Curvelet-Regularized SPDE Inversion on Piecewise-Planar Fractures with Trace-Graph Coupling

TL;DR

This paper introduces a novel regularization framework combining SPDE/GMRF priors with curvelet sparsity for reconstructing fractured media from sparse measurements, enabling efficient and accurate piecewise-planar fracture modeling.

Contribution

It develops a convex optimization method integrating trace-graph coupling, anisotropic curvelet sparsity, and connectivity regularization for fracture reconstruction from limited data.

Findings

Effective sparse-to-dense reconstruction of fractured media.

Demonstrated efficiency and convergence of the ADMM-based optimization.

Provided comprehensive benchmarks and reproducible code for validation.

Abstract

We formulate a sparse-to-dense reconstruction layer for fractured media in which sparse point measurements are mapped onto piecewise-planar fracture supports inferred from 3D trace polylines. Each plane is discretized in local coordinates and estimated via a convex objective that combines a grid SPDE/GMRF quadratic prior with an $\ell_1$ penalty on undecimated discrete curvelet coefficients, targeting anisotropic, fracture-aligned structure that is poorly represented by isotropic smoothness alone. We further define an along-fracture distance through trace-network geodesics and express connectivity-driven regularization as a quadratic form $z^\top P^\top L_G P z$, where $L_G$ is a graph Laplacian on the trace network and $P$ maps plane grids to graph nodes; plane intersections are handled by linear consistency constraints sampled along intersection lines. The resulting optimization admits efficient splitting: sparse linear solves for the quadratic block and coefficient-wise shrinkage for the curvelet block, with standard ADMM convergence under convexity. We specify reproducible synthetic benchmarks, baselines, ablations, and sensitivity studies that isolate directional sparsity and connectivity effects, and provide reference code to generate the figures and quantitative tables.