  The inversion of electromagnetic induction data to a conductivity profile is
an ill-posed problem. Regularization improves the stability of the inversion
and, based on Occam's razor principle, 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 in
the wavelet domain has both temporal as spatial resolution, and penalizing
small-scale coefficients effectively reduces the complexity of the model.
Depending on the expected conductivity profile, an optimal wavelet basis
function can be chosen. The scheme is thus adaptive. 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.
