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.

翻译：本文提出了一种针对裂缝介质的稀疏到稠密重建层，该层将稀疏点测量映射到由三维迹线多边形推断出的分段平面裂缝支撑上。每个平面在局部坐标系中离散化，并通过一个凸目标函数进行估计，该函数将网格SPDE/GMRF二次先验与对未抽取离散曲线波系数的$\ell_1$惩罚项相结合，旨在捕捉各向同性平滑性难以充分表征的各向异性、与裂缝对齐的结构。我们进一步通过迹线网络测地线定义了沿裂缝距离，并将连通性驱动的正则化表达为二次型$z^\top P^\top L_G P z$，其中$L_G$是迹线网络上的图拉普拉斯矩阵，$P$将平面网格映射到图节点；平面交线通过沿交线采样的线性一致性约束处理。所得优化问题允许高效分裂求解：二次项块采用稀疏线性求解，曲线波块采用系数收缩，在凸性条件下具有标准ADMM收敛性。我们制定了可复现的合成基准测试、基线方法、消融实验和敏感性研究，以分离方向稀疏性与连通性的影响，并提供了生成图表和量化表格的参考代码。