Inversion of nuclear magnetic resonance echo data based on maximum entropy

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.