## 3 Experimental Results and Analysis
### Analysis on the Premonitory Information of Rock in AE Time-Dependent Dominant Frequency  
In this study, parameters including AE count [22; 23], RA-AF value, visual density [16], Fourier dominant frequency, and amplitude [24; 25] were extracted and analyzed.  
According to the relationship between the dominant frequency and the amplitude of red sandstone specimens with prefabricated cracks with different inclination angles, the dominant frequency is divided into three ranges, i.e., 330-400 KHz, 400-460 KHz, and 460-520 KHz, based on the distribution characteristics of the dominant frequencies. They are defined as low frequency (LF), intermediate frequency (IF), and high frequency (HF), respectively (Figure 12). Meanwhile, in the hope of distinguishing different types of AE-dominant frequency signals, five types of AE signals are defined, namely, low frequency-low amplitude (LF-LA), low frequency-high amplitude (LF-HA), intermediate frequency-low amplitude (IF-LA), intermediate frequency-high amplitude (IF-HA), and high frequency-low amplitude (HF-LA) AE signals.  
Figure 12: Band classification of the dominant frequency for red sandstone specimens with prefabricated cracks: (**a**) No. 8–30\({}^{\circ}\) and (**b**) No. 9–60\({}^{\circ}\).  
Figures 13 and 14 depict the distribution of the AE time-dependent dominant frequency and amplitude at different stages in the loading process of red sandstone specimens with 30\({}^{\circ}\) and 60\({}^{\circ}\) prefabricated cracks. At the initial stage of loading (compaction stage and elastic deformation stage), the primary micro-cracks and pores are compacted and closed, while new micro-cracks continuously emerge, develop, and expand. At this stage, the AE dominant frequency signals are distributed relatively sparsely and are mainly composed of IF-LA and LF-LA signals.  
Figures 13a and 14a mainly provide three-dimensional diagrams established based on the corresponding relationships among the dominant frequency, amplitude, and time. The dominant frequency and amplitude are frequency-domain signals converted from time-domain signals through FFT. Because frequency-domain signals do not vary with time, they appear quite similar along the time axis in Figures 13 and 14. Time-domain analysis can intuitively reflect the dominant frequency and amplitude of AE signals, while frequency-domain analysis allows for a clear observation of the frequency at which the highest amplitude occurs. In Figures 13b and 14b, the dominant frequency does not change over time. That is, both the high and low values of the dominant frequency occur throughout the loading process. However, high amplitudes only appear when the specimen is about to fail, which is a detail that cannot be obtained from time-domain signals.  
Figure 13: Distribution of AE time-dependent dominant frequencies and amplitudes in red sandstone specimens with a 30\({}^{\circ}\) prefabricated crack: (**a**) dominant frequencies, (**b**) amplitude, and (**c**) 3D distribution points.  
From the yield stage to the final failure, micro-cracks in rock continuously emerge, develop, and expand. They keep squeezing each other and finally connect into macro-cracks. Resultantly, the LF, IF, and HF dominant frequency signals are all concentrated at this stage. After stress reaches its peak strength in red sandstone, massive internal cracks expand and interconnect into macro-cracks, and macroscopic damage and deformation occur, finally leading to overall failure. Additionally, this stage primarily witnesses the appearance of a small number of IF-HA and LF-HA signals.  
In summary, different failure stages of red sandstone exhibit different types of AE dominant frequency signals during the cracking process. Thus, the types of AE dominant frequency signals can be taken as effective references for predicting rock instability deformation.