Manifold embedding in geostatistical inversion: Redefining optimality in subsurface characterization

Optimality in geostatistical inversion is commonly defined within a Euclidean framework, often oversimplifying complex and nonstationary geological features. Consequently, conventionally derived “optimal” models may fail to adequately represent geological reality. To address this limitation, we incorporate manifold embedding—a non-Euclidean approach designed to capture the true complexity and heterogeneity of geological structures—directly into the geostatistical inversion process. Using both Kalman filtering (KF) and a geostatistical principal component adaptation evolution strategy (GPCA-ES), established but conventional inverse methods, we compare Euclidean and manifold-based frameworks for estimating hydraulic conductivity in a synthetic aquifer. Our results illustrate that while Euclidean-based methods typically yield a single “optimal” solution, manifold embedding provides a broader “spectrum” of geologically reasonable models, each producing similarly accurate hydraulic responses at observation points. These findings challenge the traditional assumption of uniqueness and highlight the importance of adopting geometric perspectives beyond conventional Euclidean metrics. Rather than seeking incremental numerical improvements in inverse algorithms, this study embodies a fundamental conceptual departure, demonstrating that Euclidean geometry represents merely a special case within a broader class of more realistic, structurally informed non-Euclidean frameworks. By replacing the quest for a single best-fit model—defined by conventional metrics such as minimal RMSE—with a structurally faithful spectrum of plausible solutions, we redefine optimality in hydrogeological modelling under realistic data constraints.