Joint inversion of T1–T2 spectrum combining the iterative truncated singular value decomposition and the parallel particle swarm optimization algorithms

Abstract With more information than the conventional one dimensional (1D) longitudinal relaxation time ( T 1 ) and transversal relaxation time ( T 2 ) spectrums, a two dimensional (2D) T 1 – T 2 spectrum in a low field nuclear magnetic resonance (NMR) is developed to discriminate the relaxation components of fluids such as water, oil and gas in porous rock. However, the accuracy and efficiency of the T 1 – T 2 spectrum are limited by the existing inversion algorithms and data acquisition schemes. We introduce a joint method to inverse the T 1 – T 2 spectrum, which combines iterative truncated singular value decomposition (TSVD) and a parallel particle swarm optimization (PSO) algorithm to get fast computational speed and stable solutions. We reorganize the first kind Fredholm integral equation of two kernels to a nonlinear optimization problem with non-negative constraints, and then solve the ill-conditioned problem by the iterative TSVD. Truncating positions of the two diagonal matrices are obtained by the Akaike information criterion (AIC). With the initial values obtained by TSVD, we use a PSO with parallel structure to get the global optimal solutions with a high computational speed. We use the synthetic data with different signal to noise ratio (SNR) to test the performance of the proposed method. The result shows that the new inversion algorithm can achieve favorable solutions for signals with SNR larger than 10, and the inversion precision increases with the decrease of the components of the porous rock.