Smooth Reconstruction Of Complex Geological Surface Based On Partial Differential Equations

Geological surface reconstruction is the important basis for three-dimensional modeling of the geological structure and has important applications in geological and mineral resources exploration and other fields. Geological surface reconstruction methods are commonly divided into interpolation and fitting method, including the triangulation method, Kriging interpolation method and the weighted least-squares fitting method and so on. However, due to the specificities of the geological conditions, geological exploration data is sparse, uneven and contains a large amount of geological faults, including normal faults, vertical faults and reverse faults, etc., the above methods have certain limitations in complex geological surface reconstruction. On this account, the application of surface modeling method with geometric partial differential equation(PDE) in computational geometry in complex geological surface smooth reconstruction was researched and a complex geological surface reconstruction method with geometric PDE was proposed. The method firstly selects a suitable PDE, discretizes the geometry differential operators involved, and discusses the stability of the discrete format. On this basis, build the spatial topology of geological data in a work area which to meet the discrete format of selected PDE and the geological faults are treated as constraint boundary conditions. By adopting the idea of evolution to solve the selected geometric PDE iteratively, the steady-state solution of the discrete equation is regarded as an approximate solution of the PDE to fit the original surface. A smooth geological surface containing complex fault constraints expressed with discrete meshes is constructed.In this paper, the problem of smooth surface reconstruction with complex geological fault constraints was studied. With the surface modeling technology of geometric PDEs, the smooth reconstruction of a geological surface with complex fault constraints was achieved. Details are as follows:The selected PDE was discretized on a rectangular grid to construct corresponding difference equation and its stability was analyzed. By solving the difference equation with finite difference method iteratively, the steady-state solution of the discrete equation is approximated as the solution of the corresponding partial differential equation to fit the original surface, a complex geological surface expressed with rectangular grids is reconstructed. The method has some advantages, such as fast computation and easy computer programming. The disadvantages are that the rectangular grids cannot adapt to the arbitrary topology of geological fault polygons, which leads to that the reconstructed complex geological surface is not smooth enough at fault constraints.In order to adapt the arbitrary topology of geological fault polygons, triangular mesh topology of the geological sampling data was built. The spatial scattered data was projected to a two-dimensional plane and the fault constraints are joined to do Delaunay triangle subdivision with constraints, and then map the planar triangulation to three-dimensional space to realize the spatial topology. By constructing the discrete formats of differential operators on the triangulation and solving selected partial differential equation discretely, a smooth geological surface with complex fault constraints expressed by triangulation was remodeled. This method based on triangulation solved the problem of smooth surface reconstruction at fault constraints. But the stability conditions of discrete formats of the differential operators are harsh and the calculation speed is relatively slow.Combined with advantages of partial differential equation surface reconstruction methods on rectangular grids and triangular meshes, a method for complex geological surface smooth reconstruction expressed by hybrid grids was proposed. At the non-fault regions, by solving geometric partial differential equation with finite difference method on rectangular grids iteratively, the reconstructed surface achieved the specified smoothness quickly. While at the fault areas, solve selected partial differential equation on triangulation network to reconstruct the smooth surface that match to fault topology. This approach realized the rapid reconstruction of geological surface with complicated constraints, while maintaining the characteristics of the geological faults.With the actual exploration data, three kinds of geological surface reconstruction method proposed in this paper are verified and analyzed. The application examples show that these methods consider the various fault constraints in geology adequately and can reconstruct smooth geological surface containing complicated faults quickly. The complex geological surface reconstruction method with geometric PDE is expected to further research and application in geological field.