Reference evapotranspiration modeling using radial basis function neural network in different agro-climatic zones of Chhattisgarh

Precise estimation of evapotranspiration (ET) is extremely essential for efficient utilization of available water resources. Among the empirical models, FAO-Penman-Monteith equation (FAO-PM) is considered as standard method to determine reference evapotranspiration (ET ). In developing countries 0 like India, application of FAO-PM equation for ET estimation has certain limitations due to unavailability of 0 specific data requirements. Several empirical models such as Hargreaves, Turc, Blaney-Criddle etc., are also considered for ET estimation. However, ET estimates obtain with these models are not comparable 0 0 with benchmark FAO-PM ET . To address this issue, potential of radial basis function neural network 0 (RBFNN) is investigated to estimate FAO-PM ET . Result obtained with proposed RBFNN models are 0 compared with equivalent multi-layer artificial neural network (MLANN) and empirical approach of Hargreaves, Turc and Blaney-Criddle. Lower RMSE values obtained with RBFNN and MLANN models is 2 an indication of improved performance over empirical models. Similarly, higher R and Efficiency Factor obtained with RBFNN and MLANN models also approves the superiority of machine learning techniques over empirical models. Among the two machine learning techniques, RBFNN models performed better as compared to MLANN. In a nut shell, proposed RBFNN models can simulate FAO-PM ET even with limited 0 meteorological parameters and consistence degree of accuracy level.

like India, application of FAO-PM equation for ET estimation has certain limitations due to unavailability of 0 specific data requirements. Several empirical models such as Hargreaves, Turc, Blaney-Criddle etc., are also considered for ET estimation. However, ET estimates obtain with these models are not comparable Evapotranspiration (ET) is considered as one of most important parameter for agro-climatic analyses such as determination of crop water requirement and computation of water balance parameters. ET being an extremely complex non-linear process in nature, it is always very difficult to measure it. Therefore, consumptive use of water from a reference crop under non limiting conditions with weather parameters being the only factor affecting the process is computed by means of empirical models and termed as reference ET (Et ). Several empirical models have been 0 developed in the past to determine ET . Among the empirical 0 models, Food and Agricultural Organization has recommended Penman-Monteith equation (FAO-PM) as standard method for ET estimation (Allen et al. 1998). FAO-0 PM equation requires meteorological parameters such as temperature, humidity, wind speed, sunshine hours and net radiation to determine ET . Empirical models like Hargreaves, 0 Turc, Open Pan, Blaney-Criddle and Christianson etc., have also been used by several working as they require less number of meteorological parameters (Dar et al. 2017;Phad et al. 2019). However, ET estimates obtained using these models is 0 not comparable with FAO-PM as these methods yield large error and hence their application becomes limited.
Application of data-driven machine learning techniques such as fuzzy logic, artificial neural network and evolutionary computation are increasingly gaining popularity in the recent decade and emerged as alternate approach for FAO-PM ET estimation with higher order accuracy as 0 compared to equivalent empirical methods. (Chauhan and Shrivastava 2009;Mallikarjuna et al. 2012, Khedkar et al. 2019. A study conducted by Landeras et al. (2008) on comparison between ANN models and empirical equations for daily ET estimation in the Basque Country (Northern Spain) 0 confirms the superiority of ANN models over empirical equations. Study conducted by Feng et al. (2016) for estimating FAO-PM ET in humid region of Southwest China 0 reveals that extreme learning machine (ELM) and artificial neural network optimized by genetic algorithm (GANN) based models produced better estimates than wavelet neural network (WNN) and empirical approaches of Hargreaves,Vol. 21,No. 3 and considered as benchmark for model calibration and validation purposes.

Radial basis function neural network (RBFNN) based estimator
RBFNN is a category of feed forward network with single hidden layer and an output layer formulated by Broomhead and Lowe (1988). Pictorial representation of the RBFNN is given in Fig.1. Each processing unit termed as neuron in the hidden layer is associated with centers c = c , c , Output of i hidden layer neuron is basically a Gaussian function and is represented by: where, represents the Euclidian distance between input data and corresponding centers and . The Gaussian function used in the each hidden layer neuron is actually a category of radial basis function. Finally the response of the RBFNN for a given set of input data at the output layer neuron is linear in terms of weights and computed using the following expression.
Calibration of the RBFNN network for each instant of input data and its corresponding output {x, y} is done in recursive manner by updating the network parameters {w , c i i, s } to minimizing the following instantaneous error cost i function.
The weight update rules to optimize the network parameters {w , c s } at time t is given by following equations which are derived using gradient descent algorithm.
Makkink, Priestley-Taylor and Ritchie models. Gocić et al. (2015) has reported support vector machine-wavelet (SVM-Wavelet) as the best method for ET prediction. SVM-Wavelet 0 and support vector machine-firefly algorithm (SVM-FFA) methods produced higher correlation coefficient with ET as 0 compared to Artificial Neural network (ANN) and Genetic programming (GP) computational methods. Shiri et al. (2014) has computed ET estimates using heuristic data driven 0 (HDD) models such as ANN, ANFIS, SVM and gene expression programming (GEP) for a wide range of weather stations in Iran and compared the same with corresponding empirical models (Hargreaves-Samani, Makkink, Priestley-Taylor and Turc). They have found that HDD models generally outperformed empirical models, whereas among the HDD models GEP-based model produced higher accuracy.
Literature review discussed above, motivated us to conduct present investigation with an objective to demonstrate the ability of radial basis function neural network (RBFNN) for computing weekly FAO-PM ET estimates in three agro-0 climatic zones (ACZs) of Chhattisgarh. The material and methods section provides the details of the study area, architecture of RBFNN and MLANN and performance evaluation criteria. The results obtained based on simulation studies are discussed in the subsequent section. Significant finding of the study are given in the concluding section.

Study area
The present investigation is aimed at estimating weekly FAO-PM ET using RBFNN for three representative stations, 0 i.e. Raipur, Jagdalpur and Ambikapur located in Chhattisgarh Plains, Bastar Plateau and Northern Hills ACZs respectively in the Chhattisgarh state of east central India with available climatic data. Climate of Chhattisgarh is moist sub humid in general with an average annual rainfall of 1200-1400 mm and ET losses between 1400-1600 mm in different ACZs. Long 0 term weekly meteorological data of Raipur (1981Raipur ( -2015, Jagdalpur  and Ambikapur(1999Ambikapur( -2015 have been collected from respective meteorological observatories and their descriptive statistics in terms of weekly averages of maximum temperature (°C) -T , minimum temperature (°C) max -T , relative humidity during morning hours (%) -RH ,  (Hargreaves et al., 1985) Turc (Turc, 1961) Blaney-Criddle (Doorenbos and Pruitt, 1977) FAO Penman -Monteith (Allen et al., 1998)

Multi-layer artificial neural network (MLANN)
MLANN is a feed forward neural network suggested by Haykin (1998) with an input layer, one or more hidden layer and an output layer. A N-5-1 structure of MLANN (N=2,3 & 6 represents number of input data, 5 neurons in hidden layer and one neuron at the output layers) is used in this study with different input combinations. The training of the network is done by back-propagation algorithm which is based on the error-correcting learning rule to update the weights and biases of each neuron in different layers. Hyperbolic tangent (tanh) is used as the activation function.

Empirical models
The PET Calculator V3.0 software developed by AICRPAM, CRIDA, Hyderabad is used to estimate ET by 0 different empirical approaches. This software is a freeware and computes daily, weekly and monthly ET using different 0 input combination of climatic data. More details regarding empirical approaches considered for this investigation may be obtained from the references listed in Table 2.

Performance evaluation measures
Validation performance of the predictive models are evaluated by computing root mean square error (RMSE), 2 coefficient of determination (R ) and efficiency factor proposed by Nash and Sutcliffe, 1970 (EF)

RESULTS AND DISCUSSION
Proposed RBFNN and corresponding MLANN models are developed in MATLAB. Simulations studies are carried out to investigate the potential of proposed RBFNN models as compared to corresponding MLANN and equivalent empirical models (Hargreaves, Turc and Blaney-Criddle) for estimating FAO-PM ET . Input combinations 0 used in proposed RBFNN models are similar to that of equivalent empirical models as shown in Table 3. Long term weekly meteorological data of Raipur (1981-2010), Jagdalpur (1993-2012 and Ambikapur (1999Ambikapur ( -2012) is used for model calibration, whereas recent 3 to 5 years of data of Raipur (2011Raipur ( -2015, Jagdalpur (2013)(2014)(2015)(2016)(2017) and Ambikapur (2013Ambikapur ( -2015 is used for model validation. To calibrate the model, input and desired output data is normalized between -1 to 1. Model parameters of the RBFNN i.e., weights, centers and corresponding width {w , c s } are i i, i initialized to random numbers between -1 to 1. Centers have the same dimension as that of input data. Input patterns are given to the input layer of the model in a sequential manner and corresponding estimated output is obtained at the output layer after completion of forward pass for each set of input patterns (Fig. 1). Estimated output is compared with the target output to compute the instantaneous error which is the cost function for the proposed model. Real time update of the model parameters is done in each instance to minimize the squared error. The process continues till all the available input patterns for model calibration gets exhausted. This completes one cycle called epoch. At the end of each epoch, mean square error is computed. The iterative process is repeated several times until MSE is minimized to a desired low value nearly close to zero. This completes the supervised calibration process and model parameters are then fixed to constitute proposed model. Similar calibration process is adopted for MLANN model with corresponding model parameters. The set of parameters which produces optimum results during the simulation study are shown in Table 4 325 Vol. 21,No. 3