Hydrogeological Insights Through Complex Kriging of Groundwater Head Gradients

Understanding groundwater head gradients is crucial for hydrogeology and groundwater management. Together with hydraulic conductivity, it provides valuable insights into the speed and direction of groundwater flow within an aquifer, which is important for various applications such as water resource management, contamination assessment, and aquifer characterization. Complex kriging is designed to estimate complex-valued random fields and has found novel applications in various domains, including the estimation of vectorial datasets in two-dimensional space, such as wind velocities and ocean currents. In a pioneering effort, the capability of complex kriging is employed in the present work to estimate groundwater gradients for the years 2002 and 2007, focusing on the upper aquifer of the southern part of the Basin of Mexico aquifer system, which is characterized by high water stress. This study presents a comparative analysis of two kriging methodologies for estimating hydraulic head gradients. The first method, referred to as the "traditional kriging approach," uses ordinary kriging to estimate hydraulic head values. Subsequently, hydraulic gradients are derived from these values by computing slopes in both the x and y directions. In contrast, the second method, termed the "complex kriging approach," directly uses hydraulic head data to compute gradients. This is achieved through a triangulation process based on the spatial positions of the data, utilizing the Delaunay triangulation method. In this approach, each triangle's hydraulic head gradient is approximated by the gradient of the plane that passes through the hydraulic head values at each vertex of the triangle. The resulting hydraulic gradients are then assigned to the triangle centroids and represented as vectors. Complex ordinary kriging is applied to analyze these vectors and generate vectorial estimates across the same grid utilized in the traditional approach. Overall, the empirical evidence demonstrates that the use of the complex approach improves the estimate's reliability compared to the traditional one. Indeed, the goodness-of-fit of the complex covariance model for assessing the spatial correlation of the investigated vectorial components is confirmed by the statistics results, expressed in terms of Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) between the actual values and the estimated ones for the two components.