Kinematic equations

This section describes the kinematic equation that transforms a subsurface offset-domain CIG to an opening-angle-domain CIG, for the converted-mode case. The derivation follows the well-known equations for apparent slowness in a constant-velocity medium in the neighborhood of the reflection/conversion point. Our derivation is consistent with those presented by Fomel (1996);Sava and Fomel (2000); and Biondi (2005).

The expressions for the partial derivatives of the total traveltime with respect to the image point coordinates are as follows Rosales and Rickett (2001a):

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

Where S_s and S_r are the slowness (inverse of velocity) at the source and receiver locations. 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. Through these set of equations, we obtain:

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

We define two angles, $\alpha$and $\gamma$, to relate $\beta_s$ and $\beta_r$ as follows:  

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

 

angles Figure 1 Angle definition for the kinematic equation of converted mode ADCIGs

The meaning of the angles $\alpha$ and $\gamma$ will become clear later in the paper; for now, we will refer to $\gamma$ as the full-aperture angle. Through the change of angles presented on equation (3), and by following basic trigonometric identities, we can rewrite equations (2) as follows:

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

where,

$$\mathcal{S}=\frac{S_r-S_s}{S_r+S_s}=\frac{\phi-1}{\phi+1},$$ (5)

and $\phi$ is the velocity ratio, as for example the P-to-S velocity ratio. This leads to quadratic equations for $\tan{\alpha}$ and $\tan{\gamma}$ as follows:

$$\begin{eqnarray} \left [ \frac{\partial z_\xi}{\partial m_\xi} \mathcal{S}- \fr... ...l h_\xi} \mathcal{S}- \frac{\partial z_\xi}{\partial m_\xi} &=& 0.\end{eqnarray}$$ (6)

Each equation has two solutions, which are:

$$\begin{eqnarray} -\tan{\gamma} &=& \frac{\mathcal{S}^2 -1 \pm \sqrt{ (1-\mathca... ...S}- \frac{\partial z_\xi}{\partial m_\xi} \mathcal{S}^2 \right]}.\end{eqnarray}$$ (7)

The first of equation (7) provides the transformation from subsurface offset-domain CIG into angle-domain CIG for the converted-mode case. This theory is valid under the assumption of constant velocity. However, it remains valid in a differential sense in an arbitrary-velocity medium, by considering that $h_\xi$ is the subsurface half offset. Therefore, the limitation of constant velocity is on the neighborhood of the image. For $\mathcal{S}(m_\xi,z_\xi)$, it is important to consider that every point of the image $I(z_\xi,m_\xi,h_\xi)$is related to a point on the velocity model with the same coordinates.

In order to implement this equation, we observe that this can be done by an slant-stack transformation as presented on Figure . Note that the contribution along the midpoints is a correction factor needed in order to perform the transformation. This allows us to do the transformation from SODCIGs to ADCIGs including the lateral and vertical variations of $\mathcal{S}$.

 

sketch Figure 2 Slant stack angle transformation from SODCIGs to ADCIGs. This transformation allows lateral and vertical variation of $\mathcal{S}$.

 

• A Fourier domain look
• Transformation into independent angles