Electromagnetic imaging is modelled as an inverse problem for the 3D system of Maxwell's equations of which the isotropic conductivity distribution in the domain of interest has to be reconstructed. The main application we have in mind is the monitoring of conducting contaminant plumes out of surface and borehole electromagnetic imaging data. The essential feature of the method developed here is the use of adjoint fields for the reconstruction task, combined with a splitting of the data into smaller groups which define subproblems of the inversion problem. The method works iteratively, and can be considered as a nonlinear generalization of the algebraic reconstruction technique in x-ray tomography. Starting out from some initial guess for the conductivity distribution, an update for this guess is computed by solving one forward and one adjoint problem of the 3D Maxwell system at a time. Numerical experiments are performed for a layered background medium in which one or two localized (3D) inclusions are immersed. These have to be monitored out of surface to borehole and cross-borehole electromagnetic data. We show that the algorithm is able to recover a single inclusion in the earth which has high contrast to the background, and to distinguish between two separated inclusions in the earth given certain borehole geometries.

We present a detailed sensitivity analysis for a nonlinear electromagnetic inversion method which was introduced earlier by the authors. Whereas the earlier work was restricted to the 3D imaging of isotropic structures in the earth from cross-borehole electromagnetic data, the analysis presented here is focused on the imaging of anisotropic structures which often have to be taken into account in practical situations. The inversion scheme considered can be described as a single-step adjoint field scheme. It avoids calculating huge sensitivity matrices (which we call linearized residual operators) during the inversion and uses only the data corresponding to one source position at a time. Doing so, the action of the adjoint linearized residual operator on the corresponding (filtered) residual vector can be calculated very efficiently by just running one forward and one adjoint Maxwell problem on the most recent best guess for the parameters. The outcome of these two runs is combined to find a correction to the latest best guess. The anisotropic sensitivity functions have the property that they decompose the linearized residual operator as well as the corresponding adjoint linearized residual operator. Playing this dual role, they provide useful information about how sources and receivers should be arranged in a given experiment, and which structures in the earth can be expected to be resolved in the inversion from a given data set. In the paper, we present numerical examples of 3D anisotropic sensitivity functions for homogeneous as well as for inhomogeneous background parameter distributions, and discuss their dual role in the nonlinear adjoint field inversion scheme.

Summary This work deals with reconstruction of non‐smooth solutions of the inverse gravimetric problem. This inverse problem is very ill‐posed, its solution is non‐unique and unstable. The stable inversion method requires regularization. Regularization methods commonly used in geophysics reconstruct smooth solutions even though geological structures often have sharp contrasts (discontinuities) in properties. This is the result of using a quadratic penalization term as a stabilizing functional. We introduce the total variation of the reconstructed model as a stabilizing functional that does not penalize sharp features of the solution. This approach permits reconstruction of (non‐smooth) density functions that represent blocky geological structures. An adaptive gradient scheme is shown to be effective in solving the regularized inverse problem. Numerically simulated examples consisting of models with several homogeneous blocks illustrate the behaviour of the method.

Summary. In this paper we give an overview of a recently developed method for solving an inverse Maxwell problem in environmental and geophysical imaging. Our main focus is on low-frequency cross-borehole electromagnetic induction tomography (EMIT), although related problems arise also in other applications in nondestructive testing and medical imaging. In typical applications (e.g. in environmental remediation), the isotropic or anisotropic conductivity distribution in the earth needs to be reconstructed from surface-to-borehole electromagnetic data. Our method uses a backpropagation strategy (based on adjoint fields) for solving this inverse problem. The method works iteratively, and can be considered as a nonlinear generalization of the Algebraic Reconstruction Technique (ART) in x-ray tomography, or as a nonlinear Kaczmarz-type approach. We will also propose a new regularization scheme for this method which is based on a proper choice of the function spaces for the inversion. A detailed sensitivity analysis for this problem is given, and a set of numerically calculated sensitiviy functions for homogeneous isotropic media is presented.

Reconstruction of shallow subsurface structure from geophysical data is a central problem for many environmental and engineering applications. We observe that for shallow soil distributions, seismic data alone or electrical data alone may provide a good reconstruction of the subsurface. We show that using joint seismic and electrical data will improve the reconstruction of shallow structure. Our results emphasize that the availability of techniques for making laboratory measurements of ultrasonic velocities at low pressures in unconsolidated materials and the ability to measure complex impedance in similar samples are key elements of this approach.