Iranian Water Research Journal

Iranian Water Research Journal

Application of copula functions in groundwater studies (A review)

Document Type : Review Article

Authors
Rasoul Mirabasi Najaf Abadi 1
Mohammad Nazeritahroodi 2
1 Associated Professor, Water Resources Engineering, Faculty Agriculture, Shahrekord University, Iran.
2 Department of Water Engineering, Faculty of Agriculture, Lorestan University, Khorramabad, Iran.
10.22034/iwrj.2025.15130.2670
Abstract
Introduction:
Groundwater is a vital resource for human livelihoods and agricultural productivity, especially in arid and semi-arid regions. Therefore, the quantitative and qualitative management of these resources is of great importance. Traditional methods of analyzing groundwater levels often overlook the complex interdependencies among various influencing factors, such as precipitation, temperature, and land use. Over the past two decades, copula functions have emerged as a powerful statistical tool for modeling such dependencies, enabling the joint distribution of multiple variables affecting a given phenomenon. This paper reviews existing research on the application of copula functions in groundwater-related studies and attempts to highlight gaps in the current literature while suggesting potential directions for future research.
Methods:
Copula functions are mathematical tools that link univariate distribution functions to form a multivariate distribution. A copula can describe the dependence structure between two or more random variables and has gained significant attention over the past two decades as a practical and efficient method for modeling the overall dependence structure of variables in multidimensional phenomena. Three main approaches are used to construct copula functions in three or more dimensions: (1) symmetric multidimensional copulas, which are limited by mathematical constraints and symmetric dependence requirements; (2) nested (hierarchical) copulas, where bivariate copulas sequentially link pairs of variables or their joint distributions in a stepwise manner until reaching the desired dimensionality, and (3) vine copulas, which offer greater flexibility through structured pair-copula constructions.
Results:
In this study, methods based on copula functions in groundwater studies were investigated from different aspects. The results showed that one of the biggest challenges in this field is the selection of a suitable copula function that can accurately describe the dependencies between the variables. An incorrect choice of copula function can lead to poor modeling results. Also, the complexity of the calculations and the need for data with adequate accuracy and length are further limitations when using copula functions in groundwater studies, as access to groundwater monitoring data is limited in many areas. However, copula functions offer significant advantages in modeling groundwater systems. These functions provide the necessary flexibility to model nonlinear dependencies between variables, and allow the separation of marginal distributions from the dependency structure. These features help to improve the risk assessment and management of groundwater resources. To realize the full potential of copula functions in hydrological research, it is necessary to conduct more case studies in different environments. Priority should also be given to studying the long-term effects of climate change and the more complex interactions between socio-economic and environmental variables. Finally, integrating copula functions with innovative modeling approaches can help improve modeling accuracy, and ultimately lead to a better understanding and management of groundwater resources. The results of a review of published studies in various fields (including joint frequency analysis of quantitative and qualitative variables, joint simulation, spatial analysis, effects of other variables on groundwater quantity and quality, groundwater drought, etc.) indicate the favorable performance of copula functions in groundwater studies. This effectiveness is primarily attributed to their distribution-oriented nature of this method. Despite recent advances in the use of copula functions in hydrology, their application to complex groundwater systems, involving spatially distributed variables and structural heterogeneities, has yet remained underexplored. The development of basic copula methods to account for non-stationarity in groundwater data could also be an area for future study. In addition, the integration of copula functions with more advanced modeling methods, such as Bayesian hierarchical models, could open up new opportunities for future research. This integration could help to improve the modeling of complex dependencies between hydrological variables, especially in groundwater systems.
Conclusion:
A review of current literature on climate change impacts reveals that most studies focus on specific geographic regions, limiting the generalizability of their findings. There is a need for more extensive case studies across diverse environments to validate the effectiveness of copula functions in various hydrogeological contexts. While some research has begun using copulas to assess climate change effects on groundwater levels, comprehensive studies examining long-term trends and their implications for water resource management remain scarce. Although copulas can effectively capture dependencies among multiple variables, existing studies often concentrate on a limited set of factors. More complex interactions—such as those involving socio-economic variables, land-use changes, and policy impacts—require further exploration. Additionally, most studies assume stationarity in time series data, leaving room for future research to incorporate non-stationarity in both marginal distributions and copula functions.
Keywords
Bivariate copulas
Dependency analysis
Groundwater
Multivariate analysis
Vine copulas
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  • Receive Date 25 February 2025
  • Accept Date 02 June 2025
  • Publish Date 22 June 2025