Introduction

For successful water resource management and implications modeling on evapotranspiration variability, the rainfall, temperature, and solar radiation are fundamental components and biotic keys1,2. Multiple uncertainties have existed about key drivers of water management in worldwide in the case that this problem is keeping in touch with meteorological parameters and evapotranspiration being the most driver of flooding risk while it’s associated with prolonged drought3,4. For this propose, evapotranspiration earns universal and methodical study because of the direct signal and important relationship with drought and flooding risk5.However, evapotranspiration regimes are under bluster totally from meteorological impacts. Under climatic conditions, evapotranspiration (ET) measurements play the most imperative role for a better understanding of water management, such as flooding risk monitoring and hydrologic modeling6,7,8,9,10,11. Recent works exist to model evapotranspiration variability. All selected studies conclude that the used methods are suitable for decision-making in drought and hydrological cycle analysis and management12,13,14,15. In KSA, several studies exist to develop meteorological data effects on evapotranspiration variability. These all studies concluded that temperature increasing affects 1–4.5% of evapotranspiration values16,17. Several stochastic methods are used to model observed and estimated values of any phenomena by calibration-optimized parameters. Yang and Gandomi in 2012, use Bat Algorithm as a novel approach for global engineering optimization to fix some multi-objective problems18. A stochastic method, namely the Bat-Inspired Optimization Approach, is used to model the Brushless DC Wheel Motor Problem19. Also; the metaheuristic bat algorithm is used to model the relationship between tuning of power system stabilizer and signal stability enhancement20. LCBA (Local Centroid Bat Algorithm) to model the protocol of improved low energy adaptive clustering hierarchy21. In power systems interconnection, frequency loading is controlled by a Bat-inspired algorithm22. Bat-quasi-Newton coupling is used to study Node localization of wireless sensor networks23. In unconstrained scenarios, facial feature is selected using an A membrane-inspired bat algorithm24. Almufti in 2017 use swarms Intelligence for solving NPHard Problems. This research applies some stochastic models such as Bat algorithm (BA), Artificial Bee Colony (ABC), and Particle Swarm Optimization (PSO) to find a best solution for TSP, which is one of the most well-known NP-Hard problems in computational optimization25. Data mining, classification, and wireless all of these operations are used to improve DV-Hop performance for cyber physical systems in research article created by Cui et al. in 2017 using novel oriented cuckoo search algorithm. Chaotic Bat Algorithm is used to describe weighted support vector data26. Both big data sensing and malicious code variants detection are analyzed respectively in the Internet of things and deep learning using stochastic models27,28. Multi-objective optimization of vector machines using a pigeon-inspired optimization algorithm29. In all of these researches, stochastic methods and its extensions and modifications have been applied in practically every case study of modeling and optimization in economic sciences, industrial engineering area, deep learning, internet of things, etc. In our previous studies Newton Method (NM), Genetic Algorithm (GA), Brid Swarm Algorithm (BSA) and Chicken Swarm Optimization Algorithm were used to model the rainfall-runoff relationship and climate change impact on runoff values with acceptable and successful methodology in hydrological science30,31,32,33. Hence, the aim of this work was twofold: the first one is evaluating the change in meteorological data of arid and semi-arid zones in Saudi Arabia using a stochastic model based on hybrid metaheuristic optimization. The second fold for evaluating the rainfall, temperature, and solar radiation implications, change-induced future modeling on evapotranspiration variability areas in Mekkah and Medinah basins located in Saudi Arabia. This work pioneers the use of hybrid metaheuristic optimization methods in modeling climate change impacts on evapotranspiration in Saudi Arabia. This innovative approach leverages the strengths of different algorithms to improve the accuracy and robustness of predictions. This study incorporates methods to quantify uncertainty, such as sensitivity analysis or probabilistic modeling using statistical metrics. This helps us to understand the range of potential outcomes and the associated risks. Finally, this work advances our understanding of climate change impacts on ET in Saudi Arabia and provides valuable insights for policymakers and researchers. By focusing on Saudi Arabia, the research can provide a more detailed and nuanced understanding of climate change impacts on evapotranspiration in this region, leading to more effective and targeted policy responses.

Materials and methods

Study area

In this work, data from three basins of Saudi Arabia from several hydrometric areas are downloaded from Data Downloads Tools in the Climate Toolbox (https://climatetoolbox.org/tool/data-download). Mekkah and Medinah basins with climatological characteristics were considered in this work. These basins were selected based on the uncertainty of evapotranspiration and precipitation relationships where non-stationarity in rainfall, Temperature, and solar radiation-evapotranspiration relationships were observed. The two basins were Mekkah Basin, MEEKA city, where the stream gauge is located at 21.25° N; 39.49° E with a drainage area of 1725 km2, and Medinah Basin, MEDINAH city, where the stream gauge is located at 24.28° S; 39.36° E with a drainage area of 104,679 km2 (Fig. 1a,b)34. Figure 1 was generated by coupling QGIS software and Google map, whereas QGIS will always be, available free of charge if downloaded from https://qgis.org/download/.

Fig. 1
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Mekkah and Medinah basins locations (coupling QGIS software and Google map; https://qgis.org/download/).

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Meteorological data and evapotranspiration processing

In this work, meteorological data processing is based on data organization, transformation, and analysis for further use in decision-making on evapotranspiration variability using some techniques and tools or software. Databases, sensors, and customer surveys are sources of meteorological data collection where these data confirm accuracy and have a good relationship with goals or analysis. Meteorological data preparation is one of the key drivers of data subsequent processing to avoid incorrect or incomplete data. After the data collection stage, the clean and prepped data is fed into a processing and algorithm system to model stochastically meteorological data and evapotranspiration relationship.

Meteorological data analysis

The meteorological process is the key driver in all data analysis approach35. For that, in both the Makkah and Medinah basins, hydrological variability is essential. The case studies are characterized as poorly gauged basins. Both gauging and delving are important stages for any data analysis (eg: meteorological data) (Table 1) when it lets hydrologist experts to clarify all anomalies in data series and the good proposal statistical techniques. In hydrology study, spatial and temporal data analysis understanding can interpret the results, thus this analysis is very important36,37,38.

Table 1 Meteorological characteristics of study area.
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Evapotranspiration modeling

In water cycle process, evapotranspiration (ET) is a vital and crucial key, produced by phenomena transferring from land surface to atmosphere reservoirs. The main ET components are Evaporation and Transpiration. Regarding Evapotranspiration modelling, ET measurement is difficult. For that, several models estimate Evapotranspiration using auxiliary technics whereas considering different factors (eg: implications of rainfall, temperature and solar radiation) influencing ET39,40.

Flowchart and pseudo-code

The used hybrid model is a large coupling of Metaheuristic optimsation methods : CSO-BAT,BAT-GA,BSA-GA,BSA-NM,BAT-NM,GA-NM,CSO-NM and CSO-GA.For ET estimation , meteorological data such as rainfall, temperature and solar radiation are used and collected from meteorological data sources. Data quality is very important for a good ET estimating and forecasting and should be normalized for model performance improvement (Fig. 2a–c).

Fig. 2
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Flowchart and pseudo-code of hydride model.

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Model calibration

BAT calibration indicates successive comparing operations of the model’s outputs (estimated evapotranspiration) with observed values of evapotranspiration and making corrections to meteorological data (rainfall, temperature, and solar radiation) to get suitable results41. Model inputs are deduced from observed evapotranspiration when it creates a calibration stage to adjust meteorological data42,43,44.

Model validation

Validating and objective function analysis is crucial for accurate hydrological cycle management and hydrological studies. In this work, evaluation metrics come into play during the validation period and are mathematically written as follow:

$$NSCE = 1 - \frac{{\mathop \sum \nolimits_{{i = 1}}^{n} \left( {Et_{i} - {\check{E}t}_{i} } \right)^{2} }}{{\mathop \sum \nolimits_{{i = 1}}^{n} \left( {Et_{i} - \overline{{Et}} _{i} } \right)}}$$
(1)
$$MAPE = \frac{1}{n}\mathop \sum \limits_{{i = 1}}^{n} \left| {\frac{{Et_{i} - {\check{E}t}_{i} }}{{Et_{i} }}} \right| \times 100\%$$
(2)
$$NMSE = \frac{1}{n}\mathop \sum \limits_{{i = 1}}^{n} \left( {\frac{{Et_{i} - {\check{E}t}_{i} }}{{Et_{i} }}} \right)^{2}$$
(3)
$$IA = 1 - \frac{{\sum \left( {Et_{i} - \widetilde{Et}_{i} } \right)^{2} }}{{\sum \left( {\left| {Et_{i} - mean_{Et} } \right| + \left| {Et_{i} - mean_{Et} } \right|} \right)^{2} }}$$
(4)

where \(Et_{i} \;{\text{and}}\;{\check{E}t}_{i}\) denote the desired and estimated evapotranspiration. ET evaluation metrics are key drivers to confirm used hybrid models in this case study. In hydrological cycle and water science management, hybrid ET models can be leveraged by typical statistical parameters using and understanding to confirm a highlight decision-making in water fields such as Accurate ET estimates allow for efficient irrigation scheduling, minimizing water waste and optimizing crop yields, help drought management and forecasting.

Result

Validation of hybrid models with parameter

Hybrid model validation is a crucial stage to ensure the accuracy and reliability of the model in simulating evapotranspiration. It essentially checks how well the model replicates observed evapotranspiration processes45. Regarding estimated values of ET, hybrid models yields good results. Modeled value of ET is evaluated by metrics cited in previous subsection (2.4.3) such as Nash–Sutclife coefficient of efficiency (NSCE), normalized mean squared error (NMSE), root mean squared error (RMSE). We can see that metrics values yields good interpretation regarding the relative agreement and average squared difference between observed and modeled ET value using NM in hybrid system almost every stochastic method used in this work whereas NMSE is the objective function (Fig. 3).

Fig. 3
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Objective function and its variability for hybrid model in both Mekkah and Madina basins.

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Hybrid models calibration

In this work, hybrid model calibration is the process of fine-tuning a model’s parameters (optimized parameters, eg: meteorological data) to achieve the best possible agreement between desired and estimated evapotranspiration whereas we can deeper dive into calibration model processing by keeping in touch the real study of spatial and temporal variations that evapotranspiration model can’t perfectly capture whereas calibration with coupling mode between NM, BAT, BSA, CSO and GA helps the model adapt to these specific conditions with acceptable value of correction factors (eg:GA accuracy).

Scenario precipitation, temperature and solar radiation

All meteorological data-evapotranspiration research concludes that solar radiation is a key energy source that directly influences temperatures and evaporation, with warmer temperatures accelerating evaporation and increasing evapotranspiration in other ways. As a result, higher precipitation leads to increased potential evapotranspiration (PET)46,47.

Simulations with a rain, temperature and solar radiation

In water resource and climate change management evapotranspiration predicting, understanding and simulation are key drivers whereas ET is influenced by rain, temperature, and solar radiation. In this case during and after raining soil moisture is replenish in which reduces evapotranspiration need. other than, temperature increase evapotranspiration rates and more available energy for evapotranspiration is translated by higher solar48,49. For Mekkak and Madinah cities, modeled and evapotranspiration are shown in Fig. 4.

Fig. 4
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Modeled evapotranspiration for Mekkah (a–h) and for Madinah (1–8).

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Discussion

Accurately estimating evapotranspiration is crucial in both hydrology and ecology. This work proposes hybrid models that combine meteorological data preprocessing, machine learning, and meta-heuristic optimization methods. Hybrid model accuracy is greater than traditional physically based methods, although they can be complex to develop50,51.

Modeled and observed evapotranspiration

Understanding ET is a key driver in water management studies. To assess this parameter, observed data is required to calibrate and validate estimated ET values52,53. At this point, modeled and observed ET are tested and compared to evaluate the accuracy of the hybrid model and improve the performance of coupling between used stochastic methods (NM, BAT, BSA, and GA) under limit conditions (Fig. 5).

Fig. 5
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Scatter plots for hybrid model for Mekkah (a–p) and for Madinah (1–16).

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Figure 5 showed that hybrid model was able to track the monthly variations of evapotranspiration. We can see that coupling of three methods can give good performance of optimization those coupling tow methods in both Mekkah (arid zone) and Madina (semi-arid zone). The evaluation metrics take values and ranges for the studies area (Table 2). Modelling was evaluated with estimated evapotranspiration using stochastic models cited in previous subsections. Both Mekkah and Medinah basins had positive correlation between desired and estimated evapotranspiration with acceptable values of evaluation metrics (Table 3).

Table 2 Evaluation metrics values in calibration and Validation periods for Mekkah basin.
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Table 3 Evaluation metrics values in Calibration and Validation periods for Madina basin.
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In both Mekkah and Madinah, Several findings emerged from the evaluation metrics. Newton method model (NM) combined with data preprocessing Genetic Algorithm (GA), Brid Swarm Algorithm (BSA), Bat algorithm (BAT) and Chicken Swarm Optimization Algorithm (CSO) achieved acceptable performance for ET estimation, as shown by their lower evaluation metrics compared to other coupling (Tables 2 and 3). However, hybrid models based on coupling with NM significantly outperformed other models, with hybrid BAT–GA–NM, BSA–NM, BAT–NM, GA–NM, CSO–NM, CSO–BAT–NM, BSA–BAT–NM models demonstrating superior performance to NM-based models whereas classic and old methods are existed (eg: GA) (Tables 2 and 3). Compared to BSA–NM, BAT–NM, GA–NM, CSO–NM, CSO–BAT–NM, BSA–BAT–NM models achieved substantial improvements, with BAT–GA–NM reducing MAPE from 38.88 to 4.16% for Mekkah and from 14.94 to 2.04% for Medina and NSCE from − 0.17 to 0.97 for Mekkah and from 0.82 to 0.99%. These results highlight the exceptional capability of BAT–GA–NM and other models besd on NM coupling for monthly ET estimation. The depth statistical analysis, and limitations in comparison with other existing methods is also very important to improve this research quality54,55.

Impact of rain, temperature and solar radiation on evapotranspiration

Evapotranspiration is influenced by all key factors (rain, temperature and solar radiation). Soil is fully saturated by rainfall. This saturation create the main source of evapotranspiration with proportional rainfall-evapotranspiration relationship. Kinetic energy was increased whereas temperatures rise in such water molecules gains this energy and translated by important volatility when plants transpire this energy or making them more likely to evaporate. The entire ET process is driven by solar radiation. Water and plant evapotranspiration were influenced by sunlight hitting in conclusion that solar radiation-ET relationship was created. The three key drivers of evapotranspiration were modeled in this work using hybrid model whit metaheuristic optimization methods.

Evapotranspiration and discharge relationship

Discharge and evapotranspiration are key drivers of hydrological cycle whereas their relationship modelling is crucial for water resources managing, climate change impact and water availability understanding56,57. Looking at hydrographs series of discharges and of evapotranspiration (which depends on rainfall, temperature and solar radiation), promote enhanced differences between desired and estimated evapotranspiration are found when discharge peaks occur, and when evapotranspiration affected by meteorological data. This relationship translate the evident discharge when the drought has a long period occur (Fig. 6). Fig. 6 is constructed using mathematical impact of evapotranspiration on runoff values.

Fig. 6
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Calculated and measured discharge for Mekkah (a–p) and Madinah (1–16).

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According to Fig. 6 we can say that the hybrid models CSO–BAT–NM, BSA–CSO–BAT CSO–BAT–GA give thousands of results by contributing the methods which use Genetic Algorithms, as well as the good superposition of the curves blades of water flow calculated and measured. The calibration appears correct for the MEKKAH and MADINA basins and translates into the very significant variation in the production reservoir.

The validation correlation of the simulated flow rates related by evapotranspiration rates (Fig. 7) gives fairly significant statistical parameter values. Consequently we can say that evapotranspiration-discharge modeling using the hybrid model gives acceptable and very encouraging results for the basin in arid and semi-arid zones in Saudi Arabia.

Fig. 7
figure 7

Scatter plots of modeled and estimated discharge for Mekkah (a) and for Madinah (b).

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Climate change impact on evapotranspiration

Several key trends and research gaps in the broader scientific literature are aligns with the study of climate change impacts on evapotranspiration (ET) in Saudi Arabia using hybrid metaheuristic optimization methods. Firstly, this research addresses the critical issue of understanding climate change impacts in arid regions. Saudi Arabia, as an arid region, is particularly vulnerable to climate change due to its limited water resources. Furthermore, climate change is expected to increase the frequency and intensity of extreme events like heat waves and droughts, which can significantly impact ET and water availability. Secondly, the study advances modeling techniques for ET. ET is influenced by a complex interplay of factors, including temperature, rainfall, wind speed, and solar radiation. Traditional modeling approaches may struggle to capture these interactions accurately. The use of hybrid metaheuristic optimization methods in this study represents a significant advancement in modeling techniques. These methods combine the strengths of different algorithms to improve accuracy and efficiency, making them well-suited for complex systems like ET modeling. Thirdly, the research addresses the challenge of data limitations in arid regions. Saudi Arabia often suffers from limited and sparse meteorological data, making it difficult to develop accurate models. The hybrid metaheuristic methods used in this study can help address this issue by optimizing the use of available data and incorporating expert knowledge. In conclusion, this research contributes to the broader scientific literature by addressing a critical research question in a vulnerable region, employing advanced modeling techniques, and providing valuable insights for policymakers and researchers. The results of this study can inform the development of effective water resource management strategies and agricultural planning in Saudi Arabia, helping to mitigate the impacts of climate change on ET.

Conclusion

This work quantified a rainfall, temperature and solar radiation implications modeling on evapotranspiration variability in arid and semi-arid zones in Saudi Arabia using hybrid model over the 20 year period (2001–2021) in a region experiencing both meteorological data values and evapotranspiration. The results of this work provide further evidence of using stochastic methods in making decision on water management. Furthermore, the study provides a new showing of how rainfall, temperature and solar radiation data can be utilized for estimating of evapotranspiration. However, these results would likely increase the number case studies in worldwide and different hydrological regimes datasets required for robust hybrid model calibration to avoid divergence in algorithm parameters. For this propose, a multi-objective function is preferred given the proposed methodology. Hybrid model performs robustly in terms of both the amplitude and temporal dynamics of ET and meteorological data at many scales in both Mekkah and Madinah, as validated by in-situ desired and estimated evapotranspiration. For investigations needing synchronized meteorological data response to evapotranspiration values limitations, this seamless dataset is especially valuable because of the hybrid model’s thorough definition of modeled evapotranspiration path. By featuring long-term time modeling of evapotranspiration and study aera coverage with high spatial and temporal resolution, this hybrid model will provide a dependable solution and data support for implications modeling and analysis on evapotranspiration variability in arid and semi-arid zones in Saudi Arabia and future drought related studies.