# Unveiling Accurate Seismic Imaging through the Advanced Target-Oriented Kirchhoff Migration Method
## 2 Methodology
### Target-Oriented Kirchhoff Migration
Hence, due to this, it subsequently changed Equation (1) to become target-oriented where the image point \(P\) is now known as \(P_{to}\):  
\[P_{to}(x,z,t_{to})=-\frac{1}{2\pi}\iint_{Ao}W\frac{\partial}{\partial t}U_{r}(x _{so},x_{ro},t_{Do})dx_{so}dx_{ro}, \tag{2}\]  
The coordinate of \(P_{to}\) is still at \((x,z)\) because we are referring to same image point. Since the position of source and receiver is altered nearer to the image point which represented by \(x_{so}\), \(x_{ro}\), the time is also changed to \(t_{o}\), \(t_{Do}\) for total traveltime and \(Ao\) for the aperture for target-oriented area. Now, the wavefield corresponds to the redatumed data, denoted as \(U_{r}\), which is retrieved through seismic redatuming method. As for the implementation of target-oriented imaging, the proposed workflow is illustrated as in Figure 2. The idea is to develop a reliable and efficient migration approach that specifically focusing on reservoir level or targeted areas, while concurrently achieving a migrated image quality that is at least equivalent. The work conducted in this study pertains to the 2D scenario, yet it can be seamlessly extended to 3D implementation. Overall, the aim is to reduce the total computational costs with 3D pre-stack depth migration, rendering it more practical and advantageous in practice. The first step involved establishing the virtual depth of the target area. The region above this virtual depth, often referred to as the datum, is termed the overburden area, while the region below is referred to as the underlying area. Next, the process involves retrieving the redatumed shot gathers from the initially recorded shot gathers through the application of seismic redatuming. For this step, any redatuming method can be employed to obtain the redatumed shot gathers, which will then use as input for the migration process. The subsequent step involved computing the migration for the target area situated below the datum. Despite being recognized for its straightforward algorithm, Kirchhoff migration is said to have expensive computation because of the travel-time calculation. In addition, we used the fast sweeping method [52; 53; 54; 55] for better accuracy of travel-time calculation for Kirchhoff migration. Indeed, conducting full-scale travel-time calculations  
Figure 2: The proposed workflow for target-oriented seismic migration, combing the seismic redatuming with Kirchhoff migration for better target-oriented imaging. This study only focused on the kinematic correction of the target-oriented migrated image while the angle-domain CIG will be used in the next study related to dynamic correction. can undoubtedly be a resource-intensive computation. However, the proposed target-oriented migration approach has the advantage of mitigating the overall computational cost for Kirchhoff migration. Given the smaller size of the target area, there was consequently a reduction in the computation cost required for travel time calculations. In the final step, we illustrate that there were two approaches available, as shown in Figure 3. One option is to migrate the entire underlying area, while another alternative is to simultaneously migrate several small target areas. Here, we effectively utilized the advantages of Kirchhoff migration for target-oriented imaging. By incorporating target-oriented imaging using seismic redatum, we effectively addressed the limitations of Kirchhoff migration, particularly in the realm of complex and challenging structural imaging scenarios. In this study, our focus was on the kinematic correction aspect of target-oriented imaging, while the dynamic correction for target-oriented imaging will be addressed in a subsequent study.