Anisotropic {Bending} {Energy} for {Geological} {Data} {Interpolation}.

Ayoub Belhachmi. ( 2026 )
in: 2026 {RING} meeting, pages 85, ASGA

Abstract

Regularization energies are essential in implicit geological modeling, where a scalar function is used to interpolate sparse and uncertain geological observations. However, conventional regularization energies are generally isotropic, which can make them inadequate for geological structures exhibiting strong thickness variations and may lead to strong oscillations. We introduce an auto-adaptive anisotropic regularization energy based on the bending equation of an anisotropic thin plate. In this framework, the implicit function is the solution of a stochastic thin-plate bending equation. The proposed method generalizes classical bending regularization through a parameterized rigidity tensor, constrained to preserve convexity and updated iteratively in order to capture and encode local geological anisotropy. This PDE-based energy formulation also opens the way to future stochastic extensions, in which uncertainties in the observations could be incorporated to quantify geological uncertainty and generate ensembles of plausible implicit geological models rather than a single deterministic solution. The energy is discretized on tetrahedral meshes conforming to geological discontinuities. While C1 elements, such as Powell–Sabin elements, allow a direct discretization of the bending energy in two dimensions, no standard C1 construction is available on general tetrahedral meshes. This motivates a mixed finite element formulation based on standard C0 approximation spaces. In particular, we introduce a quadratic mixed discretization, which provides a richer approximation than linear elements and reduces the sensitivity of the solution to mesh quality. Applications to two- and three-dimensional geological models with strong anisotropy and complex fault networks show that the proposed approach yields more geologically consistent implicit models than isotropic regularization, exhibiting higher smoothness and regularity.

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BibTeX Reference

@inproceedings{Belhachmi2026RM2,
 abstract = {Regularization energies are essential in implicit geological modeling, where a scalar function is used to interpolate sparse and uncertain geological observations. However, conventional regularization energies are generally isotropic, which can make them inadequate for geological structures exhibiting strong thickness variations and may lead to strong oscillations. We introduce an auto-adaptive anisotropic regularization energy based on the bending equation of an anisotropic thin plate. In this framework, the implicit function is the solution of a stochastic thin-plate bending equation. The proposed method generalizes classical bending regularization through a parameterized rigidity tensor, constrained to preserve convexity and updated iteratively in order to capture and encode local geological anisotropy. This PDE-based energy formulation also opens the way to future stochastic extensions, in which uncertainties in the observations could be incorporated to quantify geological uncertainty and generate ensembles of plausible implicit geological models rather than a single deterministic solution. The energy is discretized on tetrahedral meshes conforming to geological discontinuities. While C1 elements, such as Powell–Sabin elements, allow a direct discretization of the bending energy in two dimensions, no standard C1 construction is available on general tetrahedral meshes. This motivates a mixed finite element formulation based on standard C0 approximation spaces. In particular, we introduce a quadratic mixed discretization, which provides a richer approximation than linear elements and reduces the sensitivity of the solution to mesh quality. Applications to two- and three-dimensional geological models with strong anisotropy and complex fault networks show that the proposed approach yields more geologically consistent implicit models than isotropic regularization, exhibiting higher smoothness and regularity.},
 author = {Belhachmi, Ayoub},
 booktitle = {2026 {RING} meeting},
 language = {en},
 pages = {85},
 publisher = {ASGA},
 title = {Anisotropic {Bending} {Energy} for {Geological} {Data} {Interpolation}.},
 year = {2026}
}