Forward modeling is a powerful tool in the design and simulation of electrical imaging surveys. Forward modeling codes are commonly based on finite-difference or finite-element solutions of the Poisson equation (Equation 1 for ER only and Equation 5 for ER and IP). Forward-modeling codes can simulate “synthetic data” for different spatial distributions of electrical conductivity (and phase or intrinsic chargeability for IP), survey geometries, and random noise levels for a given discretization of Equation 1 or 5 (Figure 7a). Common software packages allow users to set the size of finite-difference cells or finite elements. These predicted data can be inverted to generate tomograms (Figure 7b and Section 4). Such exercises provide insight into the resolving power of different survey geometries, noise levels, and inversion approaches.
Forward-model accuracy is thus another factor to consider in designing survey geometries. We should eliminate quadripoles that are expected to have large modeling errors from our measurement sequence. We can quantitatively evaluate the forward-model accuracy by modeling the apparent resistivity for all candidate quadripoles assuming a homogeneous Earth (i.e., a single conductivity) and comparing them to the apparent resistivities based on the geometric factors calculated analytically above. Quadripoles that do not model well (i.e., the apparent resistivity calculated numerically differs substantially from that computed analytically) should not be collected in the field, or the discretization of the mesh and the location of the boundary conditions should be refined if problems exist or too many data are eliminated in this manner. This exercise can only be performed in the simple case where an analytical solution to Equation 1 exists. For example, in the presence of a heterogenous subsurface or topography, analytical solutions are not generally available, and assessment of numerical model accuracy is cumbersome for thousands of quadripoles. Hence test criteria are based on simple analytical models, assumptions of homogeneity, or criteria based on experience. As a rule of thumb, grid spacing near electrodes where potential gradients are large should be finer than one quarter of the electrode spacing. Grid spacing can be coarser further from electrodes where voltage gradients are smaller. Commonly, grid spacing is increased by a factor of less than 1.5 from one grid row or column to the next, deeper or neighboring row in finite-difference models. In finite-element models, unstructured meshes are typically refined about the electrodes with similar discretization.
In the example of Figure 7, a cross-well ER survey is conducted for a cross section containing a single 25-cm fracture zone and no other heterogeneity. Assuming a low-noise dataset, 2 percent random, normally distributed errors are added to the data to introduce noise as might be expected in the field. The resulting tomogram provides only a blurry and blunted image of the true electrical conductivity distribution, and interpretation of the location and extent of the fracture zone is complicated by the limited resolution. If another heterogeneity existed in the cross section (for example, lithologic or porosity variation), or if the fracture zone was a small discrete fracture (perhaps 2.5 mm instead of 25 cm), it might not be possible to identify the fracture at all. By considering different input models, or targets, it is also possible to gain insight into how resolution varies spatially over a tomogram. Indeed, conducting such synthetic modeling exercises prior to field surveys represents a best practice. Many ER modeling and inversion software packages can be used for this purpose. Open-source and free solutions for testing scenarios include:
- the United States Geological Survey spreadsheet-based Scenario Evaluator for Electrical Resistivity (SEER), which approximates the inversion of ER data and is particularly aimed at the non-expert (see Figure 8 and also Terry et al., 2017); and,
- ResIPy (Blanchy et al., 2020), an intuitive graphical user interface for the family of ER and IP inversion codes written by Andrew Binley (Lancaster University).
Box 1 of this book provides an opportunity for readers to experiment with the electrical resistivity modeling software SEER and explore the impact of some survey design parameters on the resulting electrical image.