2DNMR data inversion using locally adapted multi-penalty regularization

Geologists and Reservoir Engineers routinely use time-domain nuclear magnetic resonance (NMR) to learn about the porous structure of rocks that hold underground fluids. In particular, two-dimensional NMR (2DNMR) technique is now gaining importance in a wide variety of applications. Crucial issue in 2DNMR analysis are the speed, robustness and accuracy of the data inversion process. This paper proposes a multi-penalty method with locally adapted regularization parameters for fast and accurate inversion of 2DNMR data. The method solves an unconstrained optimization problem whose objective function contains a data-fitting term, a single L1 penalty parameter and a multiple parameter L2 penalty. We propose an adaptation of the Fast Iterative Shrinkage and Thresholding (FISTA) method to solve the multi-penalty minimization problem, and an automatic procedure to compute all the penalty parameters. This procedure generalizes the Uniform Penalty principle introduced in [Bortolotti et al., Inverse Problems, 33(1), 2016]. The proposed approach allows us to obtain accurate 2D relaxation time distributions while keeping short the computation time. Results of numerical experiments on synthetic and real data prove that the proposed method is efficient and effective in reconstructing the peaks and the flat regions that usually characterize 2DNMR relaxation time distributions.