Bayesian subsampling of time-domain electromagnetic data using kernel density product

The Bayesian method is a powerful tool to estimate the resistivity distribution and associate uncertainty from time-domain electromagnetic (TDEM) data. As the forward simulation of the TDEM method is computationally expensive and a large number of samples are needed to globally explore the model space, the full Bayesian inversion of TDEM data is limited to layered models. To make high-dimensional Bayesian inversion tractable, we propose a divide-and-conquer strategy to speed up the Bayesian inversion of TDEM data. First, the full datasets and model spaces are divided into disjoint batches based on the coverage of the sources so that independent and highly efficient Bayesian subsampling can be conducted. Then, the samples from each subsampling procedure are combined to get the full posterior. To obtain an asymptotically unbiased approximation to the full posterior, a kernel density product method is used to reintegrate samples from each subposterior. The model parameters and their uncertainty are estimated from the full posterior. The proposed method is tested on synthetic examples and applied to a field dataset acquired with a large fixed-loop configuration. The 2D section from the Bayesian inversion revealed several mineralized zones, one of which matches well with the information from a nearby drill hole. The field example shows the ability of Bayesian inversion to infer reliable resistivity and uncertainty.

Convergence of the uniaxial PML method for time-domain electromagnetic scattering problems

In this paper, we propose and study the uniaxial perfectly matched layer (PML) method for three-dimensional time-domain electromagnetic scattering problems, which has a great advantage over the spherical one in dealing with problems involving anisotropic scatterers. The truncated uniaxial PML problem is proved to be well-posed and stable, based on the Laplace transform technique and the energy method. Moreover, the $L^2$-norm and $L^{\infty}$-norm error estimates in time are given between the solutions of the original scattering problem and the truncated PML problem, leading to the exponential convergence of the time-domain uniaxial PML method in terms of the thickness and absorbing parameters of the PML layer. The proof depends on the error analysis between the EtM operators for the original scattering problem and the truncated PML problem, which is different from our previous work (SIAM J. Numer. Anal. 58(3) (2020), 1918-1940).

1D Stochastic Inversion of Airborne Time-Domain Electromagnetic Data with Realistic Prior and Accounting for the Forward Modeling Error

Airborne electromagnetic surveys may consist of hundreds of thousands of soundings. In most cases, this makes 3D inversions unfeasible even when the subsurface is characterized by a high level of heterogeneity. Instead, approaches based on 1D forwards are routinely used because of their computational efficiency. However, it is relatively easy to fit 3D responses with 1D forward modelling and retrieve apparently well-resolved conductivity models. However, those detailed features may simply be caused by fitting the modelling error connected to the approximate forward. In addition, it is, in practice, difficult to identify this kind of artifacts as the modeling error is correlated. The present study demonstrates how to assess the modelling error introduced by the 1D approximation and how to include this additional piece of information into a probabilistic inversion. Not surprisingly, it turns out that this simple modification provides not only much better reconstructions of the targets but, maybe, more importantly, guarantees a correct estimation of the corresponding reliability.

Three-dimensional transient electromagnetic inversion with optimal transport

Transient electromagnetic method (TEM), as one of the essential time-domain electromagnetic prospecting approaches, has the advantage of expedition, efficiency and convenience. In this paper, we study the transient electromagnetic inversion problem of different geological anomalies. First, Maxwell’s differential equations are discretized by the staggered finite-difference (FD) method; then we propose to solve the TEM inversion problem by minimizing the Wasserstein metric, which is related to the optimal transport (OT). Experimental tests based on the layered model and a 3D model are performed to demonstrate the feasibility of our proposed method.