Numerical estimation of choice of the regularization parameter for NMR T2 inversion

Nuclear Magnetic Resonance (NMR) T-2 inversion is the basis of NMR logging interpretation. The regularization parameter selection of the penalty term directly influences the NMR T-2 inversion result. We implemented both norm smoothing and curvature smoothing methods for NMR T-2 inversion, and compared the inversion results with respect to the optimal regularization parameters (aopt) which were selected by the discrepancy principle (DP), generalized cross-validation (GCV), S-curve, L-curve, and the slope of L-curve methods, respectively. The numerical results indicate that the DP method can lead to an oscillating or oversmoothed solution which is caused by an inaccurately estimated noise level. The aopt selected by the L-curve method is occasionally small or large which causes an undersmoothed or oversmoothed T-2 distribution. The inversion results from GCV, S-curve and the slope of L-curve methods show satisfying inversion results. The slope of the L-curve method with less computation is more suitable for NMR T-2 inversion. The inverted T-2 distribution from norm smoothing is better than that from curvature smoothing when the noise level is high.