Determining Uncertainty in NMR T2 Distribution Using Frequentist Method

Abstract

Petrophysicists use the T₂ distribution from nuclear magnetic resonance (NMR)-logging data inversion to evaluate reservoir. However, low SNR NMR-logging data lead to a great uncertainty of T₂ distribution, and therefore, analyzing the uncertainty of T₂ distribution has a guiding significance for NMR-logging evaluation. For regularized solution from linear inversion, the mean square error which jointly determined by variance and bias can be used to measure its uncertainty. The variance of regularized T₂ distribution can be calculated from the variance of the noise level of measured data. Since the true T₂ distribution is unknown, the bias of regularized T₂ distribution cannot be accurately calculated. This paper calculated the upper and lower boundaries of the bias of regularized T₂ distribution for analyzing the uncertainty of regularized T₂ distribution based on the frequentist method, using the NMR echo amplitudes, T₂ distribution amplitudes, T₂ distribution slopes, and T₂ distribution curvatures as prior information to restrict the range of T₂ distribution amplitude. The results indicate that as the noise level of measured data increases, the T₂ distribution uncertainty increases and the confidence interval of T₂ distribution broadens. The confidence interval width of T₂ distribution from the norm-smoothing method is narrower than that of from the curvature-smoothing method.