## 1 Introduction  
Rock burst refers to a phenomenon in which the sudden and violent release of energy accumulated within rock formations occurs in deep underground mining or areas subjected to high tectonic stress, leading to the abrupt and catastrophic failure of overlying rock masses [1, 2]. In underground engineering, particularly in mining engineering, the release of energy is often unavoidable during excavation, which can lead to the rapid occurrence of rock bursts, posing a threat to the safety of underground personnel and impacting the safety of mine operations [3]. Therefore, the short-term prediction of rock bursts is one of the crucial research areas for the sustainable development of underground engineering and the protection of underground facilities and personnel safety [4]. Acoustic emission (AE) is a non-destructive testing technique widely employed for material damage monitoring. This geophysical monitoring method has become a significant tool for the monitoring and prediction of coal-rock material damage due to its non-destructive, non-contact, and real-time transmission characteristics [5, 6, 7, 8, 9]. AE not only encompasses spatial information regarding internal fractures in complex materials but also conveys the temporal and frequency information associated with the evolution of material fracture. The temporal patterns and frequency characteristics of AE time-frequency feature parameters have been extensively studied and reported [10, 11, 12, 13]. In the field of seismology, the famous Gutenberg-Richter Law is well-known, and its \"\(b\)-value\" is considered an effective indicator for earthquake prediction [14]. In recent years, many scholars have integrated the \"\(b\)-value\" with AE technology for analyzing the damage and fracture instability of solid materials. For example, [PERSON] et al. [15] investigated the precursor characteristics of failure of weathered granite, combining the critical slowing down theory with the synchronous analysis of the \(b\)-value, which can accurately grasp the failure characteristics of rock. [PERSON] et al. [16] conducted laboratory direct shear tests and AE tests on standard cylindrical specimens with different levels of rock-surface roughness. The evolution of AE \(b\)-values indicates that the macroscopic -fracture surface of the bonded surface is the result of repeated conversion between large and small microcracks. The fluctuation of the AE \(b\)-value is not affected by the roughness of the rock interface. [PERSON] [17] studied the fractal structure of seismic activity in terms of spatial, temporal, and magnitude distributions, which are represented by fractal dimension \(D\), Omori exponent \(p\), and \(b\)-value, respectively. [PERSON] et al. [18] used an AE system to carry out cyclic loading and continuous monitoring of concrete specimens and compared the \(b\)-value with the applied load, damage parameters, and cracks appearing on the beam. Some quantitative conclusions are obtained by studying the whole loading cycle and the \(b\)-value calculation of each channel. The AE \(b\)-value reflects the relationship between amplitude and frequency within a time series, following a power-law distribution within a specific time range. A sudden decrease in the \(b\)-value can be regarded as a precursor indicator of impending instability in solid materials [19]. However, due to the complexity of rock as a porous material, relying solely on extracting the \(b\)-value from time series to assess the damage process and criteria in rock is not entirely reliable. This is because the calculation of the AE \(b\)-value is entirely dependent on amplitude and often overlooks the crucial frequency-domain information. In the frequency domain, the Flicker Noise Spectroscopy (FNS) method is used to quantify the characteristics of signal frequency distribution and is effective in predicting instability in nonlinear signals [19,20]. The characteristic of \(1/f\) is presented in unstable signal analysis, where f is the signal frequency.