Manifold Embedding Based on Geodesic Distance for Non-stationary Subsurface Characterization Using Secondary Information

In geological characterization, the traditional methods that rely on the covariance matrix for continuous variable estimation often either neglect or oversimplify the challenge posed by subsurface non-stationarity. This study presents an innovative methodology using ancillary data such as geological insights and geophysical exploration to address this challenge directly, with the goal of accurately delineating the spatial distribution of subsurface petrophysical properties, especially, in large geological fields where non-stationarity is prevalent. This methodology is based on the geodesic distance on an embedded manifold and is complemented by the level-set curve as a key tool for relating the observed geological structures to intrinsic geological non-stationarity. During validation, parameters ρ and β were revealed to be the critical parameters that influenced the strength and dependence of the estimated spatial variables on secondary data, respectively. Comparative evaluations showed that our approach performed better than a traditional method (i.e., kriging), particularly, in accurately representing the complex and realistic subsurface structures. The proposed method offers improved accuracy, which is essential for high-stakes applications such as contaminant remediation and underground repository design. This study focused primarily on two-dimensional models. There is a need for three-dimensional advancements and evaluations across diverse geological structures. Overall, this research presents novel strategies for estimating non-stationary geologic media, setting the stage for improved exploration of subsurface characterization in the future.