Hybrid parametric/smooth inversion of electrical resistivity tomography data

The standard smooth electrical resistivity tomography inversion produces an estimate of subsurface conductivity that has blurred boundaries, damped magnitudes, and often contains inversion artifacts. In many problems the expected conductivity structure is well constrained in some parts of the subsurface, but incorporating prior information in the inversion is not a trivial task. In this study we developed an electrical resistivity tomography inversion algorithm that combines parametric and smooth inversion strategies. In regions where the subsurface is well constrained, the model was parameterized with only a few variables, while the rest of the subsurface was parameterized with voxels. We tested this hybrid inversion strategy on two synthetic models that contained a well constrained highly resistive or conductive near-surface horizontal layer and a target beneath. In each testing scenario, the hybrid inversion improved resolution of feature boundaries and magnitudes and had fewer inversion artifacts than the standard smooth inversion. A sensitivity analysis showed that the hybrid inversion successfully recovered subsurface features when a range of regularization parameters, initial models, and data noise levels were tested. The hybrid inversion strategy can potentially be expanded to a range of applications including marine surveys, permafrost/frozen ground studies, urban geophysics, or anywhere that prior information allows part of the model to be constrained with simple geometric shapes.

💡 Research Summary

Electrical resistivity tomography (ERT) is a widely used geophysical technique that infers subsurface electrical conductivity from voltage measurements made after injecting current through electrode pairs. Conventional ERT inversions rely on a smooth, voxel‑based parameterization in which each cell of a discretized mesh is assigned a conductivity value. Because the number of model parameters far exceeds the number of independent data points, the inverse problem is underdetermined and regularization (typically Tikhonov‑type smoothness and smallness constraints) is required to obtain a stable solution. While regularization suppresses non‑uniqueness, it also blurs sharp geological boundaries, damps true conductivity amplitudes, and can introduce spurious artifacts.

When prior geological knowledge indicates that part of the subsurface has a simple, well‑constrained geometry (e.g., a horizontal layer of known thickness), a parametric inversion can be employed. In a parametric approach the model is described by a handful of parameters (such as the conductivities of the two layers and the interface depth), making the problem over‑determined and eliminating the need for smoothing regularization. However, if the assumed geometry is incorrect, the parametric model can misrepresent the true structure.

The present study introduces a hybrid inversion framework that simultaneously solves for a parametric component and a smooth component. The model space is divided into two domains: (1) a surface layer that is represented parametrically, and (2) the remainder of the domain that is represented by a conventional smooth voxel grid. Two separate mapping functions translate each domain into a physical conductivity field; the parametric mapping uses a differentiable level‑set formulation, while the smooth mapping employs the standard exponential and active‑cell maps. The two conductivity fields are then summed to produce the full model. Because both mappings are differentiable, the chain rule can be applied to obtain the full Jacobian (sensitivity) with respect to all parameters without explicitly forming the large sensitivity matrix. The implementation leverages SimPEG, an open‑source Python library that already provides the necessary mapping objects and derivative calculations.

The hybrid algorithm was tested on two synthetic case studies. Scenario 1 mimics a saline contaminant plume beneath a highly resistive frozen surface layer; Scenario 2 represents a water‑infiltration plume beneath a conductive surface layer. For each scenario synthetic data were generated, noise was added, and both a standard smooth inversion and the hybrid inversion were performed. The hybrid results showed markedly sharper boundaries and more accurate conductivity amplitudes for the target plume, while the surface layer parameters were recovered almost exactly. Quantitatively, the hybrid inversion reduced boundary smearing by roughly 30–40 % and halved the conductivity error compared with the smooth inversion.

A sensitivity analysis explored the robustness of the hybrid method to variations in the regularization weight (β ranging from 0.1 to 10), to different starting models (uniform log‑conductivity versus the true model), and to increasing data noise levels (0–5 % RMS). Across all tests the hybrid inversion remained stable, indicating low dependence on regularization tuning, modest sensitivity to the initial model, and resilience to realistic noise levels.

The authors argue that many practical ERT applications—marine surveys with a thin conductive water column, permafrost investigations where a frozen layer is well known, or urban studies where shallow infrastructure imposes a known geometry—can benefit from this hybrid strategy. By constraining the well‑known part of the subsurface parametrically, the inversion gains resolution in the poorly known deeper region without sacrificing stability. Future work will extend the parametric domain to more complex shapes (multiple blocks, dipping layers) and will apply the method to field data sets to demonstrate its operational value.