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

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.