The Use of a Semi-Structured Finite-Element Mesh in 3-D Resistivity Inversion

Abstract

Calculating the electric potentials 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 the 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 to 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, that 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.

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

“…There is also an interesting diffused low resistivity linear anomaly below the mine shaft which could be a mined-out mineralized sheet. This anomaly was less clear in a previous inversion using a subset of the data set (Loke et al, 2021).…”

“…Loke et al. (2021) used a subset of the data set with 418,331 electrode positions and 207,997 data points from the central part of the survey area with more uniform data coverage to test the semi‐structured mesh method. In this paper, we used the entire data set that has significant areas without data coverage at the corners and sides of the model grid (Figure 10).…”

“…The limitation of the rectangular mesh is that the same fine mesh is used in the deeper levels where the potentials vary gradually. Figure 1b shows a semi‐structured mesh (Loke et al., 2021) with a coarser mesh at the deeper levels using triangular elements to connect levels with a different number of nodes. For most data sets, the semi‐structured 2‐D mesh reduces the number of nodes by about half and the calculation time by about four times.…”

“…This section describes in more detail the semi‐structured mesh technique introduced by Loke et al. (2021). Figure 2a shows three mesh levels in a 3‐D structured rectangular mesh at different depths.…”

“…We also used the wavelet compression method to reduce the computer memory required by the Jacobian matrix and the time to solve the least-squares equation (Davies & Li, 2013;Loke et al, 2020). Loke et al (2021) used a subset of the data set with 418,331 electrode positions and 207,997 data points from the central part of the survey area with more uniform data coverage to test the semi-structured mesh method. In this paper, we used the entire data set that has significant areas without data coverage at the corners and sides of the model grid (Figure 10).…”