Monte Carlo inversion of 3-D magnetic resonance measurements

The surface nuclear magnetic resonance (SNMR) method is a geophysical method designed for non-invasive groundwater investigations. Inversion of experimental data provides the spatial distribution of the water content in the subsurface. However, SNMR inversion is ill-posed and admits many solutions because of the imaging equation properties that are compounded by experimental error. SNMR data sets are conveniently presented as complex numbers, thus possessing phase and amplitude components. Subsurface electroconductive formations and fluctuations of the Earth's magnetic field cause non-negligible phase shifts. Consequently, the forward modelling of the SNMR signal generated by 3-D water saturated formations is achieved in the complex domain. Nevertheless, in many cases, phase measurements are less reliable than amplitude measurements and water content rendering cannot be carried out using the complex SNMR signal. This problem is resolved by performing inversion using complex forward modelling whose resulting signal amplitude is used for comparison with the data. Along with water content boundaries ranging from 0 to 1, this property turns the linear initial value inversion problem into a non-linear one. In such a situation, the comprehensive analysis of inversion uncertainties is achieved by performing a solution space exploration based on a Monte Carlo approach. An adapted Metropolis-Hastings algorithm has been used on SNMR 3-D data sets to perform such an exploration. Computing time depends on the problem dimensions. With a standard laptop computer about 10 hr were necessary for the inversion of our field data set. The resulting model collection is used to calculate the probability density functions of the water content. From there, it is possible to estimate the uncertainty of the water content imagery. Using both synthetic and experimental data, we show that our routine provides robust estimates of the spatial distribution of the water content for the SNMR 3-D initial amplitude inversion.