A Wasserstein Distance and Bernstein Copula Framework for Modeling the Spatial Variability of Hydraulic Conductivity

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Abstract
Knowledge of the spatial distribution of hydraulic conductivity within an aquifer is required for groundwater flow modeling. The number of hydraulic conductivity measurements is usually scarce, making it necessary to interpolate available data. Commonly, a larger number of hydraulic head values measured in wells are available; therefore, they have been used as a secondary variable to improve hydraulic conductivity estimates. However, this relationship is clearly non-linear, leading to poor performance of models based on linear correlations, such as co-kriging. To address this limitation, we model the joint distribution of these variables using nonparametric Bernstein copulas within a bivariate cosimulation framework, where hydraulic head is incorporated as a supporting variable to improve the estimation of hydraulic conductivity. However, copula-based approaches often produce non-unique solutions, requiring selection criteria. Common strategies based on the mean or median produce overly smooth realizations that fail to preserve the aquifer's natural variability. To overcome these limitations, we propose a tri-objective function that integrates spatial structure, interpolation accuracy, and distributional consistency. The first term minimizes the discrepancy between reference and simulated experimental variograms, the second minimizes the interpolation error at known locations, and the third incorporates the Wasserstein distance to enforce distributional similarity. To further capture spatial dependence, the copula is estimated locally within defined neighborhoods. The tri-objective function is optimized using a two-stage genetic algorithm. To evaluate our proposal, we compared our approach against the mean and median reference values derived from Bernstein copula‐based spatial stochastic cosimulation. Our method achieved substantially better performance across all metrics: Wasserstein distance of 0.031 compared to 0.417 (mean) and 0.344 (median); interpolation NRMSE of 0.131 compared to 0.529 (mean) and 0.484 (median); and Pearson correlation of 0.991 compared to 0.856 (mean) and 0.882 (median). These results suggest that the proposed framework yields a more faithful representation of aquifer heterogeneity by optimizing spatial structure, point accuracy, and distributional fidelity.

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