An inversion method of 2D NMR relaxation spectra in low fields based on LSQR and L-curve

The low-field nuclear magnetic resonance (NMR) inversion method based on traditional least-squares QR decomposition (LSQR) always produces some oscillating spectra. Moreover, the solution obtained by traditional LSQR algorithm often cannot reflect the true distribution of all the components. Hence, a good solution requires some manual intervention, for especially low signal-to-noise ratio (SNR) data. An approach based on the LSQR algorithm and L-curve is presented to solve this problem. The L-curve method is applied to obtain an improved initial optimal solution by balancing the residual and the complexity of the solutions instead of manually adjusting the smoothing parameters. First, the traditional LSQR algorithm is used on 2D NMR T1-T2 data to obtain its resultant spectra and corresponding residuals, whose norms are utilized to plot the L-curve. Second, the corner of the L-curve as the initial optimal solution for the non-negative constraint is located. Finally, a 2D map is corrected and calculated iteratively based on the initial optimal solution. The proposed approach is tested on both simulated and measured data. The results show that this algorithm is robust, accurate and promising for the NMR analysis.