Manifold embedding based on geodesic distance for nonstationary spatial estimation in higher dimensions

Advances in geostatistical modeling are critical for accurate subsurface characterization, a domain where traditional methods often fall short due to the inherent complexity of geological media. This study presents a manifold embedding method extended to three-dimensional space, revolutionizing the representation of complex, nonstationary spatial correlations. The proposed method incorporates an additional dimension to account for variable spatial properties, allowing for a more detailed and accurate representation of subsurface heterogeneity. Although direct numerical comparisons with traditional two-point geostatistics are not possible due to the enhanced data integration of manifold embedding, the method clearly surpasses traditional approaches by allowing the incorporation of complex structural information. This transformative approach, while computationally demanding and reliant on extensive secondary data, proves particularly valuable for geological repository design and contaminant remediation. These results underscore manifold embedding as an invaluable tool in hydrogeology and geostatistics, setting new standards for high-precision subsurface modeling. The capability of the developed method to finely resolve spatial correlations makes it an indispensable tool for hydrogeological exploration and environmental management.