Conclusions

We introduced the extension of Stolt prestack residual migration for converted waves. Our new operator involves the selection of three parameters in order to update the image.

To help in the memory and disk space necessary for the implementation of our operator, we also derived approximations that reduce the number of free parameters to a two. The most appropriate way of reducing the number of parameters is by freezing one of them. Our experience suggests freezing $\gamma_0$at the RMS value of the ratio between the P and the S migration velocities.

In constant velocity, we proved that we can recover the image obtained with an initial migration that uses an inaccurate velocity model. Therefore, we can update a migration with constant two-velocities model using our new operator. We can also update an image obtained with a depth variant velocity. However, the refocusing performed by residual migration is only approximate.

The advantages of having an operator to update converted waves images allow us to extrapolate our ability to handle multiple mode data. We hope that it will lead to more accurate methods for performing velocity analysis for converted waves.

A

Solving the first equation of:

 

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

for $\omega^2$ we have:

$$2k_z = \sqrt{{\omega^2}{s_{p_0}^2}-k_s^2}+\sqrt{{\omega^2}{s_{s_0}^2}-k_g^2},$$

and

$$\left( 2k_z-\sqrt{\omega^2 s_{p_0}^2 -k_s^2} \right)^2 = \left( \sqrt{\omega^2 s_{s_0}^2 -k_g^2} \right)^2.$$

Squaring the previous equation and isolating the remaining square root we obtain:

$$4k_z^2 +{\omega^2}{s_{p_0}^2}-{\omega^2}{s_{s_0}^2}+k_g^2 -k_s^2 =4k_z \sqrt{\omega^2 s_{p_0}^2 -k_s^2}.$$

Squaring the previous equations, grouping common terms, and setting equal to zero, we get:

$$\omega^4 \left( s_{p_0}^2 -s_{s_0}^2 \right)^2 + \omega^2 \l... ...ht] + \left(4k_z^2 +k_g^2 -k_s^2 \right)^2 + 16k_z^2 k_s^2 = 0.$$

Solving for $\omega^2$ we obtain

 

$$\omega^2 = \frac{s_{s_0}^2 \left(4k_z^2 +k_g^2 -k_s^2 \right... ..._g^2 - s_{s_0}^4 k_s^2}}{\left(s_{p_0}^2 -s_{s_0}^2 \right)^2}.$$ (6)

We select the negative sign of the radical as the final solution for $\omega^2$,as discussed in Appendix B.

Substituting the result of $\omega^2$ in the second equation of relation (7), we obtain the relationship for residual prestack migration for converted waves.

In order to demonstrate this fact, we need to simplify the dispersion relation for $\omega^2$ in terms of $\gamma_0 = \frac{v_{p_0}}{v_{s_0}}$, s_p0 and s_s0 depending on the source or receiver SSR equation.

$$\left(s_{p_0}^2 -s_{s_0}^2 \right)^2 = s_{p_0}^4 \left(1- \gamma_0^2 \right)^2$$

therefore,

$$\omega^2 = \frac{\gamma_0^2 \left(A \right) + \left( B \righ... ...- \gamma_0^4 k_s^2}}{s_{p_0}^2 \left(1- \gamma_0^2 \right)^2} ,$$

since

$$k_{z_m}=\frac{1}{2} \left ( \sqrt{\omega^2 s_{p_m}^2 -k_s^2}+\sqrt{\omega^2 s_{s_m}^2 -k_g^2} \right ) ,$$

calling

 

$$\kappa_0^2 = \frac{\gamma_0^2 A+B-4k_z^2 \sqrt{\gamma_0^2 C-k_g^2 - \gamma_0^4 k_s^2}}{\left(1- \gamma_0^2 \right)^2} ,$$ (7)

we finally note that $\omega^2 = \frac{\kappa_0^2}{s_{p_0}^2}$.

We then have equation (3), which is:

$$k_{z_m}=\frac{1}{2} \left ( \sqrt{\gamma_0^2 \rho_s^2 \kappa... ...k_g^2} + \sqrt{\rho_p^2 \kappa_0^2 - k_s^2} \right ). \nonumber$$

B

We want to evaluate equations (3), (5) and (6) when v_p = v_s, or equivalently, when $\gamma_m = \gamma_0 = 1$.

It is possible to see from equation (9) that for the particular case of $\gamma_0 = 1$ we have a division by zero. Since we have a division by zero, we need to analyze the equation when we approach to $\gamma_0 \rightarrow 1$.For this purpose, we are going to use L'Hôpital. Therefore, we need to have a zero also in the numerator, which is possible for any value of k_z if, and only if, we choose the negative sign as a solution in equation (8).

Referring to equation (9) as $\kappa_0^2 = f(\gamma_0) / g(\gamma_0)$, and applying the L'Hôpital, we calculate the derivative with respect to $\gamma_0$ to $f(\gamma_0)$ and $g(\gamma_0)$.

We derive

$$\frac{\partial}{\partial \gamma_0} f(\gamma_0) = 2 \gamma_0 ... ...left(4k_z^2 +k_g^2 + k_s^2 \right) - k_g^2- \gamma_0^4 k_s^2}}.$$

On the other hand, the derivative of the denominator is:

$$\frac{\partial}{\partial \gamma_0} \left ( \left (\gamma_0^2 -1 \right)^2 \right ) = 4 \left (\gamma_0^2 -1 \right) \gamma_0.$$

Analyzing the limit for $\gamma_0 \rightarrow 1$ of the $f\prime (\gamma_0) / g\prime (\gamma_0)$,we still have a $\frac{0}{0}$ relation, which means we must re-apply L'Hôpital,

$$\frac{\partial}{\partial \gamma_0} f^{\prime}(\gamma_0) = 2 ... ...s^2 \right) - k_g^2 - \gamma_0^4 k_s^2 \right )^{\frac{1}{2}}}.$$

On the other hand, the second derivative of the denominator is:

$$\frac{\partial}{\partial \gamma_0} g^{\prime}(\gamma_0) = 8 \gamma_0^2.$$

We finally have

$$\lim_{\gamma_0 \to 1} \frac{f(\gamma_0)}{g(\gamma_0)} = \lim... ...\gamma_0)}{\frac{\partial^2}{\partial \gamma_0^2} g(\gamma_0)}.$$

Therefore, we have $\kappa_0^2$ for $\gamma_0 \rightarrow 1$ reduces to:

$$\left [ 4k_z^2 + \left ( k_g - k_s \right )^2 \right ] \left[ 4k_z^2 + \left ( k_g + k_s \right )^2 \right ]$$

which is the expression for the conventional case of PP waves.