SUMMARY: The Born approximation offers a computationally efficient alternative to full electromagnetic (EM) forward modelling, but suffers from limited accuracy due to its reliance on a fixed background conductivity. In this work, we develop an adaptive Born approximation that treats the background medium as a tunable parameter to enhance accuracy in a goal-oriented manner. The background conductivity is selected locally for each measurement configuration using spatial sensitivity functions, enabling accurate modelling in both isotropic and anisotropic media. In this study, we primarily focus on horizontally layered earth models penetrated by a vertical well to investigate the fundamental behaviour of the approximation in a simplified setting. We formulate our approach to be applicable to general anisotropic media by using the Green’s function defined for a homogeneous medium. Furthermore, the approach extends to cases where the background conductivity is isotropic while the actual medium is anisotropic. For a layered medium, the orientation of induced current densities relative to the layering provides physical intuition for background selection, drawing analogies to Voigt- and Reuss-type bounds. While these analogies offer useful guidance, our numerical results do not always conform to the expectations derived from them. Among the averaging schemes evaluated, arithmetic averaging generally yields the most accurate results. Numerical experiments indicate that the adaptive approach significantly outperforms fixed-background models across a range of frequencies, spacings and conductivity contrasts. Furthermore, an example with a 3-D structure illustrates the method’s broader applicability beyond the horizontally layered earth setting. This framework provides a principled and efficient path toward fast, accurate EM borehole modelling for real-time well geosteering and subsurface electrical imaging.

Structural information about the subsurface near the borehole can be obtained from reconstructed conductivity distributions. These distributions may be reconstructed via the inversion of deep-sensing electromagnetic induction log data. Unfortunately, these complex media often display anisotropy and structural variations in both horizontal and vertical directions, making the three-dimensional inversion computationally demanding and ill-posed. To address these challenges, we introduce a sequential inversion strategy of deep-sensing electromagnetic induction logging data that is measured while drilling. For the inversion at each logging position, we employ a matrix-free implementation of the adjoint integral equation method and a quasi-Newton algorithm. To tackle the ill-posed nature of the problem, we regularize the inverse problem by employing a multi-dimensional inversion parameter technique that shifts from zero- to three-dimensional parameterization. The model derived from the inversion of the data at multiple positions is incrementally integrated by utilizing the sensitivity data at each logging position. To validate our approach, we tested our method on simulated data using an anisotropic model. These experiments show that this approach produces a good reconstruction of the true conductivity for the whole track while only doing the inversion at a single position at a time.

We present an iterative nonlinear inverse scattering algorithm for high-resolution acoustic imaging of density and velocity variations. To solve the multi-parameter nonlinear direct scattering problem, the acoustic wave equation for inhomogeneous media in the frequency domain is transformed into a vectorial integral equation of the Lippmann-Schwinger type for the combined pressure and pressure-gradient field. To solve the multi-parameter nonlinear inverse scattering problem, we use the Newton-Kantorovich method in conjunction with matrix-free representations of the Fréchet derivative operators and their adjoints. The approximate Hessian information that is accounted for in our iterative solution of the (nonlinear) multi-parameter inverse scattering problem is essential for the mitigation of multi-parameter cross talk effects. Numerical examples related to seismic and medical ultrasound breast imaging illustrate the performance of the new algorithm for multi-parameter acoustic imaging.

Transcranial ultrasound has been used to image the brain since 1942. Currently, it is regaining interest and full-waveform inversion (FWI) methods are now employed to reconstruct speed-of-sound profiles of the brain. Many of these methods require a good starting model. Here, we test the applicability of contrast source inversion (CSI) as a FWI method to reconstruct two-dimensional speed-of-sound profiles of the soft brain tissue enclosed by the skull. The advantage of CSI is that it can handle large acoustic contrasts without the need for a good starting model. To test the performance of CSI, we first compute synthetic data. The resulting pressure field clearly shows a significant amount of multiple scattering caused by the skull that acts as a hard acoustic contrast. Next we invert the resulting synthetic data within the Born approximation as well as by applying CSI as a FWI method. The results clearly show that Born inversion can only image the soft brain tissue in the absence of the skull whereas it generates erroneous results when the skull is present. On the other hand, with CSI it is feasible to reconstruct both the skull and the soft brain tissue accurately. Importantly, as compared to other methods CSI does not require any a priori information about the contrast, a mask or a heterogeneous starting model to reconstruct the soft tissue enclosed by the skull.