Research on Maximum Likelihood b Value and Confidence Limits Estimation in Doubly Truncated Apparent Frequency-Amplitude Distribution in Rock Acoustic Emission Tests

The b value deduced from the Gutenberg-Richter law is an important parameter for sequence and precursory analyses, both in laboratory acoustic emission tests and seismology. As the b value is a statistical value, the maximum likelihood estimation is mostly used to estimate the b value. However, traditional singly truncated maximum likelihood estimation in seismology only considers the minimum magnitude, while the acquisition device in rock acoustic emission tests will set the threshold value and maximum value of the amplitude; therefore, maximum likelihood estimation will estimate the b value in a doubly truncated size distribution, and its confidence limits need to be discussed. Here, in this study, we derive the calculation equations of the b value and the corresponding confidence limits for the maximum likelihood estimation with a narrow amplitude span in a doubly truncated frequency-amplitude distribution. The maximum likelihood b values estimated by the scheme of a singly and doubly truncated frequency-amplitude distribution are compared through acoustic emission data with the known underlying distribution. The results show that the maximum likelihood b value and confidence limits estimation scheme derived for rock acoustic emission tests with a narrow amplitude span is more reasonable. Then, the derived estimation scheme is applied to the rock dilation rupturing test; the results confirm its applicability.