3D CSEM modeling and inversion algorithms for a surface‐to‐borehole survey

Abstract

Summary We have developed efficient three-dimensional controlledsource electromagnetic modeling and inversion algorithms for a frequency-domain surface-to-borehole survey. In the forward modeling, Maxwell’s equations of the secondary electric field are discretized using edge-based finite elements, which can exclude spurious solutions caused in the general node-based finite element method. The linear system of equations for the secondary field is solved using a direct solver specialized to a sparse matrix while the primary field is calculated analytically for a background layered-earth model. The inversion algorithm which is based on a regularized Gauss-Newton method calculates Jacobians using an adjoint equation approach and exploits the Active Constraint Balancing scheme and the logarithmic parameter to stabilize the inversion process. In order to reduce the computation time, we adopt OpenMP for solving the huge matrix as well as the Message Passing Interface parallelization for the frequency. The forward calculation results are validated by comparison with integral equation solutions for various types of the source. The direct solver has shown the superior efficiency to iterative solvers, especially in the calculation of multisource problems. The calculation of Jacobians using an adjoint equation approach and solving the linear system of equations using a direct solver suit the configurations of three-dimensional surface-to-borehole surveys which are usually performed with multiple sources and limited receiver positions.