Low-field NMR inversion based on low-rank and sparsity restraint of relaxation spectra

In this paper, we proposed a novel method for low-field nuclear magnetic resonance (NMR) inversion based on low-rank and sparsity restraint (LRSR) of relaxation spectra, with which high quality construction is made possible for one- and two-dimensional low-field and low signal to noise ratio NMR data. In this method, the low-rank and sparsity restraints are introduced into the objective function instead of the smoothing term. The low-rank features in relaxation spectra are extracted to ensure the local characteristics and morphology of spectra. The sparsity and residual term are contributed to the resolution and precision of spectra, with the elimination of the redundant relaxation components. Optimization process of the objective function is designed with alternating direction method of multiples, in which the objective function is decomposed into three subproblems to be independently solved. The optimum solution can be obtained by alternating iteration and updating process. At first, numerical simulations are conducted on synthetic echo data with different signal-to-noise ratios, to optimize the desirable regularization parameters and verify the feasibility and effectiveness of proposed method. Then, NMR experiments on solutions and artificial sandstone samples are conducted and analyzed, which validates the robustness and reliability of the proposed method. The results from simulations and experiments have demonstrated that the suggested method has unique advantages for improving the resolution of relaxation spectra and enhancing the ability of fluid quantitative identification.