A statistical learning perspective on the inversion of NMR relaxation data

Abstract

T2 distribution is a powerful tool in the low-field nuclear magnetic resonance technique. The T2 distribution obtained from time-domain data involves an ill-posed inverse Laplace transformation. Tikhonov regularization with an L2 penalty term is most commonly used in this kind of problem, and the discrepancy principle, generalized cross-validation, L-curve, and S-curve methods are widely used in the selection of the regularization parameter. However, these selection approaches require prior knowledge, such as an accurate estimation of the threshold level of noise or setting a default value. In this paper, we propose a new method—the stability-enhanced k-fold cross-validation (SECV) approach—to perform a robust automatic search for the regularization parameter from a statistical learning perspective. In addition to considering test set residuals, additional terms—the Pearson’s correlation coefficients of the solutions of the disjoint subsets—are put forward to enhance the stability of the solution and make a trade-off between its imitative effect and interpretability. A bimodal T2 distribution model was constructed, and abundant echo trains with different noise levels were generated for the validation of the proposed method. The relative error of the estimates is used as a measure to evaluate the performance. The inversion results from the SECV method were compared with the solutions from the conventional methods, and the results showed that the proposed method is robust without manual intervention and suitable for both low- and high-signal-to-noise ratio data. Finally, mercury injection and nuclear magnetic resonance experiments were carried out on rock core samples to verify the correctness of our method.