## 2 Theory and Methodology
### Theory of Common-Receiver Reverse Time Migration  
In contrast to conventional RTM methods, the common-receiver reverse time migration (CRRTM) method utilizes primary reflections instead of seismic wavelets to propagate at source locations. To ensure the validity of the results, forward propagation was replaced with back-propagation. Simultaneously, seismic wavelets are forward-propagated from the receiver location to generate a receiver wavefield. Finally, the source and receiver wavefields were calculated based on the imaging conditions to obtain the imaging results [28]. Figure 3 presents a simple model to demonstrate the effectiveness of CRRTM. The model consists of a horizontal reflector, with the source location represented by a red dot and the observation well and receiver indicated by a green rectangle and a triangle, respectively. Without a loss of generality, the source-normalized cross-correlation imaging condition is applied in this study to process the wavefields, as expressed in Equation (1) [29]. Figure 1: Schematic diagram of the propagation path for primary and multiple waves in the VSP survey. Figure 2: Mechanism of artifact formation during the multiples’ migration. \[I(x,z)=\frac{\int S(x,z,t)\cdot R(x,z,T_{max}-t)dt}{\int S(x,z,t)\cdot S(x,z,t)dt}. \tag{1}\]  
According to [PERSON]'s principle, the actual path of elastic wave propagation in a medium is the one that minimizes the travel time from the starting point to the endpoint. This implies that the propagation path from the source to the receiver remains unchanged for the wavefield components that contribute to accurate imaging when the excitation positions of the seismic wavelet and seismic data are interchanged. That is, in both conventional RTM and CRRTM, the seismic wavelet and seismic data travel the same propagation path and take the same amount of time to travel from the source location to the reflection point R1 and from R1 to the receiver. This ensures that, in the CRRTM method, the source and receiver wavefields at the reflection point also satisfy the imaging condition described in Equation (1). In contrast, the seismic wavelet was excited at the source location and propagated downward to the reflection point R1. This process can be expressed as follows:  
\[D(z_{1},z_{0})=X(z_{1},z_{0})S(z_{0}) \tag{2}\]  
Here, \(D(z_{1},z_{0})\) represents the seismic data at location R1, \(X(z_{1},z_{0})\) denotes the response matrix of the subsurface medium for seismic wave propagation from the source to the reflection point, and \(S(z_{0})\) represents the source matrix. Subsequently, the seismic data at R1 are reflected and propagated upward, where they are received at the receiver location:  
\[D(z_{2},z_{1})=D(z_{1},z_{0})R(z_{1})X(z_{2},z_{1}) \tag{3}\]  
Here, \(D(z_{2},z_{1})\) represents the seismic data received at the receiver location, \(X(z_{2},z_{1})\) denotes the response matrix of the subsurface medium for seismic wave propagation from the reflection point to the receiver, and \(R(z_{1})\) is the reflection coefficient of the reflector. The effect of the geophone on the seismic data is ignored in this context. Inserting Equation (3) into Equation (2) yields the following:  
\[D(z_{2},z_{0})=X(z_{1},z_{0})S(z_{0})R(z_{1})X(z_{2},z_{1}) \tag{4}\]  
During the reverse extrapolation of the seismic data, the seismic data received at the receiver location were moved to the source location for reverse propagation. Theoretically,  
Figure 3: Propagation path for seismic wave in a simple model.