# Inspiration for Seismic Diffraction Modelling, Separation, and Velocity in Depth Imaging
## 3 Results and Discussion
### Importance of Velocity for Seismic Depth Imaging  
Velocity plays a key role in various aspects of seismic methods, especially in migration, as a correct velocity is needed for proper stacking of the diffracted energy from the flanks of the hyperbola to the apex, as well as for the reflection. A velocity field is essential for field development planes, including but not limited to:  
Figure 8: **(a)** Marmousi velocity model with a velocity range from 1650 to 4600 m/sec. Three major faults containing unconformity and overburden cause complexity in the velocity model for anticline structures. **(b)** Synthetic shot gather data are generated using finite difference modeling.  
Figure 9: Seismic gather data **(a)** before the NMO correction and **(b)** after the NMO correction. Events are slightly straight at the true reflection points.  
1. Gain derivation of wavefield divergence and attenuation;
2. Multiple eliminations, wavefield continuation, or Radon demultiple processes;
3. Seismic imaging for structures and reservoirs;
4. Lithology prediction through Vp/Vs (Velocity of P-wave and S-wave) and elastic inversion;
5. Pressure prediction;
6. Porosity prediction from acoustic impedance (AI);
7. Time-epth (T-X) conversion;
8. Volumetric calculation;
9. Net sand map plots.  
Velocity estimation is a key processing parameter. The sensitivity and accuracy can be significantly improved with the use of a long spread survey design and by consuming higher-order terms. An anisotropic model may be required in certain geologic situations.  
To enhance the processing and imaging accuracy, a velocity analysis was performed using two different methods:  
1. Semblance analysis;
2. Hybrid travel time.  
Semblance analysis is the conventional way of calculating velocity during the NMO correction. It allows for the refinement of seismic data. This is done by developing a velocity spectrum display to determine the velocity through different layers at depth. The velocity data are used to correct the curves of the hyperbolas and create a flat line, where all points are at an equal depth.  
The hybrid travel time method is described in Figure 10, which shows the raytracing phenomena in a complex and heterogeneous medium with 100 rays. Paraxial optics can be defined as ray-tracing accomplished within the limits of very small ray angles and heights. This allows us to make a number of simplified assumptions that make the raytracing arithmetic considerably easier.  
The eikonal differential equation is used for the elementary mathematical models, which describe the travel time (eikonal) propagation in a velocity model. The benefits of this method in comparison with ray tracing techniques are the capability to work in consistent model grids, the wide-ranging coverage of the receiver space, and the reasonable numerical robustness. A common implementation of the finite-difference eikonal equation is calculating the first-arrival travel times though frequency-dependent enrichment.  
The eikonal equation, describing the travel time propagation in an isotropic medium, has the following form:  
\[(\
abla\tau)^{2}=n^{2}(x,y,z) \tag{12}\]  
Figure 10: Trajectories of 100 rays emitted by a source point in a heterogeneous medium. Shadow zone and multipathing effects can be seen [30].  
where \(\tau(x,y,z)\) is the travel time from the source to the receiver point with coordinates (\(x\),\(y\),\(z\)) and n is the slowness at that point (the velocity v equal to 1/n) [31].  
This method calculates each travel time table using paraxial raytracing, then in the shadow zone (Figure 10) the travel times are computed by solving the eikonal equation.