The use of a semi‐structured finite‐element mesh in 3‐D resistivity inversion

Abstract

Calculating the electric potential for 3‐D resistivity inversion algorithms can be time consuming depending on the structure of the mesh. There have been generally two approaches to generating finite‐element meshes. One approach uses a structured rectangular mesh with hexahedral elements on a rectangular model grid. The distribution of model cells can be designed to follow known boundaries, and directional roughness constraints can be easily imposed. A 1‐D wavelet transform that takes advantage of the regular arrangement of the model cells can also be used to reduce the computer time and memory required to solve the smoothness‐constrained least‐squares equation. However, the structured rectangular mesh uses an unnecessarily fine mesh in parts of the model that are far away from the electrodes where the potential changes gradually. A second approach uses an unstructured mesh with tetrahedral elements created automatically by a mesh generation program with finer elements nearer the electrodes and coarser elements in the more remote regions. This generates a mesh with a much smaller number of nodes. The disadvantage is that an irregular model grid is normally used. We examine an alternative approach that combines structured and unstructured meshes. We employ a regular model grid with a finer mesh near the surface and a coarser mesh in deeper regions using a combination of hexahedral and tetrahedral elements. The semi‐structured mesh reduces the calculation time by more than three times compared with a structured mesh. An adaptive semi‐structured mesh that also uses a coarser mesh for model cells near the surface if they are more than one unit electrode spacing from the nearest electrode was also developed for surveys with non‐uniform data coverage. For the Bonsall Leys field survey, which used a capacitively coupled mobile system and collected a data set with nearly a million electrode positions, the adaptive mesh reduces the calculation time by about 80%. The calculation time can be further reduced by about 93% when it is combined with a mesh segmentation method.

Forward and inversion approach for direct current resistivity based on an unstructured mesh and its application to tunnel engineering

“…In the DC resistivity field, Loke et al. (2022) used a combination of hexahedral and tetrahedral meshes to achieve three‐dimensional resistivity inversion and successfully applied it, using finer meshes near the surface and coarser meshes in deeper regions, which largely improved the computational efficiency.…”

“…To solve the problem of the complex and computationally intensive topography of unstructured mesh simulations, Meng et al (2020) combined an unstructured mesh with a structured mesh in gravity inversion to efficiently obtain the density features of ores. In the DC resistivity field, Loke et al (2022) used a combination of hexahedral and tetrahedral meshes to achieve three-dimensional resistivity inversion and successfully applied it, using finer meshes near the surface and coarser meshes in deeper regions, which largely improved the computational efficiency.…”