Inversion of nuclear magnetic resonance echo data based on maximum entropy

Abstract

Nuclear magnetic resonance (NMR) [Formula: see text] inversion is an ill-posed problem in which regularization techniques are usually adopted to suppress the oscillations caused by noise in the solutions. The maximum entropy concept provides an unbiased way to obtain information from incomplete data, and it implicitly imposes a positive constraint on probability distribution, so we used the maximum entropy method to invert NMR echo data. We have developed a simple and effective method for solving the objective function of the maximum entropy method. First, the solution was replaced by a positive function to achieve the positive constraint of the solution, the objective function was converted to an unconstrained one, and then the Levenberg-Marquardt method was used to solve the newly obtained unconstrained objective function. To suppress the highly tilted tail at the short relaxation time of the [Formula: see text] distribution, a modified or normalized Shannon entropy function was used to replace the standard Shannon entropy function as the penalty term. Furthermore, the S-curve method was used to select the regularization parameter and the formula of the slope of the S-curve was developed. We have determined that the maximum entropy method was better able to separate the peaks of short and long relaxation times in the [Formula: see text] distribution in comparison with the truncated singular value decomposition method. This was true for low signal-to-noise ratio data derived from numerical simulation and the NMR log. In addition, the short relaxation peak caused by the norm smoothing method can also be reduced.

Nuclear Magnetic Resonance Characterization of Petrophysical Properties in Tight Sandstone Reservoirs

“…Many empirical methods have been proposed for the estimation of permeability based on the characteristic parameters of the T 2 distribution in downhole NMR applications, including the Schlumberger-Doll Research (SDR) method and the Timur-Coates method (Daigle & Dugan, 2009;Liu et al, 2017;Liu et al, 2019;Timur, 1968). However, existing NMR-based methods for S wb and permeability estimations generally require the inversion of the NMR echo data to obtain the T 2 distribution, which is an ill-posed problem of the Fredholm equation of the first kind (Li & Misra, 2017;Venkataramanan et al, 2002;Venkataramanan et al, 2010;Zou et al, 2015). Small changes in the measured NMR echo data due to noise can result in large differences in the T 2 distribution.…”

Moisture localisation in unsaturated materials using nuclear magnetic resonance experiments, X-ray computed tomography and pore network modeling

“…where M 0 is the initial amplitude of the signal and T 2,k is the pore-dependent transverse relaxation time belonging to pore k. For both signals measured in this paper, the signal-tonoise ratio, calculated as the initial signal amplitude (M 0 ) divided by the standard deviation of the noise, exceeds 100. Using 3 different regularization schemes based on the maximum entropy principle, Tikhonov regularization and the CONTIN method [8], the ill-posed inversion problem represented by equation 2 is solved to calculate the distribution of transverse relaxation times as shown in Figure 1. Only the result based on the maximum entropy principle is shown due to the different solutions' similarity.…”

FPGA-Based Real-time T2 Spectrum Inversion Algorithm for LWD-NMR Logging with Periodic Thermal Management

“…The original echo train signal measured by the LWD-NMR instrument using the CPMG sequence is the result of multiple transverse relaxations, and the signal waveform is multi-exponential decay [30][31][32]. The discrete expression of the T2 spectrum signal is…”