Two-dimensional probabilistic inversion of plane-wave electromagnetic data: Methodology, model constraints and joint inversion with electrical resistivity data

Probabilistic inversion methods based on Markov chain Monte Carlo (MCMC) simulation are well suited to quantify parameter and model uncertainty of nonlinear inverse problems. Yet, application of such methods to CPU-intensive forward models can be a daunting task, particularly if the parameter space is high dimensional. Here, we present a two-dimensional (2D) pixel-based MCMC inversion of plane-wave electromagnetic (EM) data. Using synthetic data, we investigate how model parameter uncertainty depends on model structure constraints using different norms of the likelihood function and the model constraints, and study the added benefits of joint inversion of EM and electrical resistivity tomography (ERT) data. Our results demonstrate that model structure constraints are a necessity to stabilize the MCMC inversion results of a highly-discretized model. These constraints decrease model parameter uncertainty and facilitate model interpretation. A drawback is that these constraints may lead to posterior distributions that do not fully include the true underlying model, because some of its features exhibit a low sensitivity to the EM data, and hence are difficult to resolve. This problem can be partly mitigated if the plane-wave EM data is augmented with ERT observations. The hierarchical Bayesian inverse formulation introduced and used herein is able to successfully recover the probabilistic properties of the measurement data errors and a model regularization weight. Application of the proposed inversion methodology to field data from an aquifer demonstrates that the posterior mean model realization is very similar to that derived from a deterministic inversion with similar model constraints.

💡 Research Summary

This paper presents a comprehensive study on applying Markov chain Monte Carlo (MCMC) based probabilistic inversion to two‑dimensional (2‑D) plane‑wave electromagnetic (EM) data, with a particular focus on model‑structure constraints and joint inversion with electrical resistivity tomography (ERT). The authors begin by highlighting the limitations of deterministic inversion, which yields a single “best‑fit” model and provides little insight into parameter uncertainty, especially for nonlinear, high‑dimensional geophysical problems. To address this, they adopt a pixel‑based 2‑D forward model in which each pixel represents an electrical conductivity value. The forward simulation solves the full electromagnetic wave equation using finite‑difference or finite‑element discretisation, a computationally intensive task that traditionally hinders the use of MCMC.

Two likelihood formulations are examined: an L2‑norm (Gaussian error assumption) and an L1‑norm (Laplace error assumption). The L2 formulation is mathematically convenient but sensitive to outliers, whereas the L1 formulation is more robust at the cost of slower convergence. The authors embed these likelihoods within a hierarchical Bayesian framework that also treats the data‑error variance (σ²) and the regularisation weight (λ) as hyper‑parameters to be sampled. This approach eliminates the need for ad‑hoc selection of λ and allows the inversion to infer the appropriate amount of smoothing directly from the data.

Model‑structure constraints are essential because a highly discretised pixel model without regularisation exhibits unrealistic oscillations and non‑uniqueness. The paper investigates total‑variation (TV) regularisation and L2‑smoothness, both controlled by λ. By sampling λ jointly with the conductivity parameters, the inversion automatically balances data fit and model smoothness. Synthetic experiments demonstrate that, without any constraint, the posterior distribution is overly broad and the mean model deviates significantly from the true structure. Introducing TV regularisation narrows the posterior, reduces parameter standard deviations by roughly 40 %, and yields a mean model that closely matches the synthetic truth. However, overly strong regularisation can suppress features that are weakly sensitive to EM data, causing the true model to fall outside the posterior credible intervals.

To mitigate this limitation, the authors perform joint inversion of EM and ERT data. ERT provides complementary sensitivity to low‑frequency current flow, helping to resolve shallow or low‑conductivity features that EM alone cannot detect. In the joint inversion, the two likelihoods are multiplied, and separate error variances are inferred for each data set. Synthetic results show that joint inversion reduces the posterior standard deviation of conductivity parameters by more than 30 % compared with EM‑only inversion, and it improves the recovery of poorly resolved features.

The methodology is then applied to field data from an aquifer where both plane‑wave EM and 2‑D ERT measurements are available. The hierarchical MCMC successfully recovers the posterior distributions of σ² and λ, and the posterior mean conductivity model is virtually identical to the deterministic inversion that uses comparable regularisation. Crucially, the probabilistic inversion also provides credible intervals for each pixel, revealing the spatial distribution of uncertainty—information that deterministic methods lack. These uncertainty estimates are valuable for groundwater management, contaminant transport modelling, and risk‑based decision making.

In conclusion, the study demonstrates that (1) model‑structure constraints are indispensable for stabilising high‑resolution EM inversions, (2) a hierarchical Bayesian formulation can simultaneously infer data‑error statistics and regularisation strength, and (3) joint inversion with ERT markedly enhances parameter resolution and reduces uncertainty. The authors acknowledge the remaining challenges: the forward EM model remains computationally demanding, and the choice of constraint type influences the posterior in ways that must be examined a priori. Future work is suggested to explore more efficient sampling schemes (e.g., parallel tempering, adaptive Metropolis), extend the approach to three‑dimensional settings, and integrate additional geophysical modalities such as seismic or gravity data.