Study On 3D Resistivity Modeling Based On Multi-grid Method

Since the multi-grid method proposed in the 1960 s, after 40 years of research and development, has a complete theory system, algorithm step by step toward maturity, is one of the algorithm, considered to be one of the efficient algorithms in the field of numerical calculation. With its fast convergence speed, higher calculation precision applied in the field of exploration geophysics. In recent years, with the realization of the dc resistivity of 3D data acquisition, large-scale resistivity modeling and inversion calculation become an urgent need to solve the problem. In this paper, the multi-grid method and finite difference method combined to achieve a three-dimensional resistivity forward modeling.In this paper, starting from the basic idea of multi-grid method, this paper introduces the multi-grid method‘s calculation steps and derive the theoretical formula of the V cycle under two layers of grid. Based on the basic theory of dc resistivity method, derived calculation formula of point source in 3D steady electric field, and combined with the boundary conditions, using the finite difference method get the discrete equations under non-uniform subdivision border coefficient equations of each node. Then apply multi-grid method to limit and interpolation the discrete equations, according to the smooth iteration to get linear equations, to solve the equation can get each node voltage value. Validated from three aspects: 1) Compared with the other iteration method by solving one-dimensional Poisson equation; 2) Solving three dimensional elliptic partial differential equations from different angles; 3) Validation by geo-electric model of two layers of level; The results show that the multi-grid method has some characteristics: the number of iterations does not increase with the increase of computing scale, convergence speed is fast, high precision, need less number of iterations, etc.Finally this paper simulate the Wenner device and Dipole-dipole device under the three dimensional geologic body profile response characteristics, through calculation shows that the line direction, the trend of the abnormal body length and other factors have serious influence on 2D resistivity profile data. When designing a 2D measuring line, as far as possible with abnormal body to vertical, at the same time the trend of abnormal body have the enough length; otherwise consider to acquisition three-dimensional resistivity and inversion.