## 3 Method  
Due to the low signal-to-noise ratio of the actual collected passive seismic data, various seismic signals are mixed together ([PERSON] et al., 2018) [37]. Direct cross-correlation without extracting surface wave noise and body wave noise cannot yield high-quality virtual seismic records of surface and body waves. Additionally, surface waves and low-frequency noise in passive seismic data typically have lower frequencies, whereas body waves and high-frequency noise have higher frequencies ([PERSON] et al., 2015) [38]. Conventional bandpass filtering methods struggle to effectively separate surface wave noise from body wave noise, often damaging one part to preserve the quality of the other, and they are still prone to high-frequency or low-frequency interference. Given the excellent time-frequency characteristics of wavelet transforms, which are commonly used in seismic data processing, we choose wavelet transform for the analysis and processing of passive seismic data in this study. Additionally, the CWT lacks sufficient resolution in the time-frequency domain, resulting in inadequate precision for data processing and analysis ([PERSON], 2021) [39]. To address this limitation, we utilize SWT, which provides higher resolution in the time-frequency domain, thereby facilitating more effective identification of signals and noise and enhancing the thresholding effect. After successfully separating high signal-to-noise ratio body wave noise and surface wave noise from the raw passive data, wavelet-domain cross-correlation is applied to obtain high-quality virtual seismic records for both surface waves and body waves (Figure 2). In this study, after transforming the original passive seismic data using SWT, single-trace seismic records are selected for SWT processing. The high-frequency and low-frequency components in the time-frequency spectrum are separated, corresponding to body wave noise and surface wave noise, respectively. The background noise from the high-frequency component is selected, as surface wave energy is weak in the body wave-dominated high-frequency band, and there are also some interference events that contribute minimally to the cross-correlation results. These can be removed using wavelet thresholding, allowing the body wave signal to be preserved with minimal surface wave interference. The same approach is applied to surface wave processing. Common wavelet thresholding functions include hard and soft thresholding functions. The hard thresholding function sets all coefficients below the threshold to zero and retains those above the threshold, which often leads to significant errors. In contrast, the soft thresholding function provides smoother processing, and its formulation is as follows:  
\[\eta_{\lambda}^{\text{sfsf}}(W_{s})= \begin{cases}W_{s}-\lambda&\text{if}\quad W _{s}\geq\lambda,\\ 0&\text{if}\quad|W_{s}|<\lambda,\\ W_{s}+\lambda&\text{if}\quad W_{s}\leq-\lambda.\end{cases}  \tag{10}\]  
where \(W_{f}\) represents the wavelet coefficients and \(\lambda\) is the threshold. The threshold selection is based on the universal threshold proposed by [PERSON] (1994) [PERSON] (1994), given by \(\lambda=\sigma\sqrt{2\text{log}_{\sigma}n}\). At the same time, in order to retain more valid signals, Gaussian filtering is applied to smooth the data before thresholding:  
\[WS_{0}(\omega_{i},\tau)=(\omega_{i},\tau)*_{\omega_{i},\tau}WS_{xy}(\omega_{i },\tau) \tag{11}\]  
where \(WS_{xy}(\omega_{i},\tau)\) represents the wavelet coefficients after SWT transformation and \(*\) is the 2D convolution operator. \(WS_{0}(\omega_{i},\tau)\) is the wavelet coefficient after Gaussian filtering. Wavelet thresholding is then applied to the Gaussian-filtered wavelet coefficients to better preserve the effective signals. Through the above operations, we can obtain the processed body wave noise and surface wave noise. By cross-correlating these two components, high-quality body wave and surface wave signals can be obtained.