3-D media

In 3-D media, the rayparameters for each of the sources and receivers have two components: p_x and p_y. For general anisotropic media, this suggests that we need to solve for two parameters for each of the source and receiver rays in resolving a 2-D stationary point problem. This results in a fairly complicated process that can be avoided by relying on polar coordinates.

In VTI media, unlike more complicated anisotropies, the group and phase angles for a given ray are confined to the same vertical plane that includes the source or the receiver and the image point. Simply stated, the VTI model with respect to the horizontal plane is isotropic. We can simplify the 3-D problem by using azimuth instead of multi-component rayparameters. As a result, only one parameter need to be solved for each of the source and receiver rays, and this rayparameter has the same form given in the 2-D case [equation (12)].

The four stationary points in 3-D media are:

$$p_{sx} = p_s \cos\phi_s, \,\,\,\, p_{sy} = p_s \sin\phi_s,$$

$$p_{gx} = p_g \cos\phi_g, \,\,\,\, p_{gy} = p_g \sin\phi_g,$$

where $\phi_s$ and $\phi_g$ is the source-to-image-point and receiver-to-image-point azimuth, respectively. The polar rayparameters (p_s and p_g) are computed using equation (12) with

$$y=2\sqrt{\left(h+(x-x_0)\right)^2+\left(y-y_0\right)^2}$$

for p_g, and

$$y=2\sqrt{\left(h-(x-x_0)\right)^2+\left(y-y_0\right)^2}$$

for p_s.

Therefore, the total traveltime is given by  

$$t(\tau,x,h,v,\eta) = {\frac{\tau}{2} \,\left( {\sqrt{1 - {... ...\,{{{p_s}}^2}}}}} \right) } + 2\,{p_g}\,y_g + 2\,{p_s}\,y_s,$$ (16)

where $y_s=2\sqrt{\left(h-(x-x_0)\right)^2+\left(y-y_0\right)^2}$, and $y_g=2\sqrt{\left(h+(x-x_0)\right)^2+\left(y-y_0\right)^2}$.Equation (16) can be used to perform prestack time migration on 3-D datasets.