Literature review on optimization of transboundary water for irrigation

ANN

Artificial Neural Network

BRT

Boosted Regression Tree

CFA

Cooperative Framework Agreement

CHIRPS

Climate Hazards Group InfraRed Precipitation with Station data

CH_K2

Channel's Hydraulic Conductivity

CNN

Convolution Neural Networks

CPP

The Chance-Constrained Programming

DL

Deep Learning

DQN

Deep Q-Network

DSFP

Double-Sided Stochastic Fractional Programming

EC

Electrical Conductivity

EVI

Enhanced Vegetation Index

FNN

Feed-Forward Neural Network

FPP

Fuzzy Possibility Planning

GEE

Google Earth Engineering

GPM

Global Precipitation Measurement

GSOD

Global Surface Summary of the Day

GQI

Groundwater Quality Index

GWL

Groundwater Levels

GWO

Gray Wolf Optimization

HEC-HMS

Hydrologic Engineering Center's Hydrologic Modelling System

HEC-RAS

Hydrologic Engineering Center's River Analysis System

ICPR

International Commission for the Protection of the Rhine

IGWO

The Improved Gray Wolf Optimization

IPP

Interval Parameter Programming

IRS-P6

Indian Remote-Sensing Satellite

IWQI

Irrigation Water Quality Index

IWR

Irrigation Water Requirement

KNN

k-nearest Neighbors

LCBC

Lake Chad Basin Commission

LINDO

Linear Interactive and Discrete Optimizer

LISS

Linear Imaging and Self Scanning

LST

Land Surface Temperature

LSTM

Long Short-Term Memory

LU/LC

Land Use/Land Cover

LVBC

Lake Victoria Basin Commission

MARS

Multivariate Adaptive Regression Spline

MCW

Monthly Climatic Data

MIKE SHE

Mike Surface Water – Groundwater Hydrologic Model

MLP

Multilayer Perceptron

MODFLOW

Modular Groundwater Flow

MODSIM

Modular Modelling System

MRC

Mekong River Commission

NDVI

Normalized Difference Vegetation Index

NDWI

Normalized Difference Water Index

NGOs

Non-Governmental Organizations

NOAA

National Oceanic and Atmospheric

PCA

Principal Component Analysis

PI

Permeability Index

PSO

Particle Swarm Optimization

RF

Random Forest

RNN

Recurrent Neural Networks

RSC

Residual Sodium Carbonate

SAVI

Soil Adjusted Vegetation Index

SDGs

Sustainable Development Goals

SOL_AWC

Soil Layer's Available Water Capacity

SOL_K

Saturated Hydraulic Conductivity

SVM

Support Vector Machine

SVR

Support Vector Regression

SWAT

Soil and Water Assessment Tool

TSCCP

Two-Stage Chance-Constrained Programming

TWS

Terrestrial Water Storage

UAVs

Unmanned Aerial Vehicles

USGS

United States Geological Survey

VIC

Variable Infiltration Capacity

WAEN

Water-Agriculture-Energy Nexus Systems

WEAP

Water Evaluation and Planning System

WFE-NEXUS

Water-Food-Energy Nexus

ZAMCOM

Zambezi Watercourse Commission

The escalating demand for water resources is expected to exacerbate the ongoing growth of both the population and the economy. Urbanization intensifies water scarcity problems, particularly during periods of drought, and burdens already strained water resources, resulting in increased conflicts among various water consumers. Furthermore, climate change poses a threat to the availability of water resources by disturbing the normal patterns of the water flow and causing more frequent and intense extreme weather events (Bhaga et al. 2020; Basuki et al. 2022). Therefore, it is essential to apply adaptation approaches to efficiently manage unpredictable water resources and mitigate potential risks (Mishra et al. 2021). However, the management of transboundary water resources presents difficulties because of the differing needs and interests of the countries concerned.

Transboundary water resources are essential for maintaining the long-term viability of agriculture and fostering regional growth. These resources are in accordance with the sustainable development goals (SDGs) of the United Nations and the water-food-energy nexus (WFE-NEXUS) methodology (Tsani et al. 2020; Zarei 2020; Elsayed et al. 2022). Effective management of transboundary water for irrigation is essential in achieving the SDGs, particularly Goals 2, 6, 7, and 13 which eliminate hunger (Mogomotsi et al. 2020), ensure clean water and sanitation (Ladel et al. 2020), promote affordable and clean energy (Saklani et al. 2020), and take climate action (Rajosoa et al. 2021). The WFE-NEXUS plan emphasizes the interconnectedness of water, energy, and food systems, as well as the intricate relationships between these sectors, which are influenced by the success of transboundary irrigation (Botai et al. 2021). Effective governance of transboundary water resources for irrigation is essential for not only attaining specific SDGs but also for promoting integrated and sustainable development across multiple sectors worldwide.

Transboundary water bodies, such as the Nile, Mekong, Danube, and Amazon rivers, exert a substantial influence on irrigation systems spanning entire continents (Dogaru et al. 2019; Merem et al. 2020; Paredes-Trejo et al. 2021; Sridhar et al. 2021). However, the utilization of water for irrigation across different countries faces several challenges. These include conflicting allocation (Gorgoglione et al. 2019), upstream–downstream dynamics and climate change impacts (Munia et al. 2020), population growth (Merem et al. 2020), economic expansion (Tian et al. 2020), and management complexity along with governance issues (Mogomotsi et al. 2020). Disagreements frequently cause conflicting water allocation, resulting in unequal distribution and endangering biodiversity (Merem et al. 2020; Rasul & Neupane 2021). Dam development and other upstream interventions have an impact on downstream water availability, aggravating water stress (Ahmed et al. 2019; Kazemi et al. 2022). Population growth and economic expansion drive up demand, increasing scarcity, particularly during droughts (Liu et al. 2020; Munia et al. 2020). Climate change changes hydrological patterns, resulting in more intense occurrences with many uncertainties (Basheer et al. 2024). Furthermore, managing transboundary water requires balancing several interests, which necessitates strong governance and international cooperation. Thus, the lack of widely recognized frameworks impedes dispute resolution and sustainable resource management.

The progress in knowledge has led to the development of transboundary water for irrigation optimization strategies, which can offer resolutions to the challenges associated with the management of shared water resources. By taking into account technical and socioeconomic factors, these optimization techniques can be adopted to attain both efficiency and equity in transboundary irrigation water management (Hatamkhani & Moridi 2021; Santos et al. 2023). However, earlier research often restricted optimization efforts to certain areas that need attention. The objective of this study is to analyse and consolidate the current approaches and provide a comprehensive overview of the integrated optimization concept, encompassing both technical and socioeconomic elements. This study conducted a comprehensive review of 370 articles obtained from reputable academic sources such as Google Scholars, Publish and Perish, and other relevant sources. The articles were published between 2014 and 2024 and were selected based on their relevance to the topic. The study focused on keywords such as ‘Transboundary River,’ ‘Transboundary watershed,’ ‘Transboundary basin,’ ‘Transboundary water,’ ‘Transboundary irrigation,’ ‘Water optimization,’ ‘Irrigation optimization,’ and ‘Irrigation.’ After eliminating duplicate and irrelevant articles, this study analysed a total of 189 articles. The results of this study are advantageous for developing plans to enhance the administration and utilization of shared water resources for irrigation, ultimately promoting the sustainable growth of agriculture and regional collaboration, especially in developing countries like Indonesia and other similar regions.

Irrigation is essential in agriculture because it ensures food security, increases crop yields, and promotes economic growth. Irrigation may greatly enhance crop yields compared to rainfed agriculture, adding to food production and security (Kumar & Sen 2023). Transboundary water is a vital supply of irrigation in many countries, considerably increasing agricultural output and water security (Merem et al. 2020).

The Nile River Basin is one example of transboundary water for irrigation in Africa, which includes Egypt, Sudan, Ethiopia, and Uganda, which is one of the world's largest transboundary river basins (Abdullah et al. 2020). The Blue Nile, a significant tributary of the Nile River, has enormous transboundary irrigation potential (Tedla et al. 2022). With rising limits on the Blue Nile's water resources due to population and economic growth, as well as climate change, integrated water resource planning and management is required to assure the basin's water, energy, and food security (Basheer et al. 2024). The Grand Ethiopian Renaissance Dam is an example of transboundary water used for irrigation in the Nile River Basin that is expected to considerably boost irrigation opportunities in the region, not only by supplying water for irrigation to millions of hectares of agricultural land but also by providing hydropower generating in Sudan and Egypt (Kahsay et al. 2019).

The Mekong River Basin in Southeast Asia provides another example of transboundary water for irrigation. The Mekong River flows through China, in particular Yunnan Province, Myanmar, Laos, Thailand, Cambodia, and Vietnam providing critical water resources for cultivation in the region (Zhao et al. 2021). With its large network of tributaries and significance in providing livelihoods for millions of people, the Mekong River Basin has tremendous potential for transboundary irrigation projects (Sridhar et al. 2021). Countries in the Mekong area have collaborated to create joint irrigation systems and manage water resources to ensure sustainable agricultural practices and food security for their citizens (Mallick 2022). The Nam Theun 2 Hydropower Project in Laos is a major project in the Mekong River Basin that has helped irrigation and agricultural growth in downstream areas of Thailand and Cambodia, illustrating the favourable effects of transboundary water cooperation for regional irrigation (Khan et al. 2017).

In Europe, the Danube River Basin is another example of transboundary water for irrigation. The Danube River spans 19 countries and provides water for agriculture along its banks (Habersack et al. 2016). The water resources of the Danube are critical for irrigation in the rich plains of Hungary and Serbia, enabling the development of crops such as wheat, maize, and sunflowers (Arsic et al. 2015). Countries in the Danube River Basin work together to improve irrigation infrastructure, increase water efficiency, and promote sustainable agriculture practices (Izakovičová et al. 2020).

In North America, the Colorado River Basin is a well-known example of transboundary water utilized for irrigation. The Colorado River, which runs through several states in the United States and Mexico, provides water for irrigation of farmlands in the region (Fleck & Castle 2022). The river's water sources benefit agriculture in California, Arizona, and Colorado, as well as the Mexicali Valley in Mexico. Several agreements and treaties between the United States and Mexico govern the allocation of water from the Colorado River for irrigation, emphasizing the importance of collaboration in managing transboundary water resources for sustainable agriculture (Wilder et al. 2020; Blumstein & Petersen-Perlman 2021). Aside from that, the Amazon River Basin acts as a model for South America. The Amazon River and its large network of tributaries flow through several countries, including Brazil, Peru, Colombia, Bolivia, and Ecuador. The river and its basin have an abundance of water resources, allowing for transboundary irrigation projects and making the region's agricultural practices more sustainable (Mujica et al. 2021).

Despite its great promise, using transboundary water for irrigation comes with a host of issues, as shown in Table 1. Included in this list are factors such as water distribution, climate change, hydropower, agriculture, and pollution. The water scarcity problem is exacerbated when countries' competing interests and goals lead to disputes over the distribution of water resources (Katz & Nagabhatla 2023). Because water shortage disturbs ecosystems and lowers habitats in the downstream areas, the continuing conflict increases tensions and threatens the river's biodiversity. The management of transboundary rivers is inherently intricate due to the intricate interplay between nations located upstream and downstream (Hossen et al. 2023). Dams and other extensive irrigation projects disrupt the regular water flow patterns in communities located downstream (Hoque et al. 2022). This alteration will lead to water scarcity and diminished resources for agriculture, industry, and ecosystem well-being and will have a significant effect on the individuals residing downstream (Ingrao et al. 2023), especially when that area is contaminated by severe pollution issues like that of the Rhine due to industrial activities (Chen et al. 2021). Key pollutants such as toxic chemicals from industrial discharge can significantly degrade the river's water quality, which also contributed to a form of water scarcity. This is because polluted water becomes unusable for various critical needs, including drinking, agriculture, and industry.

Effective coordination and cooperation are necessary to achieve a harmonious balance between the competing needs of agriculture, urbanization, industry, and hydropower generation. The task of managing transboundary water resources becomes extremely difficult when there is a lack of effective leadership and collaboration. International cooperation is essential for the establishment of equitable allocation processes and governance structures, including data sharing. For example, the absence of formal agreements and structured data exchange mechanisms within the Tigris-Euphrates Basin lead to the hesitancy of upstream countries, particularly Turkey, to share hydrological data with downstream nations, which complicates cooperative management (Rougé et al. 2018). Furthermore, the lack of generally accepted frameworks makes it difficult to resolve conflicts and manage shared water resources in the long run (Hossen et al. 2023). Despite the efforts of river organizations such as the Mekong River Commission (MRC), there is no overarching international agreement that fully binds all six riparian countries included to specific data-sharing and water management protocols (Yu et al. 2019). Likewise, the differences in methodologies between the countries complicate data standardization. In Quaraí (Cuareím) River Basin, Uruguay does not use the Otto–Pfafstetter method for basin delineation, which Brazil applies, and this disparity complicates attempts to integrate data for cohesive transboundary management, as varying data scales and resolutions can result in mismatched borders (Garrick 2017). Thus, policies that prioritize cooperation, investments in infrastructure, and equitable distribution processes are crucial for the effective long-term management of transboundary water resources (Santos et al. 2023).

As previously noted, a lack of data such as hydrological, geographical, topographical, and socioeconomic elements is one of the very first problems with transboundary water for irrigation optimization (Degefu et al. 2018; Al-Addous et al. 2023). The analysis of water availability and distribution across borders relies heavily on field data, including rainfall and river flow rates (de Oliveira et al. 2023). Similarly, topography data aid in identifying natural water flow channels and prospective infrastructure locations, and land use data reveal how water resources are used across varied landscapes. Furthermore, socioeconomic statistics, such as water usage and population trends, provide information about the demand for irrigation resources and the economic activities that rely on them (Santos et al. 2023; Li et al. 2024). The lack of comprehensive data makes it impossible to develop effective management plans and allocate resources efficiently, which impedes collaborative decision-making and long-term transboundary water management (Saha et al. 2021).

Addressing this data gap is critical for increasing collaboration and ensuring equitable and long-term usage of transboundary water resources. Direct field measurement is one step towards achieving this aim, but it requires significant time and financial investment (ter Horst et al. 2023). As a result, in the current situation, remote sensing appears to be a viable option for overcoming data constraints, as shown in Table 2. Remote sensing provides several advantages, including the ability to efficiently cover large geographic areas, provide regular and consistent data updates, and monitor dynamic environmental changes over time (Popescu et al. 2021). Water resource management employs several remote sensing technologies, including satellite imagery and aerial drones (Abdulraheem et al. 2023). These groups can obtain information from satellites like the United States Geological Survey (USGS), Google Earth Engineering (GEE), the Sentinel program of the European Space Agency, the Landsat program of the National Aeronautics and Space Administration, and other private companies (Hemati et al. 2021). In addition, unmanned aerial vehicles are being made to gather data that is more specific to a certain area, which will provide more leeway and accuracy when tracking irrigation systems and water resources (Peladarinos et al. 2023).

Remote sensing technologies such as Landsat, MODIS, Sentinel, and SAR data, as well as digital elevation model (DEM) information, are critical in data collection for transboundary water for irrigation optimization. Landsat and MODIS data are commonly utilized for land cover monitoring, cropping pattern analysis, and detecting irrigated crop areas because they offer both detailed categorization and extensive spatial coverage (Tariq et al. 2023). Sentinel data track vegetation health and detect responses to environmental changes for assessing agricultural conditions over time (Snevajs et al. 2022). Park et al. (2020) propose employing SAR data to generate high-resolution backscatter images that are unaffected by atmospheric conditions for accurate monitoring of water bodies and determining water levels along reservoir shorelines. DEM data aid in terrain analysis by detecting demand nodes, measuring heights, and precisely estimating water levels, all of which are necessary for effective water resource distribution across diverse landscapes (Wei et al. 2023).

Landsat and Sentinel data have been used to track land cover change and assess agricultural practices in transboundary areas. For example, Landsat 8 data were used to estimate land cover changes over 33 years, focusing on anthropogenic activities in Armenia and the Nakhchivan Autonomous Republic (Khoshnoodmotlagh et al. 2020). Landsat 8, being the primary component, provides critical information for understanding land use changes and finding patterns of anthropogenic impact. Fujihara et al. (2020) also employed Landsat 8 data to analyse agricultural practices within the watershed, finding a preference for 2-year rotation patterns in the top block and 3-year rotation patterns in the bottom block. Furthermore, Sentinel data have been used in conjunction with Landsat to track and assess normalized difference vegetation index (NDVI) values over time, providing information about vegetation health and stress levels (Kempf & Glaser 2020). The accurate delineation of irrigated areas using NDVI from Landsat 8 is critical for monitoring groundwater usage in agriculture, especially during dry seasons when groundwater sources are heavily relied on (Nhamo et al. 2020).

Data forecasting and decision-making for irrigation optimization in transboundary water systems have come a long way with the advent of extremely potent technologies like deep learning and machine learning (Saha et al. 2021). Understanding subtle patterns and representations from vast and complex datasets is one of the several advantages of deep learning. Thus, it does not require explicit feature engineering to discover minute connections (Taye 2023). Deep learning does particularly well with high-dimensional, nonlinear data, which are common in transboundary irrigation networks, as shown in Table 3. As new data are collected, deep learning models can be modified and improved upon. This improves accuracy in predicting irrigation demand and optimizing resource allocation (Aldoseri et al. 2023).

LSTM was found to be a highly effective deep learning technique and is utilized for predictive modelling in many water resource management applications (Omar & Kumar 2021). LSTM outperforms RNN, FNN, ANN, and CNN (Malakar et al. 2021; Alibabaei et al. 2022). LSTM predictive modelling and CPP-FPP-IPP programming techniques are used in the hybrid LSTM-CPP-FPP-IPP model to improve irrigation water allocation. An investigation has shown that this strategy outperforms traditional deterministic planning techniques (Jia et al. 2022). The incorporation of ambiguous components including rainfall patterns, crop output projections, and climatic fluctuations is made possible by this integration. Because these uncertainties are included into the decision-making process, the hybrid model enables decision-makers to look into a wider range of possible scenarios and look for more suitable solutions in ambiguous situations. However, accuracy can be further boosted by combining LSTM with other approaches, such as particle swarm optimization (PSO)-LSTM, which optimizes parameters for LSTM, resulting in more exact and reliable forecasts (Kilinc 2022).

Machine learning approaches offer significant benefits for optimizing transboundary water usage in irrigation (Drogkoula et al. 2023). Machine learning algorithms possess the capacity to effectively process a wide range of data kinds and objectives, enabling its application in numerous scenarios and decision-making contexts within transboundary irrigation management (Peladarinos et al. 2023). Furthermore, machine learning models are recognized for their computational efficacy, allowing for rapid decision-making in relation to changing environmental circumstances and irrigation requirements (Shams et al. 2024).

The management of transboundary water resources extensively uses machine learning in predictive modelling. The applications include, as shown in Table 4, the estimation of runoff coefficient, analysis of land cover change, river flow prediction, evaluation of groundwater quality, computation of irrigation water quality index (IWQI), and categorization of cultivated land and water resources resilience. Machine learning techniques can be used to increase the precision of mapping and model accuracy, including water potential (Sarkar et al. 2024). A study looked at and predicted spring growth using RF, SVM, MARS, and BRT among other data mining methods (Al-Shabeeb et al. 2023). The study concluded that the RF model demonstrated the highest level of accuracy. The utilization of elevation, precipitation, and soil data in the RF model enhances the accuracy of predicting flow coefficients for water resource management and land use planning in water-scarce locations (Yan et al. 2019). Furthermore, the application of machine learning methods can augment the precision of data downscaling procedures, resulting in enhanced accuracy of models and maps. In the previous study, the XG Boost model outperformed the ANN model in successfully fitting anticipated TWS anomalies to GRACE data (Ali et al. 2023). These improved calculations allow for more thorough evaluations utilizing the water storage deficiency index and water storage deficiency analysis, which are essential for assessing the severity and the frequency of droughts.

Hydrological and hydraulic models can be used with enough data to gain a thorough understanding of water dynamics and improve transboundary water for irrigation system management (Wang et al. 2023). These models provide precise estimates of water flow, distribution, and behaviour throughout the basin and irrigation infrastructure (Garcia-Espinal et al. 2024). These models, which incorporate high-quality data on precipitation, soil quality, land use, and terrain, can provide information on water availability, predict probable flood or drought scenarios, and optimize water distribution systems (Xiao et al. 2024). Hydrological and hydraulic modelling can also aid in the identification of irrigation network vulnerabilities, estimate the effects of climate change, and develop adaptive management techniques (Kypreos et al. 2024).

Depending on the water management aims and study fields, a variety of advanced modelling approaches can be utilized to conduct hydrological and hydraulic modelling as shown in Table 5. Tools such as the Hydrologic Engineering Center's Hydrologic Modelling System (HEC-HMS), Hydrologic Engineering Center's River Analysis System (HEC-RAS), MIKE SHE, MIKE 11, soil and water assessment tool (SWAT), variable infiltration capacity (VIC), and modular groundwater flow (MODFLOW) are utilized.

HEC-HMS is hydrological cycle modelling software that simulates surface runoff, surface runoff, and base flow in a drainage basin (Davamani et al. 2024). It enables the investigation of rainfall, infiltration, evapotranspiration, and runoff formation, all of which are critical for a thorough understanding of the hydrological cycle in a specific area (Bronstert et al. 2023). Meanwhile, HEC-RAS is a hydraulic modelling programme used mostly along rivers and waterways to investigate water flow in open channels by computing surface elevations, flow velocities, flow distributions, and flood potentials (Chomba et al. 2021). HEC-RAS is widely used to assess how hydraulic constructions such as dams, bridges, and irrigation channel networks affect water flow patterns (Kamran et al. 2021). Furthermore, integrated systems that use multiple software packages, such as HEC-HMS and HEC-RAS, improve flood prediction accuracy. This is critical for arid region urban development and flood prevention since it improves flood hazard comprehension and aids in urban infrastructure planning and protection (Salazar-Briones et al. 2018).

MIKE SHE, created by the DHI Group, is a hydrological modelling software required for optimizing transboundary irrigation (Rahim & Yusoff 2023). MIKE SHE fully incorporates surface and groundwater dynamics, allowing for comprehensive modelling of hydrological systems, which makes it easier to simulate important processes as precipitation, evapotranspiration, infiltration, surface runoff, groundwater flow, and surface–groundwater interaction (Prucha et al. 2016). Furthermore, when modelling nitrogen dynamics in hydrological systems, tools like MIKE SHE combined with MIKE 11 are particularly effective (Waseem et al. 2018). These models allow for detailed simulation of nitrogen transport and transformation processes. Other than that, MODFLOW Model can also simulate a wide range of hydrogeological phenomena, including groundwater flow, interactions between groundwater and surface water, and recharge and discharge in porous medium aquifers (Condon et al. 2021; Davamani et al. 2024). MODFLOW's ability to estimate groundwater levels, evaluate pollutant movement, and assess the effects of land use changes makes it a valuable tool for optimizing transboundary irrigation techniques and guaranteeing sustainable water resource management (Davamani et al. 2024). Lesser-used software, such as HYDRUS, can help with transboundary irrigation optimization by simulating water transport and successfully managing groundwater resources (Mirlas et al. 2022). By tying water demand to sustainable practices, HYDRUS assures a consistent water supply for agriculture while reducing the risk of groundwater depletion. Despite its low profile, HYDRUS delivers critical insights into groundwater dynamics that are essential for effective irrigation control.

Moreover, SWAT and VIC models are widely used in hydrological and water quality modelling, of which SWAT is a physical model designed to simulate long-term impacts of agricultural activities on water quality and quantity in large watersheds, covering parameters such as evapotranspiration, erosion, and nitrogen dynamics (Waseem et al. 2018). It is particularly useful for analysing nitrogen pollution in intensively farmed areas and assessing human interventions. VIC, on the other hand, focuses on simulating the hydrological cycle by incorporating key inputs like precipitation and air temperature, allowing for sensitivity and uncertainty analysis in water processes, particularly in cold or snow-covered regions (Lilhare et al. 2020). VIC is suitable for predicting streamflow variations and improving water management in cold climates, making it an essential tool for transboundary irrigation systems (Safeeq et al. 2014).

With enough information and thorough data, transboundary water for irrigation optimization is possible by modelling the water resource and management (De Keyser et al. 2023). Typically, this optimization process involves using a variety of approaches geared to the complexities of cross-border water resource management (Derepasko et al. 2021; Santos et al. 2023). Machine learning, stochastic methods, linear programming, and particular water resource management and optimization models are examples of methodology and modelling tools used in transboundary water for irrigation optimization (Agastya et al. 2024). Aside from data predicting, machine learning can also be used for optimization, which frequently involves stochastic modelling or linear programming.

Stochastic modelling adds randomness and uncertainty into the optimization process, resulting in more accurate representations of complex systems, and this technique is especially effective in dynamic and uncertain conditions, when typical deterministic approaches may fail. For example, two-stage chance-constrained programming (TSCCP) approach blends pre-emptive decision-making based on expected situations with tailored adjustments in reaction to random events, resulting in a dynamic solution for handling uncertainties such as weather changes. In the initial step of the optimization model, decision variables are determined before any random occurrences occur, functioning as preventative actions based on projected scenarios within the agricultural irrigation system, considering the field water cycle. In the second step, decision variables are established after random events have occurred, and they are modified to handle the details of the uncertainties that have been identified, such as weather changes or water supply. Integrating these two steps into the decision-making process can thus optimize agricultural irrigation water allocation, improving the effectiveness and sustainability of water resource management in agriculture while considering the field water cycle (Yan & Li 2018). In addition, the double-sided stochastic fractional programming (DSFP) model handles uncertainty in water-agriculture-energy nexus (WAEN) systems, with a specific emphasis on climate change consequences (Zhang et al. 2024). The strategy is utilized to build an optimal water allocation methodology for a WAEN system, with a focus on mitigating the effects of climate change. To determine the best water allocation plans for a transboundary river basin, a DSFP-WAEN model was developed and applied to 48 different scenarios to analyse the long-term planning implications of various factors such as climate change, improved irrigation efficiency, and varying levels of system risk. The overarching goal is to optimize water allocation schemes by balancing economic advantages against system hazards, all while effectively managing the inherent uncertainties and intricate interrelationships of water supplies for agricultural and energy output.

In addition to forecasting and decision-making, stochastic modelling can optimize transboundary irrigation. Copula-based stochastic multi-objective programming, for example, optimizes irrigation plans by maximizing economic advantages while conserving water in drought-prone locations (Zhang et al. 2023). To effectively analyse and manage the effects of seasonal agricultural drought, the Copula model is used, which combines regional net irrigation water demand and stream runoff. This model addresses the uncertainty and complexity of drought scenarios by optimizing three core objectives: maximizing economic benefits while considering the cost-effectiveness of water uses and conservation, ensuring the highest crop yield during drought conditions through efficient water allocation, and improving irrigation strategies for greater water savings, which is critical in areas experiencing water scarcity. Similarly, Ortiz-Partida et al. (2019) employed a two-stage stochastic optimization model to propose robust procedures for a multifunctional reservoir. The first part of the project comprises developing a robust set of reservoir releases that address a variety of objectives, including water supply, flood control, hydropower generation, and ecological considerations across a wide range of hydroclimatic situations. The goal of this stage is to plan for efficient and cost-effective water management in the face of climate and water availability uncertainties. The second stage evaluates these planned releases by comparing their performance to a wide range of future hydroclimatic conditions, allowing for any necessary adjustments to maximize economic advantages while reducing potential dangers from changeable hydroclimatic events.

Linear programming, on the other hand, is a mathematical optimization technique for determining the optimal outcome in a model according to particular limitations. Linear programming can be used to reduce water use by assessing precipitation efficacy and estimating crop irrigation water requirements, with the goal of maintaining agricultural yields (Difallah et al. 2017). The study's linear programming model combines the ‘knapsack’ problem decisional form to analyse the efficacy or ineffectiveness of precipitation and to forecast the amount of irrigation water needed to optimize its use in agricultural activities. Their proposal addressed worries about water scarcity while also promoting good agricultural water management. Linear programming, like stochastic modelling, seeks to maximize economic advantages by maximizing, reducing, or combining the two to accomplish desired results within given constraints (Kumar & Sen 2020). Other studies sought to maximize net economic benefits by distributing water to urban and agricultural consumers while conforming to supply and regulatory restrictions (Kucukmehmetoglu & Geymen 2014). Their research began with a linear programming model but grew into a mixed integer programming model to better handle linear, binary, and integral decisions, allowing for trade-off evaluation and collaborative efforts among countries sharing water resources. This improvement facilitates the assessment of trade-offs and cooperative efforts among countries that share these water resources. Furthermore, linear programming maximizes agricultural production benefits by balancing planting patterns and water requirements against seasonal water availability (Juwono et al. 2018). The model aimed to maximize yield or profit by efficiently utilizing water resources within the restrictions of land availability, water supply, and crop requirements, hence increasing the benefits of crops grown in irrigated zones.

More than that, transboundary optimization can also be achieved using water resource management modelling tools as shown in Table 6. Modelling tools, including the water evaluation and planning (WEAP) System, SWAT, modular modelling system (MODSIM), MIKE HYDRO, and linear interactive and discrete optimizer (LINDO), have been employed in research to provide comprehensive frameworks for monitoring and managing water resources across borders, incorporating a diverse set of data sources and modelling methodologies to aid decision-making. Water resource management modelling tools assist in the optimization of transboundary irrigation systems by simulating water allocation, usage, and distribution scenarios. This promotes sustainable water use while maximizing economic benefits and lowering environmental costs (Dinar 2024).

WEAP is a commonly used tool for managing water resources for the sustainable management and allocation of available surface water to meet competing demands such as irrigation, livestock, and domestic use (Esraa et al. 2023). When combined with HEC-HMS, WEAP optimizes surface water allocation by balancing current and projected future water demands, reducing potential water use conflicts, encouraging efficient water use, and promoting economic development in the face of water scarcity and competing water uses (Genjebo et al. 2023). SWAT is another important hydrological modelling method, notably for determining the effects of land management practices on water, sediment, and agricultural chemical yields (Wang et al. 2019; Nda et al. 2020; Al Khoury et al. 2023). SWAT is frequently coupled with other modelling methodologies, including hydrological and hydraulic modelling, as well as water resource management modelling (Nda et al. 2020). For example, it can be combined with HEC-RAS or WEAP to provide complete water resource and flood management (Psomas et al. 2016; Taraky et al. 2021). MODSIM is a versatile modelling programme that may be used for hydroeconomic analysis as well as strategic water resource management and allocation, which is frequently combined with other models, such as SWAT or PSO, to improve irrigation techniques and water distribution systems (Emami & Koch 2018; Fadaeizadeh & Shourian 2019). MIKE HYDRO consists of the MIKE HYDRO River and MIKE HYDRO Basin modules (Vashist & Singh 2022). MIKE HYDRO River simulates river hydraulics, whereas MIKE HYDRO Basin models basin-level water availability (Yimere & Assefa 2022). These techniques are frequently integrated with MIKE SHE to provide comprehensive insights into water management strategies that account for both surface and groundwater dynamics (Malamataris et al. 2020). LINDO is a mathematical optimization tool for linear, integer, and nonlinear programming (Paramjita et al. 2019). Its strength stems from its capacity to efficiently address complicated optimization problems, particularly those connected to water resource management, such as water efficiency and allocation models (Ramadan et al. 2021). The benefits of using LINDO include the ability to quickly uncover optimal solutions, improve decision-making processes, and develop cost-effective resource allocation approaches (Babel et al. 2023).

Furthermore, socioeconomic aspects must be included as input data to make informed water management decisions. Policies and governance components are critical socioeconomic factors to consider when integrating technology characteristics (Shao et al. 2020). First, a clear legal and legislative framework is essential for sustainable water use, as evidenced by the 1964 Columbia River Treaty between the United States and Canada. This treaty defines water allocation and flood control measures, establishing a systematic framework for collaborative management of the Columbia River Basin (Zhang et al. 2021). Similarly, the MRC, created in 1995, ensures equitable use and management of the Mekong River Basin among member nations by legally binding agreements (Feng et al. 2019). Second, effective stakeholder engagement and empowerment are critical for inclusivity and support for water management initiatives, as demonstrated in the Rhine River Basin by the International Commission for the Protection of the Rhine (ICPR), which engages diverse stakeholders in collaborative decision-making (Booth et al. 2020). Furthermore, the Lake Chad Basin Commission (LCBC), which was created in 1964, actively engages communities and non-governmental organizations in decision-making processes (Mbaigoto et al. 2023). Third, increased information sharing and data openness promote trust and informed decision-making, as seen by the Danube River Basin Commission's extensive data-sharing platform (Lipponen 2020). Fourth, institutional coordination and capacity building are essential, as demonstrated by the Lake Victoria Basin Commission (LVBC) and the Zambezi Watercourse Commission (ZAMCOM) (Paisley & Curlier 2019). Finally, dispute resolution and consensus building are critical, as evidenced by the Indus Water Treaty (IWT) between India and Pakistan and the Nile Basin Cooperative Framework Agreement (Rigi & Warner 2020; Wehling 2020).

The IWT of 1960, mediated by the World Bank, governs water sharing between India and Pakistan, primarily allocating the Indus Basin's rivers while providing dispute resolution mechanisms like the Permanent Indus Commission and international arbitration. Both countries have at times linked water issues to broader geopolitical concerns, such as terrorism, complicating cooperation. Pakistan's alliance with China on water projects further affects negotiations, as these agreements bolster Pakistan's position and reduce India's leverage. The securitization of water resources and issue-linkage strategies have reinforced the strategic and contentious nature of transboundary water diplomacy between the two nations. Other than that, the Nile Basin Cooperative Framework Agreement was developed as a legal framework for managing the Nile waters equitably among the riparian states. Despite its intent to foster cooperation, only a subset of the 11 basin countries initially signed the CFA in 2010. The CFA aims to align with international principles, such as equitable and reasonable utilization and the no-harm rule, adapted from the 1997 UN Watercourses Convention. Egypt and Sudan have refrained from signing due to concerns over water security and existing rights, while other nations like Ethiopia ratified it, reflecting varied regional perspectives on shared water management.

Despite the importance of initiatives, as previously stated, difficulties still exist in these areas. This suggests that studying socioeconomic criteria alone does not guarantee the success of irrigation projects. Likewise, relying solely on technical aspects will not ensure success, as effective irrigation requires a balanced consideration of both technical and socioeconomic aspects. As a result, it is vital to integrate these two dimensions with all stakeholders, including the community and farmers, fulfilling their tasks with utmost dedication (Shao et al. 2020). To ensure the long-term management of irrigation water resources, stakeholders may need to work effectively together, obey regulations, and actively participate in decision-making processes (Barua et al. 2019; Rivera-Torres & Gerlak 2021).

With good input data and the integration of processes involving forecast data and decision-making factors, simulations of water quantity and quality, and optimization itself, yield significant outcomes (Xue et al. 2021). The medium-term outputs of this optimization are adaptive capacity, sustainable practices, and resilient communities. Irrigation systems can dynamically react to changing environmental circumstances and water demand by leveraging predictive data and decision-making criteria (Nikolaou et al. 2020). This enables quick responses to significant weather changes, water quality decreases, or policy alterations, improving the irrigation system's ability to manage new difficulties. Furthermore, optimization of transboundary water for irrigation can result in more sustainable agricultural practices (Guo et al. 2022; Ruan et al. 2023). Decisions regarding water usage and land management can be made by considering environmental, economic, and social aspects comprehensively (Li et al. 2020). More sustainable agriculture can be achieved by effective water usage, environmentally friendly irrigation systems, and enhanced soil conservation practices (Velasco-Muñoz et al. 2019). Furthermore, by increasing adaptive ability and implementing sustainable agriculture practices, irrigation-dependent areas, the countries, and communities will become more robust to environmental and social change, and thus, they will be more prepared to address issues such as droughts, floods, policy changes, water resource sustainability, and food security (Srivastav et al. 2021). These three outcomes are extremely related to the SDGs, particularly eradicating hunger, providing clean water and sanitation, affordable and clean energy, climate action, and the WFE-NEXUS idea. All of these efforts eventually yield long-term economic benefits, demonstrating the connection of water, food, energy, and economic development.

This study underscores the critical need for a comprehensive approach to transboundary water management for irrigation, addressing challenges like competing water demands, climate change impacts, and governance complexities. Key technologies, such as remote sensing and predictive modelling, are essential for enhancing data collection and improving water allocation strategies. Integrating these technical solutions with socioeconomic policies promotes adaptive capacity and fosters sustainable development, aligning with global goals like the SDGs and the Water-Food-Energy Nexus.

Future research should focus on refining predictive models that incorporate both environmental and socioeconomic variables, as well as exploring ways to strengthen cross-border cooperation frameworks. In addition, advancing the use of machine learning in water management could improve accuracy in forecasting and decision-making under uncertainty. For practical applications, continued emphasis on building resilience through sustainable practices will be crucial to mitigating the effects of water scarcity and variability, thereby supporting long-term resource sustainability and regional prosperity.

The authors would like to thank Jember University as a funder and IDB Internal Support Grant provider.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.