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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 calculat

New developments in mobile resistivity meter instrumentation have made it possible to survey large areas with dense data coverage. The mobile system usually has a limited number of electrodes attached to a cable that is pulled along behind an operator so that a large area can be covered within a short time. Such surveys can produce three‐dimensional datasets with hundreds of thousands of electrodes positions and data points. Similarly, the inverse model used to interpret the data can have several hundred thousand cells. It is impractical to model such large datasets within a reasonable time on microcomputers used by many small companies employing standard inversion techniques. We describe a model segmentation technique that subdivides the finite‐element mesh used to calculate the apparent resistivity and Jacobian matrix values into a number of smaller meshes. A fast technique that optimizes the calculation of the Jacobian matrix values for multi‐channel systems was also developed. A one‐dimensional wavelet transform method was then used to compress the storage of the Jacobian matrix, in turn reducing the computer time and memory required to solve the least‐squares optimization equation to determine the inverse model resistivity values. The new techniques reduce the calculation time and memory required by more than 80% while producing models that differ by less than 1% from that obtained using the standard inversion technique with a single mesh. We present results using a synthetic model and a field dataset that illustrates the effectiveness of the proposed techniques.

Resistivity monitoring surveys are used to detect temporal changes in the subsurface using repeated measurements over the same site. The positions of the electrodes are typically measured at the start of the survey program and possibly at occasional later times. In areas with unstable ground, such as landslide‐prone slopes, the positions of the electrodes can be displaced by ground movements. If this occurs at times when the positions of the electrodes are not directly measured, they have to be estimated. This can be done by interpolation or, as in recent developments, from the resistivity data using new inverse methods. The smoothness‐constrained least squares optimisation method can be modified to include the electrode positions as additional unknown parameters. The Jacobian matrices with the sensitivity of the apparent resistivity measurements to changes in the electrode positions are then required by the optimisation method. In this paper, a fast adjoint‐equation method is used to calculate the Jacobian matrices required by the least squares method to reduce the calculation time. In areas with large near‐surface resistivity contrasts, the inversion routine sometimes cannot accurately distinguish between electrode displacements and subsurface resistivity variations. To overcome this problem, the model for the initial time‐lapse dataset (with accurately known electrode positions) is used as the starting model for the inversion of the later‐time dataset. This greatly improves the accuracy of the estimated electrode positions compared to the use of a homogeneous half‐space starting model. In areas where the movement of the electrodes is expected to occur in a fixed direction, the method of transformations can be used to include this information as an additional constraint in the optimisation routine.

Adaptive optimal experimental design methods use previous data and results to guide the choice and design of future experiments. This paper describes the formulation of an adaptive survey design technique to produce optimal resistivity imaging surveys for time-lapse geoelectrical monitoring experiments. These survey designs are time-dependent and, compared to dipole-dipole or static optimized surveys that do not change over time, focus a greater degree of the image resolution on regions of the subsurface that are actively changing. The adaptive optimization method is validated using a controlled laboratory monitoring experiment comprising a well-defined cylindrical target moving along a trajectory that changes its depth and lateral position. The algorithm is implemented on a standard PC in conjunction with a modified automated multichannel resistivity imaging system. Data acquisition using the adaptive survey designs requires no more time or power than with comparable standard surveys, and the algorithm processing takes place while the system batteries recharge. The results show that adaptively designed optimal surveys yield a quantitative increase in image quality over and above that produced by using standard dipole-dipole or static (time-independent) optimized surveys

Adaptive optimal experimental design methods use previous data and results to guide the choice and design of future experiments. This paper describes the formulation of an adaptive survey design technique to produce optimal resistivity imaging surveys for time-lapse geoelectrical monitoring experiments. These survey designs are time-dependent and, compared to dipole–dipole or static optimized surveys that do not change over time, focus a greater degree of the image resolution on regions of the subsurface that are actively changing. The adaptive optimization method is validated using a controlled laboratory monitoring experiment comprising a well-defined cylindrical target moving along a trajectory that changes its depth and lateral position. The algorithm is implemented on a standard PC in conjunction with a modified automated multichannel resistivity imaging system. Data acquisition using the adaptive survey designs requires no more time or power than with comparable standard surveys, and the algorithm processing takes place while the system batteries recharge. The results show that adaptively designed optimal surveys yield a quantitative increase in image quality over and above that produced by using standard dipole–dipole or static (time–independent) optimized surveys.

The ‘Compare R’ method is used to automatically generate optimized arrays for two-dimensional resistivity surveys with electrodes arranged along parallel lines on the surface and the subsurface. The resolution at depth is greatly improved by carrying out measurements with at least one line of electrodes below the surface using a direct push installation technique. The performance of the optimized arrays is compared with a standard measurement sequence created manually that was used for previous surveys. Tests were conducted using a synthetic model and a field survey in an area with karst geology that has some ground truth. Both tests show that the optimized arrays have significantly better resolution compared with a standard measurement sequence. We also show that artefacts in the inversion model can be reduced by using higher damping factors near the positions of the subsurface electrodes.