## 5 Results  
The above section discusses the use of the frequency-Bessel transform algorithm to calculate the multi-mode dispersion energy spectrum of noise data, and this section mainly verifies the feasibility and practical application effect of this method by taking 45 sets of noise data measured in the Wuhan karst development area as an example. Figure 8 shows the dispersion energy spectrum of nine positions, from point 0 to point 40 calculated at equal intervals, with seven seismic stations as a group. In the dispersion spectrum, there is a significant development of fundamental and higher-mode energy, with a wide range of dispersion energy distribution where the fundamental dispersion is mainly developed in the frequency band below 10 Hz, and the frequency band above 10 Hz is mainly of a higher-mode energy type. On the basis of the dispersion energy spectrum, the multi-mode dispersion curves can be extracted for inversion. In the paper, the known geological data were used to establish a constrained parameter model for stratum P-wave velocity, S-wave velocity, and density, and the damped least squares algorithm was used to iteratively invert the extracted multi-mode dispersion curves, obtaining 45 sets of data curves of stratum S-wave velocity changing with depth. Finally, a S-wave velocity profile was produced. The survey line was located in the karst area, where karst is relatively developed, and the dissolution of the rock is relatively serious. The pores of karst after dissolution will be filled with a large amount of water, which is shown in the lower S-wave velocity value in the dissolved area, and not in the intact and undissolved area. As shown in Figure 9, the black curve marks the bottom interface of the Quaternary, with a maximum burial depth of 32 m and a minimum burial depth of 16 m. The S-wave velocity of the bottom interface is about 260 m/s. By tracking the value of S-wave velocity, the undulating shape of the bottom of the Quaternary can be sketched. The inversion profile marks four areas of low shear wave velocity anomalies, namely anomaly a at 25 m, anomaly b at 50 m, anomaly c at 100 m, and anomaly d at 100 m to 140 m. It can be inferred that the karst fissures in the underlying rock layers are relatively developed. At the same time, low-speed strips appear near the sections 155 m and 185 m, showing the characteristics of large burial depth in the longitudinal direction and small width in the transverse direction, which are different from the above-mentioned four anomalies. This can be assumed to be the location of karst strip development, and karst fissures are very developed. Figure 7: Joint inversion results of fundamental and higher-mode curves. (**a**) S-wave velocity structure of fundamental and higher-mode inversion. (**b**) Dispersion curve of the inversion in the fundamental and first higher-mode fitting the measured data dispersion curve. Figure 8: Multi-mode dispersion energy spectrum of measured data. Figure 9: Shear-wave velocity profile of strata by multi-mode inversion. According to the borehole data at the 110 m position of the S-wave velocity profile, it was revealed that within 4 m of the stratum, there is a miscellaneous fill layer, 4 m to 10 m is a silty clay layer, 10 m to 27 m a silty sand layer without water content, 27 m to 28 m lime dolomite (strongly weathered, mixed with purplish red gray dolomite, with a developed karst structure and numerous karst fissures forming pores in the rock layer with a pore diameter of 2-7 mm), and 28 m to 45 m gray-yellow lime dolomite with a developed karst structure and numerous dissolution fissures. It can be seen that the borehole data and the inverted shear wave velocity anomaly d in Figure 9 correspond well with each other, verifying the correctness of the interpretation.