Nuclear magnetic resonance T-T inversion with double objective functions.

We report an effective and robust method for nuclear magnetic resonance (NMR) longitudinal relaxation time-transverse relaxation time (T-T) inversion with double objective functions. First, we develop the first objective function based on L1 regularization, proposed an effective method to choose the optimum L1 regularization parameter, and solve the objective function employing a two-step iterative shrinkage/thresholding algorithm. Subsequently, we update the kernel matrix based on the solution of the first objective function, and then develop the second objective function using the measured data and updated kernel matrix based on the least-squares principle, and we use the conjugate gradient algorithm for the first time to solve the objective function about NMR data inversion. To improve the speed of NMR T-T inversion, we present a Gaussian-based random SVD method. Finally, numerical and experimental examples are done to test the robustness of the proposed inversion method. The results indicate that the proposed inversion method can effectively achieve NMR T-T inversion at a low data SNR.