Multi-level error propagation of distance models in Gaussian random field generation over manifolds

Abstract

This study examines the practical consequence of distance model selection in Gaussian random field generation over manifolds. An exponential correlation model is considered on three representative surfaces: a half-cylinder, a hemisphere, and a saddle surface. For each manifold, three distance models are compared: the reference geodesic distance computed on the triangulated surface using the MMP algorithm, the approximate geodesic distance obtained by the heat method, and the Euclidean distance computed from Cartesian coordinates. Errors are evaluated successively at the distance matrix, covariance matrix, and random field levels. The results show that higher distance-matrix accuracy does not necessarily lead to lower field-level discrepancy. Although the heat-method distance is consistently closer to the reference geodesic distance than the Euclidean distance, this geometric advantage is not preserved after covariance construction and field generation. In all investigated cases, the Euclidean distance model produces smaller field-level discrepancies and requires the lowest computational cost. These results suggest that distance model selection for manifold based random field simulation should be based on the propagated effect on the covariance structure and generated random fields, rather than on geometric distance accuracy alone.