Numerical inversion methods for recovering negative amplitudes in two-dimensional nuclear magnetic resonance relaxation-time correlations

摘要

Two-dimensional nuclear magnetic resonance measurements are ubiquitous in the literature, with correlations of longitudinal T_1 and transverse T_2 relaxation times used extensively to characterize porous media. Decomposition of the signal acquired in the time domain to a pseudocontinuous distribution of relaxation times is achieved using numerical inversion. A popular technique to generate a stable solution to this ill-posed problem in the presence of noise is Tikhonov regularization with a non-negativity constraint imposed on the output. However, coupling of the longitudinal and transverse eigenfunctions can generate eigenvalue pairs with apparent T_2 > T_1 and negative amplitude. Such apparent signal components are encountered in the classic example of Brownstein- Tarr "slow" diffusion in an isolated pore, and in weakly coupled pores governed by different relaxation rates. We show that when negative-amplitude components comprise ≥1% of the total signal, the solution achieved by non-negative Tikhonov regularization is sufficiently distorted to prevent robust interpretation. We demonstrate two alternative inversion methods that recover the negative-amplitude components: (1) half-bound Tikhonov regularization assigns a negative amplitude to any peak with apparent T_2 > T_1, and (2) the optimization problem is expressed as a ?_2 regression with ?_1 penalization and a solution estimated using a primal-dual algorithm without constraint on the output sign. These methods are applicable to T_1-T_2 experiments on porous materials characterized by a hierarchy of length scales, such as biological cells, cement, and limestone.