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.