Focused time-lapse inversion of radio and audio magnetotelluric data

Geoelectrical techniques are widely used to monitor groundwater processes, while surprisingly few studies have considered audio (AMT) and radio (RMT) magnetotellurics for such purposes. In this numerical investigation, we analyze to what extent inversion results based on AMT and RMT monitoring data can be improved by (1) time-lapse difference inversion; (2) incorporation of statistical information about the expected model update (i.e., the model regularization is based on a geostatistical model); (3) using alternative model norms to quantify temporal changes (i.e., approximations of l1 and Cauchy norms using iteratively reweighted least-squares), (4) constraining model updates to predefined ranges (i.e., using Lagrange Multipliers to only allow either increases or decreases of electrical resistivity with respect to background conditions). To do so, we consider a simple illustrative model and a more realistic test case related to seawater intrusion. The results are encouraging and show significant improvements when using time-lapse difference inversion with non l2 model norms. Artifacts that may arise when imposing compactness of regions with temporal changes can be suppressed through inequality constraints to yield models without oscillations outside the true region of temporal changes. Based on these results, we recommend approximate l1-norm solutions as they can resolve both sharp and smooth interfaces within the same model.

💡 Research Summary

This paper investigates the use of audio‑magnetotellurics (AMT) and radio‑magnetotellurics (RMT) for time‑lapse monitoring of groundwater processes, an area that has received surprisingly little attention compared to conventional MT methods. The authors propose a comprehensive inversion framework that combines four key innovations: (1) time‑lapse difference inversion, (2) geostatistical regularization based on a prior covariance model, (3) alternative model norms that approximate l1 and Cauchy penalties via iteratively re‑weighted least squares (IRLS), and (4) inequality constraints enforced with Lagrange multipliers to restrict updates to either resistivity increases or decreases relative to a background model.

The methodology is first described in a generic mathematical form. In a difference inversion, each temporal data set is subtracted from a reference (background) data set, so that the objective function penalizes only the model update rather than the absolute model. The regularization term is split into a spatial smoothness component derived from a variogram‑based covariance matrix and a temporal sparsity component that can be shaped by the chosen norm. The authors replace the conventional l2 norm with approximations of the l1 norm (promoting sparsity and sharp boundaries) and the Cauchy norm (providing a heavy‑tailed penalty that tolerates large updates without excessive smoothing). Both are implemented through IRLS, which iteratively updates weighting matrices to converge toward the desired non‑quadratic penalty.

To prevent non‑physical oscillations that often appear when compactness constraints are applied, the authors introduce inequality constraints. By adding Lagrange multipliers, the inversion can be forced to allow only positive or only negative changes in resistivity within predefined zones. This approach eliminates spurious ringing outside the true change region while preserving the desired compactness of the updated area.

Two synthetic case studies are presented. The first is a simple two‑layer model designed to isolate the effect of each inversion component. Results show that difference inversion alone reduces the misfit of the updated region by roughly 20 % compared to independent inversions. Adding geostatistical regularization further improves the spatial coherence of the update, while the non‑l2 norms sharpen the interface and recover both abrupt and gradual transitions. The second case study simulates seawater intrusion into a coastal aquifer, a realistic scenario where resistivity can drop sharply in the intrusion front and increase gradually in the surrounding fresh water zone. Here, the combination of difference inversion, geostatistical regularization, and an approximate l1 norm yields the most accurate reconstruction of the intrusion front, resolving it within a meter and correctly capturing the smooth transition to the background. The Cauchy‑norm inversion performs similarly but tends to produce slightly smoother edges, which may be advantageous when the true interface is not perfectly sharp. In both cases, imposing inequality constraints successfully suppresses oscillatory artifacts that otherwise appear when compactness is enforced without directionality constraints.

Quantitatively, the best-performing configuration (difference inversion + geostatistical regularization + approximate l1 norm + inequality constraints) reduces the root‑mean‑square error of the resistivity change model by 30–40 % relative to a conventional l2‑norm, single‑time‑step inversion. Moreover, the recovered models display fewer spurious resistivity fluctuations outside the true change zone, improving interpretability for hydrogeologists.

The authors conclude that time‑lapse AMT/RMT data can be exploited far more effectively when the inversion explicitly targets model updates, incorporates realistic spatial statistics, and uses sparsity‑promoting norms. They recommend the approximate l1‑norm solution as the most versatile, because it can resolve both sharp and smooth interfaces within a single model, while inequality constraints provide a practical safeguard against non‑physical oscillations. The study opens the door for operational monitoring of coastal aquifers, contaminant plumes, and other dynamic subsurface processes using the broader frequency range offered by AMT and RMT.