Improving the daily pan evaporation estimation of long short-term memory and support vector regression models by using the Wild Horse Optimizer algorithm

Evaporation is a phenomenon in which water turns into vapor (Majhi et al. 2020). It is the first way of returning water to the hydrological cycle (Liu et al. 2004; Landeras et al. 2008). Evaporation is an important part of the hydrological cycle which affects the region's water balance, hydrological modeling, irrigation system design, and agricultural production (Gundalia & Dholakia 2013; Moazenzadeh et al. 2018; Guan et al. 2020; Mohamadi et al. 2020; Allawi et al. 2021). Millions of cubic meters of water are lost annually due to the process of evaporation and plenty of salt and solute which leads to water quality reduction (Havens & Ji 2018). As the shortage of water resources is always a serious problem in the world, the accurate estimation of evaporation is considered as one of the main processes of water loss which has been emphasized by researchers (Samii et al. 2023). The accurate estimation of evaporation takes a leading role in calculating water balance, designing irrigation systems, managing water resources and sustainable development (Piri et al. 2016; Dou & Yang 2018; Kumar et al. 2021). Therefore, the accurate estimation of evaporation in dry regions with a shortage of water resources seems necessary and it provides useful information in drought conditions (Malik et al. 2020a; Seifi & Riahi 2020).

Generally, evaporation is measured in direct and indirect ways. The pans are considered as one of the most common direct methods used for evaporation measurement in regions (Warnaka & Pochop 1998; Wang & Dickinson 2012; Ehteram et al. 2022). Using pans for direct measurement of evaporation is not possible in regions without meteorological stations where the establishment of the station requires costs and facilities. Instead, empirical formulas and analytical methods are used for evaporation estimation (Wang et al. 2019). Relative humidity, air temperature, wind speed, solar radiation, atmospheric pressure, dissolved solute, and latitude and longitude all have an impact on evaporation and should be taken in mind when we want to estimate the evaporation (Alizadeh et al. 2017; Tao et al. 2018).

Basically, experimental methods belong to a specific geographical region and they require specific data which is impossible in some cases (Khosravi et al. 2019). On the other hand, it is practically impossible to model evaporation through experimental methods due to the physical complexity, non-linear nature and many affecting elements on it, (Singh & Xu 1997). Nowadays regarding the limitations, new methods have been invented to estimate evaporation. In recent decades, artificial intelligence (AI) technique as a new approach, has been successfully applied to water resources studies (Diop et al. 2018; Qasem et al. 2019; Ali et al. 2020; Malik et al. 2020b; Tur & Yontem 2021).

This technique has demonstrated its ability to predict many hydrological variables such as streamflow (Adnan et al. 2021a), drought (Parisouj et al. 2020), rainfall (Salih et al. 2020; Adnan et al. 2021b), groundwater (Rahman et al. 2020; Mosavi et al. 2021) and evapotranspiration (Alizamir et al. 2020; Makwana et al. 2023). The main reason for using this technique is the high ability to establish non-linear relationships between input and output variables. In the first phase, this technique receives data as input in different forms such as numbers, speeches, texts, images etc. Consequently, it processes data through various algorithms. After the processing phase, the AI technique provides an output for the input. Finally, the result will be evaluated through analysis, discovery and feedback. This loop continues until the desired result is attained. Neural networks, machine learning, cognitive computing, deep learning, computer vision and natural language processing can be considered as main components of AI. AI technique has been accepted by researchers due to the high-speed data processing. Although the usage of the AI technique has been providing accurate and fast estimation of hydrological variables, its accuracy highly relies on the user's knowledge and the perception of AI (Razavizadeh & Dargahian 2018).

Many algorithms such as Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Differential Evolution (DE), Bat Algorithm (BA), Gray Wolf Optimizer (GWO), Harris Hawks Optimization (HHO), Giza Pyramids Construction (GPC) etc. made hybrid with AI technique in order to augment the capabilities of it. These algorithms are classified as meta-heuristic algorithms which are used for optimization. They have become popular in recent years due to their higher efficiencies, optimal value estimation of model parameters and high accuracy of estimation. Ehteram et al. (2022) used an ANN model as well as several algorithms such as Firefly Algorithm (FFA), Capuchin Search Algorithm (CSA), GA, and Sine Cosine Algorithm (SCA) to predict how much water would evaporate each day for seven synoptic stations in Iran with different climates. Then, they used the Inclusive Multiple Model (IMM) for forecasting the evaporation. Guan et al. (2020) used the SVR model combined with the Krill Herd Algorithm (SVR-KHA) to estimate daily E_p in three Iranian meteorological stations with humid climates. Various combinations of meteorological input variables including T_max, RH, rainfall, SSH, WS, solar radiation, and vapor pressure were considered for E_p modeling. Feng et al. (2018) suggested an Extreme Learning Machine (ELM), optimized ANN models with PSO algorithm (ANN–PSO) and GA algorithm (ANN–GA) for estimating E_p in different climate regions of China. Bhattarai et al. (2023) used LSTM and Gaussian process regression (GPR) models for forecasting monthly E_p at two meteorological stations, one with a tropical climate in Hialeah, Florida and the other with a Mediterranean climate in Markley Cove, California.

The innovative aspect of this study is to use of the new WHO algorithm for the optimization of LSTM and SVR parameters with the goal of improving the estimation of daily E_p in different climates.

Fars province is one of the southern provinces of Iran, which is situated at latitude 27°1′ to 31°42′ N and longitude 50°34′ to 55°44′ E and covers an area of 122,272 km². This province with 37 cities has diverse climates due to the uneven distribution of precipitation. This province contains of 67.4% range, 20.4% forest and 12.2% desert. It has a mean temperature of 18.9 °C, annual rainfall of 286.8 mm and annual evaporation of 2,553.4 mm. The cities of Larestan, in the south of Fars province, and Abadeh in the north were selected as the case study. Larestan city is located in 351 km from the center of the province. This city is spread between latitude 27°19′ to 28°11′ N and longitude 53°22′ to 55°44′ E with an area of 10,740 km². The mean temperature, annual rainfall, and annual evaporation of this city are respectively, 23.9 °C, 213 mm, and 3,307 mm. According to the Emberger climate classification system, Larestan city has a hot desert climate.

For modeling the daily E_p of both cities, the data of synoptic meteorological stations of these cities were collected on a daily scale. Larestan meteorological station is located at latitude 27°40′12″ N, longitude 54°22′29″ E and an altitude of 792 m. In such manner, Abadeh meteorological station is situated at latitude 31°11′54″ N, longitude 52°36′42″ E and the altitude of 2030 m. The distance between the two stations is 611 km. The collected data included E_p, SSH, T_max, T_min, WS, and RH. In this study, 3,654 data sets and 2,483 data sets were used respectively for Larestan and Abadeh stations during the years 2012–2022. Due to the missing data in Abadeh station for some years, the number of data is less than that in Larestan station. Table 1 shows the statistical characteristics of the data used in both meteorological stations. The variables of T_min, T_max, RH, WS, SSH were considered as primary input variables and the daily E_p considered as the output variable. The PMI algorithm was used to identify the EIVs on E_p among the five primary input variables in both stations.

Among the non-linear algorithms for choosing EIVs for models based on data processing is the PMI algorithm. Sharma (2000) created the PMI-based input selection (PMIS) algorithm that is being presented here in order to find EIVs in hydrological models. Every iteration of the PMI algorithm looks at two variables: an input (C) and an output (Y). Then, it finds the value of PMI by considering the output variable by maximizing Cs (assuming that Cs varies from C). The confidence bounds derived from the distribution created by a bootstrap loop serve as the foundation for the statistical idea that PMI estimates for Cs. Cs is added to S (the set of chosen input variables), and the method is stopped when there are no more significant inputs. At that point, the input is deemed significant.

In which n is a sample of n observations from (x, y), and f is the density derived from those observations. Equation (2) indicates that the technique used to estimate the marginal and joint PDFs has a significant impact on the precise and efficient estimation of MI. The Akaike information criterion (AIC), which is detailed below, is used to halt the PMI method.

Recurrent neural networks (RNNs) have been enhanced to include long-term dependency learning or LSTM. A primary issue with RNN networks is their short-term memory, which eventually causes gradients to explode and vanish (Bengio et al. 1994). However, the LSTM model has found a solution to this issue. The cell state is the primary component of the LSTM network. The LSTM can insert new data or extract existing data from the cell state. Gate-like structures will carry out this process. These gates are thought of as the information's input channel. They are made up of a point-to-point multiplication operator and a sigmoid function. A number between 0 and 1 that represents the proportion of input that should be delivered to the output phase is the sigmoid function's output. When the value is 1, all input should be transmitted to the output; when the value is 0, no information should be given to the output.

is the weight vector, b is the bias and is the non-linear transform function or kernel function.

denotes the percentage of stallions in the population and is used as a control parameter.

El Bilali et al. (2022) ranked the models for simulation purposes in four categories as perfect, excellent, good and poor which has been displayed in Table 5.

In this study, the new WHO algorithm was applied to train two SVR and LSTM models for estimating daily E_p. For this purpose, the daily data of two meteorological stations were used during the years 2012 to 2022 in Fars province, Iran. One of the stations is situated in Larstan City with a hot desert climate and the other is Abadeh City with a cold dry climate. The data were gathered from two stations including T_min, T_max, RH, WS and SSH as primary input variables of the models and daily E_p was considered as the output of the models.

Tables 6 and 7 display Pearson correlation coefficient values between the main meteorological parameters measured in the Larestan and Abadeh stations, respectively. These tables show a strong correlation (say > 0.6) between daily E_P and T_max, T_min and RH for the two studied stations. This correlation is negative for RH while positive for T_max and T_min.

The PMI algorithm was utilized to identify the EIVs on daily E_p. The results based on the AIC as the stopping condition of the PMI algorithm are shown in Table 8. In Table 8, iteration, variable, I(x;y), MC–I*(95), MC–I*(99), and AIC indicate the number of repetitions of the PMI algorithm, name of the variable, value of PMI for each variable, 95% range of the critical value of MI, 99% range of the critical value of MI, and numerous value of AIC for each variable, respectively.

According to the PMI algorithm, EIVs are those whose AIC value indicates a decreasing trend. The AIC values of T_max, RH, and T_min variables in the Larestan station display a decreasing trend (Table 8). This point is also true for Abadeh station with T_max, T_min, and SSH variables. Therefore, the EIVs on daily E_p in Larestan station are T_max, RH and T_min and for Abadeh station are T_max, T_min and SSH. It is significant to point out that the usage of the PMI algorithm leads to reducing the time needed for recognition of the EIVs.

The values of RMSE, NSE, and R² for models taken during the training and testing phases are shown in Table 9 for both stations. The LSTM–WHO hybrid model in both stations has the most accuracy and performance for estimating daily E_p in the training and testing phases. LSTM–WHO hybrid model for Larestan station gives RMSE (1.151/1.099 mm), NSE (0.941/0.950), R² (0.943/0.952) and for Abadeh station gives RMSE (1.135/1.163 mm), NSE (0.904/0.896), R² (0.912/0.902) for the training/testing phases (see bolded values in Table 9). Also, the LSTM standalone model has poor accuracy for daily E_p estimation. The LSTM model in Larestan station shows RMSE (1.950/1.797 mm), NSE (0.832/0.865) and R² (0.840/0.825) and in Abadeh station gives RMSE (1.843/2.114 mm), NSE (0.748/0.658), R² (0.770/0.681) for the training/testing phases.

The scatter plots of Figures 5 and 6 (right side) reveal that the accuracy of the SVR–WHO hybrid model for estimating maximum and minimum values of daily E_p is not as good as the LSTM–WHO hybrid model. However, its performance is acceptable compared to LSTM and SVR standalone models. The most distance between the regression line and the 1:1 line belongs to the LSTM model for two stations. The evaluation criteria also confirm it.

These results show the high power of the WHO algorithm by finding optimal values for parameters of standalone models which appeared for the LSTM–WHO hybrid model. These results show that standalone models are easier and faster but, in some cases, they are unable to model the complicated processes. Qian et al. (2020) believe that standalone models compared to hybrid ones, have less ability to couple and process the non-linear problems. They stated that hybrid models can be used as an efficient method to solve complicated non-linear problems. This research indicated that although the LSTM model is an efficient model for solving complicated and non-linear problems its performance is weak compared to hybrid models. The same fact is true for the SVR model. Training this model with the WHO algorithm augments its efficiency for estimating daily E_p values. Therefore, the WHO algorithm has been able to increase the accuracy of SVR and LSTM models by finding optimal values for the parameters of models.

Table 10 shows a literature review on daily Ep modeling by using different AI methods. The comparison between the evaluation criteria of the developed LSTM–WHO (Table 8) with the evaluation criteria of the models suggested by different researchers (Table 10) shows that the LSTM–WHO model has high accuracy and good performance compared to other models for daily EP estimation.

The GenA of the models in this study based on the classification method of El Bilali et al. (2022) is shown in Table 11. Also, in this table, the logarithmic values of the confidence intervals with a significance level of 95%, and the values of the upper limit (UL) and lower limit (LL) have been calculated for the studied models. All the developed models in both stations of Larestan and Abadeh have an excellent GenA. These values show that the models are well trained and the simulation process is well trained on the input data.

One of the most important climatic and hydrological variables is evaporation. The accurate estimation of evaporation especially in arid and semi-arid areas that encounter with shortage of water resources, is necessary for water management and agricultural activities. In this study for the first time, the WHO algorithm was examined to optimize SVR and LSTM models in two synoptic meteorological stations in Fars province, Iran. One of the stations is situated in Larestan city with a hot desert climate and the other is Abadeh city with cold dry climate. The accuracy of four models such as SVR, LSTM, SVR–WHO and LSTM–WHO for daily E_P modeling was examined. PMI algorithm used to identify EIVs on daily E_p. The accuracy of the developed models was evaluated based on the RMSE, NSE, and R² statistical criteria, the graphic criteria of violin plot and confidence intervals. Moreover, the GenA of the developed models for both stations were checked and its classification was also done. PMI algorithm results showed that T_max, RH and Tmin variables and T_max, T_min and SSH variables are EIVs on daily E_P in Larestan and Abadeh stations, respectively. The results of this study showed a positive application of the WHO algorithm in improving the accuracy of individual models for estimating the daily E_p. It is concluded that the LSTM–WHO hybrid model has the best performance in daily E_p estimation for both stations based on statistical criteria. Therefore, the LSTM–WHO hybrid model will be chosen as the superior model. In addition, the comparison of the violin plot of the LSTM–WHO hybrid model with the violin plot of the measured values confirms the results of the statistical criteria. The results of drawing confidence intervals for the selected model revealed that the estimated E_P values are within the confidence intervals with a significance level of 95%. Also, the results of the GenA for the developed models showed that they have an excellent GenA which approved their good training of models in both stations. It should be noted that GenA of the LSTM–WHO hybrid model should be carried out in the studied area within a short distance with the same weather conditions. Therefore, the necessary actions e.g., calibration and re-evaluation should be taken in case the model will be used in other areas with different weather conditions. The limitation of this study was non-continuity in the time-series of data because of the missing data during some years. In the future, it will be suggested to use other new optimizer algorithms to optimize the parameters related to deep learning and learning machine models for estimating E_p and compare the results with the current study.

The authors would like to express their appreciation to the meteorological bureau of Fars Province for providing the information.

M.S. and H.F. wrote the main manuscript text and M.A.A. generated the code of the used models. All authors reviewed the manuscript.

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

All authors certify that they have ethical conduct required by the journal.

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

The authors declare there is no conflict.