# Efficient Method for Enhancing Reverse-Time Migration Images Using Vertical Seismic Profiling Data
## 2 Theory and Methodology
### Theory of Etc-Ic  
The choice of imaging conditions also significantly influences the final imaging results. Currently, the imaging conditions used in pre-stack RTM imaging are primarily of two types: excitation time and cross-correlation. The cross-correlation imaging condition was first proposed by [PERSON] (1971) [25]. This involves performing zero-lag cross-correlation between the source and receiver wavefields at each time step to obtain the imaging result for each point. The specific imaging condition is as follows:  
\[I_{(x,z)}=\int S(x,z,t)\cdot R(x,z,T_{max}-t)dt, \tag{1}\]  
where \(S(x,z,t)\) and \(R(x,z,t)\) represent the source and receiver wavefields at location \((x,z)\) and time \(t\), respectively, \(t\) represents time, and \(T_{max}\) represents the maximum recording time at the receiver. From a numerical perspective, the result of the cross-correlation imaging condition is the product of the energies of the source and receiver wavefields at the imaging point. However, this product does not have any direct physical significance. To address this issue, a source-normalized cross-correlation imaging condition can be used as follows:  
\[I_{norm}(x,z)=\frac{\int S(x,z,t)\cdot R(x,z,T_{max}-t)dt}{\int S(x,z,t)\cdot S (x,z,t)dt}. \tag{2}\]  
By compensating for the attenuation of seismic waves during propagation, the imaging results of the source-normalized cross-correlation imaging condition become more interpretable and accurately reflect subsurface features, thereby improving the reliability of seismic imaging. Another widely used imaging condition is the excitation-time imaging condition. This method does not require storing the entire source wavefield but does require obtaining the first-arrival travel time from the source wavefield to each imaging point. During back-propagation of the receiver wavefield, the amplitude at the first-arrival travel time is taken as the imaging result for that point. The specific process is defined as follows:  
\[I_{(x,z)}=R(x,z,T_{max}-t), \tag{3}\]  
where \(t\) represents the travel time of the first arrival from the source wavefield to the imaging point. The excitation-time imaging condition can be considered a special case of the cross-correlation imaging condition. Because the source wavefield is not stored, its value in the cross-correlation imaging condition is replaced with a unit amplitude. In this case, the excitation-time imaging condition can be expressed as  
\[I_{(x,z)}=\int_{t}^{t+\epsilon}R(x,z,T_{max}-\tau)\cdot 1d\tau, \tag{4}\]  
where \(\epsilon\) represents an arbitrarily small number and \(t\) represents the first-arrival travel time from the source wavefield to the imaging point. In the RTM process, only a few sources generate effective seismic records that are reflected back to the receiver position for each imaging point. Most sources do not contribute to the imaging. In surface seismic methods, this does not result in the loss of imaging quality because the sources and receivers are randomly distributed. However, because of the specificity of the VSP observation system, multiple imaging processes can lead to noise accumulation. Therefore, it is necessary to introduce additional methods to eliminate the noise effects. Based on the above analysis, it is essential to propose optimized imaging conditions suitable for WVSP RTM. Building on the two imaging conditions mentioned earlier, this study proposes an excitation-time-limited cross-correlation imaging condition expressed as follows:  
\[I_{ETC-IC}(x,z)=\frac{\int_{t-\Delta}^{t-\Delta+T}S(x,z,t)\cdot R(x,z,T_{max}- t)dt}{\int_{t-\Delta}^{t-\Delta+T}S(x,z,t)\cdot S(x,z,t)dt} \tag{5}\]  
Here, \(T\) represents the period of the seismic wavelet used, and \(Delta\) is a customizable window to balance the potential errors in the first-arrival travel times. The ETC-IC retains only portions of the source and receiver wavefields that are related to the first-arrival travel times of the seismic wavelet to each imaging point, typically within one period.