Multivariate spatial analysis of groundwater quality using copulas

The quality and quantity of groundwater resources are often assessed by sampling a limited number of wells drilled sparsely across a plain. So far, various methods have been developed for assessing and modeling groundwater quality, each of which has its capabilities and limitations. In the present study, the copula functions were applied to multivariate analysis of groundwater quality variables (including SAR, K, Mg, Na, Ca, Cl, EC, pH, TDS, SO4, TH and HCO3) in Shahrekord plain, Iran. For this purpose, the quality data from 24 wells distributed across the Shahrekord plain during the period of 1990-2020 were used. For bivariate analyses of groundwater quality, first the fitness of some common distributions in hydrology were examined to the quality variables and the appropriate marginal distribution was determined by Kolmogorov-Smirnov test (K-S). The results revealed that the Generalized Extreme Value distribution has an acceptable fitness on groundwater quality variables of Shahrekord plain. The results of correlation analysis indicated that the highest correlation based on Spearman Rho, Kendall Tau and Pearson correlation coefficient is related to the paired variables of SAR-Na and EC-TDS with correlation coefficient greater than 0.9. Therefore, these variables were selected for further analysis. In the next step, bivariate distribution of two different quality variables in a well (point analysis) and also bivariate distribution of a quality variable in two different wells (inter-well analysis) were created using copula functions. To do this, the fitness of different copulas (including Plackett, Joe, Clayton, Frank, Farlie-Gumbel-Morgenstern, Ali-Mikhail-Haq, Gumbel, Gumbel-Hougaard, Gumbel-Barnett, Philip-Gumbel and Galambos) was tested to construct a bivariate distribution of quality variables. For choosing the best fitted copula on studied paired variables, the theoretical copula values was compared with the corresponding values of empirical copula based on the goodness of fit criteria. Based on the results of point analysis, the Joe copula function was chosen as the best fitted copula function for multivariate analysis of two quality variables in a well. For inter-well analysis (IWA), the Clayton copula function was selected for creating bivariate distribution of a quality variable in two different wells. Comparing the performance of the proposed IWA method with geostatistical methods showed that in addition to the IWA method having acceptable accuracy, it has a higher efficiency than geostatistical methods, especially in areas where the number of sampling wells is less.