## Abstract

This paper investigated how the meta-heuristic models can be used to facilitate the estimation of evapotranspiration (ET) images. Focusing on estimating daily ET directly from received images of the electromagnetic bands of Landsat 8 satellite utilizing metaheuristic models, authors used daily ET images estimated by the SEBAL algorithm to calibrate and verify these models. The results of this research showed that the ANN model with DC and RMSE of 0.98 and 0.09025 mm/day, respectively, is more accurate compared to the ACO (with DC = 0.65 and RMSE = 1.45 mm/day) and PSO (with DC = 0.23 and RMSE = 1.60 mm/day) models in the verification stage in estimating daily ET images. The ACO model compared to the PSO model is more accurate in estimating ET images with DC of 0.65 and 0.23 in the verification step, respectively. While removing half of the training data, the accuracy of the PSO model surpasses the ACO model with DC of 0.85 and 0.80, respectively. Also, the ANN model is more accurate than the other two models in estimating ET, both when considering all the data and half of the training data (with DC = 0.98 and RMSE = 0.09 mm/day).

## Highlights

A new methodology is proposed for more efficient ET estimation.

The applications of meta-heuristic models are evaluated in estimating ET.

Meta-heuristics models are calibrated and verified using remotely sensed data.

The most efficient meta-heuristic model in ET images estimation is the ANN model.

The proposed methodology improves the process of ET estimation in SEBAL.

## INTRODUCTION

Until now, by systematizing the acquisition of wells and installing smart meters, steps have been taken in relation to the sustainable management of groundwater resources. Measuring the amount of water withdrawal from a well by a smart meter is not suitable for measuring the actual consumption of water resources, because the amount of water withdrawal and the efficiency of management scenarios need to consider the type of irrigation and especially the amount of return flow. Therefore, it is necessary to use water based on the rate of evapotranspiration (ET) and assign its main task to an expert and trusted group. Regarding this, remote sensing of ET can be helpful for monitoring real water consumption and saving irrigation water in water-scarce regions (Cetin *et al*. 2023).

So far, the use of data obtained from satellite images and remote sensing methods have always been used in various areas of environmental and natural resource management and monitoring (Field *et al*. 1995; Gao *et al*. 2013). As an example, Fang *et al*. (2012) measured and investigated the growth stages of plants in pastures using remote sensing data. In another research, Shakoor *et al.* (2006) introduced the application of remote sensing techniques for water resources management and planning in a basin in Pakistan. This research described the importance and capabilities of new methods such as remote sensing and geographic information system (GIS) as tools for monitoring, protecting and managing water resources. The method of remote sensing analysis and GIS can show where water enters the basin and how it leaves the basin through ET and runoff. In this way, using this information, planners can identify places that have potential for the development of new water sources. This development can include redistribution and transfer of water between basins. As a result, the main goal of aforementioned research was to accurately calculate the cultivation pattern and water requirement of agricultural products in the target basin by combining a hydraulic model and data obtained from remote sensing. Psomas *et al*. (2016) discussed the concept of sustainable management of water resources in the agricultural sector using remote sensing and the concept of water, energy, land, and food nexus in the Pinios river basin of Greece. They extracted the historical data related to the cultivation and land use pattern in the desired area using satellite images and remote sensing and hired it as input information in a hydrological model. In the research of Hamzeh *et al*. (2016), by using the fuzzy-hierarchical decision-making structure and satellite images, including images related to land topography, salinity, alkalinity, moisture and soil texture, to make a decision to choose suitable areas for barley cultivation in the Shavor plain of Khuzestan.

Recently, satellite-based ET estimation models have been developed (Allen *et al*., 2007; Fisher *et al*., 2008; Leuning *et al*., 2008; Jung *et al*., 2009; Mu *et al*., 2011; French *et al*., 2015; Shekar & Raju 2022). Regarding the estimation of ET using satellite images, Ramos *et al*. (2009) calculated the amount of ET in the Ebro plain in Spain for a period of 4 years by satellite data and based on the energy balance model namely the Surface Energy Balance Algorithm for Land (SEBAL). And then the estimated ET was compared with the calculated ET values obtained from the experimental formulas. This comparison indicates a 20% difference between the results of SEBAL algorithm and the results of experimental equations in estimating ET. In Bellvert *et al*. (2018), actual ET and plant growth coefficient were calculated for almond and pistachio fields located in the Central Valley in California during their growing period. These calculations were done by combining a simple crop ET model and satellite data. Rahimzadegan & Janani (2019) investigated the efficiency of the SEBAL algorithm to estimate the ET of the pistachio plant, which is one of the most important agricultural products in Iran. In this research, they used 29 images of Landsat 8 in a period of 3 years for pistachio orchards in Sarkheh city area in Semnan province. The remotely sensed ET was compared with the ET calculated by the iMetos-pessl device. The results of this research proved that this algorithm is very accurate for calculating the actual ET for pistachio during its growing period and can be a good tool for this purpose in water resources management. Badiehneshin *et al*. (2019) estimated the actual ET in pistachio orchards using Landsat 7 satellite images for 2000 and 2001 using the SEBAL algorithm. The obtained results were compared with the available lysimetric data. The results of this research indicated an error of 20% equivalent to 0.6 mm in the estimation of real ET. They reached the conclusion that in order to better manage the irrigation deficiency of pistachio plant, it is possible to improve the performance of pistachio orchards by reducing irrigation in the pistachio growth stage and increasing irrigation in the third stage. Sane *et al*. (2020) calculated the actual daily ET in the Verdij area in the north of Tehran using the image of Landsat 8 in 2019 and the SEBAL algorithm. They compared the obtained result with the ET calculated by the experimental equations of Fao Penman-Monteith, Samani Hargreaves, and Blaney Kridel. The result of this research indicated the high efficiency of the SEBAL algorithm in estimating daily ET. Safari *et al*. (2023) developed a market-based mechanism for long-term groundwater management using remotely sensed data of ET. This paper aims to improve the efficiency of the water market mechanism by monitoring the withdrawal from the Rafsanjan Plain aquifer. For this purpose, the authors successfully used SEBAL algorithm for remote sensing of the ET of pistachio orchards. Nevertheless, in the aforementioned research, the large model uncertainties and too many input variables of the SEBAL algorithm were pointed out as the main challenges of this method including determining the reference crop ET, calculating the surface temperature, determining the appropriate hot and cold pixels, etc.

The fundamental goal of using machine learning models for estimating ET with the help of remotely sensed ET images data is to accurately predict and quantify the amount of water that is evaporated from the Earth's surface and transpired by plants. ET is an important component of the Earth's water cycle and plays a crucial role in various fields such as agriculture, hydrology, and climate modeling. By utilizing machine learning models (meta-heuristic models), researchers aim to leverage the power of algorithms to analyze and interpret the complex relationships between remotely sensed ET images and other relevant variables. These models can learn from the calibration and verification data to establish patterns and correlations, enabling them to estimate ET in areas where direct measurements may be limited or unavailable.

The use of machine learning models offers several advantages, including the ability to handle large volumes of data, capture non-linear relationships, and adapt to changing environmental conditions. By accurately estimating ET, these models can provide valuable insights into water resource management, irrigation scheduling, drought monitoring, and climate change studies, ultimately contributing to more sustainable and efficient water management practices (Yang *et al*. 2021).

In recent years, machine learning techniques have advanced significantly in estimating ET. In order to estimate daily reference ET, Hai *et al*. (2018) introduced the innovative hybridized fuzzy model with frefy algorithm (ANFIS-FA). They showed that the ANFIS-FA model performed better than the other ANFIS-based model. Six artificial intelligence (AI) models were used by Sanikhani *et al*. (2019) to estimate reference ET using temperature-based climate data from two stations; all of the AI models showed good accuracy and outperformed the Hargreaves-Samani (HS) equation. In order to forecast reference evaporation using weather factors as input variables, Khosravi *et al*. (2019) used nine models (including data mining techniques and hybrid ANFIS models). The findings showed that hybrid models performed better than the individual models. To predict ET from meteorological data, Salih *et al*. (2019) presented the co-active neuro-fuzzy inference system (CANFIS) model. They revealed that the CANFIS gave higher accuracy compared with AI models. Mohammadrezapour *et al*. (2019) explored the possibility of predicting ET using three different heuristic methods, i.e., support vector machine (SVM), adaptive neuro-fuzzy inference system (ANFIS), and gene expression programming (GEP). Monthly weather data were used to calculate the potential ET at four synoptic stations. SVM, ANFIS, and GEP models were used to estimate potential ET based on the weather data. In Malik *et al*. (2020), multiple learning AI models (MM-ANN, MGGP, MARS, SVM, and M5Tree) were used to predict monthly pan evaporation based on climatological information. Yang *et al*. (2021) coupled four machine learning models with four energy-based ET estimation methods using Bayesian model averaging method (BMA). Among all eight single models, the BMA method outperformed all of them, reducing uncertainties among them and taking different intrinsic advantages of empirical and physical models into account.

As discussed, machine learning techniques have been increasingly used in remote sensing of ET estimation, offering several advantages over traditional methods. Recent papers have highlighted the positive impact of machine learning in this field.

Improved accuracy: Machine learning algorithms can effectively capture complex relationships between input variables and ET, leading to improved accuracy compared to traditional empirical models. For example, a study by Yang

*et al*. (2021) used random forest regression to estimate ET from satellite data and achieved higher accuracy compared to the widely used SEBAL.Reduced data requirements: Machine learning algorithms can handle large datasets efficiently and extract relevant features automatically, reducing the need for extensive preprocessing or manual feature selection. This capability is particularly valuable when dealing with multi-temporal and multi-sensor remote sensing data (Agrawal

*et al*. 2022; Zhang*et al*. 2022; Yong*et al*. 2023).Transferability: Machine learning models trained on one region or dataset can often be transferred to other regions or datasets with minimal adaptation, making them more versatile than traditional models that require recalibration for each new location or dataset. This transferability is crucial for large-scale applications of ET estimation using remote sensing data (Diaz-Gonzalez

*et al*. 2022).Real-time monitoring: Machine learning algorithms can provide near real-time estimates of ET by processing satellite imagery as it becomes available, enabling timely monitoring of water resources and agricultural practices (Amani & Shafizadeh-Moghadam 2023).

In conclusion, recent papers demonstrate that machine learning techniques have a significant positive impact on remote sensing of ET estimation. They offer improved accuracy, enhanced spatial resolution, reduced data requirements, transferability across regions/datasets, and real-time monitoring capabilities. Despite these advantages, challenges remain in the application of machine learning techniques for remote sensing of ET. These include the need for large and diverse training datasets, potential overfitting issues, interpretability of complex models, and the requirement for domain knowledge to ensure accurate model selection and validation. Further research is needed to address challenges and refine these techniques for more robust and reliable ET estimation.

In spite of this promising attempts, there is still a lack of studies in evaluating the application of machine learning models in getting rid of the difficulty of calculating various parameters of energy balance-based ET measurement methods and removing the uncertainties resulting from these parameters. Utilizing remote sensing data as independent variables of machine learning models rather than incorporating additional meteorological parameters is another factor that needs to be studied more (e.g., estimating ET based on electromagnetic band images from satellite images). Accurate, user-friendly and real time estimation of daily ET by using new techniques such as machine learning can help to monitor net water consumption, irrigation management and drought management in fields.

The use of traditional methods and field data to monitor ET and evaluate water resources real consumption is very time-consuming and expensive, and in most cases it gives a poor estimate of the amount of water in the region. The use of satellite images and remote sensing technology is very economical and reasonable due to saving time and money. Nevertheless, on the one hand, the accuracy of remote sensing models is always in doubt, and on the other hand, the complexity, volume, and computational cost of these models are significant. Based on this, the question arises as to how meta-heuristics models can be used to pave the way for the use of remote sensing models of ET. The information received from satellite images has several electromagnetic bands and each band (each image) contains countless information pixels. Therefore, the accuracy and power of meta-heuristics models in calibrating and validating an ET estimation model using satellite images is discussed. Therefore, in this research, it is supposed to measure the daily ET using SEBAL algorithm. Then, using the input data of electromagnetic bands of Landsat 8 as predictors and output of daily ET images obtained in the previous stage as predictant, various meta-heuristics models were calibrated and verified, and the efficiency of these models were evaluated and compared.

According to the reviewed researches, the novelty of the paper is the use of images of the electromagnetic bands of Landsat 8 and daily ET estimated by SEBAL algorithm as independent and dependent variables, respectively, for calibrating and verifying of meta-heuristic models. In this regard, for the first time, the ant colony optimization (ACO), particle swarm optimization (PSO), and artificial neural networks (ANNs) were used to estimate the daily ET images of the study area from electromagnetic band images from Landsat 8.

The other parts of this article are organized as follows. In the material and methods section, a brief introduction of the SEBAL algorithm and meta-heuristics models is given. Then, the study area is introduced and in the next part, the discussion about the results obtained from the introduced models is discussed. In the final part of this research, the final conclusion and summary of the research is discussed.

## MATERIALS AND METHODS

- (1)
Acquisition of Landsat 8 images of the study area (including seven short wavelength bands and one thermal band) on different dates.

- (2)
Applying the SEBAL algorithm for estimating daily ET images on the desired dates.

- (3)
Using the data obtained from step (1) as independent variables (predictors) and the ET estimated in step (2) as dependent variables (responses) for calibration and verification of the meta-heuristic models (i.e., ACO, PSO, and ANN).

- (4)
Evaluating and comparing the efficiency and accuracy of the above-mentioned models in estimating the ET images.

In the following, the SEBAL algorithm and the ACO, PSO, and ANN models are discussed.

### SEBAL algorithm

*H*, and

*G*represent latent heat flux (W/m

^{2}), net radiation (W/m

^{2}), sensible heat flux (W/m

^{2}) and soil heat flux (W/m

^{2}), respectively.

*R*) using the input and output radiation fluxes is as follows (Figure 3 and Equation (2)):where , , and denote incoming shortwave radiation, incoming longwave radiation and outgoing longwave radiation (W/m

_{n}^{2}), respectively. and are surface albedo and surface thermal emissivity, respectively.

*G/R*ratio is calculated in the middle of the day using the empirical Equation (3) (Allen

_{n}*et al*. 2002).where and are surface temperature (°C) and normalized difference vegetation index, respectively.

The value of *G* is obtained by multiplying the above ratio by *R _{n}*. For access to other computational details of the above relationship, refer to Safari

*et al*. (2023).

*et al*. 2023), the instantaneous value of latent heat flux for the time of the satellite overpass the (

*λ*ET) is calculated using Equation (4). Using the instantaneous latent heat flux obtained, the amount of instantaneous ET is calculated in the form of Equation (4) (Allen

*et al*. 2002).where and are instantaneous ET (mm/h) and latent heat of vaporization or the heat absorbed when a kilogram of water evaporates (J/kg), respectively.

*Λ*parameter, which is known as the evaporation fraction, should be calculated. This fraction (Equation (6)) shows the ratio of the energy consumed for the ET process on the total amount of energy available for this process.

^{3}) and average daily net radiation (W/m

^{2}), respectively.

### Meta-heuristic models

In the following sub-sections, a brief introduction of simulation and optimization models will be discussed. It is supposed to estimate ET from the band data received from Landsat 8 satellite by using these algorithms, and the accuracy of each of these methods will be evaluated and compared.

#### ACO model

*et al*. (1991) presented several articles in the introduction of a computational algorithm called ant system. The basis of this theory is inspired by the behavior of searching for food by the community of ants. The sense of sight of ants is very weak and even a group of them are completely blind, for this reason the most communication between all ants is through the chemical substances (pheromone) left by them. Ants use this chemical to mark the route from the nest to the target (food). The shorter the path chosen by an ant the greater the amount of pheromone that the ant leaves on the ridges forming the path. In order to prevent all ants from choosing only one path, pheromone evaporation is considered. In this algorithm, the amount of pheromone of each path is calculated based on the following equation (Cordon

*et al*. 2002):

*τ*

_{ij}(t*+*

*1)*and

*τ*are the amount of pheromone remaining in route ij before and after updating, respectively.

_{ij}*ρ*is the evaporation parameter. Δ

*τij (t)*represents the amount of pheromone left by ants on the mentioned path at time

*t*, and its value can be calculated according to the following equation (Cordon

*et al*. 2002):

*m*represents the number of ants that search the decision space to find the optimal solution. Based on this, the amount of pheromone left by the mentioned ant in the

*t*th repetition on the path

*ij*is obtained from the following equation (Cordon

*et al*. 2002):

*Q*is a constant number and

*L*is the length of the path traveled by the

_{k}*k*th ant. The probability that the

*k*th ant chooses the path

*i*and

*j*is according to the following equation (Cordon

*et al*. 2002):

In this equation, represents the reverse of the distance between route *ij*; *a* and *β* are two parameters that determine the relative effect of *τ _{ij}* and .

*i*is the node that the

*k*th ant wants to choose from the nodes it has not met yet. Finally, the path that has the highest amount of pheromone is selected as the optimal path (the general optimal solution of the problem).

#### PSO model

The PSO model was first proposed in the mid-1990s by Kennedy & Eberhart (1995). This method is inspired by the movement of a group of birds to find the best path among other birds. In the PSO method, the set of random solutions is used as an initial solution in the decision space in order to reach the optimal solution through an iterative process.

*s*and

*v*, respectively. Correcting the position of the operator is done using position and speed information. Flocks of birds optimize a specific objective function. Each agent knows its best value so far (

*Pbest*) and its position

*s*. Also, each agent is aware of the best value ever obtained in the group (

*Gbest*). In other words, in an optimization problem, each particle chooses a location in the decision space as a desirable position (

*Pbest*). This position contains the answers to the problem (the decision variables of the problem). Using the velocity parameter (

*v*), each particle modifies its location many times to reach the best and most desirable point (general optimal point). In other words, one memory is allocated to store the best position of each particle in the past (

*Pbest*) and one memory to store the best position among all particles (

*Gbest*). With the experience of these memories, the particles decide how to move in the next turn. In each iteration, all the particles move in the n-dimensional space of the problem until finally the general optimal point is found (Figure 4).

*et al*. 2007):

In this equation, and represent the best location experienced by a particle and the best location experienced by the entire population in the *k*th displacement iteration, respectively. *c*, *r*, and *ω* are the acceleration coefficient, random number between 0 and 1, and speed guiding coefficient, respectively. According to Equation (14), the new position of a particle is the previous position of that particle plus the velocity vector . The term itself consists of three separate terms, the sum of all three vectors which are a coefficient of three vectors (the vector from the primary location ends to the secondary location of the particle). The spatial changes of the particles will eventually be such that all the particles will be concentrated in the optimal position and the coordinates of this position will be the solution to the optimization problem. The criteria of the number of repetitions of the problem and the optimality of the solutions are determined based on the definition of an objective function that should be minimized or maximized based on the position of the particles.

#### Artificial neural networks

*X*is the input variables,

_{i}*y*is the output variable,

*w*is the weight coefficients in the network,

*n*,

*m*and

*k*are the number of neurons in the input layer, middle layer and output layer, respectively. This structure can be expressed in the form of the following equation (Nourani

*et al*. 2009):

In this regard, the ANN model estimates the weight coefficients of the last equation using the training (observation) data during the training (calibration) process. By estimating these weight coefficients, the simulation function is calculated and can predict and calculate the response of the model for the input data.

In the present study, to determine the accuracy of the modeling of daily evaporation-transpiration images, the determination coefficient (DC) (dimensionless) and root mean square errors (RMSE) (cubic meters per second) are used (Equations (16) and (17)) (Nourani *et al*. 2009).

In the above equations, *N* is the number of observational data, *Q*_{obs} is the observational data, *Q*_{com} is the estimated value and is the average of the observational data.

### Case study

^{2}, of which the plain area is 300.7 square kilometers and the rest is related to the highlands of the study area. Its area spans between 33° 26′ to 33° 27′ longitude and 47° 44 ′ to 47° 28 ′ latitude. The mean, minimum, and maximum elevation of the plain are 1,330; 1,261; and 1,396 m, respectively. The mean annual precipitation is 454 mm and the mean, minimum and maximum annual temperature are 16.7, 5, and 30 °C, respectively. The mean annual evaporation of the plain is 2,128 mm.

In the plain of Kohdasht, the average level of underground water varies between 1,150 m located in the lands of Malmir, Katuleh and Boghlan located in the south of the plain to 1,180 m in the lands of the alluvial cone located in the west of the plain. In the last decade, indiscriminate exploitation of underground water resources has caused a sharp drop in the underground water level in this plain. In the alluvial cone lands of the plain, the hydraulic slope of the underground water is more than 5 per thousand, which decreases toward the center of the plain and reaches less than 4 per thousand in the southern lands of the plain.

In the alluvial cone lands of the plain, the depth of contact with the underground water level is more than 75 m, and in the southern lands of the plain, it is less than 25 m.

## RESULTS AND DISCUSSION

The input images of the SEBAL algorithm (seven short range bands and one thermal band obtained from Landsat 8 satellite) and the output images estimated by SEBAL algorithm (daily ET) as, respectively, independent and dependent variables are used for calibration and verification data of meta-heuristic models, respectively.

At this stage, the received images of the electromagnetic bands of Landsat 8 and remotely sensed ET images are, respectively, used as input and output response in meta-heuristic models. The input pixel data set for each of the 8 bands is 164,528 pixels equal to 164,528 × 8 = 1,316,226 pixel data for each shooting date. The number of pixels of remotely sensed transpiration evaporation images is the same. Finally, 70% of these data are used for calibration and the remaining 30% for the verification of ACO, PSO, and ANN models.

In the following, the results of calibration and verification of the above-mentioned models will be discussed. Due to the large amount of training data, ACO and PSO models did not perform well when faced with a large amount of calibration and verification data. Therefore, the last two models were executed once for all the data and once for only half of the data. In these models, the objective function is to minimize the RMSE error resulting from the comparison of computational data (ET images calculated in meta-heuristic models) and observational data (ET images obtained from the SEBAL algorithm).

Table 1 compares the accuracy and efficiency of the ACO, PSO and ANN models. In the comparison of ACO and PSO models in the simulation of daily ET images, it can be said that when calibration and verification are done considering all the data, the ACO modeling accuracy (with DC and RMSE of 0.65 and 1.45 mm/day in the verification stage, respectively) is higher than the PSO model (with DC and RMSE of 0.23 and 1.60 mm/day in the verification stage, respectively). While removing half of the calibration and verification data, the accuracy of the PSO (with DC and RMSE of 0.85 and 0.93 mm/day in the verification stage, respectively) model has surpassed the ACO (with DC and RMSE of 0.80 and 1.09 mm/day in the verification stage, respectively) model. It can be concluded that by reducing the training data, the efficiency of the ACO and PSO model in estimating daily ET images can be increased by 23 and 269%, respectively.

Assessment criteria . | ACO . | PSO . | ANN . | ||||
---|---|---|---|---|---|---|---|

Calibration . | Verification . | Calibration . | Verification . | Calibration . | Verification . | ||

Using all data | DC | 0.42 | 0.65 | 0.64 | 0.23 | 0.98 | 0.98 |

RMSE mm/day | 2 | 1.45 | 1.57 | 1.60 | 0.09 | 0.09 | |

Using half of the data | DC | 0.74 | 0.80 | 0.87 | 0.85 | 0.99 | 0.98 |

RMSE mm/day | 1.38 | 1.09 | 0.96 | 0.93 | 0.09 | 0.09 |

Assessment criteria . | ACO . | PSO . | ANN . | ||||
---|---|---|---|---|---|---|---|

Calibration . | Verification . | Calibration . | Verification . | Calibration . | Verification . | ||

Using all data | DC | 0.42 | 0.65 | 0.64 | 0.23 | 0.98 | 0.98 |

RMSE mm/day | 2 | 1.45 | 1.57 | 1.60 | 0.09 | 0.09 | |

Using half of the data | DC | 0.74 | 0.80 | 0.87 | 0.85 | 0.99 | 0.98 |

RMSE mm/day | 1.38 | 1.09 | 0.96 | 0.93 | 0.09 | 0.09 |

By comparing the ET images modeled by the ANN model with the observed ET images obtained by the SEBAL algorithm and calculating the DC = 0.98 and RMSE = 0.09 mm/day criteria in the verification step for the conditions where both half of the data and all the data are considered in model training, we have come to the conclusion that the accuracy and efficiency of the ANN model in modeling and estimating the ET images is significant. The accuracy of estimating daily ET in the ANN model has increased by 50.7 and 326% compared to the ACO and PSO models, respectively.

Therefore, the ANN model among other meta-heuristic models such as ACO and PSO can be a suitable choice for estimating ET images and getting rid of the complex and time-consuming process of algorithms based on energy balance such as SEBAL.

Last but not least, in the field of ET estimation using meta-heuristic models many studies have used meta-heuristic models for estimating ET and this paper aligns with this approach (Agrawal *et al*. 2022; Zhang *et al*. 2022; Yong *et al*. 2023). In addition, SEBAL is a widely accepted method for estimating ET, so the use of training data obtained from this algorithm is in agreement with previous work (Elsayed *et al*. 2022; Ma *et al*. 2023). Regarding the efficiency of ANN Models, ANNs have been recognized for their high efficiency in various applications, including ET estimation. The previous papers have also found this, as a point of agreement (Dimitriadou & Nikolakopoulos 2022; Elbeltagi *et al*. 2022). Furthermore, Krishnashetty *et al*. (2021) found that ANN approach outperforms support vector machine (SVM) and genetic programming (GP) for ET estimation that is in line with the results of this paper. On the other hand, previous papers have used different sources of training data like FAO Penman–Monteith equation, crop growth stage and weather data (including maximum, minimum and mean air temperature and diurnal temperature range, solar radiation, sunshine duration, relative humidity, and wind speed) instead of the SEBAL algorithm, this could be a point of disagreement (Krishnashetty *et al*. 2021; Qin *et al*. 2022).

## CONCLUSION AND RECOMMENDATIONS

ET is considered as one of the most important components of the hydrological balance of a region, which represents net water consumption. Based on this, with the development of ET estimation methods in the field, it is possible to monitor consumption and harvests. So far, new methods have been introduced to estimate ET in the watershed based on energy balance using satellite image analysis such as the SEBAL algorithm. The SEBAL algorithm can appropriately and accurately estimate daily ET with appropriate temporal and spatial resolution (every 16 days and 30 m, respectively). However, the process of estimating daily ET satellite images using the electromagnetic bands of Landsat 8 satellite in the SEBAL algorithm has many complications and calculation steps. Therefore, the question is raised, how to use the remote sensing daily ET images by the SEBAL algorithm to calibrate and verify meta-heuristic models in order to reduce the cost and complexity of the SEBAL algorithm.

Machine learning models have proven their capacity to effectively delineate intricate relationships between environmental variables and ET, thereby enhancing accuracy in contrast to traditional empirical models. Nevertheless, the deployment of machine learning methodologies for remote sensing of ET is not without its challenges. These include the necessity for comprehensive training datasets, the potential for overfitting, the interpretability of sophisticated models, and the demand for domain-specific knowledge to ensure precise model selection and validation. The innovative aspect of this paper lies in the employment of Landsat 8′s electromagnetic band images and daily ET derived from the SEBAL algorithm as independent and dependent variables, respectively, for the calibration and verification of meta-heuristic models. In this regard, for the first time, the ACO, PSO and ANNs were used to estimate the daily ET images of the study area.

By comparing the results of estimating daily ET images (as computational data) and remotely sensed daily ET by SEBAL algorithm (as observational data), it was concluded that the ANN model compared to others afore mentioned models, namely ACO and PSO, have very good efficiency and accuracy (with DC and RMSE of 0.98 and 0.0902 (mm/day) respectively in the verification stage). It can also be concluded that by reducing the training data, the efficiency of the ACO and PSO model in estimating daily ET images can be increased by 23 and 269%, respectively. Meanwhile, when considering all the training data, the accuracy of estimating daily ET in the ANN model has increased by 50.7 and 326% compared to the ACO and PSO models, respectively.

The results of this research will enable future researchers to take steps in the direction of verifying these models by being aware of the strengths and weaknesses of the new meta-heuristic models. Also, the machine learning models presented in this research in estimating daily ET can be turned into user software for farmers for irrigation management by cloud computing researchers.

Although this research has evaluated the meta-heuristic models in the modeling of data related to satellite images, one of the main limitations of this research is the lack of access to lysimetric data to evaluate the accuracy of the SEBAL algorithm in the study area. Therefore, it is recommended to pay attention to solving this limitation in future researches. Meanwhile, the ANN model has significant efficiency and accuracy for estimating daily ET in the study area. However, the accuracy of this model in estimating daily ET in other study areas with different hydroclimatological conditions and cultivation patterns is doubtful. Therefore, it is suggested to pay attention to this issue in future researches. Finally, the primary objective of this paper was to explore the potential of using black box (machine learning) models for estimating daily ET based on electromagnetic band images from Landsat 8. As such, the emphasis was placed on utilizing remote sensing data as independent variables rather than incorporating additional meteorological parameters. However, we recommend that future research endeavors specifically address wind speed, sunshine hours, temperature, and evaporation alongside remote sensing data to gain a more comprehensive understanding of their combined influence on ET.

## DATA AVAILABILITY STATEMENT

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

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Regionalization of surface flux densities and moisture indicators in composite terrain A remote sensing approach under clear skies in mediterranean climates*

*PhD Dissertation. CIP Data Koninklijke Bibliotheek, Den Haag, The Netherlands*