Theory

Stolt 1996 first introduced prestack residual migration. Sava 1999 reformulated Stolt residual migration in order to handle prestack depth images. This section presents the extension of Sava (1999) for two different wavefields, therefore, two different velocities. We present this extension for converted waves data, where the P to S conversion occurs at the reflector. Although the formulation involves P-velocities and S-velocities, its application is not limited to converted waves only. Rosales and Biondi (2001) present a possible application for imaging under salt edges.

Residual prestack Stolt migration operates in the Fourier domain. Considering the representation of the input data in shot-geophone coordinates, the mapping from the data space to the model space takes the form

 

$$k_z=\frac{1}{2} \left ( \sqrt{\frac{\omega^2}{v_p^2}-k_s^2}+\sqrt{\frac{\omega^2}{v_s^2}-k_g^2} \right ),$$ (1)

where k_s, k_g, v_p, and v_s stand for, respectively, the source and geophone wavenumbers, and the P and S velocities.

In residual prestack Stolt migration for converted waves, we attempt to simultaneously correct the effects of migrating with two inaccurate velocity fields.

Supposing that the initial migration was done with the velocities v_0p and v_0s, and that the correct velocities are v_mp and v_ms, we can then write

 

$$\left\{\begin{array} {l} k_{z_0}=\frac{1}{2} \left ( \sqrt{\... ...2}{v_{ms}^2}-k_g^2} \right ) . \\ \nonumber\end{array} \right.$$

Solving for $\omega^2$ in the first equation of (2) and substituting it in the second equation of (2), we obtain the expression for prestack Stolt depth residual migration for converted waves [equations (3) and/or (4)] (see Appendix A, for details in derivation)

 

$$\begin{array} {r} k_{z_m}=\frac{1}{2} \sqrt{ \rho_p^2 \overl... ...ma_0}^2 \overline{\overline{{\kappa_0}^2}} - k_g^2},\end{array}$$ (2)

 

$$\begin{array} {r} k_{z_m}=\frac{1}{2} \sqrt{ \rho_p^2 \overl... ...ma_m}^2 \overline{\overline{{\kappa_0}^2}} - k_g^2},\end{array}$$ (3)

where $\overline{\overline{{\kappa_0}^2}}$ is the transformation kernel and is defined as

$$\overline{\overline{{\kappa_0}^2}}= \frac{4({\gamma_0}^2 +1)... ...2 - {k_g}^2)+4{\gamma_0}^2 {k_{z_0}}^2}}{({\gamma_0}^2 - 1)^2},$$

and $\rho_p=\frac{v_{0p}}{v_{mp}}$, $\rho_s=\frac{v_{0s}}{v_{ms}}$, and $\gamma=\frac{v_p}{v_s}$.

In equation (2) it appears that Stolt residual migration for converted waves depends on four parameters: v_0p, v_0s, v_mp, v_ms. These four degrees of freedom can be reduced to three ($\rho_p$, $\rho_s$ and $\gamma$), as seen in equations (3) and (4) and demonstrated in Appendix A. This is important, because a three parameters search for updating converted waves images is simpler than a four parameters search. However, it would be useful to further reduce the number of parameter to two. Assuming that the v_p/v_s ratio is the same after and before the residual migration process, it is possible to simplify equations (3) and/or (4) into a two parameter equation:

 

$$\begin{array} {r} k_{z_m}=\frac{1}{2} \sqrt{ \rho_p^2 \overl... ... \rho_p^2 \gamma^2 \overline{{\kappa_0}^2} - k_g^2},\end{array}$$ (4)

where the transformation kernel takes the form

$$\overline{{\kappa_0}^2}= \frac{4(\gamma^2 +1){k_{z_0}}^2 + (... ...2 {k_s}^2 - {k_g}^2)+4\gamma^2 {k_{z_0}}^2}}{(\gamma^2 - 1)^2}.$$

If we just specify two different ratios in both square roots of Sava's 1999 formulation we have

 

$$\begin{array} {r} k_{z_m}=\frac{1}{2} \sqrt{ \rho_p^2 \kappa... ...^2} +\frac{1}{2} \sqrt{ \rho_s^2 \kappa_0^2 -k_g^2},\end{array}$$ (5)

where the transformation kernel has the same form as the one of PP waves:

$$\kappa_0^2 = \frac{\left[4k_{z_0}^2+(k_g-k_s)^2\right]\left[4k_{z_0}^2+(k_g+k_s)^2\right]}{16 k_{z_0}^2}.$$

Equation (6) shows another way of doing prestack residual migration for converted waves. Although equations (5) and (6) may look similar because they depend on only two parameters, the transformation kernels ($\kappa_0$ and $\overline{{\kappa_0}}$)are different. Equation (6) has a similar transformation kernel as the conventional PP prestack residual migration, while equation (5) presents a kernel deduced for the case of converted waves.

It is important to note that all three equations (3), (5), and (6) reduce to the same expressions in the limit when of v_s tends to v_p, or $\gamma$ tends to 1. All of them reduce to the simple case of prestack residual migration for conventional PP data. Appendix B demonstrates this result.