Artificial neural networks for daily rainfall-runoff modelling

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1 Hydrologkal Sciences-Jo umai-des Sciences Hydrologiques, 47(6) December 2002 g 5 Artificial neural networks for daily rainfall-runoff modelling M. P. RAJURKAR Water Resources Development and Training Centre, Indian Institute of Technology (IIT) Roorkee (formerly University of Roorkee), Roorkee , India U. C. KOTHYARI Department of Civil Engineering, I IT Roorkee, Roorkee , India umeshfceffiiitr.ernet.in U. C. CHAUBE Water Resources Development and Training Centre, IIT Roorkee, Roorkee , India Abstract The application of artificial neural network (ANN) methodology for modelling daily flows during monsoon flood events for a large size catchment of the Narmada River in Madhya Pradesh (India) is presented. The spatial variation of rainfall is accounted for by subdividing the catchment and treating the average rainfall of each subcatchment as a parallel and separate lumped input to the model. A linear multiple-input single-output (MISO) model coupled with the ANN is shown to provide a better representation of the rainfall-runoff relationship in such large size catchments compared with linear and nonlinear MISO models. The present model provides a systematic approach for runoff estimation and represents improvement in prediction accuracy over the other models studied herein. Key words artificial neural network; multiple-input single-output models; nonlinear models; rai nfal l-runo ff model 1 ing Modélisation pluie-débit journalière à base de réseau de neurones artificiel Résumé Nous présentons une application de la méthodologie des réseaux de neurones artificiels à la modélisation des écoulements journaliers en période de crues de mousson, d'un grand bassin versant de la rivière Narmada, en Madhya Pradesh (Inde). La variation spatiale de la pluie est prise en compte grâce à une subdivision du bassin versant, puis grâce à un traitement en parallèle des pluies moyennes sur les sousbassins versants, considérées comme des variables d'entrée globales du modèle. Il est ainsi mis en évidence qu'un modèle linéaire entrées multiples-sortie unique, couplé avec le réseau de neurones artificiel, fournit pour de tels bassins versants de grande taille une meilleure représentation de la relation pluie-débit que les modèles linéaires et non-linéaires entrées multiples-sortie unique. Ce modèle fournit une approche systématique pour estimer l'écoulement et présente une amélioration par rapport aux autres modèles étudiés jusqu'à présent, en terme de précision. Mots clefs réseau de neurones artificiel; modèles entrées multiples-sortie unique; modèles nonlinéaires; modélisation pluie-débit INTRODUCTION The transformation of rainfall into runoff over a catchment is a complex hydrological phenomenon, as this process is highly nonlinear, time-varying and spatially distributed. A number of models have been developed to simulate this process. Depending on the complexities involved, these models are categorized as empirical, black-box, conceptual or physically-based distributed models. In operational hydrology, the system-theoretic Open for discussion until I June 21)03

2 866 M. P. Rajurkar et al. black-box and conceptual models are usually employed for rainfall-runoff modelling because the physically-based distributed models are too complex, data intensive and cumbersome to use. The conceptual models attempt to represent the known physical process occurring in the rainfall-runoff transformation in a simplified manner by way of linear/nonlinear mathematical formulations. While conceptual models have proved their importance in understanding hydrological processes, their implementation and calibration present various difficulties. In fact, there are many situations demanding the use of simple, system theoretic models such as linear black-box models, which establish a relationship between input and the output functions without considering the complex physical laws governing the natural process such as rainfall-runoff transformation. The unit hydrograph, which is a linear rainfall-runoff model, is one well-known example of such a relationship. However, these simpler models normally fail to represent the nonlinear dynamics inherent in the process of rainfall-runoff transformation. The inability of the unit hydrograph to represent the catchment operation, which is a nonlinear process, promoted the development of nonlinear black-box models. Amorocho (1973) summarized his previous works demonstrating that the functional series nonlinear models could be employed to simulate rainfall-runoff transformation. Diskin & Boneh (1973), Papazafiriou (1976) and many others have also explored the functional series as a catchment model for nonlinear black-box analysis. Muftuoglu (1984) proposed a physically realizable nonlinear runoff model, which was later applied to a larger catchment by Kothyari & Singh (1999). The multiple-input singleoutput linear and nonlinear black-box models have also been studied in detail by Liang & Nash (1988), Papamichail & Papazafiriou (1992), Liang et al. (1994), among others. A new dimension has been added to the system-theoretic modelling approach through the adoption of the artificial neural network (ANN) technique for rainfallrunoff modelling. The distinct advantage of an ANN is that it learns the previously unknown relationship existing between the input and the output data through a process of training, without a priori knowledge of the catchment characteristics. The ANN is also described as a mathematical structure, which is capable of representing the arbitrary complex nonlinear process relating the input and the output of any system. Presently more and more researchers are utilizing ANNs because these models possess desirable attributes of universal approximation, and the ability to learn from examples without the need for explicit physics. In the present study, the results come from an investigation carried out on the simultaneous application of the system-based methods and the ANN for rainfall-runoff modelling. BRIEF REVIEW Artificial neural networks offer a relatively quick and flexible means of modelling and thus applications of ANN-based modelling are widely reported in the hydrological literature (French et ah, 1992; Raman & Sunilkumar, 1995; Maier & Dandy, 1996; Coulibaly et al, 2000; Persson et al, 2001). An exhaustive review investigating the role of ANNs in various branches of hydrology and a comparison of the ANN and other modelling philosophies in hydrology is reported in a two-part publication by the American Society of Civil Engineers (ASCE) Task Committee on the Application of Artificial Neural Networks in Hydrology (ASCE, 2000a,b). The details on the ANN structure are also available in these references; hence these are not repeated here.

3 Artificial neural networks for daily rainfall runoff modelling 867 The work by French et al. (1992) demonstrates the capability of ANNs in handling the complex processes of the space-time evolution of rainfall. Artificial neural networks have also been used for streamflow forecasting and have been shown by Karunanithi et al. (1994), Dawson & Wilby (1998), Campolo et al. (1999), Zealand et al. (1999), Imrie et al. (2000), Thirumalaiah & Deo (2000) and Hu et al. (2001) to perform much better than conventional models. Halff et al. (1993) used a three-layer feed-forward ANN for the prediction of hydrographs, which opened up several possibilities for the application of ANNs to such problems. Smith & Eli (1995) used the back-propagation artificial neural network model to predict peak discharge and time to peak. This study was carried out on a synthetic catchment using simulated data. Hsu et al. (1995) presented a procedure called Linear Least Squares Simplex (LLSSIM) for identifying the structure and parameters of a three-layer feed-forward ANN, which involved multiple random starts in weight space and thus reduced the probability of finding local minima. Lorrai & Sechi (1995) reproduced river flow data using a two hidden layer network applying both mean areal and point rainfall along with temperature data. Carrière et al. (1996) developed a virtual runoff hydrograph system using the recurrent back-propagation method while Mason et al. (1996) used a radial basis function (RBF) network for rainfall-runoff modelling which provided faster training compared with the regular back-propagation technique. A study by Minns & Hall (1996) points out the importance of standardizing data as the ANN they used failed to extrapolate when required to predict out-of-range values. Shamseldin (1997) used the conjugate gradient method to train the network using data from six catchments from different climates and the ANNs showed a better performance compared with other models. Sajikumar & Thandaveswara (1999) used the temporal back-propagation neural network (TBP-NN) for monthly rainfall-runoff modelling in scarce data conditions. The results indicated an improved performance of the TBP-NN compared with the Volterra type functional series model. Tokar & Johnson (1999) reported that their ANN model had better prediction accuracy and flexibility than statistical regression and simple conceptual models. Their study also demonstrates the impact of the selection of training data on the accuracy of daily runoff prediction, while Tokar & Markus (2000) extended this study for two more catchments and compared the ANN's performance with that of conceptual models. Campolo et al. (1999) made use of distributed rainfall data, observed at different raingauge stations, for the prediction of water levels at the catchment outlet. The model results obtained by them were poor when only rainfall observations were used as the input. The model accuracies were found to improve when the water levels observed in the recent past were also used as input. Zhang & Govindaraju (2000) used a modular neural network (MNN) for the prediction of catchment runoff, and utilized Bayesian concepts in deriving the training algorithm. The performance of the MNN showed improved results compared with the standard ANN. In the present study, an attempt has also been made to model the physical process of rainfall-runoff within the framework of the ANN. For this puipose, a response function that relates rainfall with runoff has been derived first using the rainfall-runoff data in the framework of a linear multiple-input single-output (MISO) model. The values predicted using the convolution of the derived response function with the rainfall are then refined by means of the ANN so that the final computed runoff values

4 868 M. P. Rajwkar et al. Antecedent and current rainfall 1 Linear MISO Model Response function and multiple input rainfall values Multi-layer feed forward neural network Computed runoff values Fig. 1 Schematic representation showing linkages between a linear MISO and an ANN model. compare well with their corresponding observed values. A standard multi-layer, feedforward ANN is used for this, as it is well known as a universal function approximator. The schematic diagram showing the linkages between the linear MISO model and the ANN in the computational process applied here is shown in Fig. 1. DATA USED Daily rainfall and runoff data for the Narmada catchment at the Jamtara gauge and discharge site (area knr), located in the central Indian State of Madhya Pradesh, are used. A total of 10 storm events that occurred during the monsoon season of the years were selected. Table 1 shows the duration and dates of these storms, out of which the first six events were used for calibration/training of the model and the remaining four events were used for verification or testing purposes. The data selected here have also been used by Kothyari & Singh (1999) in the development of a nonlinear MISO rainfall-runoff model. The locations of the raingauge stations and catchment subdivisions are given in Kothyari & Singh ( 1999). Table 1 Selected storm events for study. Catchment Calibration period Verification period Narmada at Jamtara (1)13 July-31 August 1981 (7) 22 August-29 September 1987 Madhya Pradesh, India (2) 8 August-18 September 1982 (8) 20 July-22 September 1988 (area km 2 ) (3) 25 July-15 September 1983 (9) 12 July-30 July 1990 (4) 7 August-18 September 1984 (10) 27 August-11 September 1990 (5) 16 July-7 October 1985 (6) 6 JuIy-9 September 1986 NORMALIZATION OF DATA The most commonly used transfer function for neurons in hidden layer(s) and the output layer of an ANN is the sigmoid function, which has a bounded output range between zero and one. The actual observed outputs of the network, being outside this bounded range of the neuron transfer function, need to be normalized or rescaled such that they fall within the bounded output range. The training also slows down, making the learning ineffective if the rescaled values are near to the bounds (Ooyen & Nichhuis, 1992). Keeping this in mind, a logistic sigmoid is used here as the transfer function and the observed discharges are normalized using the equation:

5 Artificial neural networks for daily rainfall-runoff modelling 869 g = x' Qi s^nrax where Q is the normalized discharge, Q max is the maximum observed discharge, and Qi is the observed discharge. This transformation bounded the discharges in the range [0.1,0.9]. (1) METHODOLOGY As the data used are from a large-sized catchment, catchment representation was done in three different ways following Kothyari & Singh (1999). Firstly, the whole catchment was considered as a single unit and average areal rainfall values were determined using the Thiessen polygon method. Secondly, the catchment was divided into two, and then lastly into three sub-areas to consider the spatial variation of rainfall. This division of catchment into sub-areas was based on the hydrophysiographical homogeneity indicated by vegetation, slope, soil type and the long-term rainfall isohyetal maps for the catchment. Also, the average areal rainfall for these subareas was calculated by the Thiessen polygon method. MODEL DESCRIPTION Linear MISO The output function, Q h of a lumped linear system is related to the input function P, received by it, by the causal relationship (Liang & Nash, 1988; Singh, 1988): m where U-, denotes the discrete series of pulse response ordinates, which sum up to yield the gain factor (equal to one for mass conserving systems); m is the memory length of the system; t denotes the time of the sampling and e, designates the model output errors (i.e. the residuals). Equation (2) represents the classical unit hydrograph type of model, used in the present case, between total rainfall and total runoff. The parametric forms of relationships representing the pulse response ordinates are derivable from the measurable catchment characteristics (Nash, 1957). The above model is sufficient for small catchments where the rainfall distribution can be assumed to be uniform. The larger the catchment, the higher is the probability of the assumption of uniform rainfall being violated. So, the large catchments are considered as assemblies of sub-catchments, each assumed to have a uniform rainfall distribution. Equation (2) can be generalized for the catchment, divided into J sub-areas as:./ m S, = II Ptx Uf'+e, (3) where j = 1, 2,..., J designates the sub-area. Conventionally, equation (3) is used

6 870 M. P. Rajurkar et ai. without the inclusion of the error term and is written in matrix form: [Q] = [P] [U\ (4) where Q is the vector of runoff values (outputs), U is the vector of response function ordinates and P is the matrix of effective rainfall values (inputs). The solution of equation (4) for U values can be obtained using the ordinary least squares method. The method of smoothed least squares described by Bruen & Dooge (1984) can be used when some of the ordinate values of U computed by the ordinary least squares method are negative. When used for scenarios having more than one input, equation (4) is known as the linear MISO model. Nonlinear MISO Kothyari & Singh (1999) proposed a multiple-input single-output nonlinear rainfallrunoff model which accounted for the spatial variation of rainfall in a large catchment. This model is referred to as the nonlinear MISO model. ANN-MISO It is well known that the rainfall-runoff transformation cannot be adequately described by a simple, linear, and time-invariant system like the one represented by equation (4). Amorocho (1973), Muftouglu (1984), Kothyari & Singh (1999), among others, proposed that the vector of coefficients U of equation (4) is not constant but varying as a function of current and antecedent rainfall. This indicates that the relationship between rainfall and runoff is nonlinear. Therefore the residuals, i.e. the differences between the observed discharges and the discharges estimated by equation (4), exhibit evidence of persistence and seasonality (Kachroo & Natale, 1992). Thus equation (3), when used without the inclusion of the error term, computes the values, which are termed here as RI,: in Rl, = X p u- M, U > ( 5 ) The RI, values are also considered to be related nonlinearly to the corresponding observed values of runoff. The ANN technique has been found to be very successful in representing this complex nonlinear relationship. It is therefore used for transforming the RI, values, obtained by means of equation (5), into Q, values. The values of response function ordinates, /,-, were calculated separately for the one, two and three parallel input scenarios. The neural network was applied next for those three different input scenarios, each of which used RI, computed by equation (5) as input. The procedure can be referred to as the ANN-MISO model. MEMORY OF THE CATCHMENT Determination of the memory of the catchment is the critical part of a rainfall-runoff study in which current and antecedent rainfall values are used as inputs. Response

7 Artificial neural networks for daily rainfall runoff modelling 871 functions obtained for the final selected memory lengths for the three cases studied are plotted in Fig. 2(a)-(c). It can be seen from these figures that a memory of 6 days provided more realistic shapes of the response function ordinates in one- and two-input cases, but, when the catchment was divided into three sub-areas, the memory period combination of 6, 5, and 6 days respectively for the three sub-areas gave the best reproduction of the response function ordinates with the shape of a unit hydrograph. (a) 0.5 (b) (c) 3 4 Memory () Fig. 2 Computed response unction ordinates for (a) single input, (b) two inputs, and (c) three inputs. It should also be mentioned here that response function shapes as shown in Fig. 2 can be parameterized using the gamma function of Nash (1957). Values of the para-

8 872 M P. Rajurkar et al. meters of such a function can be estimated by making use of the catchment characteristics (NERC, 1975). In this context, the model schematized in Fig. 1 could also be applicable to an ungauged catchment. MODEL APPLICATION AND RESULTS For each input scenario, various configurations of ANN were trained by varying the number of nodes in the hidden layer. For each ANN configuration, the mean squared error (MSE) values were worked out first during the training and then with the same weights during the testing. Next, the target MSE value was lowered and the network was trained again and the corresponding MSE value in testing was worked out. This procedure was continued until the MSE value in testing was found to increase instead of decrease. This indicated that the network was becoming overtrained and would continue to do so, if trained further with a lower MSE target. Such an ANN would perform very well in training but would fail to generalize when given an unknown input. The final selection of the ANN configuration was based on the fact that it should have minimum complexity and maximum possible performance. The ANN results were transformed back to the original domain and the Nash-Sutcliffe efficiency (E') values were worked out for training as well as test data. Table 2 shows the details of the best performing network configuration along with the values of the performance indicators, namely root mean square error (RMSE) and E~, for the different input scenarios considered. As can be observed from this table, the maximum value of E~, both in training and testing, was obtained for the scenario in which the catchment was divided into three hydro-physiologically homogeneous sub-areas. A comparison of the relative performances of a linear MISO model, a nonlinear MISO model and the ANN-MISO model is presented in Table 3. It can be seen from this table that performance of the ANN-MISO model is much better for the three cases considered and the efficiency of all the models increased with the increase in the number of inputs. Table 2 Variation of and RMSE values for selected ANN configurations. Case Description ANN structure: f-h-o* 1 Single input Two inputs Three inputs * /: number of neurons in input layer; output layer. Training: RMSE E 1 (%) Testing: RMSE E 2 (%) H: number of neurons in hidden layer; O : number of neurons in Table 3 Comparative performance of various model s. Case Description Linear MISO model: Calibration Validation 1 One input 70.6% 2 Two inputs 71.0% 3 Three inputs 71.5% * Source: Kothyari & Singh, % 69.3% 75.3% Nonlinear MISO model*: ANN-MISO model: Calibration Validation Training Testing 73.9% 78.4% 79.1 % 74.5% 78.4% 79.2% 78.0% 80.6% 80.2% 78.9% 80.2% 83.2%

9 Artificial neural networks for daily rainfall-runoff modelling 873 ANN-MISO (Single Input) ANN MISO (Two Inputs) ANN MISO (Three Inputs) 8 8 ( a) Observed Runoff (b) Observed Runoff (c) Observed Runoff Linear MISO (Three Input) Nonlinear MISO (Three Input) 8000 (d) Observed Runoff (e) Observed Runoff Fig. 3 Scatter plots for different models (validation). Figure 3(a)-(c) shows the scatter plots of observed and computed runoff values for the validation periods for the ANN-MISO model. The scatterplot is well spread over the ideal line for the three-input scenario (Fig. 3(c)), whereas, in the case of the one- and two-input scenarios, the plot is shifted towards one side. The shift from the ideal line indicates the possibility of systematic errors (Aitken, 1973). This justifies the division of a large catchment into smaller, homogeneous sub-areas, in which the rainfall is subsequently considered as an independent and parallel input signal to the system. Similar scatter plots for the linear MISO model, shown in Fig. 3(d), exhibit a larger scatter and shift from the ideal line, thus indicating the existence of persistence and time variance in the error values. The scatter for the nonlinear MISO model (Fig. 3(e)) is obviously less than that of the linear MISO model (Fig. 3(d)), but is greater than that of the ANN-MISO model (Fig. 3(c)). These disadvantages are considered to have been accounted for by the application of the ANN-MISO model as is revealed by relatively more symmetrical scatter in Fig. 3(c). As expected, a very good match is obtained between the observed runoff values and those computed by the ANN-MISO model for the training data in all the input scenarios. The plots between the corresponding observed storm runoff hydrographs and those computed using the ANN-MISO model for validation data are presented in Fig. 4. The performance of the ANN model for the validation period using a single rainfall input is found to be less satisfactory as the time to peak is poorly estimated, whereas the magnitude and time to peak are both predicted to a greater degree of accuracy for the three-input scenario; the performance of the ANN-MISO model for the two-input case lies in between. The results of the three-input scenario for the linear MISO, nonlinear MISO and the ANN-MISO models for the validation periods are compared graphically with the

10 874 M. P. Rajurkar et al. Storm No. 7 Storm No. 8 Storm No. 10 Fig. 4 Comparison of observed and computed runoff for ANN-M1SO model (validation period). corresponding observed runoff values in Fig. 5. These and other such figures for training storm events (not shown here) revealed that the time to peak of the hydrographs was well predicted by all the models. The linear MISO model always underpredicted the hydrograph peak values. However, the peak discharge was well predicted by both the nonlinear MISO and the ANN-MISO models. Also, it was noticed that the runoff values predicted by the ANN-MISO model were closest to the corresponding observed values in both the rising and the falling limbs of the hydrographs. Overall, the performance of the ANN-MISO model was found to be the best, as also revealed by the E 2 values presented in Table 3. CONCLUSIONS The application of an ANN and a system-based model is demonstrated for the transformation of rainfall into runoff in a large size catchment. The catchment was represented first as a single unit and then it was divided into two and three sub-areas respectively to account for the spatial variation of rainfall. It is observed that coupling of the ANN with a multiple-input single-output model predicted the daily runoff values

11 Artificial neural networks for daily rainfall-runoff modelling Storm No. 7 -Observed - -Linear M ISO Nonlinear M ISO X ANN-MISO Storm No, 8 Observed Linear y ISO Nonlinear M ISO * ANN-yiSO a 3ooo Storm No Observed Linear M ISO Nonlinear M ISO 4000 * ANN-MISO Storm No. 10 Observed - - Linear M ISO Nonlinear M ISO -K ANN-MISO uays uays Fig. 5 Comparison of observed and computed runoff by various models for three-input scenario (validation period). with high accuracy, both in the training and the validation periods. The generalization capability of the ANN model improved with an increase in the number of inputs from one to three, for which the RMSE is a minimum (358.43) and the corresponding E~ is a maximum (83.21%), as can be noted from Table 2. The linear response functions derived from the data were found to have physically realizable shapes that can be parameterized using gamma type functions. Therefore, it is hoped that the methodology of runoff estimation using the ANN can be extended to catchments for which the gauge and discharge records are nonexistent. Acknowledgements The authors wish to sincerely thank the anonymous reviewers whose comments greatly improved the quality of this paper. REFERENCES Aitken, A. P. (1973) Assessing systematic errors in rainfall-runoff models. J. Hydro!. 20, ASCE Task Committee on Application of Artificial Neural Networks in Hydrology (2000a) Artificial neural networks in hydrology, 1: preliminary concepts. J. Hydrol. Engng ASCE 5(2),

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13 Artificial neural networks for daily rainfall-runoff modelling S77 Tokar, A. S. & Markus, M. (2000) Precipitation runoff modeling using artificial neural networks and conceptual models. J. Hydrol. Engng ASCE 5(2), Zealand, C, Burn, D. 11. & Simonovie, S. P. (1999) Short term streamflow forecastine using artificial neural networks. J. Hydrol. 214, 32^18. Zhang, B. & Govindaraju, R. S. (2000) Prediction of watershed runoff using Bayesian concepts and modular neural networks. Wal. Resour. Res. 36(3), Received 30 May 2001; accepted 17 March 2002

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