Regularization-Based Structural Constraints in Two-Dimensional Magnetotelluric Inversion: Implications for Non-Uniqueness and Uncertainty

In geophysical inversion problems, the model–data misfit between the theoretical responses f(m) and observational data d is quantified by a data misfit function:  Φ(m) =C_d^1/2(d- f(m))²₂, and the inverse problems are inherently non-unique.

To reduce the non-unique, regularization is commonly introduced by adding structural constraint terms that favor smooth models consistent with the data and a prescribed error tolerance. This leads to the minimization of an augmented objective function, Φ(m) =C_d^-1/2(d- f(m))²₂+λC_m^-1/2(m-m₀)²_2. However, such approaches may suppress legitimate model variability and fail to adequately characterize the inherent non-uniqueness of geophysical inverse problems. Bayesian inversion provides a probabilistic framework to address these challenges by characterizing the posterior probability distribution p(md) through the combination of data likelihood p(dm) and prior information p(m) , p(md)∝ p(dm)  p(m), with p(dm) ∝exp[-Φ(m)]. The posterior distribution can be efficiently explored using reversible-jump Markov chain Monte Carlo (rj-MCMC) methods, which allow both model parameters and model dimensionality to be inferred from the data.

This study examines the impact of smoothing-based structural constraints on two-dimensional magnetotelluric (MT) inversion through a comparison of conventional regularized and Bayesian approaches, using a wavelet-domain, tree-based trans-dimensional  MCMC sampling. Two numerical examples are designed to systematically examine the effects of smoothing-based regularization. In the first example, a synthetic model with anomalies of varying sizes and burial depths is used to compare a Bayesian inversion constrained only by model parameterization and weakly informative priors, without smoothness-based regularization, with a conventional nonlinear conjugate gradient (NLCG) inversion that enforces structural constraints through regularization. In the second example, a single high-conductivity anomaly is inverted to directly compare Bayesian inversions without and with regularization-based structural prior information, where the structural prior is explicitly introduced through smoothness constraints. The  structural prior can be expressed as :p_structure(m)=(1/2πλ²)^-Mexp[-λ(C_m^-1/2(m-m₀)²₂)].

Results from the first example show that the NLCG inversion produces a smooth conductivity model in which the recovered anomalies are larger than the true anomalies, reflecting the strong influence of smoothness regularization. In contrast, the Bayesian inversion recovers the main anomaly locations while yielding rougher boundaries and a background field that is no longer uniformly smooth, indicating that multiple model realizations are consistent with the observed data. While the NLCG solution provides a stable and easily interpretable model, it may underestimate uncertainty, whereas the Bayesian inversion without regularization-based structural priors offers a more complete characterization of model non-uniqueness through marginal probability density distributions. In the second example, introducing smoothness-based structural priors within the rj-MCMC framework produces smoother posterior samples with reduced uncertainty and improved convergence stability, but at the cost of diminishing the relative contribution of the data in constraining the solution.

Overall, our results demonstrate that prior information plays a critical role in Bayesian MT inversion. While structural priors can reduce non-uniqueness and improve convergence in high-dimensional problems, they must be selected with caution to avoid excessive prior-driven bias when interpreting real data.