## 2 Materials and Methods  
### Theory of the Frequency-Bessel Transform  
The spatial autocorrelation theory has been derived in detail in many references [30; 31; 32], which is considered mature for extracting the fundamental dispersion cure; this section mainly introduces the derivation process of the frequency-Bessel transform. Previous research [33] proved that the Fourier transform of the cross-correlation function between the noise data records observed at any two stations \(x_{1}\) and \(x_{2}\) on the surface at the same time is proportional to the imaginary part of [PERSON]'s function; the difference between the two is only in amplitude, and the relationship between them can be expressed as follows:  
\[C(r,\omega)\approx\text{A}\cdot\text{Im}\{G_{zz}(r,z=0,\omega)\} \tag{1}\]  
where \(C(r,\omega)\) is the Fourier transform of the cross-correlation function; A is a constant; \(r\) is the distance between the two stations; \(\omega\) is the angular frequency; and \(G_{zz}(r,z=0,\omega)\) is the surface Green's function of the vertical component. Therefore, the theoretical formula of the frequency-Bessel function transform is derived as follows:  
\[\int_{0}^{+\infty}C(r,\omega)J_{0}(kr)rdr\approx\text{A}\cdot\text{Im}\{g_{z} (z=0,\omega,k)\} \tag{2}\]  
where \(J_{0}(kr)\) is the first type of zero-order Bessel function; \(g_{z}(r,\omega,k)\) is the kernel function of [PERSON]'s function; and \(k\) is the wave number. According to the nature of the kernel function, when the wave number \(k\) is at the dispersion point, the dispersion point of the kernel function approaches infinity, the dispersion point is the singularity of the kernel function, and the value corresponds to the dispersion curve [34; 35]. The integral operation on the left side of Equation (2) can be simplified as the sum of finite terms; \(C(r,\omega)\) is expressed as:  
\[C(r,\omega)\approx a_{i}+b_{i}r \tag{3}\]  
where coefficient \(b_{i}=[C(r_{i},\omega)-C(r_{i-1},\omega)]/(r_{i}-r_{i-1})\); coefficient \(a_{i}=C(r_{i-1},\omega)+[C(r_{i},\omega)-C(r_{i-1},\omega)]r_{i-1}/(r_{i}-r_{i -1})\); and \(r_{i}\) is the distance between stations of the cross-correlation spectrum \(C(r_{i},\omega)\) in the frequency domain. According to the recursive property of Bessel function:  
\[\left\{ \begin{array}{l}\int xJ_{0}(x)dx=xJ_{1}(x)\\ \int x^{2}J_{0}(x)dx=x^{2}J_{1}(x)+xJ_{0}(x)-\int J_{0}(x)dx\end{array} \right.. \tag{4}\]  
By using Equation (4), the integral term on the left side of the frequency-Bessel transform Equation (2) can be derived as [36]:  
\[ \begin{array}{l}\int_{0}^{+\infty}C(r,\omega)J_{0}(kr)rdr\approx\sum\limits \limits_{i=1}^{N}\frac{a_{i}c}{\omega}\left\{J_{1}(\frac{\omega}{c}r)\right\} \Bigg{|}\begin{array}{l}r_{i}\\ r_{i-1}\end{array}+,\\ \frac{b_{i}c^{2}}{\omega^{2}}\{kr^{2}\}_{1}(\frac{\omega}{c}r)+rJ_{0}(x)\} \Bigg{|}\begin{array}{l}r_{i}\\ r_{i-1}\end{array}-\frac{b_{i}c^{2}}{\omega^{2}}\int_{r_{i-1}}^{r_{i}}J_{0}( \frac{\omega}{c}r)dr.\end{array}  \tag{5}\]  
According to Equation (5), the theoretical calculation of the frequency-Bessel function of the noise data is carried out, and the dispersion curves can be obtained by picking up the energy maximum in the dispersion diagram. The implementation process of this method mainly consists of the following aspects:  
Firstly, for a given seismic observation array, the spectral cross-correlation functions for all possible station pairs are calculated after preprocessing the ambient seismic records. Secondly, the spectral CCFs are ordered as a function of the inter-station distance \(r_{i}\) and subjected to the frequency-Bessel transform. The frequency-Bessel spectral map is calculated by the discrete summation formula for each given \(\omega\). Finally, with the help of image recognition algorithms, dispersion curves can be identified from images. The frequency-Bessel transform is an effective method to extract multimode dispersion curves from ambient background noise. Several local and regional applications have demonstrated the applicability of the method approach [37].