3-D forward modelling of controlled-source frequency-domain electromagnetic data using the meshless generalized finite-difference method

Abstract

SUMMARY This paper proposes a procedure of forward modelling 3-D frequency-domain wire-source electromagnetic data using the meshless generalized finite-difference (MGFD) method. This method is based on Taylor series expansions and the weighted least-squares method, and its basic principle is to express the partial derivatives of the unknown function on a particular central point by a linear combination of function values on the adjacent points. The advantages of the method over mainstream forward-modelling methods, for example, the regular finite-difference (FD) method, or the finite-element (FE) method, is that mesh generation is not needed: a discretization in the form of just points is applied instead. This allows the points to be distributed freely to fit the arbitrary shape of the structures in the model, which is helpful in the modelling of complex earth structures. It makes the MGFD method more suitable to deal with complex model than FD method. Also, unlike that in the FE method, interpolation functions are not required and no integral needs to be calculated in MGFD method. This results in high computational efficiency and a concise forward-modelling process. In this paper, the particulars of the MGFD method are introduced, the discretized MGFD system of equations (for an ${\boldsymbol{A}} - {\rm{\ }}\varphi $ potential decomposition of the fields, with the Coulomb gauge condition enforced and a primary–secondary separation approach to deal with the singularity of the source) are solved using a direct solver, and the forward-modelling code are programmed. To test the method and code, we compare the MGFD solutions for three 3-D earth models with the equivalent solutions calculated by other methods, and verify the correctness of the MGFD solution by the good agreement between the corresponding results (with relative error of the electric field ${{\boldsymbol{E}}}_{\boldsymbol{x}}$ smaller than 4.89 per cent). We also investigate the performance of this method when applying different discretizations of points, and when using different weighting functions, to assess the influence of these two factors on the forward-modelling accuracy and efficiency. Results indicate that denser point distributions and straightforward weighting functions result in better accuracy and efficiency.