Introduction

The evapotranspiration is a component of hydrological cycle. For determination of agricultural water demand and the useful storage of dam, considering this natural phenomenon is very necessary. The transpiration value of different plants and water available in each region distinguish type of cultivation of that region. In addition, in design and calculation of the useful storage volume of dams, the evaporation value must be considered. This considerate helps in supplying of different water demands.

Iran locates in the north subtropical high-pressure belt or horse latitude (between 30° and 35° N). Therefore, the value of precipitation is low and the value of evapotranspiration is high in Iran. Among different provinces of Iran, the Khuzestan Province has the highest temperature; the value of evapotranspiration is very high and the value of precipitation is very low in this province. Because of location of this province in outlet of the Persian Gulf watershed, this province has six large rivers and the largest surface water resources in Iran. In this province, the mean annual precipitation and temperature are 284.3 mm and 25.3 °C, respectively. The mean annual minimum and maximum temperatures are 18.2 °C and 32.4 °C, too. Therefore, the Khuzestan Province is the most important agricultural region in Iran. Because of the lack of measurement of reference evapotranspiration (ET0), using a standard method for estimation of ET0 is necessary. The FAO Penman-Monteith (FPM) method is sole standard method that uses meteorological data such as the maximum temperature; wind speed; average relative humidity; sunny hours. However, this method needs lots of meteorological data. For reduction in number of effective meteorological parameters, this study utilizes different machine learning (ML) methods and selects the best method for estimation of ET0.

Shirmohammadi-Aliakbarkhani and Saberali (2020) applied FPM method and eight other empirical equations for estimating ET0 in arid regions of Iran and showed superiority of FPM method. Zarei and Mahmoudi (2021) used FPM, Hargreaves–Samani (HS), Jensen–Haise (JH), and Blaney–Criddle (BC) equations for estimating ET0 in Iran. They observed the good fitness between results of these equations with results of FPM method. Rafiee and Mahmoodi-Eshkaftaki (2021) applied seven empirical equations for estimating ET0 in the Bushehr province of Iran and observed superiority of FPM and Makkink methods.

Due to development of new AI methods and improvement of previous methods in recent years, researchers use these methods for estimation of evapotranspiration. They used one method or utilized several AI methods and compared their results by performance criteria such as R2 and the root mean square error (RMSE).

da Silva Júnior et al. (2019), Guven and Kisi (2013), Kiafar et al. (2017), Kisi and Guven (2010), Kumar et al. (2002) and Mattar (2018) applied a method for estimation of ET0 while Gavili et al. (2018), Granata (2017), Ladlani et al. (2014), Landeras et al. (2018), Mehdizadeh (2018), Mehdizadeh et al. (2017), Nourani et al. (2019), Sanikhani et al. (2019), Tang et al. (2018), Yassin et al. (2016), Rahimikhoob (2009), Rahimikhoob et al. (2013), Patle et al. (2020) and Zounemat-Kermani et al. (2019) used several methods for this purpose.

Among these studies, Gavili et al. (2018), Landeras et al. (2018), Sanikhani et al. (2019), Yassin et al. (2016) and Zounemat-Kermani et al. (2019) used artificial neural network (ANN) and Gene Expression Programming (GEP) methods; Granata (2017) and Rahimikhoob et al. (2013) applied M5 tree (M5T) model; Kiafar et al. (2017), Mattar (2018), Mehdizadeh (2018) and Mehdizadeh et al. (2017) utilized GEP method and Kumar et al. (2002), Rahimikhoob (2009), Patle et al. (2020) and Nourani et al. (2019) considered ANN for estimation of ET0.

The previous studies evaluated the ability of different AI methods for estimation of reference evapotranspiration and distinguished the superior method in their considered region. The region of this study is a watershed in the southwest of Iran. The Karun and Dez Rivers watershed is a large watershed. The south of the watershed has an arid and hot climatic and the south of watershed has a semi-arid and moderate climatic. This study considered two synoptic weather stations in the south and north of watershed. The novelties of this research include options:

  1. 1.

    Considering a distinguished watershed and determination of effective parameters on evapotranspiration. These parameters are dependent on location of synoptic weather station and their climatic conditions. This fact can help designers and managers that study about methods of evapotranspiration control. The most of researchers considered synoptic weather stations that are far away from each other. Their studies cannot show importance of effective parameters well. Due to the importance of these parameters can change in a short distance.

  2. 2.

    Using the FPM method for evaluation of accuracy and verification of applied AI methods. This method considered a large number of meteorological parameters. These parameters include the maximum, average, minimum temperature, wind speed, maximum, average, minimum relative humidity and sunny hours. This study considers 42 combinations of these parameters and selects the best combination for each AI method by the stepwise method.

  3. 3.

    Sensitivity analysis determines importance of each meteorological parameter. This study used sensitive analysis and expounds the obtained results based on locations of considered synoptic weather station.

Materials and methods

Case study

The Dez and Karun Rivers watershed is a large basin in the southwest of Iran. The area of this watershed is 67257 km2 and it is a sub-basin of the Persian Gulf and the Sea of Oman watershed. This study considers a sub-basin of this watershed called the downstream of the Karun River watershed. The area of this sub-basin is 5924 km2. The case studies are Ahvaz and Dezful. Ahvaz is center of the Khuzestan Province and locates in the south of watershed (downstream) and has hot arid climatic, while Dezful has semi-arid and moderate climatic and locates in the north of watershed (upstream). Dezful is close to reservoir of the Dez dam. The location of Ahvaz synoptic weather station is in 31° 20′ N and 48° 40′ E, and its elevation is 22.5 m a.s.l. The location of Dezful synoptic weather station is in 32° 24′ N and 48° 23′ E, and its elevation is 143 m a.s.l. The meteorological parameters of these stations are illustrated in Table 1:

Table 1 The values of used monthly meteorological parameters in the FPM method
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Data analysis

The used meteorological data were prepared from the Iran Meteorological Organization (IMO) and the Khuzestan Water and Power Authority (KWPA). The Mann–Kendall trend test illustrated that annual and monthly time series of meteorological parameters (the minimum and maximum temperature, average relative humidity, wind speed and sunny hours in day) have not any trend (p > 0.1 and \(\left| Z \right|\)< 1.65) (see Mann 1945; Kendall 1975). In addition, the pettitt test did not show any change point in time series of these parameters (see pettitt 1979). Tables 2 and 3 illustrate maximum obtained \(\left| Z \right|\) for these tests and the month in which the maximum amount of \(\left| Z \right|\) occurred. The Kruskal–Wallis test showed the accuracy of the estimated values. The degrees of freedom of considered time series was 11 (12 month − 1 = 11). For all meteorological parameters χ2 < 19.675 (19.675 is critical chi-square value for α level = 0.05 and degrees of freedom = 11) (Fig. 1).

Table 2 The results of the Mann–Kendall trend test for five meteorological parameters in Ahvaz and Dezful
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Table 3 The results of the Pettitt's test for five meteorological parameters in Ahvaz and Dezful
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Fig. 1
figure 1

The location of Ahvaz and Dezful in the Khuzestan province

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The correlation analysis determines coefficient of determination (R2) between five meteorological parameters and ET0. Figures 2 and 3 illustrate the values of R2 in Ahvaz and Dezful.

Fig. 2
figure 2

The value of R2 between five meteorological parameters and ET0 in Ahvaz

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Fig. 3
figure 3

The value of R2 between five meteorological parameters and ET0 in Dezful

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Figures 2 and 3 show high correlation between ET0 and maximum and minimum temperature, average relative humidity and sunny hour. However, the correlation between wind speed and ET0 is relatively low. Wind speed alone cannot significantly change the ET0 value. Wind speed and changes in other meteorological parameters can change the ET0 value. In other words, to evaluate the effects of wind speed on the ET0 value, other meteorological parameters must be constant.

Also, the coefficients of determination of the five meteorological parameters in Ahvaz and Dezful are almost equal and R2 cannot show the most effective meteorological parameter in these cities. For this purpose, a suitable analysis meted (sensitive analysis) must be applied. Sensitive analysis differentiates the effects of changes in one meteorological parameter on the value of ET0 while other meteorological parameters are constant.

The M5T model

The M5T is regression trees were introduced by Quinlan (1992). The characteristics of the applied M5T model in this study are (Table 4):

Table 4 The characteristics of the applied M5T, GEP and ANN-GA model in this study
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The GEP model

The Gene expression programming (GEP) was introduced by Ferreira (2001). The characteristics of the applied GEP model in this study are (Table 4):

The GEP and M5T models with these parameters have the best performance, and they have been selected by trial and error method.

The optimized artificial neural network by genetic algorithm (ANN-GA)

The characteristics of the applied ANN-GA in this study are (Table 4):

The Taylor diagram

The Taylor diagram was developed by Taylor (2001) and find the best method for simulating ET0 by using of the Pearson correlation coefficient (R), the root-mean-square error (RMSE), and the standard deviation (SD) (see Esmaeili-Gisavandani et al. 2022).

The FPM equation

The Penman (1948) and Monteith (1965) developed and introduced the FPM equation for calculating ET0. The equation of the FPM method is (Pereira et al. 1997 and Allen et al. 1998):

$$ {\text{ET}}_{0} = \frac{{0.408\Delta ({\text{Rn}} - G) + \gamma \frac{900}{{(T + 273)}}u_{2} ({\text{es}} - {\text{ea}})}}{{\Delta + \gamma (1 + 0.34u_{2} )}} $$
(1)

where ETo is reference evapotranspiration (mm/day), Rn and G are net radiation at the crop surface and soil heat flux density (MJ/m2 day), respectively, T is average of daily temperature at 2 m above ground level (°C), u2 is wind speed at 2 m above ground level (m/s), es is saturation vapor pressure (kPa), ea is actual vapor pressure (kPa), es–ea is saturation vapor pressure deficit (kPa), ∆ and γ are slope vapor pressure curve and psychrometric constant (kPa/°C), respectively. For using this equation, preparation of daily solar radiation (sunny hours), air temperature, humidity and wind speed time series data is necessary. Due to the use of different meteorological data, the ET0 calculated by this equation is more accurate than the ET0 calculated by other equations. For example, the Torrent White (TW) equation uses the mean air temperature alone.

The performance criteria

The applied performance criteria in this study are:

The coefficient of determination (R2):

$$ R^{2} = \frac{{\sum\limits_{i = 1}^{N} {({\text{ET}}_{{0{\text{AI}}}}^{i} - \overline{{{\text{ET}}_{{0{\text{AI}}}} }} } )({\text{ET}}_{{0{\text{FP}}}}^{i} - \overline{{{\text{ET}}_{{0{\text{FP}}}} }} )}}{{\sqrt {\sum\limits_{i = 1}^{N} {({\text{ET}}_{{0{\text{AI}}}}^{i} - \overline{{{\text{ET}}_{{0{\text{AI}}}} }} } )^{2} \sum\limits_{i = 1}^{N} {({\text{ET}}_{{0{\text{FP}}}}^{i} - \overline{{{\text{ET}}_{{0{\text{FP}}}} }} )^{2} } } }} $$
(2)

where ET0AI is the calculated value of ET0 by AI method and ET0FP is the calculated value of ET0 by the FPM equation.

The Mean Absolute Error (MAE) (mm/day):

$$ {\text{MAE}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left| {{\text{ET}}_{{0{\text{AI}}}}^{i} - {\text{ET}}_{{0{\text{FP}}}}^{i} } \right|} $$
(3)

The Nash–Sutcliffe model Efficiency coefficient (NSE):

$$ {\text{NS}}_{E} = 1 - \frac{{\sum\limits_{i = 1}^{N} {({\text{ET}}_{{0{\text{AI}}}}^{i} - {\text{ET}}_{{0{\text{FP}}}}^{i} )^{2} } }}{{\sum\limits_{i = 1}^{N} {({\text{ET}}_{{0{\text{AI}}}}^{i} - \overline{{{\text{ET}}_{{0{\text{FP}}}} }} )^{2} } }} $$
(4)

The Root Mean Square Error (RMSE) (mm/day):

$$ {\text{RMSE}} = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {({\text{ET}}_{{0{\text{AI}}}}^{i} - {\text{ET}}_{{0{\text{FP}}}}^{i} )^{2} } } $$
(5)

The Kling-Gupta efficiency (KGE) (McCuen et al. 2006; Gupta et al. 2009; Knoben et al. 2019)

$$ {\text{KGE}} = 1 - \sqrt {\left( {R - 1} \right)^{2} + \left( {\alpha - 1} \right)^{2} + \left( {\beta - 1} \right)^{2} } $$
(6)

where R is the linear correlation coefficient,\(\alpha\) is standard deviation of the computed ET0 to standard deviation of observed ET0, and \(\beta\) shows the average of calculated ET0 to average of observed ET0 (Lotfirad et al. 2021).

The values of KGE, R2 and NSE must be close to one and the values of MAE and RMSE must be close to zero.

The advantages of applied research methodology in this study are:

  1. 1.

    Using the sensitive analysis. The coefficient of determination (R2) cannot show effects of meteorological parameters on ET0 value. This coefficient cannot evaluate the effects of each parameter independently. The climatic conditions in Ahvaz and Dezful are relatively similar. However, topographic features and their distance from the Persian Gulf are different. Therefore, the sensitive analysis is a suitable tool for determine the most effective meteorological parameter on ET0 value in these regions.

  2. 2.

    This study uses three types of AI methods. The ANN-GA is a black box method, M5T is a classification and linear regression method and GEP is a nonlinear regression method. Therefore, this study evaluates abilities of different categories of AI methods for estimation of ET0.

  3. 3.

    This study uses three types of performance criteria (a) for determination correlation between estimated ET0 values with estimated ET0 by FPM method (R2 and NSE) (b) for determination accuracy of estimated ET0 values (MAE and RMSE) (c) combination of them (KGE). At the end, accuracy and correlation of estimated ET0 values by different AI methods are compared by the Taylor and boxplot diagrams. Figure 4 illustrates the flowchart of research methodology.

Fig. 4
figure 4

The flowchart of research methodology

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Results and discussion

This study includes the following steps, and the results of each step are reported as follows:

  1. 1.

    Collection of daily meteorological data during 1963 to 2018. For this purpose, it used climatic data of KWPA and IMO.

    In this research 8*54*365 = 157,680 daily meteorological data were considered. 2/3 of data were used for training, and 1/3 of data were used for testing.

  2. 2.

    Using the FPM equation for estimation of ET0 and verification of applied AI methods.

  3. 3.

    Using of the optimized artificial neural network by genetic algorithm (ANN-GA) for estimation of ET0. The GA optimized training process of ANN. The objective function of GA is minimizing difference between the calculated ET0s by ANN and the FPM equation. The type of used ANN is multilayer perceptron (MLP) and GA distinguishes architecture, transfer or activation function and training rule of ANN.

    Introduced meteorological parameters to ANN-GA are similar to parameters of the FPM equation (the minimum and maximum daily temperature, the average of daily relative humidity, the daily wind speed and sun hours).

    In Ahvaz, the optimum ANN- GA has one layer and 10 nodes. The training method is the Levenberg–Marquardt algorithm and transfer function is the sigmoid function.

    In Dezful, the optimum ANN- GA has one layer and 6 nodes. The training method is the Levenberg–Marquardt algorithm, and transfer function is the tangent hyperbolic function.

  4. 4.

    Selection of the best combination of meteorological parameters at each synoptic climatic station for introducing to AI methods. The number of considered meteorological parameters is eight parameters for these models (five parameters of the FPM equation, the average of daily temperature, minimum and maximum daily relative humidity). The applied method for selecting meteorological parameters is the stepwise regression method. In this method, the coefficient of determination between the calculated ET0 based on different combinations of meteorological parameters and the calculated ET0 by the FPM equation is determined. Then combination that has the highest value of the coefficient of determination is selected for estimation of ET0.

    The stepwise regression method showed that the best combination of introduced inputs to GEP model is maximum and average temperature, minimum and average relative humidity, wind speed and sunny hours in Ahvaz. This combination is average temperature and relative humidity, wind speed and sunny hours in Dezful.

    The stepwise regression method showed that the best combination of introduced inputs to M5T model is minimum, maximum and average temperature, minimum, maximum and average relative humidity, wind speed and sunny hours in Ahvaz. This combination is maximum and average temperature, maximum and minimum relative humidity, wind speed and sunny hours in Dezful.

  5. 5.

    Using GEP and M5T model for estimation of ET0.

  6. 6.

    Comparison of the results of three AI methods by the performance criteria. These criteria are:

    Tables 5 and 6 illustrate the performance criteria values for training and testing stages of ANN-GA, GEP and M5T methods. These tables show the superiority of the ANN-GA method over other ET0 estimation methods.

    Figures 5 and 6 show boxplot and Taylor diagrams for comparison between results of ANN-GA, GEP and M5T methods.

    Figure 5 shows that the average ET0 and the ET0 quartiles calculated by the FPM equation have the best fit with the average ET0 and the ET0 quartiles calculated by the ANN-GA method, while the results of the GEP method have the worst fit with results of the FPM equation.

    The Taylor diagram shows that the results of the ANN-GA (blue circle) method are closer to calculated ET0 by the FPM equation (white circle). Therefore, the results of this method are more accurate than results of GEP and M5T methods. The boxplot diagram shows that the mean and quantiles of calculated ET0 by ANN-GA method are closer to calculated ET0 by the FPM equation.

  7. 7.

    Sensitive analysis

    For the following two reasons, sensitivity analysis is an essential requirement for evaluating the performance of AI methods:

  8. (a)

    AI methods are black box models and cannot show the physical relationship between model inputs and outputs. Sensitive analysis eliminates this defect and shows the effects of input changes on output changes based on the physical relationship between them.

  9. (b)

    Data deficiencies, poor data quality, inaccurate measurements, and the use of undesirable methods to collect and record data are sources of uncertainty for application of AI methods. These factors reduce the level of confidence in the results of AI models. Sensitivity analysis determines the inputs that have the greatest impact on the outputs of the AI model. This can guide researchers and managers to increase the quality of the most effective parameters to improve the results of AI models.

Table 5 The performance criteria values for training stage
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Table 6 The performance criteria values for testing stage
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Fig. 5
figure 5

The boxplot diagram a Ahvaz b Dezful

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Fig. 6
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The Taylor diagram

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The results of ANN-GA (the best AI method for estimating ET0) can be used for sensitivity analysis. The sensitivity analysis can show importance of different meteorological parameters for estimation of ET0 and can show variations of ET0 based on changes in these five parameters. This matter can be applied for studying relation between location of synoptic weather station and meteorological parameters.

For sensitivity analysis, this study uses the one-at-a-time sensitivity measures method for finding the most effective parameter (referred to Hamby 1994) and the ANN-GA method (the best method for estimation of ET0). Figure 7 illustrates results of sensitivity analysis in Ahvaz and Dezful.

Fig. 7
figure 7

The results of sensitivity analysis of different parameters

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For sensitivity analysis, this study used one-at-a-time sensitivity measures method (see Hamby 1994 and Adib et al. 2019). The sensitivity is the ratio of the ET0 (after doubling the desired parameter) to initial value of ET0. For example, if this ratio is equal to 0.5, the ET0 will become half.

Figures. 8 and 9 show variations of ET0 vs variations of five meteorological parameters in Ahvaz and Dezful.

Fig. 8
figure 8

Variations of ET0 vs variations of five meteorological parameters in Ahvaz

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Fig. 9
figure 9

Variations of ET0 vs variations of five meteorological parameters in Dezful

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Figures 7, 8 and 9 show the most effective meteorological parameters for estimation of ET0. The maximum temperature is the most effective phenomena and the wind speed is the second effective parameter in estimating ET0. They have direct relationship with the value of ET0. In Dezful, the effect of the maximum temperature is more than those of Ahvaz but the effect of wind speed is less than those of Ahvaz. Because Ahvaz is more flat than Dezful (the movement of wind in Ahvaz is freer than those of Dezful). The third effective meteorological parameter is the average relative humidity in Ahvaz and the sunny hours in Dezful. The reason of this subject is the less distant of Ahvaz from the Persian Gulf (it is sources of moisture). In Ahvaz, relative humidity increases by increasing sunny hours. Therefore, ET0 decreases for more than 9 h of sunshine hours.

Figures 10 and 11 show comparison between estimated daily ET0 by three AI models with calculated daily ET0 by the FPM equation.

Fig. 10
figure 10

Comparison between estimated daily ET0 by three AI models and the FPM equation in Ahvaz a The mean of daily ET0 b The maximum of daily ET0 c The minimum of daily ET0

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Fig. 11
figure 11

Comparison between estimated daily ET0 by three AI models and the FPM equation in Dezful a The mean of daily ET0 b The maximum of daily ET0 c The minimum of daily ET0

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Figures 10 and 11 show superiority of ANN-GA method over GEP and M5T methods (similar to the Taylor and boxplot diagrams). The M5T method is a classification method. The GEP method uses nonlinear regression relations, while the M5T method utilizes linear regression relations. The ANN is a black box model and uses nonlinear equations and backpropagation technique for training. Also in this study, the GA algorithm optimized number of nodes and parameters of ANN. This matter improved accuracy of estimation of ET0 by ANN-GA method.

Figure 12 shows comparison between estimated daily ET0 by ANN-GA method with calculated daily ET0 by the FPM equation in testing stage.

Fig. 12
figure 12

Comparison between estimated daily ET0 by ANN-GA method and the FPM equation in testing stage a Ahvaz b Dezful

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Figure 12 presents good fitness between calculated ET0 by FPM and ANN-GA methods. Then, ANN-GA method is an appropriate tool for estimating ET0 by introducing meteorological parameters to it.

Conclusion

Because the measurement of evapotranspiration is very difficult, this study used the FPM method for calculation of ET0 and estimated ET0 by three AI methods. The ANN-GA method has better performance than M5T and GEP methods. The GA algorithm optimizes architecture and parameters of ANN, and this method uses nonlinear equations. This optimization procedure reduced RMSE and MAE and increased NSE and R2. The ANN-GA reduced RMSE by 183% and 41% (in Ahvaz) and 131% and 13% (in Dezful) compared to the GEP and M5T methods, respectively. Also, it reduced MAE by 168% and 39% (in Ahvaz) and 120% and 12% (in Dezful) compared to the GEP and M5T methods, respectively. However, this method increased NSE and R2 slightly (less than 3%). The stepwise regression method showed that more meteorological parameters should be introduced to AI models in Ahvaz. Therefore, the performance of AI models was better in Ahvaz, while the difference between the performances of AI models was less in Dezful. The sensitivity analysis illustrated that the effects of meteorological parameters depend on the topographic features and distance from the Persian Gulf. The most effective meteorological parameter is maximum temperature. For estimating ET0 in the Khuzestan province, Rahimikhoob (2009) used ANN and showed that R2 > 0.88 and RMSE < 1.2 mm/day and Rahimikhoob et al. (2013) used M5T model and showed RMSE = 0.5 mm/day and R2 = 0.98. This comparison states importance using GA algorithm for improving accuracy of estimating ET0.

For future studies, it is recommended using other AI models such as random forest (RF), multivariate adaptive regression splines (MARS) and support vector machine (SVM) and improving accuracy of AI models by data processing approach such as wavelet analysis method.