# Unveiling Accurate Seismic Imaging through the Advanced Target-Oriented Kirchhoff Migration Method
## 2 Methodology  
### Target-Oriented Kirchhoff Migration  
Theoretical assumptions assume that each point within the subsurface functions as a scattering point. Waves propagate from the source point, travel downward to the subsurface point, and, subsequently, propagate back to the receiver point. Consequently, we can say that the wavefield recorded at the receivers are coming from every scatter point in the subsurface. However, the recorded events are often not directly coming from reflector beneath the receivers; instead, they might come from geological formations situated away from the point of recording. Hence, the needs of seismic migration to accurately relocate the recorded events to their correct spatial position within the subsurface. Seismic migration can be performed either before (pre-stack) or after stacking (post-stack). Kirchhoff migration is acknowledged as one of the most effective methods for pre-stack migration and is widely employed within the industry [44]. This is because of its algorithmic versatility, which permits its application in both 2D and 3D data for pre-stack or post-stack processing, whether in the time or depth domains. Additionally, the Kirchhoff migration algorithm is capable of handling steep dips [29; 44; 45; 46], adapting to irregular sampling [19; 47], accommodating lateral and vertical velocity variations [48], and, most importantly, delivering dependable results while maintaining efficiency [2; 49]. However, Kirchhoff images provide less accuracy and suffer with migration artifacts compared to other migration methods, especially with complex and challenging structural-imaging problems [29; 46; 49]. Kirchhoff migration was originally introduced by [PERSON] in 1978 [50] building upon the migration algorithm framework established by [PERSON] (1954) [51]. Kirchhoff migration employs the Huygens principle as its foundation, involving the summation of amplitudes over diffraction hyperbolas and assigning the resultant summed amplitude to the apex. Mathematically, the Kirchhoff migration process is expressed as an integration over the recorded data, involving Kirchhoff summation conducted in both the source and receiver dimension. Let \(U\), the wavefield recorded at the receiver location, the image of the subsurface, \(P\) at \((x,z)\) location (for 2D) with time \(t\) is represented by:  
\[P(x,z,t)=-\frac{1}{2\pi}\iint_{A}W\frac{\partial}{\partial t}U(x_{s},x_{r},t _{D})dx_{s}dx_{r}, \tag{1}\]  
where \(A\) is the migration aperture, \(W\) is the weight function, \(x_{s}\), \(x_{r}\) are the coordinates of source and receiver, respectively, and \(t_{D}\) is the total traveltime from source to scatter point and from scatter point to receiver. In this study, we implemented target-oriented Kirchhoff migration by combining seismic redatuming with Kirchhoff migration. The Kirchhoff summation will be carried out only at the target area below the specified datum level. Figure 1 shows the change of the source and receiver geometry after seismic redatuming when we have the target area smaller than the original area. A crucial aspect to consider when working with seismic redatuming is the aperture. It is important to recognize that the aperture of the redatumed data can indeed differ from that of the original surface data. This is because the redatuming process involves changing the geometry of the acquisition surfaces. Depending on the specific approach adopted for redatuming and the imaging objectives, adjustments to the aperture may be made to optimize the imaging outcomes for the desired target depth or location. Figure 1: An illustration displays the transformation of geometry, transitioning from surface acquisition to a redatumed virtual acquisition when the target area is smaller. It is important to note that due to changes in the source and receiver geometry and size, the migration aperture size also undergoes alteration.