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

Changgen Xia, Daolong Chen, Wei He, Huini Liu and Xiling Liu
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Changgen Xia: School of Resources and Safety Engineering, Central South University, Changsha 410008, China
Daolong Chen: School of Resources and Safety Engineering, Central South University, Changsha 410008, China
Wei He: School of Mathematics and Statistics, Central South University, Changsha 410008, China
Huini Liu: School of Resources and Safety Engineering, Central South University, Changsha 410008, China
Xiling Liu: School of Resources and Safety Engineering, Central South University, Changsha 410008, China

Mathematics, 2022, vol. 10, issue 14, 1-13

Abstract: 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.

Keywords: b value; acoustic emission; maximum likelihood estimation; confidence limits (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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