PS angle-domain transformation

  This appendix presents the transformation from the subsurface-offset domain into the angle-domain for converted-wave data. The derivation follows the well-known equations for apparent slowness in a constant-velocity medium in the neighborhood of the reflection/conversion point. This derivation is consistent with the one presented by ();(); and ().

The expressions for the partial derivatives of the total traveltime with respect to the image point coordinates ($m_\xi,h_\xi,z_\xi$) are as follows ():

$$\begin{eqnarray} \frac{\partial t}{\partial m_\xi} &=& S_s \sin{\beta_s} + S_r \... ...tial t}{\partial z_\xi} &=& S_s \cos{\beta_s} + S_r \cos{\beta_r}.\end{eqnarray}$$ (1)

Where S_s and S_r are the slowness (inverse of velocity) at the source and receiver locations, respectively. Figure  illustrates all the angles in this discussion. The angle $\beta_s$ is the direction of the wave propagation for the source, and the angle $\beta_r$ is the direction of the wave propagation for the receiver.

Throughout these set of equations, we obtain:

$$\begin{eqnarray} -\frac{\partial z_\xi}{\partial h_\xi} &=& \frac{ S_r \sin{\bet... ...} + S_r \sin{\beta_r} } { S_s \cos{\beta_s} + S_r \cos{\beta_r} }.\end{eqnarray}$$ (2)

At this step, I define two angles, $\alpha$and $\gamma$, to relate $\beta_s$ and $\beta_r$as follows:

 

$$\alpha=\frac{\beta_r+ \beta_s}{2}, \;\;\;\;\;\ {\rm and} \;\;\;\;\;\ \theta=\frac{\beta_r- \beta_s}{2},$$ (3)

as I discussed in Chapter 3, the angles $\alpha$ and $\theta$,for the case of converted-wave data, are the half-aperture angle and the pseudo-geological dip, respectively.

Following the change of angles suggested on equation A-3, and by following basic trigonometric identities, we can rewrite equations A-2 as follows:

$$\begin{eqnarray} -\frac{\partial z_\xi}{\partial h_\xi} &=& \frac{ \tan{\theta} ... ...hcal{S}\tan{\theta} } { 1 - \mathcal{S}\tan{\theta} \tan{\alpha} }\end{eqnarray}$$ (4)

where,

$$\mathcal{S}=\frac{S_r-S_s}{S_r+S_s}=\frac{\gamma(m_{\xi},z_{\xi})-1}{\gamma(m_{\xi},z_{\xi})+1},$$ (5)

I introduce the following notation in equation A-4:

$$\begin{eqnarray} \tan{\theta_0}&=& -\frac{\partial z_\xi}{\partial h_\xi}, \nonumber\\ \d &=& -\frac{\partial z_\xi}{\partial m_\xi}.\end{eqnarray}$$ (6)

In the notation A-6, $\theta_0$ is the pseudo-reflection angle, and $\d$ is local image-dips. The bases for this definition resides in the conventional isotropic single-mode PP case. For this case, the pseudo-reflection angle is the reflection angle, and the field $\d$ represents the geological dip (). Using the notation A-6 into equation A-4, I present the equations

$$\begin{eqnarray} \tan{\theta_0}&=& \frac{ \tan{\theta} + \mathcal{S}\tan{\alpha}... ...cal{S}\tan{\theta} } { 1 - \mathcal{S}\tan{\theta} \tan{\alpha} }.\end{eqnarray}$$ (7) (8)

Following basic algebra, equation A-8 reduces to

 

$$\tan{\alpha} = \frac { \d - \mathcal{S}\tan{\theta} } { 1 - \mathcal{S}\d \tan{\theta} }.$$ (9)

Substituing equation A-9 into equation A-7, and following basic algebraic manipulations, we obtain equation  in Chapter 3.