A Method of Choosing the Optimal Number of Singular Values in the Inverse Laplace Transform for the Two-Dimensional NMR Distribution Function

Two-dimensional (2D) nuclear magnetic resonance (NMR) distributions as functions of diffusion coefficient and relaxation time are powerful tools in the study of porous media. We propose a practical method to perform proper truncation of singular value decomposition (TSVD) in Laplace inversion for obtaining 2D-NMR distributions from measured NMR data. By analyzing basic algorithms for Laplace inversion, it is well known that proper TSVD does not affect the inversion result for an ill-posed problem with zero-order Tikhonov regularization, but can greatly increase the inversion speed. In this new method, the optimal number of singular values for data compression is applied to each dimension separately. The method also makes full use of the redundancy nature of the data with a finite signal-to-noise ratio and well balances the tradeoff between the speed and the bias. The method does not require the stochastic information of the estimated parameters when obtaining the optimal number of singular values.