Summary The inversion of electromagnetic induction data to a conductivity profile is an ill-posed problem. Regularization improves the stability of the inversion and a smoothing constraint is typically used. However, the conductivity profiles are not always expected to be smooth. Here, we develop a new inversion scheme in which we transform the model to the wavelet space and impose a sparsity constraint. This sparsity constrained inversion scheme will minimize an objective function with a least-squares data misfit and a sparsity measure of the model in the wavelet domain. A model transform to the wavelet domain allows to investigate the temporal resolution (periodicities at different frequencies) and spatial resolution (location of the peaks) characteristics of the model, and penalizing small-scale coefficients effectively reduces the complexity of the model. The novel scale-dependent regularization term can be used to favour either blocky or smooth structures, as well as high-amplitude models in globally smooth structures in the inversion. Depending on the expected conductivity profile, a suitable wavelet basis function can be chosen. The scheme supports multiple types of regularization with the same algorithm and is thus flexible. Finally, we apply this new scheme on a frequency domain electromagnetic sounding (FDEM) dataset, but the scheme could equally apply to any other 1D geophysical method.

Summary Regularization methods improve the stability of ill-posed inverse problems by introducing some a priori characteristics for the solution such as smoothness or sharpness. In this contribution, we propose a multidimensional, scale-dependent wavelet-based ℓ1-regularization term to cure the ill-posedness of the airborne (time-domain) electromagnetic induction inverse problem. The regularization term is flexible, as it can recover blocky, smooth and tunable in-between inversion models, based on a suitable wavelet basis function. For each orientation, a different wavelet basis function can be used, introducing an additional relative regularization parameter. We propose a calibration method to determine (an educated initial guess for) this relative regularization parameter, which reduces the need to optimize for this parameter, and, consequently, the overall computation time is under control. We apply our novel scheme to a time-domain airborne electromagnetic data set in Belgian saltwater intrusion context, but the scheme could equally apply to any other 2D or 3D geophysical inverse problem.

If a magnetic dipole is placed above the surface of the earth, the Electromagnetic Induction (EMI) effect, encoded in Maxwell's equations, causes eddy currents in the soil which, on their turn, induce response electromagnetic fields. The magnetic field can be measured in geophysical surveys to determine the conductivity profile of the ground in a non-destructive manner. The forward model used in the inversion of experimental data usually consists of a set of horizontal homogeneous layers. A frequently used model, proposed by McNeill, does not include the interaction between the eddy currents, and therefore fails for larger conductivities. In this paper we construct a new forward model to estimate the magnetic field caused by a horizontally stratified earth but which approximates the interaction between eddy currents. This makes it valid for a broader range of parameters than the current state of the art. Furthermore, the error with the (numerically obtainable) exact result is substantially decreased. We also pay attention to the vertical sensitivity ("depth of exploration") of the model, for which we can report a satisfactory outcome as well. * steven.delrue@kuleuven.be † david.dudal@kuleuven.be ‡ benjamin.maveau@kuleuven.be (corresponding author)

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