Shape reconstruction in 3D low-frequency electromagnetic induction tomography using level sets and adjoint fields

Most of the techniques which are used for the electromagnetic imaging of the subsurface have in common that they mathematically define nonlinear inverse problems which are moreover severely ill-posed. Due to the ill-posedness of these problems, even a small amount of noise in the data can cause artifacts in the reconstructions that might render them useless for practical purposes. There are several established techniques for addressing this inherent ill-posedness of the inverse problem, which are usually referred to as ´regularization techniques´. A well-known regularization technique is for example the Tychonov-Phillips regularization, which imposes some smoothness constraint on the images which are reconstructed. However, since in many applications the structures which are sought are not necessarily smoothly varying and might have a high contrast to the background parameters, the reconstruction of ´blocky´ or discontinuous images might sometimes become more interesting. We present a novel shape reconstruction method which is intended to be applied in situations where approximate values of the parameters inside these non-smooth, high-contrast structures are available, but the sizes, shapes, locations and geometry of these structures is unknown. This new shape reconstruction algorithm is a nonlinear inversion technique. It combines the advantages of the so-called adjoint-field inversion technique, the level set technique for modelling propagating interfaces, and the fast forward modelling scheme EH3D for achieving fast, fully 3D inversions from relatively few electromagnetic surface-to-borehole data.