Performance evaluation of soft computing techniques for forecasting daily reference evapotranspiration

Water is the essential and most critical input for sustainable agriculture (Hyder et al. 2022). It is a valuable contribution of nature to the existence and survival of life on earth (Manikumari et al. 2017). But the distribution of water resources has changed in recent years due to climate change effects, which have been causing disaster events like floods and drought (Suwarno et al. 2021; Krisnayanti et al. 2022). Change in the occurrence and distribution of rainfall has affected the availability of freshwater resources at spatial and temporal scales (Chutiman et al. 2022). Moreover, increasing water demand due to a burgeoning population and changing climate necessitates judicious management of water resources in agriculture (Manikumari et al. 2017). Globally, about 70% of water drained from rivers and aquifers is diverted for agricultural activities (Shang et al. 2020). Therefore, evapotranspiration (ET), a significant component of the hydrologic cycle, needs special attention for accurate quantification. The reference evapotranspiration (ET₀) expresses the simultaneous occurrence of the soil water evaporation and the crop transpiration processes on a vegetated surface (Allen et al. 1998). Real-time computations of actual crop evapotranspiration (ET_c) require forecasting of daily reference evapotranspiration (ET₀). Therefore, precise ET₀ quantification is necessary for determining irrigation demands, irrigation scheduling, water resource management, environmental impact assessment, and water balance studies at regional and local levels, and for its use in various rainfall-runoff and ecosystem models (Nema et al. 2017; Poddar et al. 2021).

The Food and Agriculture Organization (FAO) of the United Nations lately embraced a standardized form of the Penman–Monteith (PM) equation to establish a globally valid benchmark for calculating ET₀, constructing crop coefficient curves for crops, and analyzing and calibrating other ET₀ methodologies in situations where weighing-type field lysimeter measurements are inaccessible (Allen et al. 1998; Cai et al. 2007; Kumar et al. 2012; Garg et al. 2016; Kushwaha et al. 2021). Several process-based and empirical equations based on radiation, temperature, mass transfer, and water budget methods have been derived to determine ET₀ with different input combinations of meteorological parameters (Majhi & Naidu 2021). These methods are physically based and require input data (Allen et al. 1998). Several authors have utilized these empirical models under data-limited conditions (Tabari 2010; Poddar et al. 2021; Pandey & Pandey) to estimate ET₀. In addition, irrigation scheduling software developed for different crops requires an accurate value of ET₀ and the crop coefficient values at different crop growth stages (Genaidy 2020).

Applications of soft computing techniques have become reliable in solving complex problems (Wagstaff 2012; Kumar et al. 2016; Hameed et al. 2021). Due to its high performance and the decisive advantage of capturing nonlinear and complex structures, machine learning is widely applied to different fields, namely, marketing for task classification, finance for forecasting, telecommunications for forecasting and task classification, and network analysis for relating different tasks (Bhandari 2021). Also, several studies have used machine learning algorithms to estimate reference evapotranspiration (ET₀) (Chia et al. 2020, 2022; Hanoon et al. 2021; Rai et al. 2022). ET₀ forecasting models were developed using soft computing tools, namely, support vector machine (SVM), gradient boosting decision tree (GBDT), particle swarm optimization (PSO) SVM, and PSO-GBDT algorithms. Results indicated that SVM, GBDT, PSO-SVM, and PSO-GBDT models generally resulted in better forecasting of ET₀, whereas the PSO-GBDT algorithm performed better for the Southwest China region (Zhao et al. 2021).

Artificial neural network (ANN) modeling permits easier translation between humans and computers for decision-making and a better way to handle imprecise and uncertain information (Fernández-López et al. 2020). Multiple linear regression (MLR) showed that the ANN and Gene Expression Programming (GEP) models were superior to the MLR models in forecasting daily reference evapotranspiration at Pantnagar, India (Heramb et al. 2022). In another study, a comparison of the Feed Forward Neural Network (FFNN), Radial Basis Function Neural Network (RBFNN), and GEP machine learning algorithms for estimating daily ET₀ in the Lower Cheliff Plain, northwest Algeria, was done (Achite et al. 2022). Results concluded that the RBFNN and GEP models showed promising performance. However, the FFNN model performed the best during training (R² = 0.9903, root-mean-square error (RMSE) = 0.2332, and EF = 0.9902) and testing (R² = 0.9921, RMSE = 0.2342, and EF = 0.9902) phases in forecasting ET₀ as compared with the PM evapotranspiration. The ANN model outperformed the linear regression model in forecasting daily evaporation (Singh et al. 2021). The ANN model with a 4-10-1 ANN structure was recommended to perform best for evaporation at the National Institute of Technology (NIH), Roorkee, India. Deep learning (DL) is also extensively used for forecasting hydrological variables. The performance of four learning algorithms, namely, DL-Multilayer Perceptron (DLMP), Generalized Linear Model (GLM), Random Forest (RF), and Gradient Boosting Machine (GBM), was assessed to forecast future ET₀ for Hoshiarpur and Patiala district, Punjab, India. The study concluded that the DLMP outperformed other models during the training, validation, and testing stages, and the statistical performance indicator values for DLMP models ranged between 0.95 and 0.98 for NSE, 0.95 and 0.99 for the coefficient of determination (R²), 85 and 95 for the loss and accuracy (ACC), 0.0369 and 0.1215 for mean squared error (MSE), and 0.1921 and 0.2691 for RMSE.

Machine learning approaches, namely, k-nearest-neighbor (KNN) and ANN models, were applied to forecast daily ET₀ using four combinations of climatic data in the Middle Anatolia region of Turkey. They concluded that the KNN had higher performances than the ANN in all combinations under full and limited data conditions (Yamaç 2021). The potential of algorithm KNN, a data mining method for estimating ET₀, was investigated using limited climatic data in a semi-arid environment in China. Results showed that the input variables of the KNN-based ET₀ forecast model, which requires maximum air temperature, minimum air temperature, and relative humidity, had the best accuracy (Feng & Tian 2021). The performance of soft computing algorithms, namely, Gaussian Naive Bayes (GNB), SVM, KNN, and ANN, were compared for ET₀ estimation against the PM model for Pakistan climatic conditions. The results showed that the KNN model was more accurate than SVM, GNB, and ANN models, with 92% accuracy (Hu et al. 2022).

The performance of the ANN and KNN soft computing techniques in forecasting different real-world problems depends on various factors. The choice of activation function and solver using ANN techniques determine the ANN performance in forecasting daily ET₀. Also, the number of neighbors (k) value using the KNN technique affects its efficacy in forecasting ET₀. Several authors have applied different machine learning algorithms to forecast ET₀ in different climatic conditions. However, less information is available on the application of different ANN learning algorithms in forecasting daily ET₀ with a sigmoid activation function. Previous studies using KNN techniques lacked the effect of the number of neighbors on the algorithm's performance in forecasting daily ET₀. Keeping in view the research gap in forecasting daily ET₀ using soft computing tools in previous studies, the present study was undertaken to assess the performance of ANN and KNN techniques for precise forecasting of daily ET₀. The novelty of this investigation is the application of the sigmoid activation function along with three activation learning algorithms for ANN techniques and the optimization of the number of neighbors for the KNN technique to forecast daily ET₀. The sigmoid activation function tested three learning algorithms, namely, L-BFGS-B (Limited memory-Broyden–Fletcher–Goldfarb–Shanno), stochastic gradient descent (SGD), and Adam, for daily forecasting of ET_0. Also, the selection of ANN techniques based on an optimum number of neurons in the hidden layer for ANN-based ET₀ forecasting models was carried out in this study. Additionally, the optimum number of nearest neighbors in the KNN algorithm, which influences the model performance in forecasting daily ET₀, has been worked out for semi-arid climates. Another novelty of this work is performing the sensitivity analysis of the best daily ET₀ forecasting model to identify the weather variables that most influence its performance. Thus, the current study was carried out with the objective of assessing the effectiveness of the ANN and KNN soft computing approaches in forecasting daily ET₀ at the Indian Agricultural Research Institute (IARI) farm in New Delhi, India, as well as to perform sensitivity analysis of the best ET₀ forecasting model to detect the most affecting input variables. The developed model may aid in accurate quantification of daily basis ET₀, which may be utilized to compute crop water requirements considering stage-wise water requirements for crops.

The activation functions (f) of the hidden layer neurons serve to transform the input variables (the activation levels of the neurons) into the needed output variable. The sigmoid and hyperbolic tangent transfer functions are the most commonly used in hydrological modeling (Dawson & Wilby 2001; Zanetti et al. 2007).

Initially, ANN models in ET research focused solely on the back-propagation learning algorithm (Kuo et al. 2011; Nazari & Band 2018). However, in recent years, these models have been further refined by exploiting alternative learning algorithms, such as the radial basis function (Wu et al. 2012; Du & Swamy 2014; Majhi & Naidu 2021), quick propagation (Landeras et al. 2008), Levenberg–Marquardt (Poddar et al. 2021), and conjugate gradient descent (Landeras et al. 2008). The present study used three learning algorithms, namely, L-BFGS-B learning algorithm, SGD learning algorithm, and Adaptive Moment Estimation (Adam).

The L-BFGS-B algorithm extends the L-BFGS algorithm to handle simple bounds on the model (Zhu et al. 1997). The BFGS–ANN algorithm enhanced the performance of the ANN model in ET₀ estimation for drought-prone arid and semi-arid regions over Gaussian process regression (GPR) and support vector regression (SVR), and long short-term memory (LSTM) models (Sattari et al. 2021).

where is the updated weight; is the previous value of the weight; is the learning rate of 0.1, and E is the output error computed by the objective function (Walls et al. 2020).

Adam optimization is applied to calculate the adaptive training rate of the parameters. The ADAM optimization algorithm was used in previous studies for ET₀ estimation (Sattari et al. 2020).

The performance of the ANN and KNN soft computing techniques was evaluated using five commonly used statistical indices to determine the best model for forecasting daily ET₀. These indices are as follows:

The input feature was selected using MLR approaches between dependent (ET₀) and independent (meteorological variables). Based on the values of statistical indices, a combination of all seven independent variables was selected for forecasting the ET₀. The input selection results showed that combining all meteorological variables resulted in the lowest MSE value of 0.13. Similarly, other statistical indices, namely, R², adjusted R², Mallows’ Cp, Schwarz information criterion (SBC), the Schwarz information criterion (SBC) and Amemia's prediction criterion (PC), were 0.96, 0.96, 8, −23,197.80, −23,139.12, and 0.04, respectively, and are shown by blue color in Table 1. For all other variable combinations, the values of input selection indices are displayed in Table 1. The Type III sum-of-square statistics indicated that variable Tmax was the most influential among the ET₀ influencing variables. At the same time, RHII was the least influential variable among all seven variables.

The statistical parameter indices, namely, MAE, MSE, r, MAPE, and d, were used to evaluate the performance of the ANN model for ET₀ forecasting. The number of neurons in the hidden layer varied from 1 to 12 and stopped after it as further increments resulted in the poor performance of the ANN in ET₀ forecasting. The performance of the algorithms is displayed in Table 2. The results indicated that the combination of sigmoid as an activation function and L-BFGS-B as a solver performed the best. The model's performance with the ANN architecture (7,11,1, i.e., M11) was the best among all 12 models. The best model reported values of indices, namely, MAE, MSE, r, MAPE, and d, were 0.075, 0.018, 0.997, 2.76%, and 0.974, respectively, during the training period. The performance of model 11 (7,11,1) was poor during the testing period compared to the training period. The values of the performance indices were found to be 0.091, 0.053, 0.984, 3.16%, and 0.969 for MAE, MSE, r, MAPE, and d, respectively, for model M11 during the testing period. The performance of model M2 (7,2,1) was found to be worst during the training and testing period based on the statistical performance indicator values among all the 12 models developed using sigmoid as an activation function and L-BFGS-B as a solver. The performance of the models significantly improved by increasing the number of neurons in the hidden layers except model M2, which showed poor performance compared to the M1 model. For the M12 model, the MAE and MSE indices improved by 72.72% compared to the M1 model during the training period. Similarly, the MAPE showed an improvement of 12.93% compared to the M1 model during the training period. However, the performance indices, namely, r and d, showed slight improvement for the M12 model over M1. The r and d had increased by 0.10 and 0.73%, respectively, during the model training period. The values of performance indicators during the model testing period showed improvement for MAE, MSE, MAPE, and d, and the performance was improved by 10.76, 12.67, 3.87, and 0.52% for the best model M12 compared to the M1 model. However, r showed a slight reduction in the model's performance and its value was reduced by 0.52%. Overall, we found that the performance of the ANN using sigmoid activation function and an L-BFGS-B solver to forecast daily ET₀ had improved by increasing the number of neurons in the hidden layer.

The sigmoid and SGD as activation functions and solver performed more poorly in the daily ET₀ forecasting compared to the combination of sigmoid and L-BFGS-B. The model M17 having ANN structure as (7,5,1) performed best compared to all 12 models developed during training and testing periods. At the same time, model M24 (7,12,1) performed worst in forecasting daily ET₀. The values of the MAE, MSE, r, MAPE, and d for the M17 model obtained were 0.268, 0.143, 0.975, 10.54%, and 0.907, respectively, during the training period and 0.242, 0.122, 0.979, 9.79%, and 0.916, respectively, during the testing period. The M24 model reported that the performance indices, namely, MAE, MSE, r, MAPE, and d resulted in values of 1.440, 2.934, 0.031, 59.12%, and 0.00, respectively, during the model training period and 1.455, 2.885, 0.003, 61.82, and 0.000, respectively, during the model testing period. The comparison between the minimum and maximum number of neurons in the hidden layer showed that M14, M23, and M24 models performed more poorly in forecasting daily ET₀. Overall, we observed that the performance of the ANN using sigmoid activation function with a SGD solver to forecast daily ET₀ had showed the effects of the number of neurons on the performance significantly.

The performance of models developed using sigmoid as an activation function and Adam as a solver was improved during the testing period over the training period, as indicated by the performance indicator values in Table 2. Among all the 12 combinations of the ANN structure-based models, model M32 (7,8,1) performed the best. The statistical performance indices, namely, MAE, MSE, r, MAPE, and d, reported values of 0.164, 0.081, 0.986, 7.22%, and 0.943 during the training period and 0.162, 0.082, 0.986, 7.10%, and 0.944, respectively, during the testing period. On the other hand, model M31 (7,7,1) performed worst among all the 12 models as reported by the low values of statistical performance indices. The performance of all 36 models is presented in Table 2. The values of MAE, MSE, r, MAPE, and d for model 31 were found to be 0.434, 0.500, 0.911, 14.37, and 0.849, respectively. The same during the testing period were 0.413, 0.421, 0.924, 14.14, and 0.857, respectively. All the remaining models gave the values of statistical performance indices better than model M31 and poorer than model M32. The increment of the number of neurons from 1 to 12 was done to determine the best model. It was found that the ANN functioned differently and there was no incremental improvement in the performance indicator values while increasing the number of neurons in the hidden layer. Therefore, among 12 models developed using sigmoid as an activation function and Adam as a solver, model M32 was found to be best for forecasting daily ET₀.

We found that among all the three solvers with sigmoid activation function, the performance of the models developed using a L-BFGS-B solver gave best results and thus we recommend employing the ANN model with sigmoid function and an L-BFGS-B solver to forecast daily ET₀ under semi-arid climatic condition.

Conversely, during the testing period of the sensitivity analysis, WS emerged as the most influential input variable for ET₀ forecasting. Furthermore, the influence of the input variables was followed by SR, Tmax, RHII, SS, RHI, and Tmin, respectively, in decreasing order. Therefore, it may be concluded that the WS is the most influential input variable and its omission in forecasting ET₀ using the developed ANN model M11 reduces the model performance during training and testing periods. Overall, the performance of the ANN model M11 outperformed all the ANN- and KNN- based models, and therefore, model M11 may be used to forecast daily ET₀ at the IARI, New Delhi.

Many researchers have employed different machine learning algorithms to forecast ET₀. In many cases, a single algorithm based on ET₀ forecasting was better, and in some cases, meta-heuristic algorithms performed better than single algorithms. For example, the deep neural network (DNN) model was superior in forecasting daily ET₀ based on a single input meteorological variable compared to RF and Extreme Gradient Boosting (XGBoost) algorithms in California (Ravindran et al. 2021). Improved machine learning algorithms such as the SVM-cuckoo algorithm (SVM-CA) outperformed genetic programming (GP), model tree (M5T), and the adaptive neuro-fuzzy inference system (ANFIS) in forecasting ET₀ at Ranichuri, Uttarakhand, India (Ehteram et al. 2019). In another study, the ANN model with the Levenberg–Marquardt training algorithm, single hidden layer, and nine neutron configurations were found to be the best for ET₀ forecasting in Dehradun, India (Nema et al. 2017). This study's findings align with other study results. Our study showed that all the meteorological variables are essential to forecast ET₀. However, Alves et al. (2017) reported that air temperature alone could be sufficient to forecast daily ET₀ accurately using the ANN algorithm under a data-limited scenario.

In comparison to Hargreaves and the FAO-PM methods, the ANN model has shown to be a reliable choice for estimating ET₀ using Tmax and Tmin for Salinas, California, United States of America, providing superior performance index, standard error of estimate, and correlation (Lucas et al. 2008). In forecasting ET₀, the performance of the KNN model was better than the SVM, GNB, and ANN models with 92% accuracy in the Multan region, Pakistan (Hu et al. 2022). However, the findings of our study contradict the findings reported by Hu et al. (2022). This could be due to various factors, including agro-climatic conditions, algorithm structure, and the activation function used in the ANN techniques. The activation function selection and the learning algorithm influenced the ANN performance. The influence of the ANN activation function and the architecture in forecasting daily ET₀ was assessed for Nissouri Creek, Oxford County, Canada (Walls et al. 2020). Study results showed that the SGD learning algorithm for 500 training epochs had performed better compared to a combination of ReLU and root-mean-square-propagation (RMSprop) learning algorithms in forecasting daily ET₀ for Nissouri Creek, Oxford County, Canada (Walls et al. 2020).

The ANN-based model showed higher efficacy in modeling daily ET as compared to kNN, RF, SVM, XGBoost, and LSTM ML as evidenced by the low RMSE (ranged between 18.67 and 21.23) and stronger r (ranged between 0.90 and 0.94) values for global cropped lands (Liu et al. 2021). The findings of the present study are in line with the study (Liu et al. 2021). The performance of the ANN model for drought-prone arid and semi-arid regions of Corum, Turkey proved to be better in ET₀ estimation compared to GPR, SVR, and LSTM approaches with full and partial dataset conditions (Sattari et al. 2021). The efficacy of the machine learning models is influenced by the data availability. For instance, Yamaç (2021) found that the kNN was superior to the ANN in modeling daily potato crop evapotranspiration under limited meteorological dataset conditions at the Mediterranean Agronomic Institute of Bari (CIHEAM-Bari), Valenzano, Southern Italy. However, under a full dataset scenario, the ANN model gave slightly better statistical indicators than the kNN. Thus, our findings are in line with the study (Yamaç 2021). The performance of the KNN was assessed to forecast daily ET₀ in the Ningxia irrigation area, China under a limited dataset scenario (Feng & Tian 2021). They reported that the KNN-based ET₀ forecast model (K = 3) performed better than other nearest-neighbor (k) values. Based on the different model's efficacy in forecasting daily ET₀, they found that the KNN-based ET₀ forecast model which requires Tmax, Tmin, and RH as input variables had the best accuracy. They concluded that the KNN-based prediction were consistent with the PM model and recommend applying it in semi-arid environments.

The performance of the ANN model is highly dependent on its structure, i.e., the number of variables, the number of hidden layers, and the number of neurons in the hidden layers. The daily evaporation was forecasted by Singh et al. (2021) and reported that the ANN with structure (4-9-1) was better than other structures and outperformed the MLR model. Many neighbors (k) in the KNN algorithm influenced its ET₀ forecasting capability, as reported by Feng & Tian 2021. They found that input variables in the KNN-based ET₀ forecast model, which requires Tmax, Tmin, and RH, resulted in higher forecasting accuracy. The optimal number of neighbors found was 3 (k = 3) for ET₀ forecasting in Ningxia, China. The findings of different algorithms in forecasting ET₀ vary under different climatic conditions. Therefore, under the prevailing climatic conditions of the study region, the performance of the ANN models was better compared to the KNN models.

Irrigation water management at the field level requires an accurate estimation of crop water requirements, for which the reference evapotranspiration is the prerequisite. Soft computing techniques like ANN and KNN are easy alternatives that can be developed using meteorological parameters. Furthermore, comparing advanced soft computing tools with standard methods enables the user to change the structure and algorithms to suit a particular climatic condition. The best selected ANN (M11) model's performance was superior to the KNN models. This study also demonstrated that the hidden layer structure influenced the performance of the ANN model in forecasting daily ET₀. The performance indices for model M11 were superior to others with MAE, MSE, r, MAPE, and d as 0.075, 0.018, 0.997, 2.76%, and 0.974, respectively, during model training and 0.091, 0.053, 0.984, 3.16%, and 0.969, respectively, during model testing. Sensitivity analysis of the best selected model revealed that the WS influenced the model's performance and, thus, an essential variable for daily ET₀ forecasting. Therefore, it can be recommended that the model M11 (7,11,1) with logistic activation function and L-BFGS-B learning algorithm can be utilized for ET₀ forecasting at the IARI, New Delhi. The research conclusions reported here provide irrigation managers and cultivators with fundamental guidance for forecasting ET₀ for irrigation planning and managing water resources in the context of available information.

The present study considered the full meteorological dataset availability condition. However, such a dataset may not be available in all the regions especially in the ungauged watershed. Therefore, under such a meteorological data limitation scenario, the developed models may not perform accurately enough to predict daily ET₀. Hence, the ANN and KNN techniques required to be tested depending on the meteorological data availability. Also, the performance of the soft computing techniques in forecasting daily ET₀ may be significantly influenced by the selection of algorithm, study location, dataset size, and partitioning of the dataset. Therefore, we suggest evaluating the application of the ANN and KNN techniques under different climatic conditions with varying scenarios of data availability and data partitioning to arrive to concrete conclusions for their application in forecasting daily ET₀ in other climatic conditions.

The authors are thankful to the Division of Agricultural Physics, ICAR-IARI, New Delhi, for providing meteorological data.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.