ahead estimates, a bidimensional model using an Ornstein-Uhlenbeck processis seen to give better results than a deterministic model presently in use. The latter model involves the estimationof 18 parameters, while the diffusion process model requires only one parameter, thus making it much simplerand cheaper to implement.
INTRODUCTION
Hydroelectric energy is a natural resource crucial to Quebecindustry in general and to the Alcan company in particular. Inorder to maximize its production of hydroelectricity, the com-pany would like to forecast with a high degree of accuracy,and at least one day ahead, the amount of water entering itsreservoirs from the hydrographic basin of the Saguenay-Lac-St-Jean area. At the present time, the company is using a de-terministic model called PREVIS, involving no less than 18parameters, including temperature, rainfall, and humidity, toforecast the amount of runoff.
Diffusion processes are used to interpret biological, eco-nomical, and physical phenomena. Gottardi and Scarso (1994)showed the importance of diffusion models as short-term pre-dictors and demonstrated their superior performance comparedto statistical (Box-Jenkins) models. Takasao et al. (1994) dis-cretized a system of stochastic differential equations to fore-cast the discharge of three river basins in Japan, each havingareas in the 150 km2 range. They used a forecast algorithmthat incorporates an exogenous variable (rainfall) in order toestimate the discharge up to two hours in advance. Moreover,by adding a Kalman filter, they slightly improved their esti-mate of the runoff. With the latter time intervals, they arrivedat an average standard error of estimation of about 100 m3/s.This average error is observed for mean flows of around 600m3/s.
The present paper directly uses a system of equations rep-resenting a diffusion process as the model for forecasting,without the use of any exogenous variable, the discharge ofthree different river basins, namely, the Mistassibi river andthe Chute-du-Diable and Lac-St-Jean basins.
DIFFUSION MODELS
The general model of a diffusion process can be describedas the n-dimensional system of differential equations givenbelow:
1Departement de mathematiques et de genie industriel, Ecole Poly-technique, C. P. 6079, Succursale Centre-ville, Montreal, Quebec, CanadaH3C 3A7.
2Departement de mathematiques et de genie industriel, Ecole Poly-technique, C. P. 6079, Succursale Centre-ville, Montreal, Quebec, CanadaH3C 3A7.
3Groupe YTC-INASTEC S.A. Ingenieurs & Conseils, Dakar, Senegal.4Departement des genies civil, geologique et des mines, Ecole Poly-
technique, C. P. 6079, Succursale Centre-ville, Montreal, Quebec, CanadaH3C 3A7.
5 Electrique Quebec, Societe d’electrolyse et de chimie Alcan,Energie1954 Davis, C.P. 1800, Jonquiere, Quebec, Canada G7S 4R5.
Note. Discussion open until June 1, 2000. To extend the closing dateone month, a written request must be filed with the ASCE Manager ofJournals. The manuscript for this paper was submitted for review andpossible publication on March 31, 1998. This paper is part of the Journalof Hydrologic Engineering, Vol. 5, No. 1, January, 2000. qASCE, ISSN1084-0699/00/0001-0001–0007/$8.00 1 $.50 per page. Paper No. 18005.
J. Hydrol. Eng
FIG. 1. Second Differences of Runoff and Normal ProbabilityPlot (Mistassibi 1995)
dx (t) = A dt 1 B dW (t)1 1 1 1
dx (t) = A dt 1 B dW (t)2 2 2 2? ? ?H ? ? ?? ? ?
dx (t) = A dt 1 B dW (t)n n n n (1)
where A1, A2, . . . , An, B1, B2, . . . , Bn are functions dependenton x1, x2, . . . , xn and thus dependent on time t. W1(t),W2(t), . . . , Wn(t) are defined as standard Brownian motions.Furthermore, we assume that W1(t), W2(t), . . . , Wn(t) are in-dependent.
The first proposed model for predicting the discharge x(t)of a river or a basin, at time t, was inspired by the fact thatwhen differentiated twice, the runoff time series appears to bestationary with a marginal Gaussian distribution. As an ex-ample, Fig. 1 depicts the second differences of the runoff time-series for the Mistassibi river along with the correspondingnormal probability plot. The 3D model considered was thefollowing:
dx(t) = y(t)dtdy(t) = z(t)dt
(2)H
dz(t) = mdt 1 sdW(t)
where m and s are, respectively, the infinitesimal mean andstandard deviation of the Brownian motion process. Let j(t)define the state of the system, i.e., the values of the runoff,and its first and second derivatives at time t; that is:
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j(t) = (x(t), y(t), z(t)) (3)
From the theory of diffusion processes (Cox and Miller 1965,p. 207), it can be shown that
[j(t)uj(0) = j] ; N(v, S) (4)
which means that if the initial state j = (x, y, z) is known[where x = x(0), y = y(0), z = z(0)] then the state at time tfollows a multinormal distribution with mean vector v andcovariance matrix S, where
2 3x 1 yt 1 zt /2 1 mt /6T 2v = y 1 zt 1 mt /2
(5)F G
z 1 mt
and4 3 2t /20 t /8 t /6
2 3 2S = s t t /8 t /3 t/2(6)
F G2t /6 t/2 1
The first component of the mean vector is the most importantone, since it represents the point estimate of the basin runoff.Thus, it is via this component that the performance of themodel will be measured. It is interesting to note that the co-variance matrix can be used to build confidence intervals.However, this would require the estimation of a novel param-eter, which will add some uncertainty to the model. Further-more, the objective of the Alcan company is to try to forecastaccurately the amount of water entering its reservoirs in orderto plan budgets accordingly. Hence, confidence intervals arenot critical for short-term planning.
Considering the last equation of the system given in (2),taking m = 0, and given the initial conditions, we obtain theexpression of the expected flow x(t) at time t, estimated by thefirst component of v:
2x(t) = x 1 yt 1 zt /2 (7)
This theoretical result does not enable us to perform directcomputations to estimate the runoff at a given point in timesince the variable x(t) is not measured continuously. In orderto do this, we have to discretize the continuous equation (7)of the discharge. By taking differences instead of derivatives,we obtain
dx(t) DX(t) X(t) 2 X(t 2 Dt)y(t) = ' = (8)
dt Dt Dt
where X(t) = exact value of the discharge at time t (i.e., t daysafter the initial readings on day 0). Thus, by taking Dt = 1day, we obtain the best approximation of y(t):
y(t) = X(t) 2 X(t 2 1) (9)
In the same manner, we have
dy(t) DY(t)z(t) = ' = Y(t) 2 Y(t 2 1) = [X(t) 2 X(t 2 1)]
dt Dt
2 [X(t 2 1) 2 X(t 2 2)] = X(t) 2 2X(t 2 1) 1 X(t 2 2) (10)
The previous equations enable us to estimate the expected dis-charge [X9(t 1 k)] for a k-day-ahead forecast. Using (7), weobtain
2kX 9(t 1 k) = X(t) 1 kY(t) 1 Z(t) = X(t) 1 k[X(t) 2 X(t 2 1)]
2
2 2k k1 [X(t) 2 2X(t 2 1) 1 X(t 2 2)] = 1 1 k 1 X(t)S D2 2
2k22 (k 1 k )X(t 2 1) 1 X(t 2 2)2 (11)
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Therefore, the 3D model of the diffusion process enables us,through (11), to compute the expected flow k days in advancewithout having to estimate any physical parameters related tothe characteristics of the hydrographic basin under study. Fur-thermore, the estimation of the runoff is done without the needof incorporating any exogenous variable to the system. It isimportant to note that the discretization of the expected flowleads to an equation similar to an autoregressive process. Onemust bear in mind that the diffusion model enables us to eval-uate the expected flow in continuous time. Therefore, regard-less of the step of time used in the discretization, which isdictated by the interval of time used in measuring the flow(i.e., days or hours), we always obtain the same coefficients.This is a powerful result since the coefficients of an autore-gressive model must be adjusted for time-dependent data. Us-ing the same method, but with a different system of stochasticdifferential equations, we can arrive at other formulas allowingus to estimate the expected discharge. This leads to the ques-tion of performance, and several criteria are chosen to comparethe overall efficiency of the proposed models.
The following section describes the diffusion model, namelythe Ornstein-Uhlenbeck process, that gives the best resultsamong the models considered for the river basins of the Sag-uenay-Lac-St-Jean area in Quebec.
ORNSTEIN-UHLENBECK MODEL
After testing 2D, 3D, and 4D stochastic models based ondifferent diffusion processes, we obtained the best results witha particular 2D system of differential equations called the in-tegrated Ornstein-Uhlenbeck process (Cox and Miller 1965, p.226). It is important to note that even though there are a va-riety of diffusion models to choose from, it is assumed thatthe data follow a Gaussian distribution, which therefore re-stricts the choice of the process. A 2D model seemed plausible,for if the second derivative follows a Gaussian distribution,then the first derivative can be assumed to follow one also.The original Ornstein-Uhlenbeck process was proposed as amodel for describing the velocity of a particle immersed in aliquid (Ross 1993, p. 481). The chosen system is given by
dx(t) = y(t)dt(12)Hdy(t) = 2my(t)dt 1 sdW(t)
where m and s are strictly positive constants; and W(t) is againdefined as a standard Brownian motion. From the theory ofthe Ornstein-Uhlenbeck process (Cox and Miller 1965, p.226), we obtain, after integration, the expectation vector
2mt
(13)
1 2 ex 1 y S DmT F Gv =
2mtye
where (x, y) is the initial state of the system. The expressionof the continuous discharge x(t) is given by
2mt1 2 ex(t) = x 1 y (14)S Dm
Similarly to what was done in the previous section, the dis-cretization of the above equation leads to
2mk1 2 eX 9(t 1 k) = X(t) 1 Y(t) S Dm
2mk1 2 e= X(t) 1 [X(t) 2 X(t 2 1)] S Dm
2mk 2mk1 1 m 2 e 1 2 e= X(t) 2 X(t 2 1)S D S Dm m (15)
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Thus, directly from (15), we can compute the expected flowfor a one-day ahead forecast:
2m 2m1 1 m 2 e 1 2 eX 9(t 1 1) = X(t) 2 X(t 2 1) (16)S D S Dm m
Therefore, the predicted discharge of tomorrow depends on thedischarge observed today and yesterday. As shown in (16), theonly parameter that we must evaluate, which is inherent to thesystem, is the positive constant m. The standard approach inevaluating a parameter of a stochastic process is by the methodof maximum likelihood. However, for the Ornstein-Uhlenbeckprocess the maximum likelihood method yields a transcenden-tal equation that is not tractable analytically. Since the param-eter m must represent a characteristic of the hydrographic ba-sin, we decided to obtain its optimal value by simulatingnumerous series of forecasting data and comparing them to theactual observed discharges. For a specific year (i.e., 1993), thecombined data of the three river basins under considerationserved as calibration. Next, the value obtained for m was val-idated against the data for the two other years (i.e., 1994 and1995). The process was then repeated twice using the otheryears as calibration periods. It was observed that the ‘‘best’’value for m oscillated in the interval (2.3, 2.8) without anysignificant changes in the forecasted versus actual data. Wethus established an empirical optimal value for m of 2.5 forthe hydrographic basins of the Saguenay-Lac-St-Jean.
In order to compare the aforementioned diffusion modelswith the deterministic model presently used by Alcan, we givea brief description of the PREVIS model in the next section.
PREVIS MODEL
PREVIS is a global-type conceptual model based on a clas-sical rainfall-runoff modeling approach. It was implemented in1979 after being first introduced by Kite (1978) and was con-stantly improved ever since. It uses a total of 18 parameterswhose values are obtained during its calibration period. Theseparameters, listed in Table 1 (Lauzon et al. 1997), are incor-porated into PREVIS in order to forecast the appropriate run-off. The forecasted runoff at time t, Rt, is given by
AbasR =t 86.4
t
? (RSM 1 RP 1 RPS 1 RSTOR )UH 1 RGWi i i i t2i11 tF O Gi=t2d 11UH
(17)
where Abas is the basin area; 86.4 is a conversion factor allow-ing to switch from km2mmd21 to m3s21; dUH is the length ofthe duration of the effect of the unit hydrograph; RSM is therunoff due to snowmelt; RP is the surface runoff; RPS is thesaturated surface runoff; RSTOR is the equivalent of the hy-podermic flow; UH is the value of the unit hydrograph; andRGW is the equivalent of the ground-water flow. The 18 pa-rameters of this deterministic model are intrinsically used inthe determination of the variables inherent in the formula giv-ing the forecasted runoff (Lauzon 1995).
The model is quite costly to implement since these variableshave to be monitored on a daily basis. The first major im-provement used in order to achieve a better forecast consistsof an updating algorithm. It was observed that the residuals(forecasting errors) of the basic deterministic model possessedan inherent dependent structure explained by a strong corre-lation (Bouchard and Salesse 1986). Therefore, an updatingprocedure is incorporated in the PREVIS model. This proce-dure involves modifying the forecasted values of the runofffor the present day t, or the following days, namely (i =Rt1i
J. Hydrol. Eng
TABLE 1. Parameters of Deterministic Model PREVIS
Parameter(1)
Definition(2)
X1 Intermediate soil layers flow coefficientX2 Snowmelt due to heat coefficientX3 Surface flow runoff coefficientX4 Snowmelt runoff coefficientX5 Ground-water flow runoff coefficientX6 Energy due to heat thresholdX7 Level of subsurface water at beginning of simulationX8 Reduction factor of snowmelt coefficient due to heatX9 Percolation runoff coefficientX10 Evapotranspiration coefficientX11 Mean temperature exponentX12 Daily evapotranspiration value in winterX13 Snowmelt due to rainfall coefficientX14 Reduction factor of snowmelt coefficient due to rainfallX15 Daily precipitation threshold for saturated surface flowX16 Saturated surface flow runoff coefficientX17 Peak flow time of unit hydrograph of watershedX18 Peak value of unit hydrograph of watershed
0, 1, 2, 3, 4, 5, 6), by adding a correction term in the followingmanner:
n oR = R 1 (R 2 R ) (18)t1i t1i t21 t21
where is the updated or ‘‘new’’ forecast and is then oR Rt1i t21
observed runoff of the previous day. This procedure is basedon the hypothesis that the deterministic model generates a con-stant error in response to the fluctuations of the physical pro-cess under consideration. This observation does not corre-spond to reality in long-term forecasting, but for up to aseven-day forecast, results show that it is quite valid (Bou-chard and Salesse 1986).
The second major improvement consists of an implemen-tation of a Kalman filter. By being able to reformulate thePREVIS model in a state-space representation form, the ad-dition of a filter initiating an iterative process adjusting thestate vector parameters leads to a more accurate forecast ofthe desired runoff. All the details pertaining to the state-spaceequations are given by Lauzon et al. (1997).
The following section compares the performance of theOrnstein-Uhlenbeck model with the deterministic model al-ready implemented by Alcan as well as with the initial 3Ddiffusion process. These comparisons are made using four per-formance criteria, also described in the next section.
RESULTS
The hydrographic basin of the Saguenay-Lac-St-Jean can bedivided into five important subbasins, three major reservoirs,and six powerhouses, as seen in Lauzon et al. (1997, p. 728).The present study is concerned with data relevant to the Chutedu Diable and Lac-St-Jean reservoirs and the Mistassibi River.Moreover, data from the 90 days of the spring season (April1st to June 30th) will be used in the evaluation of the sto-chastic models. It is during this period of the year that floodsoccur due to the thawing of the snow, which makes runoffforecasting much more difficult. The diffusion models aretested against the runoff values of the spring seasons of 1993to 1995, inclusively.
To evaluate a given model, four performance criteria areused. The first one to be considered is the standard error (STD)given by
p1 2STD = (19)(X 9(t) 2 X (t))Î i iOp 2 1 i=1
where p is equal to the number of days of forecasts (90);Xi(t) is the observed discharge at day i; and is the cor-X9(t)i
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responding forecast. Obviously, the ideal value of the standarderror is zero.
The second criterion is the correlation coefficient (r), givenby the following:
p
¯ ¯(X 9(t) 2 X 9(t))(X (t) 2 X(t))i iOi=1
r = (20)p p
2 2¯ ¯(X 9(t) 2 X 9(t)) (X (t) 2 X (t))Î i i iO Oi=1 i=1
where X9(t) and X(t) are the mean of predicted and observeddischarges, respectively. The closer to one r is, the more ef-ficient the model is.
Next, the Nash criterion (NC ) compares the predictionsgiven by the mathematical model to the mean value of theobservations. Its value is computed by
p
2(X 9(t) 2 X (t))i iOi=1
NC = 1 2 (21)p
2¯(X (t) 2 X(t))iOi=1
When the NC value is between 0 and 1, the forecasting modelis better than forecasting by the mean of observed values. Thecloser the NC index is to one, the better.
Finally, the fourth performance criterion used in the evalu-ation of the diffusion process models is the peak criterion(PC ):
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1/4N
2 2(X 9(t) 2 X (t)) X (t)i i iFO Gi=1
PC = (22)N
2Î X (t)iOi=1
where N denotes the number of values greater than one thirdof the observed mean flood peak. This criterion enables us tomeasure the quality of the discharge forecast during the floodseason. The objective is to obtain a peak criterion close tozero. These performance criteria will be used to select the bestmodel.
Figs. 2 and 3 show the observed discharges for the threedrainage basins, namely Chute du Diable, Lac-St-Jean, andMistassibi, for the spring of 1994 and 1995, respectively,against the one-day-ahead forecast obtained through (16). Inorder to analyze the goodness of fit of the curves, the fourperformance criteria are computed and tabulated below for thetwo years of interest (Tables 2 and 3). Also tabulated alongwith values from the integrated Ornstein-Uhlenbeck processare the values corresponding to the 3D diffusion process andto the Alcan deterministic model, PREVIS.
The stars in the different columns of Tables 2 and 3 identifythe mathematical model that gives the best result for each spe-cific performance criterion. It is clear that the integrated Orn-stein-Uhlenbeck process performs consistently better than thedeterministic model of the Alcan company. It is also muchcheaper to implement, since no parameter apart from the dis-charge has to be determined in order to make a forecast. Onaverage, the Ornstein-Uhlenbeck model gives a standard error
FIG. 2. Forecasted versus Observed Discharge and Box-Plot of Errors Showing 1st, 2nd, and 3rd Quartiles, 1994
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FIG. 3. Forecasted versus Observed Discharge and Box-Plot of Errors Showing 1st, 2nd, and 3rd Quartiles, 1995
TABLE 2. Performance Criteria for Models (1994)
Model(1)
Chute du Diable
STD(m3/s)
(2)r
(3)NC(4)
PC(5)
Mistassibi
STD(m3/s)
(6)r
(7)NC(8)
PC(9)
Lac-St-Jean
STD(m3/s)(10)
r(11)
NC(12)
PC(13)
Ornstein-Uhlenbeck 52* 0.97* 0.95* 0.17* 37* 0.99* 0.97* 0.09* 329* 0.95* 0.90* 0.20Brownian motion 80 0.95 0.89 0.28 44 0.98 0.97 0.12 514 0.89 0.64 0.56PREVIS (Alcan) 99 0.94 0.87 0.19 88 0.96 0.90 0.17 375 0.94 0.88 0.18*
TABLE 3. Performance Criteria for Models (1995)
Model(1)
Chute du Diable
STD(m3/s)
(2)r
(3)NC(4)
PC(5)
Mistassibi
STD(m3/s)
(6)r
(7)NC(8)
PC(9)
Lac-St-Jean
STD(m3/s)(10)
r(11)
NC(12)
PC(13)
Ornstein-Uhlenbeck 38* 0.99* 0.98* 0.05* 20* 0.99* 0.99* 0.02* 298 0.97* 0.95* 0.15Brownian motion 71 0.97 0.95 0.07 23 0.99 0.98 0.03 514 0.92 0.82 0.37PREVIS (Alcan) 74 0.98 0.96 0.14 54 0.99 0.97 0.13 287* 0.97 0.95 0.14*
that is 20% less than with the PREVIS model. This is quiteremarkable for a one-day-ahead forecast. Moreover, no exog-enous variable was involved in the computation of the forecastvalues. It is also clear that, although the 3D diffusion processgives relatively good results, it does not perform nearly as wellas the Ornstein-Uhlenbeck model.
J. Hydrol. Eng.
It is also important to mention that for the two years underconsideration, the mean errors of the Ornstein-Uhlenbeckmodel were computed and were found to oscillate around zero,which indicates that the model is not biased (Figs. 2 and 3).Thus, the strength of the model resides in the fact that it ismore reliable for a one-day-ahead prediction than a determin-
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FIG. 4. Forecasted (7-Day-Ahead) versus Observed Discharge and Performance Criteria (Mistassibi 1994)
istic model, with practically no cost of implementation due tothe presence of a single parameter to estimate.
However, Tables 2 and 3 clearly show that the Ornstein-Uhlenbeck model does not outperform the deterministic modelPREVIS when it comes to the peak criterion (PC). In fact, 2times out of 6, PREVIS forecasts the peak flows better thanthe stochastic model. This can be explained by the fact that,if updated carefully, rainfall-runoff modeling takes underlyingphysical processes into account and thus can predict unusualevents better. Hence, even though the absence of parameterestimation in our 2D model lowers the cost of implementation,it will affect the forecast of critical events specific to the en-vironment under consideration. For similar reasons, longer-term forecasting will not give as good results as one-day-aheadestimates. Figs. 4 and 5 show the actual runoff for the Mis-tassibi river basin, for the two years of interest, against theseven-day-ahead forecast given by the Ornstein-Uhlenbeckmodel. Also, plots of the performance criteria obtained byPREVIS and the stochastic model, for every day of the weekahead, are given. It is clear that, beyond two-day-ahead fore-casts, the 18-parameter deterministic model performs betterthan our model. Hence, the next logical step to improve ourestimation is to try several techniques of incorporating exog-enous variables into our model. We are currently investigatingthe implementation of exogenous variables through the use ofneural networks.
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CONCLUSIONS
A 2D integrated Ornstein-Uhlenbeck diffusion process wasused to estimate, one day in advance, the discharge of threesubcatchments of the Saguenay-Lac-St-Jean hydrographic ba-sin. The forecasts were evaluated using four different perfor-mance criteria in order to compare the Ornstein-Uhlenbeckmodel to other mathematical models used in forecasting. It wasseen to give better results than the actual model used by Alcan.Furthermore, only one parameter has to be estimated, com-pared to 18 parameters for the Alcan model, which reducestremendously the effort of implementation. Although this dif-fusion model performs very well for a one-day-ahead forecast,and gives an average forecast for two days in advance, it isless accurate than the deterministic model for three-day toseven-day forecasts. It seems, therefore, that the absence ofexogenous variables hampers longer-term forecasting. How-ever, the results obtained for the short-term predictions leadus to believe that, by incorporating judiciously one or severalexternal parameters, the Ornstein-Uhlenbeck model can even-tually perform better than PREVIS. Nevertheless, the Alcancompany will be able to benefit twofold from using the prob-abilistic model; firstly, by knowing more accurately the ex-pected amount of water in its reservoirs and adjusting accord-ingly, and secondly by reducing monitoring costs through theuse of a more parsimonious forecasting model.
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FIG. 5. Forecasted (7-Day-Ahead) versus Observed Discharge and Performance Criteria (Mistassibi 1995)
ACKNOWLEDGMENTS
This research has been supported by the Natural Sciences and Engi-neering Research Council of Canada (NSERC) and the Alcan Companythrough a collaborative research and development grant (CRDO167898).This support is gratefully acknowledged.
We wish to thank the reviewers of this paper for the appropriate com-ments that they provided in enhancing the value of the research.
APPENDIX I. REFERENCES
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APPENDIX II. NOTATION
The following symbols are used in this paper:
e = 2.718281828;m = infinitesimal mean;
NC = Nash criterion;N(v, S) = Gaussian distribution of vector mean v and covariance
matrix S;PC = peak criterion;
p = number of days of forecasts;STD = standard error;W(t) = Brownian motion;X(t) = mean of observed discharges at time t;
X 9(t) = mean of predicted discharges at time t;x(t) = discharge at time t;j(t) = state of system at time t;
r = correlation coefficient; ands = standard deviation.
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