# Efficient Method for Enhancing Reverse-Time Migration Images Using Vertical Seismic Profiling Data
## 3 Numerical Analysis  
### Noise Analysis of W-VSP RTM  
This section uses a simple model as an example to demonstrate the sources of noise in traditional VSP imaging and how noise affects the final imaging quality. The seismic data used in this study were calculated using the finite-difference method, retaining only the upgoing P-wave components and setting a perfectly matched layer (PML) to eliminate boundary reflections.  
Here is the actual velocity model used, along with the VSP observation system. The model consists of \(400\times 600\) grids with vertical and horizontal intervals of 5 m. Additionally, a 100-grid PML was added around the model to absorb boundary reflections. The velocity within the absorption layer was consistent with the velocity at the initial model boundary. The model contained a flat layer at \(z=1500\) m, and the S-wave and P-wave velocities were constant between the different layers. The density at all the positions in the model was 2000 kg/m\({}^{3}\). Figure 3 shows the velocity model with the PML absorption layer added, where the area outside the yellow dashed box represents the PML.  
The VSP observation system used was as follows. There were 300 sources spaced at fixed intervals of 10 m, located at the surface. A vertical observation well was positioned at \(x=2000\) m, containing 120 numerical geophones evenly distributed from \(z=800\) m to \(z=1400\) m at intervals of 5 m. A Ricker wavelet with a central frequency of 20 Hzwas used as the excitation source. The maximum recording time was 3 s with a sampling interval of 0.5 ms. The arrangement of sources and receivers ensured stable seismic data acquisition. The spatial and temporal intervals, along with the central frequency of the seismic wavelet, were optimized for computational efficiency and result quality. Too small an interval drastically reduces computational efficiency, while too large an interval leads to numerical dispersion and other issues. These parameters must also ensure the stability of the finite difference method.  
Figure 4a shows a shot gather of the seismic data received by the geophones at \(z=800\) m. After the signal separation, only the upgoing P-waves were retained for subsequent imaging, as shown in Figure 4b.  
In traditional W-VSP RTM methods, the seismic data recorded by the receiver are back-propagated, whereas the seismic wavelet is forward-propagated from the source position to generate the source wavefield. Figure 5 illustrates the cross-correlation between the source and receiver wavefields at various time intervals.  
Figure 4: Shot gather in the simple model: (**a**) original data; (**b**) separated upgoing data.  
Figure 3: Simple P-wave velocity model, where the yellow dotted box represents the boundary of the PML.  
During the cross-correlation process between the source and receiver wavefields, the direct wave of the source wavefield and reflected wave of the receiver wavefield exhibit a strong correlation along the entire propagation path. This is a major source of severe low-frequency noise in the imaging results. Simultaneously, the source wavefield is incorrectly correlated with the direct wave of the receiver wavefield, leading to arc-like artifacts near the observation well in the imaging results. Unlike low-frequency noise, these artifacts have the same frequency as the effective part of the signal, which makes them difficult to remove using methods such as Laplace filtering. Figure 6 shows the single-shot imaging results of traditional W-VSP RTM.