APPLICATION OF ARTIFICIAL NEURAL NETWORKS FOR RAINFALL-RUNOFF MODELLING D. NAGESH KUMAR and ABHIJIT RAY Department of Civil Engineering Indian Institute of Technology Kharagpur -721 302, India. ABSTRACT Rainfall-Runoff models are mostly empirical in nature demanding the knowledge of a large number of catchment parameters. On the contrary Artificial Neural Networks (ANN) can be deployed in cases where the available data is limited. The present work involves the development of an ANN model using Backward Propagation algorithm. The hydrologic variables used were Rainfall, Soil Moisture, Evaporation and Runoff of monsoon months for a specific period. Model 1 involved the training of the ANN model with Rainfall values only, the output being the Runoff. Whereas, in Model 2 the training set consisted of Rainfall, Soil Moisture and Evaporation values, Runoff being the desired output. Effect of number of layers in the network is also studied. A comparison of the performance of the two models is carried out. The model yielding the least error is recommended for simulating the Rainfall-Runoff characteristics of the watershed. The ANN model is applied to Lekkur watershed in Tamilnadu for which the hydrologic data were available for 9 years. With the developed ANN model Runoff values were predicted and they compared well with the observed values.

INTRODUCTION Neurons are nerve cells and neural networks are networks of these cells. The cerebral cortex of the brain is an example of a natural neural network. Somehow, such a network of neurons thinks, learns, feels and remembers. Many attempts had been made in the past to build models to study such neural networks. There are two major types of models - biological and technological. In biological modelling the goal is to study the structure and function of real brain in order to explain biological aspects such as behaviour. In technological modelling the goal is to study brains in order to extract concepts to be used in new computational methodologies. The latter viewpoint is taken by several investigators working in the area of artificial neural networks and nurocomputers (Hecht-Nielsen, 1988) The artificial neural network (ANN) approach differ from the traditional approaches in stochastic hydrology in the sense that it belongs to a class of data-driven approaches as opposed to traditional model driven approaches. In this paper, a neural network computer program was developed to carry out Rainfall-Runoff modelling of Lekkur catchment area in Tamilnadu. The neural network was developed using the generalized delta rule for a semi-linear feed forward net with error back propagation. The program code was written in C in UNIX environment. The neural network model was treated as a Black Box, as the relationships between the physical components of the catchment were not to be fed. Model 1 involved the training of the ANN model with Rainfall values only, the output being the Runoff. Whereas, in Model 2 the training pairs consisted of Rainfall, Soil Moisture and Evaporation values, Runoff being the desired output.

COMPUTATIONS IN ANN The computational process associated with an ANN is as follows: An artificial neuron (AN) receives its inputs from a number of other ANs or from the external world (Lippmann, 1987). A weighted sum of these inputs constitutes the argument of an activation function. This activation function is assumed to be nonlinear. Hard limiting threshold, i.e., either the step or signum function, and soft limiting threshold, i.e. sigmoidal, are the three most often used forms of non-linearities. The resulting value of the activation function is the output of the AN. This output is distributed along the weighted connections to other ANs. The components of an input pattern constitute the inputs to the node in layer i. The outputs of the nodes in that layer may be taken to be equal to the inputs. The net input to a node in layer j is netj = wkj oj The output of node j is oj = f (netj), where f is the activation function. For a sigmoidal function we have oj

1 1e

(net jj)/0

The parameter j serves as a threshold or bias and o is a constant to modify the shape of the sigmoidal function (Yoh-Han Pao, 1985). The semilinear feed forward net was proposed by Rumelhart et al.(1986). The back-propagation training algorithm is an iterative gradient algorithm designed to minimize the mean square error between the actual output of a multilayer feed-forward perceptron (Vemuri, 1988) and the desired output. It requires continuous differentiable non-linearities (Lippmann, 1987). The steps involved are as follows, STEP 1: Initialize Weights and Thresholds Set all weights and thresholds to random numbers between - 0.5 to 0.5. STEP 2: Present Inputs and Target Outputs Present a continuous valued input vector x0 , x1,....., xN-1 and the corresponding t0, t1,...., as the target or desired values.

tN-1

STEP 3: Calculate Actual Outputs Use the sigmoidal non-linearity from above to calculate the outputs as oo, o1, ...., oN-1 STEP 4: Adopt Weights Use a recursive algorithm starting at the output nodes and working back to the first hidden layer. Adjust weights by ûwji (n+1) = (/j oj) + . ûwji (n) where is the momentum rate and . is acceleration. Also, /k = (tk - ok) ok (1-ok) where k denotes the output layer. And /j = oj (1-oj) /k wkj where j is any internal hidden layer.

STEP 5: Repeat by Going to Step 2. Repeat the above steps till the conditions for iterations or error is satisfied.

RAINFALL-RUNOFF MODELS Existing methods used to estimate runoff from rainfall are frequently classified into two groups viz., Black Box model and Process model (Todini, 1988). In the black box modelling approach, empirical relations are used to relate runoff and rainfall, and only the input (rainfall) and the output (runoff) have physical meanings. Simple mathematical equations, time-series methods and Neural networks methods fall into this category. Process models attempt to simulate the hydrological processes in a catchments and involve the use of many partial differential equations governing various physical processes and equations of continuity for surface and soil water flow. Conceptual rainfall-runoff models (Chiew et.al., 1993) can be considered as a third group of modelling approach.

NEURAL NETWORK APPLICATION A rainfall-runoff model using ANNs for the Lekkur watershed in Tamilnadu state. The available hydrologic parameters were Rainfall (mm), Soil Moisture (%), Evaporation (mm) and Runoff (mm) values of the monsoon months (Jul-Sep) for 9 years (1975-83) (i.e. a total of 27 monthly data were available). The raw data values of rainfall etc. were standardized by zi

(xi x)/1n

where x and 1n are mean and standard deviation of the data. The zi values, which lie between -3.0 and +3.0 were transformed between 0 and 1, and fed as the training set to the ANN. The training parameter in Model 1 is only rainfall, whereas in case of Model 2 the training set consists of rainfall, soil moisture and evaporation. For each model number of hidden layers are altered i.e. Single hidden layer and Double hidden layer. The desired outputs from both the models is runoff. Following parameters were kept constant for all the ANN models, Momentum Rate Acceleration Permissible Average Absolute System Error Permissible Mean Square System Error Maximum Number of Iterations

= = = = =

0.9 0.7 0.012 0.0012 2,00,000

MODEL 1 Rainfall and runoff data of only 12 months were selected for training, as the dry spells yielded no runoff data. These values were standardized, the rainfall values were fed in as the training input. Single Hidden Layer The ANN model consisted of a single hidden layer with 3 nodes in the hidden layer. As the number of

training cycles or iterations increased the system error decreased. After 2,00,000 iterations the training data were fed in to the ANN model and the output values of the runoff were compared with the desired or field observed data as shown in Fig. 1 . Final error in the model is, Average Absolute System Error Mean Square System Error

= 0.060784 = 0.003974

Double Hidden Layer The model in this case consisted of a double hidden layer architecture, the hidden layers having 2 nodes each. Fig. 2 shows the scatter plots between the output runoff values from the ANN, model and the field observed data after 200,000 iterations. The errors obtained are as follows. Average Absolute System Error Mean Square System Error

= 0.047918 = 0.003158

MODEL 2 In model 2 the inputs for training were rainfall, soil moisture and evaporation values which were standardized and transformed between 0 and 1. Two ANN models having different internal structures i.e., one with single hidden layer and another with double hidden layer were studied. In each case the model was trained with a set of 12 data points for a maximum of 200,000 training cycles. This was followed by feeding the training set to the trained ANN model and the output thus obtained were compared with the desired output values of runoff.

Single Hidden Layer In this model the ANN had a single hidden layer with 3 nodes. A comparison of the output values with the field observed data is given in Fig. 3. The errors obtained after 200,000 iterations are, Average Absolute System Error Mean Square System Error

= 0.012000 = 0.000194

Double Hidden Layer In this case two hidden layered structure of the ANN model was adopted. Both the hidden layers consisted of 2 nodes each. The comparison between the output values and the actual field observed data is given in Fig. 4. The errors obtained after 200,000 iterations are, Average Absolute System Error Mean Square System Error

= 0.149618 = 0.092816

PREDICTION OF RUNOFF USING ANN MODEL It can be clearly observed that the ANN model yielding the least error, both in terms of average absolute

error and mean square error, is model 2 having a single hidden layer. This model is then used for predicting the runoff values for the next 4 consecutive rainfall months and the result is compared with the actual observed field data. These predictions comapred well with the observed values as can be seen from Table 1. Table 1. Runoff predicton using ANN Model (Single hidden layer) Rainfall

Soil moisture

Evaporation

Predicted runoff

Observed runoff

0.526

0.325

0.429

0.880816

0.622065

0.314

0.413

0.248

0.380117

0.271541

0.353

0.410

0.642

0.511718

0.435396

0.334

0.447

0.523

0.432945

0.246652

CONCLUSIONS Artificial Neural Networks (ANN) are promising and show good capability to model the hydrologic data. ANN with feed-forward network and backward propogtion algorithm is developed in the presetn study. For the case study, ANN model with three input nodes and single hidden layer is found to be the best for the rainfall-runoff modelling. ANN model's predictions comapred well with those observed.

Acknowledgements The funds for this work are provided by DST under 'Scheme for Young Scientists' with project no. HR/OY/O-02/97. The auhtors wish to acknowledge DST for providing funds.

REFERENCES 1 2 3 4

5 6 7

Hecht-Nielsen, R., 'Neurocomputing: Picking the human brain', IEEE Spectrum, 25(3), March 1988, pp.36-41. Vemuri, V., Edited., 'Artificial Neural Networks: Theoretical concepts', IEEE Computer society Technology series, 1988, pp.145. Yoh-Han Pao, 'Adaptive pattern recognition & Neural netowrks', Addison-Wesley Pub., 1985. Rumelhart, D.E., G.E. Hinton and R.J. Williams, 'Learning internal representations by error propagation', in D.E. Rumelhart and J.J. McClelland (Eds), Parallel distributed processing: Expolorations in the microstructure of cognition, Vol. 1: Foundations, MIT Press, 1986. Lippmann, R..P., 'An introduction to computing with Neural nets', IEEE ASSP Magagine, April 1987, pp. 4-22. Todini, E., 'Rainfall-Runoff modelling - Past, Present and Future', J. Hydrology, Vol. 100, 1988, pp. 341-352. Chiew, F.H.S., M.J. Stewardson and T.A. McMahon, 'Comparison of Rainfall-Runoff modelling approaches', Vol. 147, 1993, pp. 1-36.

INTRODUCTION Neurons are nerve cells and neural networks are networks of these cells. The cerebral cortex of the brain is an example of a natural neural network. Somehow, such a network of neurons thinks, learns, feels and remembers. Many attempts had been made in the past to build models to study such neural networks. There are two major types of models - biological and technological. In biological modelling the goal is to study the structure and function of real brain in order to explain biological aspects such as behaviour. In technological modelling the goal is to study brains in order to extract concepts to be used in new computational methodologies. The latter viewpoint is taken by several investigators working in the area of artificial neural networks and nurocomputers (Hecht-Nielsen, 1988) The artificial neural network (ANN) approach differ from the traditional approaches in stochastic hydrology in the sense that it belongs to a class of data-driven approaches as opposed to traditional model driven approaches. In this paper, a neural network computer program was developed to carry out Rainfall-Runoff modelling of Lekkur catchment area in Tamilnadu. The neural network was developed using the generalized delta rule for a semi-linear feed forward net with error back propagation. The program code was written in C in UNIX environment. The neural network model was treated as a Black Box, as the relationships between the physical components of the catchment were not to be fed. Model 1 involved the training of the ANN model with Rainfall values only, the output being the Runoff. Whereas, in Model 2 the training pairs consisted of Rainfall, Soil Moisture and Evaporation values, Runoff being the desired output.

COMPUTATIONS IN ANN The computational process associated with an ANN is as follows: An artificial neuron (AN) receives its inputs from a number of other ANs or from the external world (Lippmann, 1987). A weighted sum of these inputs constitutes the argument of an activation function. This activation function is assumed to be nonlinear. Hard limiting threshold, i.e., either the step or signum function, and soft limiting threshold, i.e. sigmoidal, are the three most often used forms of non-linearities. The resulting value of the activation function is the output of the AN. This output is distributed along the weighted connections to other ANs. The components of an input pattern constitute the inputs to the node in layer i. The outputs of the nodes in that layer may be taken to be equal to the inputs. The net input to a node in layer j is netj = wkj oj The output of node j is oj = f (netj), where f is the activation function. For a sigmoidal function we have oj

1 1e

(net jj)/0

The parameter j serves as a threshold or bias and o is a constant to modify the shape of the sigmoidal function (Yoh-Han Pao, 1985). The semilinear feed forward net was proposed by Rumelhart et al.(1986). The back-propagation training algorithm is an iterative gradient algorithm designed to minimize the mean square error between the actual output of a multilayer feed-forward perceptron (Vemuri, 1988) and the desired output. It requires continuous differentiable non-linearities (Lippmann, 1987). The steps involved are as follows, STEP 1: Initialize Weights and Thresholds Set all weights and thresholds to random numbers between - 0.5 to 0.5. STEP 2: Present Inputs and Target Outputs Present a continuous valued input vector x0 , x1,....., xN-1 and the corresponding t0, t1,...., as the target or desired values.

tN-1

STEP 3: Calculate Actual Outputs Use the sigmoidal non-linearity from above to calculate the outputs as oo, o1, ...., oN-1 STEP 4: Adopt Weights Use a recursive algorithm starting at the output nodes and working back to the first hidden layer. Adjust weights by ûwji (n+1) = (/j oj) + . ûwji (n) where is the momentum rate and . is acceleration. Also, /k = (tk - ok) ok (1-ok) where k denotes the output layer. And /j = oj (1-oj) /k wkj where j is any internal hidden layer.

STEP 5: Repeat by Going to Step 2. Repeat the above steps till the conditions for iterations or error is satisfied.

RAINFALL-RUNOFF MODELS Existing methods used to estimate runoff from rainfall are frequently classified into two groups viz., Black Box model and Process model (Todini, 1988). In the black box modelling approach, empirical relations are used to relate runoff and rainfall, and only the input (rainfall) and the output (runoff) have physical meanings. Simple mathematical equations, time-series methods and Neural networks methods fall into this category. Process models attempt to simulate the hydrological processes in a catchments and involve the use of many partial differential equations governing various physical processes and equations of continuity for surface and soil water flow. Conceptual rainfall-runoff models (Chiew et.al., 1993) can be considered as a third group of modelling approach.

NEURAL NETWORK APPLICATION A rainfall-runoff model using ANNs for the Lekkur watershed in Tamilnadu state. The available hydrologic parameters were Rainfall (mm), Soil Moisture (%), Evaporation (mm) and Runoff (mm) values of the monsoon months (Jul-Sep) for 9 years (1975-83) (i.e. a total of 27 monthly data were available). The raw data values of rainfall etc. were standardized by zi

(xi x)/1n

where x and 1n are mean and standard deviation of the data. The zi values, which lie between -3.0 and +3.0 were transformed between 0 and 1, and fed as the training set to the ANN. The training parameter in Model 1 is only rainfall, whereas in case of Model 2 the training set consists of rainfall, soil moisture and evaporation. For each model number of hidden layers are altered i.e. Single hidden layer and Double hidden layer. The desired outputs from both the models is runoff. Following parameters were kept constant for all the ANN models, Momentum Rate Acceleration Permissible Average Absolute System Error Permissible Mean Square System Error Maximum Number of Iterations

= = = = =

0.9 0.7 0.012 0.0012 2,00,000

MODEL 1 Rainfall and runoff data of only 12 months were selected for training, as the dry spells yielded no runoff data. These values were standardized, the rainfall values were fed in as the training input. Single Hidden Layer The ANN model consisted of a single hidden layer with 3 nodes in the hidden layer. As the number of

training cycles or iterations increased the system error decreased. After 2,00,000 iterations the training data were fed in to the ANN model and the output values of the runoff were compared with the desired or field observed data as shown in Fig. 1 . Final error in the model is, Average Absolute System Error Mean Square System Error

= 0.060784 = 0.003974

Double Hidden Layer The model in this case consisted of a double hidden layer architecture, the hidden layers having 2 nodes each. Fig. 2 shows the scatter plots between the output runoff values from the ANN, model and the field observed data after 200,000 iterations. The errors obtained are as follows. Average Absolute System Error Mean Square System Error

= 0.047918 = 0.003158

MODEL 2 In model 2 the inputs for training were rainfall, soil moisture and evaporation values which were standardized and transformed between 0 and 1. Two ANN models having different internal structures i.e., one with single hidden layer and another with double hidden layer were studied. In each case the model was trained with a set of 12 data points for a maximum of 200,000 training cycles. This was followed by feeding the training set to the trained ANN model and the output thus obtained were compared with the desired output values of runoff.

Single Hidden Layer In this model the ANN had a single hidden layer with 3 nodes. A comparison of the output values with the field observed data is given in Fig. 3. The errors obtained after 200,000 iterations are, Average Absolute System Error Mean Square System Error

= 0.012000 = 0.000194

Double Hidden Layer In this case two hidden layered structure of the ANN model was adopted. Both the hidden layers consisted of 2 nodes each. The comparison between the output values and the actual field observed data is given in Fig. 4. The errors obtained after 200,000 iterations are, Average Absolute System Error Mean Square System Error

= 0.149618 = 0.092816

PREDICTION OF RUNOFF USING ANN MODEL It can be clearly observed that the ANN model yielding the least error, both in terms of average absolute

error and mean square error, is model 2 having a single hidden layer. This model is then used for predicting the runoff values for the next 4 consecutive rainfall months and the result is compared with the actual observed field data. These predictions comapred well with the observed values as can be seen from Table 1. Table 1. Runoff predicton using ANN Model (Single hidden layer) Rainfall

Soil moisture

Evaporation

Predicted runoff

Observed runoff

0.526

0.325

0.429

0.880816

0.622065

0.314

0.413

0.248

0.380117

0.271541

0.353

0.410

0.642

0.511718

0.435396

0.334

0.447

0.523

0.432945

0.246652

CONCLUSIONS Artificial Neural Networks (ANN) are promising and show good capability to model the hydrologic data. ANN with feed-forward network and backward propogtion algorithm is developed in the presetn study. For the case study, ANN model with three input nodes and single hidden layer is found to be the best for the rainfall-runoff modelling. ANN model's predictions comapred well with those observed.

Acknowledgements The funds for this work are provided by DST under 'Scheme for Young Scientists' with project no. HR/OY/O-02/97. The auhtors wish to acknowledge DST for providing funds.

REFERENCES 1 2 3 4

5 6 7

Hecht-Nielsen, R., 'Neurocomputing: Picking the human brain', IEEE Spectrum, 25(3), March 1988, pp.36-41. Vemuri, V., Edited., 'Artificial Neural Networks: Theoretical concepts', IEEE Computer society Technology series, 1988, pp.145. Yoh-Han Pao, 'Adaptive pattern recognition & Neural netowrks', Addison-Wesley Pub., 1985. Rumelhart, D.E., G.E. Hinton and R.J. Williams, 'Learning internal representations by error propagation', in D.E. Rumelhart and J.J. McClelland (Eds), Parallel distributed processing: Expolorations in the microstructure of cognition, Vol. 1: Foundations, MIT Press, 1986. Lippmann, R..P., 'An introduction to computing with Neural nets', IEEE ASSP Magagine, April 1987, pp. 4-22. Todini, E., 'Rainfall-Runoff modelling - Past, Present and Future', J. Hydrology, Vol. 100, 1988, pp. 341-352. Chiew, F.H.S., M.J. Stewardson and T.A. McMahon, 'Comparison of Rainfall-Runoff modelling approaches', Vol. 147, 1993, pp. 1-36.