  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.
