Soussou Sambou1, Seni Tamba2, Clement Diatta1 and Cheikh Mohamed Fadel Kebe3
　　1. Department of Physics, Faculty of Science and Technics, The University of Cheikh Anta DIOP, Dakar 99000, Senegal
　　2. Department of Civil Engineering, The Polytechnical School of Thiès, Dakar 99000, Senegal
　　3. The Polytechnical High School, Dakar 99000, Senegal
　　Received: September 2, 2011 / Accepted: November 2, 2011 / Published: January 20, 2012.
　　Abstract: Heavy floods occur frequently in the Senegal River Basin, causing catastrophic flooding downstream the river rating station of Bakel. Anticipating the occurrence of such phenomena is the only way to reduce the resulting damages. Flood forecasting is a necessity. Flood forecasting plays also an important role in the implementation of flood management scenarios and in the protection of hydro electric structures. Many methods are applied. The most complete are based on the conservation laws of physics governing the free surface flow. These methods need a complete description of the geometry of the river and their implementation requires also huge investments. In practice the river basin can be considered as a system of inputs-outputs related by a transfer function. In this paper the authors first used a multiple linear regression model with constant parameters estimated by the ordinary least square method to simulate the propagation of the floods in the upstream part of the Senegal river basin. The authors then apply statistical and graphical criteria of goodness-of-fit to test the suitability of this model. Three procedures of parameters updating have then been added to this linear model: the Kalman filter method, the recursive least square method, and the stochastic gradient method. The criteria of goodness-of-fit used above have shown that the stochastic gradient method, although more rudimentary, represents better the flood propagation in the head basin of the Senegal river upstream Bakel. This result is particularly interesting because data influenced by Manantali Dam are used.
　　Key words: Hydrology, multiple linear regression models, Kalman filtering, recursive least squares, stochastic gradient, flood forecasting, Senegal river head basin.
　　 1. Introduction??
　　Free surface flow as it occurs in natural streams has very early attracted a great interest among the hydrologist: The proximity of rivers has always favoured the concentration of economic and human activities [1]. Free surface flow in rivers results in precipitations that falls in the upstream of watershed streams. The description of the formation and the propagation of flood require a prior understanding of the hydrological cycle. Various mathematical models have then been created [2-4]. The real time monitoring of flood propagation in river basins is primarily intended to protect people against the damages caused by floods and to mitigate their effects. It also ensures the safety of hydro power installed on the rivers, and allows to plan the different rules for managing these structures [5, 6]. The Saint-Venant equations [7] were the first tools for modelling free surface flow. The lack of analytical solutions has encouraged the hydrologists to search for numerical solutions, so that many numerical schemes have been developed [8, 9]. In practice, the Saint-Venant equations are not used for flood forecasting: The description of the river bed is generally not sufficiently accurate; the flood hydrographs for their validation are inevitably affected by errors. Linear models benefit from their simplicity and for the good knowledge of their underlying theoretical aspects. These models ignore many aspects of water flow like the geometry of the stream and the hydraulics features of the flow. They are by the way particularly useful when one is interested only by flood forecasting. This explains the development of hydrological input-output models or black-box models: stochastic processes, multilinear regression models[10, 11], auto regressive models [12]. In addition to these linear models, the connectionist models have emerged in the category of input-output models [13]. The multi layer neural networks with adjustable parameters by back-propagation are one of the most used [14]. The irregularity of the inter-annual river floods have led the Senegal river riparian states meeting within the OMVS (Organisation for the Development of Senegal River) to build Manantali and Diama Dams on the Senegal river. Manantali Dam, located on the Bafing tributary, is a storage dam. Diama Dam, located in the valley, allows to stop sea water intrusion and to raise the level of the water in the river for agricultural purposes. The regime of Senegal river have been completely modified. The management of the Manantali storage dam and the protection of the populations against the floods effects require the development of a tool for flood forecasting[5]. To account for the configuration of the hydraulic system upstream from Bakel, the authors have selected a multiple inputs one output system to represent the river basin. The inputs of the system consist of daily average flows observed in the uncontrolled tributaries (Bakoye and Faleme), and from the Manantali dam. The daily flows at Bakel are the outputs of the system. A multilinear regression model is then defined. Its parameters are initially assumed to be constant and estimated by the ordinary least squares method. In fact, the free surface flow is highly non linear, so that adaptive models are often applied [15]. This led us to add three adaptive procedures to the initial model: standard Kalman filter[12, 14, 16, 17], recursive least squares method [10, 11, 18] and the stochastic gradient method [11, 19]. While the first two are widely used in hydrology, the latter is much less.
　　 2. Materials and Methods
　　While determining support for the management of the Manantali reservoir during low flows, Bader [5] assumed that the daily flow at Bakel passing day j can be calculated from the sum of flows passing through Oualia, Gourbassi and Manantali day3 j?. This scheme is based on the model of Lamagat [20]. In the following, the forecasting network consists of three explanatory stations (Manantali, Gourbassi, Oualia) and one explained station, Bakel located at the outlet of the basin. The model of the general input-output linear system has been chosen as flood forecasting model (Eq. (1)).
　　is a (3 × 1) column vector of parameters model at time t,
　　t? is a scalar corresponding to the model error at time t.
　　The parameters of the model have first been estimated using ordinary least squares method(OLSM). Then, to account for the non-linearity of the processes generating free surface flow in river, the authors have added updating procedures to the initial model to reassess the parameters for real time forecasting. The authors have selected Kalman filtering, recursive least square, stochastic gradient methods. The authors make a brief overview of these methods.
　　2.1 Multiple Linear Regression Model without Updating Parameters Using Ordinary Least Square Method (OLSM)
　　The authors consider that at time k, the exact parameters k? and the vector kz containing all observations at explanatory stations are known. The flow at the outlet station Bakel at time k + δ can be calculated as follow:
　　?? (2) In fact, the vector k? is not yet known at that time. An estimated value, ?k, is used instead, giving the following estimation ?kx?? of kx??:?
　　(5) where, tE is a (t × 1) column vector representing the differences between values obtained from Eqs. (2) and(3), t X?? a (t × 1) column vector of observations at the outlet from time 1 + δ to time t + δ, and tZ a (t ×3) matrix of observations at explanatory stations from time 1 to time t. In the OLSM, the criterion to be minimized is the sum of the squared deviations between calculated and observed values. This criterion can be calculated as follow: derivatives of the resulting expression with respect to the components of the vector ??, the authors obtain model parameters as:
　　PZ Z?????? (8)
　　2.2 Multiple Regression Models with Updated Parameters
　　In all above, the model parameters were assumed to be constant all along the simulation of the flood forecasting. In fact the transitory nature of the flood doesn’t agree with the use of constant parameters so that the parameters of the model need to be updated at each time step. This is possible in real time. At the end of each forecast date, observations are available at the outlet of the system and at all explanatory stations. This information can be used to update the model parameters. Several adaptive procedures are used in real time flood forecasting. The more frequent are Kalman Filter and recursive least squares methods. The authors add to these the well known stochastic gradient that is much less applied in hydrological forecasting. The common feature of these methods is that the model parameters required to forecast the outlet flow at time t + δ are predicted at time t. When time t + δ occurs, the observations available at both explained and explanatory stations are used to update these parameters. The authors propose in what follow an overview of these methods.
　　2.2.1 Updating Parameters with Kalman Filter [21, 22]
　　The watershed is considered as a dynamic physical system. Temporal evolution of this system is described by the observed variables and hidden variables (corresponding here to state or internal parameters of the model.) This evolution is represented through a space state model, defined in the discrete case, by:
　　? A state equation describing how the various
　　(9)
　　? A measurement equation describing how the measurements are generated by the various states and their residuals
　　??? (10)
　　In Eqs. (9) and (10), t???and t? are column (3 × 1) vectors representing the state at time t?? and t, tA a (3 × 3) matrix relating the state at time t?? to the state at time t, tC is a (1 × 3) row vector relating the state to the measurement, tw a (3 × 1) column vector representing the state error, tv a scalar representing the measurement error. tw and tv are assumed to be independant, white and gaussian noises. The covariance and variance matrix of these errors are given by: Erepresents the expected value.
　　The error covariance matrix of the estimation of thet? is the following (3 × 3) matrix,/
　　????? (14)
　　In Eq. (14), t?is a column (3 × 1) vector representing the error on the estimation on the t?, and
　　in Eq. (13), ??.
　　Erepresents the expectation.
　　Formulation of the Kalman filter:
　　In our application, the authors let tAI? the identity matrix, and ttCz?. The Kalman Filter works in two steps. At time t, a posteriori state estimate of the system t?, /?t t, and the corresponding a posteriori error convariance matrix, /
　　P, are calculated given measurements vector tz . These quantities are used to make a priori state estimation of the system at time t?? (Eq. (15)) and to determine
　　(16)
　　An a priori prediction of the measurement, (here the flow at Bakel at time t??) is then performed by Eq.(17) using Eq. (15).
　　(17)
　　At timet??, measurements at the outlet of the river basin, tx??, and at all explanatory stations of the forecasting network, tz??, are available. The observation at the outlet is used to calculate the forecast error (Eq. (18)):
　　(18)
　　This error can be used to update the parameter vector of the model at time t?? (correction step) as follow:(19) where, tKF?? is the column vector (3 × 1) called innovation that minimizes the a posteriori error covariance matrix /ttP???? [23]. It is calculated by:
　　(21)
　　To start the calculation, the authors assume that tR and tQ are known and constant. The authors are given an a posteriori estimation of the parameters,?iiP??. The authors then apply Eq. (17) to forecast outlet flow /?iix?? at time i??. At the end of the time i??, the forecast error,(Eq. (18)), the gain (Eq. (20)), the updated parameters(Eq. (19)) and the associated error covariance matrix(Eq. (21)) are calculated using the available the calculation is initiated, and goes on at every instant. The authors indicate in Table 1 an example of a sequence of calculations corresponding to a 3?? days lead forecasting time.
　　2.2.2 Recursive Estimation of Parameters: The Recursive Least Square Method [19, 24]
　　The authors assume that the vector tz and an ?t estimation of the parameters vector t? are known at time t. An a priori prediction of flow at Bakel at time t??can be calculated as follow:?
　　?? (22)
　　At the end of time t??, vectors tz?? and tx?? are known. If t??? is the actual parameter vector at that time, the authors can write
　　(23)
　　The authors explore the relation between ?t and?t?? estimated values of t? and t??? using available information at timet??. According to the report of Binet [25], the authors can write:
　　0 0 respectively a column vector [(t + 1) × 1] and a [(t + 1)× 3] matrix containing the total information from time 1 to time t, and the measures tx?? and tz?? at time t??. By analogy with Eq. (7), ?t??, the estimation of vector t??? is given by:
　　(37)
　　The sequence of the calculation is similar to that of the Kalman filter.
　　2.2.3 Recursive Estimation of Parameters: Stochastic Gradient Method [19]
　　As for the recursive least squares method, the estimate of the t vector, using all information available up to time t is used to forecast flow ??
　　(38)
　　At the end of time t??, measurements at the outlet station, tx??, and at explanatory stations, tz??, are known. These are in relation through the actual parameters, but unknown parameters by Eq. (39) x??, the parameter vector t??? describes a hyper plane whose equation is (39). In the stochastic gradient method, the best estimate ?t?? of this vector is achieved by minimizing the sum of squares deviations between parameters ?t???and?tat time t?? and t with the constraints of equality between observations and forecasts. This last constraint is taken into account by the Lagrange multiplier ?. In the formulation the authors have chosen, the objective function is set as:
　　2.3 Quality Criteria Used for Calibration and Validation
　　The results of forecasting were tested using two types of criteria:
　　? Graphic criteria (comparison of observed hydrograph with calculated hydrographs);
　　? Statistical criteria, of which the authors have retained the coefficient of Nash [26]. This dimensionless criterion is represented by Eq. (46).
　　of the squared deviations between calculated flow rates ix and mean flow rates calculated over the period used
　　x ; N is the size of the sample. The Nash
　　criterion is as close to 1 as the model is correct, that is to say as F2 is very small.
　　To this global criterion, the authors added a local criterion, the relative error peak (Eq. (50)):
　　where max obsQ is the maximum streamflow observed for a given year, and maxQ the maximum flow calculated for the same year.
　　 3. Results and Discussion
　　 3.1 Area of Study and Data Presentation
　　The upstream part of the Senegal river basin covers an area of 218,000 km2. The climate is Sudanese in the South in the foothills of Fouta Djallon, and Sahelian downstream from Bakel. The Senegal river is formed by the association of the rivers Bafing (760 km long) and Bakoye (640 km) nearly Bafoulabe. The Bafing provides most of the Senegal runoff (40% to 60% of contributions). It raises in the Fouta Djallon mountains, in the Republic of Guinea, 800 meters above the sea level. The Bakoye river source lies near the southern limit of the Manding Plateau, 500 m above the sea level. The river system is then completed by the Faleme (650 km), and by some smaller tributaries such as Karakoro and Kolombina and Gorgol [27]. Bakel station controls almost all of the contributions in the lower valley. The river has a fairly pure tropical hydrological regime. It is characterized by a unimodal annual flood which runs from July to November, followed by a dry period from September to June.
　　The Manantali dam controls almost half of the flow through Bakel. Commissionned in 1978, it is intended to generate electricity and to regulate the river for several applications [5].
　　3.2 The Forecasting Network
　　For a river basin seen as a dynamic input-output system, the forecasting network includes:
　　One or more explanatory stations, containing the inputs of the model;
　　One explanation station at which the forecast is calculated.
　　In the case of our study, the authors have chosen as explanatory station three stations located upstream Bakel, located at main tributaries: Gourbassi, on the Faleme river, Oualia on the Bakoye river, Manantali, on the Bafing river at the outlet of the dam. Bakel station, on the Senegal is the explanation station (Fig. 1). The same network has been used by Bader [5].
　　3.3 Available Data
　　Database of ORSTOM (now IRD) and OMVS provided the average daily flow series used in this application. To account for the effect of the Manantali Dam, the period from 1988 to 2006 has been selected. This period corresponds to the artificial river regime. 3.4 Multilinear Regression Model without Updating the Parameters
　　3.4.1 Estimation of the Model Coefficients and Lead Forecasting Time
　　The model of Eq. (1) was selected. The parameters of the models are estimated using as period of calibration:
　　The observations of the year i (case 1);
　　The observations from year one to year i (case 2).
　　For each of these two cases, the authors make lead forecasting time ? vary from 1 to 10 days. Then Eqs.(7) and (46) are used to calculate the corresponding model parameters and Nash criterion. Then model parameters corresponding to maximum value of Nash criterion are considered as optimal parameters to be used for forecasting. In Table 2, the authors present results obtained for the two cases. The authors denote that:
　　For case (2) Nash criterions are generally higher and parameters more stable. Corresponding optimal lead forecasting time is constant and equal to three days;
　　For case (1), Nash criterions are lower, and lead forecasting varies from 1 to 4 days although the most common value is three days.
　　As an example, the authors show in Fig. 2 the time evolution of optimal lead forecast time versus time for calibration period 1988 to 2000. The maximum of the curve corresponds to the optimal prediction time, that is three days.
　　3.4.2 Simulation of Flood Forecasting
　　The simulation of the flood forecasting of year i was done according to the cases (1) and (2) with:
　　Parameters estimated with observation of year
　　1 i: case (1);
　　Parameters estimated with observations from year 1 to year 1i: case (2).
　　Results are presented in Table 3 (columns 2, Ordinary Least Square, OLS; column 3, Ordinary Least Squares for Cumulated years, OLSC; column 4, Recursive Least Square, RLS; column 5, Kalman Filter, KF; column 6, Stochastic Gradient, SG). Note that once again, estimated parameters corresponding to case (2) provide the best Nash criterions.
　　3.5 Multilinear Regression Model with Updated Parameters
　　3.5.1 Initial Conditions
　　Parameters estimated in case (2) and corresponding lead forecasting time have been used as initial conditions for the three updating methods
　　For recursive least square method, the covariance matrix is estimated from the 1i? years of observations (Eq. (8))
　　For Kalman filter procedure, both matrix of covariance of the error on the a posteriori estimation and matrix of covariance model errors were also determinated from observations made on the 1i? years preceding the year i. The covariance of forecast errors tR was estimated by the method of trial and error using the 1i? years of observations. The final value is the one that corresponds to the maximum value of the Nash criterion;
　　For the stochastic gradient method, the term RQ of the denominator of Eq. (45) was also determinate the trail and error method, using the observations made on the 1i? years preceding the year i. Once again the final value ist the one that maximizes the Nash Criterion. According to the Eq. (45), the dimension of RQ
　　ms.
　　3.5.2 Simulation of Flood Forecasting with Updated Parameters
　　For each of the three updating parameters methods, the initial conditions are first specified on the i-1 years of observations and then used to simulate flood of the year i. Global and local criteria are calculated for each year and presented in Table 3 for Nash critera calculated using Eq. (46), and in Table 4 for relative peak errors calculated using Eq. (50).
　　To facilitate the comparison, the authors focuse on the Nash criterion. The cumulative distribution function of this criterion and it interannual changes are shown in Fig. 3. In Fig. 3a, the authors denote that all methods of parameters updating make a significant improvement of the Nash criterion with comparison of the Ordinary Least Square Method. But compared with each other, the stochastic gradient method is by far the most effective (80%). The Kalman filter and the least squares methods have very similar performance within the meaning of the Nash criterion. In Fig. 3b, the authors note that for some years, the criteria of Nash is very high, and the gap between these criteria is very low, for all methods (e.g. year 2000). For these years, the update does not bring any significant improvement over the ordinary method of least squares. For other years against, the criteria of Nash are lower, and the differences between the values of these criteria is very significant for all methods. This is for example, the case of the flood of year 2006.
　　The authors represent in Figs. 4 and 5 the calculated and observed hydrographs at Bakel for year 2000 and 2006. Fig. 4 shows that year 2000 is a high flood at Bakel (maximum flow rate is 2.831 m3/s), and the observed hydrograph is regular. Nash criteria are generally high (above 0.96). There is a good adequation between calculated and observed flows for all methods, apart from an underestimation of peak flow by the stochastic gradient method.
　　Fig. 5 shows that for year 2006, the flood is low at Bakel (the maximum flow rate is 1.095 m3/s), and the observed hydrograph is very irregular. The Nash criterion corresponding to the Ordinary Least Square is weak for this year (0.7598). The adaptive procedure contribution is more important, particularly for the stochastic gradient method with a Nash criterion equal to 0.8581. It appears from this analysis that the importance of the flood at Bakel and the shape of the hydrograph at this gauge station play a major role in the performance of the updating procedures.
　　In Fig. 6 the authors represent the cumulative distribution function of the relative error of the peak, and the temporal evolution of this goodness of fit criterion. Fig. 6a shows that this criterion is closer to zero for the stochastic gradient (about 70%). More generally, the range of variation of this criterion is lower for this method compared to other updating methods.
　　3.6 Summary of Results
　　Compared to the ordinary least squares, all adaptive procedures provide a significant improvement of quality criteria, both global (Nash criterion) and local(relative error of peak) (Table 4). Comparison of different updating methods between themselves shows that the most rudimentory, here the stochastic gradient method, suits better the multiple regression model, where the authors expected the method of Kalman filter that is based on a more rigorous theoritical basis. To better understand this result, the authors have examined the expression of the gain for each updating procedure (Eq. (20) for the Kalman filter, Eq. (37) for the recursive least square method, Eq. (45) for the stochastic gradient method). The Kalman filter gain expression contains the statistics of the errors on the state equation, on the measurement equation and on the parameters estimation. The expression of the gain of the recursive least squares method contains only the statistics of the errors of the measures through the inverse of the matrix of covariance of these errors. The stochastic gradient method is by the way more flexible. It does not involve explicitely any of these statistics of the errors. It thus appears that the difficulty of calculating the statistics of the various errors explains the better performance of stochastic gradient method over the other methods used in this paper. The authors also plotted the time evolution of the parameters for each updating method for flood 2006 (Fig. 7). The parameters vary slightly for the recursive least squares method (Fig. 7a) and for the Kalman filter method (Fig. 7b). For the stochastic gradient method, which uses the same initial parameters, the variation of the parameters is more noticeable (Fig. 7c). It is therefore apparent that the difficulty in determining the statistics of the different errors is the reason of the disadvantage of other methods compared to the stochastic gradient method.
　　 4. Conclusion
　　The use of an adaptative procedure to update the parameters has improved the values of the Nash criterion from the ordinary least square method. Comparison of different updating methods of increasing complexity shows that a simple prediction method, even naive, the multiple linear regression method, coupled with the stochastic gradient method for updating parameters fits well in predicting short-term flood at Bakel station, using data from upstream explanatory stations. The lead forecasting
　　time of three days is satisfactory compared with the requirements of real-time management. However, a ranfall runoff model could be coupled to this multilinear regression model with stochastic gradient updating model to increase the lead forecasting time.
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