# Inspiration for Seismic Diffraction Modelling, Separation, and Velocity in Depth Imaging
## Materials and Methods  
### Diffraction Modelling  
The principle of the diffraction theory is mainly based on the subsurface behaving similarly to acoustic media. This means that shear waves are not considered for diffraction studies; only primary waves cause the production of diffraction with low reflectivity and constant velocity. The scalar wave equation is used to satisfy the wave equation. The reflector is an ensemble of individual elements and the response is the sum or integration of each element. The zero-offset solution is available and further planning to extended for the nonzero-offset.  
When the source and receiver are separated, the envelope of arrival times (diffraction curve) has a different shape due to the increased travel times. This can be explained by the \"double square root equation (DSR)\" for travel time, which is:  
\[T_{L}=\sqrt{(\frac{Tm}{2})^{2}+[\frac{L-\frac{x}{2}}{V}]^{2}}+\sqrt{(\frac{Tm }{2})^{2}+[\frac{L+\frac{x}{2}}{V}]^{2}} \tag{1}\]  
where \(Tm\) is the time, \(V\) is the velocity, and \(x\) is the separation between the normal incident (single square root equation) and the separated shot receiver (double square root equation). The curve is greatest at the observation location over the point source. At location \(L\), the velocity becomes smooth and Equation (1) simplifies into:  
\[T_{L}=\sqrt{Tm^{2}+\frac{L^{2}}{V^{2}}} \tag{2}\]  
The above is the usual expression used for reflection times from a horizontal reflector. This observation forms the basis of the velocity determination method.  
Using Equation (2) in MATLAB, we observed the behavior of diffraction hyperbola by taking different background velocity values, such as 2000, 3500, and 5000 m/sec, as shown in Figure 2.  
Figure 2: Diffraction hyperboloids with constant velocity at: (**a**) 2000 m/sec, (**b**) 3500 m/sec, and (**c**) 5000 m/sec. The curvature of the hyperboloids is spread out with an increase of velocity.  
We model the 3D diffraction curves on a point diffractor of different velocities. Figure 2 explains the hyperbolic pattern of diffraction concerning offset and depth; as the depth increases, the curvature of the hyperbola is smothered. The effect of velocity can be found by comparing Figure 2a-c. The scale of all parts of the hyperbolas is the same on the \(X\)-axis (offset) and \(Y\)-axis (depth), but there is a change of the velocity in the point diffractor. Velocity directly affects the diffraction hyperbolic curvature, such as in a high-velocity area. The diffraction pattern is more complex and more difficult to interoperate during processing. Separation of diffraction events, which are complex in deeper regions, will be more difficult because of their similarity with reflection data.  
Further, we extend our algorithm from constant velocity to variable velocity. Increasing and decreasing velocity diffraction behavior can be observed in Figure 3a,b, respectively. The shape of the diffraction curve at 1000 m/sec velocity changes as the depth changes. This analysis indicates that the diffraction curve is not only dependent on velocity but also on depth.