From 1D Independence to 3D Coherence: Geostatistical Simulation of Probabilistic TEM Inversions

Groundwater modelling relies on three-dimensional (3D) geological models as structural input to predict subsurface processes such as groundwater flow and contaminant transport. However, model uncertainty in the geological domain, originating from sparse data coverage and the inherent non-uniqueness of geophysical inverse problems, propagates into hydrological predictions and affects the model outcome. Accounting for this uncertainty is therefore essential. This requires methods that not only characterize the subsurface but also quantify and propagate uncertainty through the entire modelling workflow.

Probabilistic inversion of transient electromagnetic (TEM) data addresses the non-uniqueness of the inverse problem by yielding posterior samples containing hundreds of plausible one-dimensional (1D) models at each measurement location. This captures the range of subsurface structures consistent with the geophysical data and enables quantitative assessment of subsurface uncertainty. However, a critical challenge emerges: How do we transform these independent 1D posterior models into spatially coherent 3D subsurface realizations that preserve geological continuity? A simple approach would be to use the mode model, showing the most probable values. However, the mode is merely a statistical summary of the posterior, not an actual sample from it, and does not capture the uncertainty in the subsurface structure either. Alternatively, randomly selecting one posterior model at each location would result in geologically implausible 3D realizations due to lacking spatial structure and lateral correlation. Generating multiple internally consistent realizations is essential to capture the full range of plausible subsurface scenarios and quantify uncertainty. Yet, no standard algorithm currently exists to generate 3D realizations that both sample the posterior distribution and ensure geological continuity.

Existing geostatistical simulation methods are not well suited for this task. Gaussian-based approaches (e.g., sequential indicator simulation) cannot fully exploit the non-Gaussian posterior distributions from probabilistic inversion. Multiple-point statistical methods require training images that are difficult to obtain and may conflict with the prior information used in the inversion.

Here, we present a novel geostatistical simulation algorithm that generates spatially coherent 3D subsurface realizations directly from independent 1D posterior models. The algorithm directly combines 1D posterior realizations at data locations with 1D prior realizations elsewhere, using spatial correlation to generate coherent 3D structures without Gaussian assumptions or training images. We demonstrate the method using a TEM dataset, showing that the resulting realizations reproduce realistic spatial geological patterns and variability consistent with the underlying posterior information. The algorithm is computationally efficient, enabling generation of multiple realizations that in combination quantify subsurface uncertainty and provide a direct basis for propagating geological uncertainty into hydrological flow and transport simulations.