## 1 Introduction  
In the field of track tunnel engineering, engineering safety holds exceptional significance. Strengthening the real-time monitoring of tunnel rock mass stress is a common practice in track tunnel construction to prevent accidents, such as tunnel collapse [1]. Currently, microseismic detection technology is widely used in tunnel disaster recognition and crack monitoring. When cracks appear in rock, they release strain energy and propagate elastic waves to the surroundings [2; 3; 4; 5]. Crack detection based on visual and optical instrument assistance is currently the main means of daily inspection [6; 7], but the method is time consuming and labor intensive, and the detection data carry a certain degree of subjectivity and are not sensitive to early cracks. There have been many research works on crack localization and identification; some of them have focused on the study of elastic wave propagation in a single medium [8; 9], while others have analyzed the propagation and attenuation characteristics of R-waves in non-uniform media [10; 11]. Most of them have focused on the study of propagation characteristics of elastic waves and few work on specific crack localization detection techniques. In recent years, theoretical models for leak detection in pipelines in frequency-domain transient waves (pressure) have been studied [12]. Frequency-domain transient waves have great performance in terms of localization detection accuracy. In the research on crack identification, significant progress has been made in various aspects, including (1) analysis using the reverse transient method [13; 14; 15; 16], (2) integration of transient damping techniques [17; 18; 19], (3) utilization of reflection techniques in the time domain [20; 21; 22], and (4) identification based on frequency-domain responses [23; 24; 25; 26]. In this study, the aforementioned frequency-domain methods of transient waves were applied to crack localization and monitoring in tunnels. By conducting experimental simulations, the effect of localization detection can be simulated, which greatly simplifies the estimation problem of multiple cracks. Despite the aforementioned work, there are still many unresolved issues in multi-crack identification. Solving the identification and localization problem of multiple cracks is challenging; the number of cracks in practical applications was previously unknown. Previous studies have mainly focused on identifying and locating individual cracks to simplify the process. As the number of cracks increases, the computational complexity grows rapidly. As a result, a new low-cost approach was needed to make the estimation process less sensitive to the number of cracks. With the development of compressive sensing (CS) technology [27], it has been increasingly utilized in various fields. In simple terms, CS leverages the sparsity of signals in a domain to achieve accurate signal reconstruction at low sampling rates, thereby breaking the conventional Nyquist sampling criterion under specific conditions. Currently, CS is used successfully in image processing and communications [28; 29; 30; 31; 32; 33]. In the context of crack detection in tunnel rock mass within the field of railway transportation, there has been limited work. Due to the spatial sparseness of the crack locations in the whole tunnel, the transient wave model for multi-crack identification can be linearly represented in terms of crack size but not in terms of crack location [26]. Nonetheless, the actual cracks within the tunnel can be considered sparse; therefore, investigating the problem of multi-crack localization using a sparse structure became a viable approach. By establishing a transient wave model in the frequency domain for tunnel rock masses, using the geotechnical analysis software MIDAS GTS NX 2019v1.2, a transient wave model suitable for track tunnel rock masses was simulated and obtained the corresponding pressure transfer function through data analysis, which can reflect the propagation relationship.