Summary The b value is a critical parameter that has been used to describe spatial and temporal size distribution characteristics in data from both natural seismicity and laboratory rock acoustic emission (AE) tests. However, unlike the magnitude used for b value estimation in earthquakes which corrects for hypocentral distance, the apparent amplitude of the AE, is most commonly used for b value estimation in the rock AE test. This ignores the effect of attenuation, which may modify the frequency-amplitude distribution and affect the b value results. Here, we investigate the attenuation effect on the estimate of b value from synthetic data generated by random sampling of a doubly truncated frequency-amplitude distribution and a known underlying b value from a statistical perspective. We generated AE source amplitudes that obey an exponential distribution with different theoretical b values and applied an attenuation operator to obtain apparent frequency-amplitude data for comparison with the true underlying distribution. The results show that the b value is unchanged within a specific interval of amplitude distribution after accounting for attenuation under the conditions examined here, and that data truncation is the primary cause of modification of the frequency-amplitude distribution in the presence of attenuation, rather than attenuation alone. In addition, we employed the same synthetic data simulation scheme to estimate the minimum data volume for accurate estimation of the underlying b value from the apparent amplitude distribution. We found that the determination of the minimum data volume is influenced by the theoretical b value of the underlying distribution: the higher the theoretical b value, the larger the minimum data volume required, varying from 200 events from b<1.5 to 300 for b>1.5.

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.

The rock acoustic emission (AE) technique has often been used to study rock destruction properties and has also been considered an important measure for simulating earthquake foreshock sequences. Among them, the AE b value is an essential parameter for the size distribution characteristics and probabilistic hazard analysis of rock fractures. Variations in b values obtained in rock AE tests and earthquakes are often compared to establish analogies in the damage process and precursory analysis. Nevertheless, because the amplitudes measured on the sample boundary by an acoustic sensor (apparent amplitude) are often used to estimate the b value, which cannot descript the source size distribution, it is necessary to develop a method to obtain the size distribution characteristics of the real source from the apparent amplitude in doubly truncated distribution. In this study, we obtain AE apparent amplitudes by applying an attenuation operator to source amplitudes generated by a computer with an underlying exponential distribution and then use these simulated apparent amplitudes to perform a comparative analysis of various b value estimation methods that are used in earthquakes and propose an optimal b value estimation procedure for rock AE tests through apparent amplitudes. To further verify the reliability of the newly proposed procedure, a b value characteristics analysis was carried out on a non-explosive expansion agent rock AE test and transparent refractive index experiment with red sandstone, marble, granite, and limestone. The results indicate that mineral grains of different sizes and compositions and different types of discontinuities of rock specimens determine the rock fracture characteristics, as well as the b value. The dynamic b values decreased linearly during the loading process, which confirms that variations in the b value also depend on the stress. These results indicate that the newly proposed procedure for estimating the b value in rock AE tests based on apparent amplitudes has high reliability.