Inversion of Magnetic Resonance Sounding with Wavelet Basis Functions

Magnetic Resonance Sounding (MRS) is a geophysical method measuring the water content in the subsurface. Hydrogen protons are brought to resonance via an oscillating magnetic field. Their electromagnetic response is recorded. The MRS forward model depends linearly on the water content, so the inversion is a straight-forward task. However, we apply a nonlinear tangens transform to ensure that water content is bounded between 0 and 1. In higher dimensions the MRS problem can grow large. A reduction of the model parameters to the essential ones is useful, if not necessary, from a computational point of view. We achieve this with by finding a reduced model in the wavelet domain. Wavelets are hierarchical basis functions with compact support and good localization in frequency domain as well. Taking the water content in wavelet domain and omitting poorly resolved wavelet coefficients (mostly the fine scales) results in less model parameters to invert for. We propose a scheme to find a minimal set of coefficients to explain the data. Particularly in 2D, we can find models composed of very few model parameters. The model contains details according to the resolution of the method and the water content at hand.