1 Introduction

The accurate estimation of crop water needs is crucial for sustainable management of water resources, particularly in regions affected by climate change such as India. Crop evapotranspiration (ETcrop) is an important parameter that can help in estimating a field's net irrigation requirement, which varies depending on the crop being grown (Srivastava et al. 2017; Kumar et al. 2020; Maza et al. 2020). Evapotranspiration (ET0) is the process of water movement from soil and vegetation to the atmosphere, which occurs through the combination of plant transpiration and soil and atmospheric evaporation. It is a vital component in agriculture, land management, pollution detection, irrigation planning, hydrological balance, and watershed hydrology (Kumari and Srivastava 2020). Accurate estimation of ET0 is crucial in managing water resources for various purposes, including irrigation, drinking, industrial use, and water reserve management (Kumar et al. 2021a, b; 2023a, b). The precise calculation of ET0 can improve irrigation efficiency, water reuse, and seepage control, making it an essential factor in addressing water management problems. The quantity of water needed to irrigate crop fields for the entire period is called crop-water requirement. It can be computed by multiplying crop coefficient with ET0 (Srivastava et al. 2018). Accurate irrigation scheduling is impossible with prior knowledge of ET0. There are different methods developed for the calculation of ET0 from meteorological variables, which are temperature-based, radiation-based, and combination-based. The estimation of ET0 through Lysimeter is considered to be the most accurate, but it is labour-intensive, expensive and requires proper maintenance. The performance of different methods varies in different agro-climatic regions. The FAO-56 PM has been recommended worldwide for the calculation of ET0 and used as a reference approach (Tabari et al. 2013; Kovoor et al. 2018; Valipour et al. 2018; Fan et al. 2019; Kumar et al. 2022). FAO-56 PM method requires large amount of data for ET0 calculation (Shiri et al. 2012; Caminha et al. 2017; Feng et al. 2017; Trigo et al. 2018; Kumar et al. 2021a). Due to the paucity of weather data in developing countries like India, limited meteorological data are available for the calculation of ET0. In recent years, much attention has been given towards forecasting natural events (Khosravi et al. 2018; Yaseen et al. 2018; Naganna et al. 2019; Xiao et al. 2019). Therefore, there is a need for the calculation of evapotranspiration using machine learning algorithm. Several researchers across the globe in different agro-climatic regions have attempted to compute ET0 from different meteorological variables and majority of the method need data that are not readily available, particularly in the hilly region of Uttarakhand (Kumar et al. 2023a). In addition to this, some of these methods are applicable to certain specific climatic conditions and they cannot be used under conditions which are different from those they were originally developed. Simple model is basically based on the linear relationship between different meteorological variables. Due to rapid urbanization and industrialization, pollution can affect ET in several ways. For example, air pollution can reduce the amount of sunlight that reaches the land surface, which can reduce photosynthesis and therefore transpiration (Yao 2017). Several studies on assessing the impact of pollution on environmental conditions is conducted in recent years (Ambade et al. 2021; Gautam et al. 2021, 2023; Nepolian et al. 2021; Thapliyal et al. 2022). ET is a complex phenomenon and it is highly non-linear in nature. Various researches showed that machine learning algorithms have shown promising results in predicting ET0 (Partal 2009; Cobaner 2013; Falamarzi et al. 2014; Adamala et al. 2019; Kelley and Pardyjak 2019). Machine learning techniques are efficient in obtaining complicated relationships between input and output variables, which makes them robust tools for ET0 modelling (Abdullah et al. 2015; Marti et al. 2015; Wang et al. 2018; Ferreira et al. 2019; Nourani et al. 2019; Wu and Fan 2019). In the last decade, various researches have been explored to demonstrate the proficiency of various machine learning model for predicting ET0 (Chauhan and Shrivastava 2008; Zanetti et al. 2007; Khoob 2008; Trajkovic 2010; Feng et al. 2017). Based on review of the literature, MLP algorithm is the best applied and most efficient neural network to predict ET0. Therefore the complexity of the calculation of evapotranspiration can be solved using machine learning (ML) algorithm with limited meteorological variables. This research evaluates different ML algorithms for the estimation of ET0 in different agro-climatic zones of India. Several studies have emphasized on the estimation of accurate ET0 in different hydrological studies work. ANN and multivariate non-linear regression technique were applied in semi-arid areas of Iran for the estimation of ET0. For data-limited conditions, ML techniques have been extensively used as an alternative tool for the computation of ET0. Many of the aforementioned studies are done in same agro-climatic zone, particularly in India. Therefore, it is necessary to evaluate different ML techniques in various agro-climatic zones for the calculation of ET0. Numerous studies have been reported in the literature to test the applicability of ML techniques for the estimation of reference ET0. There are various softwares used for calculation, viz., CROPWAT, Daily ET and ET0 calculator. These softwares calculate ET0 based on empirical and semi-empirical equation which requires numerous meteorological and geographical parameter. These softwares do not yield any result if one parameter is missing. In most situations, meteorological variable is scarce and limited to very few numbers of climatic variables. Therefore, present and future estimation of ET0 is confined. Recently there have been many attempts to compute and predict ET0 with higher accuracy and at different time scales. Numerical and statistical techniques have been applied to mimic the random nature of meteorological variables. The difficulties associated with these methods forced scientists and researchers to look for alternative options such as, data-driven techniques and machine learning approach, viz., ANN (Kumar et al. 2011). Based on review of literature, it can be summarized that machine learning models are suitable for the computation and prediction of ET0 in data-limited conditions. The study area was selected because of significant wastage of surface and groundwater due to ineffective irrigation planning by local farmers. This results in water scarcity during dry season, highlighting the need for effective irrigation plans. Accurate estimates of ET are useful for this and important for transitioning to more integrated and adaptive water resources management. The current study gives a comparison of four ML-based models to discover the best model for assessing daily ET0 under the state using minimal input variables in the sub-tropical atmospheres. The objectives of the current study are as per the following: (1) to develop different ML models, SVR, LSTM, RF and GBR for modelling ET0 over Hawalbagh experimental farm weather observatory, (2) to assess the performance and stability of these models with different input combinations, and (3) to find an appropriate approach to boost the modelling performance under the limited input factors condition.

2 Study area and data used

To illustrate the idea presented in the paper, we selected ICAR-VPKAS, Hawalbagh experimental farm, observatory (29°36′N; 79°40′E at 1250 m above mean sea level) located in Kumaon division of Uttarakhand, Almora, India. The geographical location of the study area is shown in figure 1. Meteorological data (i.e., minimum and maximum temperature, minimum and maximum relative humidity, sunshine hours, and rainfall) were obtained from the meteorological observatory for the period, January 1985–December 2010, on a monthly basis, as given in table 1. The annual rainfall of the site varies from 693 to 1415 mm with an average rainfall of 1013 mm during the study period. Maximum temperature (Tmax) ranged from 24.7° to 26.7°C with average Tmax equal to 25.93°C and minimum temperature (Tmin) ranged from 9.08º to 11.5°C with average Tmax equal to 10.4°C (figure 2a–d).

Figure 1
figure 1

Location of the study area.

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Table 1 Monthly meteorological parameter dynamics for the period of 1985–2010.
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Figure 2
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Monthly average climatic parameters for the North Bihar (1985–2011). (a) Tmax and Tmin, (b) precipitation; (c) relative humidity; and (d) hours of sunshine.

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3 Methodology

This section covers a detailed description of different machine learning models applied in the present study. The complete dataset was divided into two parts. The first part which is 80% of the whole dataset is used for training the selected ML model, while the remaining 20% second part is used to test the developed ML model. This study compared the performance of four machine learning (ML) models for modelling the ET0 using meteorological data collected from the observatory. The ML models used were LSTM, GBR, SVR, and RF regressor. Models were prepared in Python using the Keras library (https://keras.io). The flowchart used in this study is shown in figure 3.

Figure 3
figure 3

Flow chart used in this study.

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3.1 Optimal input selection of input variable

The optimal input combination was selected based on seven statistical criteria: MSE, RMSE, R2, adjusted R2, Mallows' Cp, Akaike's AIC, and Amemiya's PC. The combination with the smallest and closest to zero values of MSE, Mallows' Cp, Akaike's AIC, and Amemiya's PC, and the largest values of R2 and adjusted R2 were chosen as the optimal input combination for ET0 modelling. This approach was applied at Hawalbagh experimental farm and five different input combinations were examined. The top five models were selected based on combined scores calculated from above-mentioned statistical parameters. The statistical analysis of these combinations is presented in table 2.

Table 2 The best top five models along with their input combination and statistics values.
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3.2 Model development

In order to build a reliable predictive model, selecting the appropriate predictors is a crucial step. In the current study, four different machine learning (ML) models were chosen to predict monthly evaporation: the classification and regression tree (CART), the cascade correlation neural network (CCNN), gene expression programming (GEP), and support vector machine (SVM). These models were developed using five different input combinations:

$$\text {Model} \; 1\; \left({\mathrm {M}}1\right){:}\; {ET}_{0}=f\left({T}_{\mathrm{max}},{\mathrm{RH}}_{\mathrm{mean}}\right)$$
(1)
$${\text {Model}} \; 2 \; \left({\mathrm {M}}2\right){:}\; {ET}_{0}=f\left({T}_{\mathrm{min}},{\text {RH}}_{\mathrm{mean}},{\text {RF}}\right)$$
(2)
$${\text {Model}} \; 3\; \left({\mathrm {M}}3\right){:}\; {ET}_{0}=f\left(\text {SSh},{\text {RH}}_{\mathrm{mean}}\right)$$
(3)
$$\text {Model} \; 4\; \left(\text {M}4\right){:}\; {ET}_{0}=f\left({T}_{\mathrm{min}},\mathrm {RF,SSh}\right)$$
(4)
$$\text {Model}\;5\; \left({\mathrm {M}}5\right){:}\; {ET}_{0}=f\left({T}_{\mathrm{min}},{T}_{\mathrm{max}},\mathrm {RF}\right)$$
(5)

where ET0 is the reference evapotranspiration, WS is the wind speed, RF is the rainfall, RH is the relative humidity, Tmin is the minimum temperature, Tmax is the maximum temperature, and SSh is the number of sunshine hours.

3.3 Calculation of reference ET

There are different equations developed to estimate ET0 across the globe in different agro-climatic zone and extensively evaluated using Lysimeter data. However, the standard and reliable method for computing ET0 is FAO Penman–Monteith (Allen et al. 1998). In the present study, we have used FAO Penman–Monteith to calculate ET0. It represents the response of weather parameters to environmental conditions. Kc is different for different crops and represents the crop canopy development and crop management practices through the growing season. Daily meteorological data were collected for five variables, which included maximum air temperature (Tmax °C), minimum air temperature (Tmin °C), minimum relative humidity (RHmin, %), maximum relative humidity (RHmax, %), wind speed (U2, m s−1), and solar radiation (Rs, MJ m−2 d−1). Measurements were taken at a height of 2 m for air temperature and relative humidity, and at a height of 10 m for wind speed above the soil surface. Wind speed data at 2 m (U2) were obtained using the log-wind profile equation from those collected at 10 m. Using the meteorological data available obtained from the weather station. Crop evapotranspiration is calculated as:

$${ET}_{c}={K}_{c}\times {ET}_{0}.$$
(6)

The FAO-56 PM method was used as a standard method in the present research.

$${ET}_{0}=\frac{0.408\left({R}_{n}-G\right)+\frac{900}{{T}_{a}+273}u\left({e}_{s}-{e}_{a}\right)}{\Delta +\gamma \left(1+0.34u\right)}$$
(7)

where ET0 is the potential reference crop evapotranspiration (mm \({\mathrm{day}}^{-1}\)); \({R}_{n}\) is the net radiation (MJ \({\mathrm{m}}^{-2} \;{\mathrm{day}}^{-1}\)); G is the soil heat flux (MJ \({\mathrm{m}}^{-2}\; {\mathrm{day}}^{-1}\)), which is assumed as null for daily periods; \({T}_{a}\) is the average daily temperature (\(^\circ{\rm C}\)); u is the wind speed at a height of 2 m (m \({\text {sec}}^{-1}\)); \({e}_{s}\) is the saturation vapour pressure deficit (kPa); \({e}_{s}\) is the actual vapour pressure deficit (kPa); \({e}_{s}-{e}_{a}\) is the vapour pressure deficit (kPa); \(\Delta\) is the slope of the saturation vapour pressure–temperature curve (kPa \({^\circ{\rm C} }^{-1}\)); and \(\gamma\) is the psychometric constant (kPa \({^\circ{\rm C} }^{-1}\)).

3.4 Machine learning models

3.4.1 Gradient boosting regressor

The gradient boosting regressor (GBR) is an ensemble machine learning model that uses a collection of sequentially arranged tree models to make predictions. It is designed to improve the accuracy of weak prediction models, typically decision trees, by iteratively building on the errors of previous models in the ensemble. This technique is known as ‘boosting’.

As the GBR algorithm iteratively builds each model, it focuses on improving the predictions of the previous model by learning from the errors it made. The result is a more robust and accurate model that is capable of making better predictions on the given dataset. Overall, the GBR model is a powerful tool for machine learning tasks that involve predicting numerical values. A GBR with M number of trees can be stated as:

$${f}_{M}\left({x}_{j}\right)=\sum_{m}^{M}{\gamma }_{m}{h}_{m}({x}_{j})$$
(8)

where \({h}_{m}\) is a weak learner that performs poorly individually and \({\gamma }_{m}\) is a scaling factor adding the contribution of a tree to the model.

GBR model is implemented using the gradient boosting regressor (GBR) method provided by Scikit-learn, a widely used machine learning library. The method is based on the algorithm developed by Jerome H Friedman in the late 1990s, which involves iteratively improving the performance of weak learners by adding them to an ensemble model. The resulting model is a more accurate and robust predictor than any of the individual weak learners used to build it. Overall, the GBR model is an effective approach to solving regression problems in machine learning.

3.4.2 Support Vector Regression

Vapnik (1995) introduced support vector machine algorithm, which is one of the most popular method among machine learning techniques (Wu et al. 2008). SVM is used by several researchers across the globe for the estimation of ET (Shrestha and Shukla 2015; Dou and Yang 2018; Tang et al. 2019). The most important property of SVM is inclusion of kernel, which makes it powerful to deal with nonlinear properties of the system. SVM is a supervised machine learning method used for both regression and classification analysis in agriculture. It plots data into a high-dimensional feature space using kernels and can classify nonlinearly separable datasets. The accuracy of the SVR model depends on appropriate selection of kernels and parameters, and RBF has been shown to have favourable performance in forecasting. The present study used the LIBSVM library to develop SVR models with the RBF kernel. The fundamental requirement of a kernel is that it must satisfy Mercer`s theorem. The optimization algorithm has a global optimization function that makes it different from other machine learning techniques such as ANN, which consider the local maxima. Kernels are the foundation of SVM. The typical assembly of the SVM model is shown in figure 4. The SVM (Support Vector Machine) technique applies the SRM (structural risk minimization) principle to minimize the risk of overfitting and find the optimal decision boundary between different classes.

Figure 4
figure 4figure 4

Scatter plots of the FAO-56 ET referred as ET0 and those estimated by the ML models. (a) SVR, (b) RF, (c) GBR, and (d) LSTM with all parameters of input for during the testing period at Hawalbagh experimental farm.

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3.4.3 Random forest

RF is an ensemble machine algorithm, which is efficient in both classification and regression. It is generally used for predictive analysis. In predicting the final output, combination of all decision trees is considered rather than depending on individual decision trees. The foundation of RF is based on supervised machine learning approaches, which are constantly used in machine learning and popularly used in hydrology. Observed and predicted values are used to calculate the sum of square error (SSE). This process occurs repeatedly until the whole set of data is covered. Mathematically RF can be expressed as:

$$u\left(x\right)={u}_{0}\left(x\right)+{u}_{1}\left(x\right)+{u}_{2}\left(x\right)+{u}_{3}\left(x\right)+\cdots$$
(9)

where the resulting function u is the addition of individual base model ui, where each individual base repressor is the individual decision tree. The fundamental principle behind RF model is to integrate several decision trees in estimating the final result rather than individual decision trees.

Random forests or random decision forests are a machine learning technique used for classification, regression, and other tasks. They work by creating multiple decision trees during training and then using them to make predictions by taking the mode of the classes for classification or the mean prediction for regression. Random forests help to reduce the overfitting of decision trees to their training set (Smith et al. 2013; Misra and Li 2020). Extra trees (extremely randomized trees) is an ensemble learning method that builds on the Random Forest algorithm (Breiman 2001). Each decision tree in the extra trees forest is constructed from the original training sample. However, at each test node, each tree is provided with a random sample of features from the feature set, and each decision tree must select the best feature to split the data based on some mathematical criteria, typically the Gini Index. This process of using a random subset of features at each test node results in the creation of multiple de-correlated decision trees, which can improve the accuracy and reduce overfitting.

3.4.4 Long short-term memory

Long short-term memory (LSTM) is a type of recurrent neural network (RNN) that is designed to address the vanishing gradient problem and the inability of traditional RNNs to handle long-term dependencies in sequential data.

LSTM networks are comprised of memory cells that can selectively remember or forget information based on inputs from different gates. These gates are made up of different layers, including the input gate, forget gate, output gate, and cell state layer. The interactions between these layers allow the LSTM network to selectively retain or discard information from previous time steps, making it more effective for modelling long-term dependencies. The structure of an LSTM model is composed of an input gate, an output gate and a forget gate. The gate’s value in LSTM is calculated using the previous cell value \({C}_{t-1}\), previously hidden values \({h}_{t-1}\) and input \({x}_{t}\).

$${i}_{t}=F\left({W}_{xi}{x}_{t}+{W}_{hi}{h}_{t-1}+{W}_{ci}{C}_{t-1}+{bias}_{i}\right)$$
(10)
$${O}_{t}=F\left({W}_{xo}{x}_{t}+{W}_{ho}{h}_{t-1}+{W}_{co}{C}_{t-1}+{bias}_{o}\right)$$
(11)
$$f_{t} = F\left( {W_{xf} x_{t} + W_{hf} h_{t - 1} + W_{cf} C_{t - 1} + bias_{f} } \right).$$
(12)

And the cell value is calculated using

$${C}_{t}={f}_{t}{C}_{t-1}+{i}_{t}F\left({W}_{xc}{x}_{t}+{W}_{hc}{h}_{t-1}+{bias}_{c}\right),$$
(13)
$$h_{t} = O_{t} \tan^{ - 1} \left( {C_{t} } \right).$$
(14)

3.5 Model assessment

To examine the accuracy of ET0 predicted by different algorithms with observed values, different statistical indicators coefficient of determination (R2), root mean squared error (RMSE), mean square error (MSE) and Nash–Sutcliffe Efficiency (NSE) were used. The mathematical expressions are as follows:

$${R}^{2}=1-\frac{\sum {\left({ET}_{\mathrm{Pre}}-{ET}_{\mathrm{Obs}}\right)}^{2}}{\sum {\left({ET}_{\mathrm{Obs}}-{ET}_{\mathrm{Mean}}\right)}^{2}}$$
(15)
$${\text {RMSE}}=\sqrt{\frac{\sum_{i=1}^{n}{\left({ET}_{\mathrm{Pre}}-{ET}_{\mathrm{Obs}}\right)}^{2}}{n}}$$
(16)
$${\text {MSE}}=\frac{1}{n}\sum_{i=1}^{n}{\left({ET}_{\mathrm{Pre}}-{ET}_{\mathrm{Obs}}\right)}^{2}$$
(17)
$${\text {NSE}} = 1 - \left[ {\frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {ET_{{{\text{Obs}}}} - ET_{{{\text{Pre}}}} } \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{n} \left( {ET_{{{\text{Pre}}}} - ET_{{{\text{Mean}}}} } \right)^{2} }}} \right]$$
(18)

where \({ET}_{\mathrm{Pre}}\) (mm/day) is predicted ET0 using different machine learning algorithm at instant i, \({ET}_{\mathrm{Obs}}\) (mm/day) is observed ET0 at instant i, \({ET}_{\mathrm{Mean}}\) (mm/day) is the average ET0 at instant i, and n is the number of total observations.

4 Results and discussion

The study analysed the performance of LSTM, SVR, GBR, and RF models in estimating daily ET0 using four pre-processing data methods: Pearson correlation, PCA, Lasso model, and random forest. The climate variables considered were daily maximum temperature, minimum temperature, relative humidity, and solar radiation. The optimal input combinations and each model's performance in terms of R2, RMSE, NSE, and MSE were presented for Hawalbagh experimental farm observatory. The Grid Search CV python library module was used to optimize SVR parameters. The performance of different machine learning algorithms in terms of statistical parameters MSE, RMSE, MSE, and R2 were summarized in table 3.

Table 3 Performance of random forest (RF), support vector regressor (SVR), gradient boosting regressor (GBR), and long shortterm memory (LSTM).
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4.1 ML models performance with various input combinations

The study evaluated the performance of LSTM, SVR, GBR, and RF models for estimating daily ET0 using different pre-processing methods to select the best set of input variables. The four pre-processing methods evaluated were Pearson correlation, principal component analysis (PCA), Lasso model, and random forest. The optimal input combinations and the performance of each model in terms of R2, RMSE, NSE, and MSE were listed in table 3 for Hawalbagh experimental farm observatory. Table 3 showed the summary of the LSTM, SVR, GBR, and RF model performances for Hawalbagh experimental farm observatory. Taking everything into account, the RF and GBR models are the most robust among the four ML models regardless of under which input combination, trailed by LSTM and SVR models, which could generally accomplish agreeable accuracy. Figure 4 showed the comparisons between observed ET0 and model-estimated values in the form of scatter plots of the FAO-56 ET0 and those estimated by the ML models with all input parameters during the testing periods. All scatter plots of different ML models showed various distributions. The study compared the performance of LSTM, SVR, GBR, and RF models in estimating daily ET0 using different input combinations and pre-processing methods. The RF and GBR models showed closer agreement with observed ET0 and the RF model performed marginally better than the GBR model, with an estimated R2 value of 0.99 for all model input combinations. The RF model achieved the best performance among the evaluated models with all input combinations at the Hawalbagh experimental station, followed by GBR. The RF model achieved excellent performance, with an RMSE of 0.11 mm/day, MSE of 0.01 mm/day, and R2 of 0.99, followed by GBR with an RMSE of 0.17 mm/day, MSE of 0.03 mm/day, and R2 of 0.98 at the Hawalbagh experimental farm observatory (figure 4). Figure 5 shows the comparisons between observed ET0 and model-estimated values in the form of a line plot. The model results were also compared with ET0 calculated using the Penman–Monteith equation, and the results were found to be very similar to the FAO-56 ET0 based comparison, indicating that the model results were not overstated. In summary, the RF model is the most robust among the four ML models, followed by the GBR and SVR models. The comparison of RMSE values for Hawalbagh experimental farm for different input combinations is shown in figure 6. The RF and GBR models achieved acceptable accuracy across different input combinations of meteorological variables, which makes them more cost-effective and practical for development and application. These models can simulate ET0 where meteorological information is limited.

Figure 5
figure 5figure 5

Comparison of observed and estimated ET referred as ET0 by different models (GBR, LSTM, RF, and SVR) with varying parameters of input for the validation period at Hawalbagh experimental farm.

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Figure 6
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Comparison of RMSE values for Hawalbagh experimental farm for different input combinations.

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4.2 Comparison with similar studies

The paper investigated the performance of four ML models in estimating daily reference evapotranspiration, including LSTM, GBR, RF, and SVR, and found that RF performed the best, with GBR, SVR, and LSTM also producing accurate results. This is consistent with previous studies that have found ANN, LSTM, and SVR to be effective for ET0 estimation. It has also shown that this study resembles the assessment of reference evapotranspiration found by Raza et al. (2020) and Heramb et al. (2023), where SVR was found to be used as an alternative ET0 estimation model to the subsistence of conventional methods. Based on table 3, the RF and GBR models performed well in terms of accuracy and computational demand, with the RF model being particularly efficient and achieving low test RMSE values regardless of input combination. The study also found that the two input combinations of maximum temperature and mean relative humidity worked comparatively well with the RF model, followed by the GBR model. The RF and GBR model performed well in estimating ET0 with limited inputs, and therefore, it can be suggested for ET0 estimation in situations with limited meteorological parameter availability.

5 Conclusions

This study aimed to compare the performance of four machine learning models, namely LSTM, SVR, GBR, and RF, in estimating ET0 using four different input combinations at two different stations.

The study concluded that using all input variables provided the best performance, but also found that using a combination of three variables (temperature, wind speed, and relative humidity) or two variables (temperature and relative humidity, temperature and wind speed) can provide nearly identical results to using all variables. Therefore, the study suggests that using all variables should be the first priority, followed by three and two variable combinations.

The study found that the RF and GBR model had the best performance among the four tested models, regardless of station or input combination. The LSTM and SVR models were able to achieve good performance using only temperature, relative humidity, and wind speed data, making them more practical and useful for application. The input combination of temperature and wind speed followed by temperature and relative humidity also showed good results, while other models did not perform as well as the LSTM and SVR models with the mentioned input combinations.

The study concluded that using a combination of three or two meteorological parameters, such as temperature, wind speed, and relative humidity, can still provide accurate estimation of reference ET0, even if not all parameter information is available. The LSTM and SVR models were found to be highly relevant for modelling ET0 at spatiotemporal scales, even when meteorological parameter information is limited or fragmented, and are strongly recommended for such cases.

The findings of this study can be applied to model ETcrop for various crops grown in the region, including rice, wheat, and pulses. Additionally, this study provides a useful reference for researchers to develop similar models for other study areas. Since machine learning techniques are data-driven techniques, performance of these ML techniques would deteriorate in case of limited meteorological data. Further, the application of different optimizers in conjunction with machine learning and deep learning models with cross-station data in different agro-climatic zone can be employed in ET0 modelling.