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This paper describes an interdisciplinary approach to geometry modeling of geospatial boundaries. The objective is to extract surfaces from irregular spatial patterns using differential geometry and obtain coherent directional predictions along the boundary of extracted surfaces to enable more targeted sampling and exploration. Specific difficulties of the data include sparsity, incompleteness, causality and resolution disparity. Surface slopes are estimated using only sparse samples from cross-sections within a geological domain with no other information at intermediate depths. From boundary detection to subsurface reconstruction, processes are automated in between. The key problems to be solved are boundary extraction, region correspondence and propagation of the boundaries via contour morphing. Established techniques from computational physics, computer vision and signal processing are used with appropriate modifications to address challenges in each area. To facilitate boundary extraction, an edge map synthesis procedure is presented. This works with connected component analysis, anisotropic diffusion and active contours to convert unordered points into regularized boundaries. For region correspondence, component relationships are handled via graphical decomposition. FFT-based spatial alignment strategies are used in region merging and splitting scenarios. Shape changes between aligned regions are described by contour metamorphosis. Specifically, local spatial deformation is modeled by PDE and computed using level-set methods. Directional predictions are obtained using particle trajectories by following the evolving boundary. However, when a branching point is encountered, particles may lose track of the wavefront. To overcome this, a curvelet backtracking algorithm has been proposed to recover information for boundary segments without particle coverage to minimize shape distortion.