next up previous print clean
Next: Synthetic example Up: Sava and Biondi: WEMVA Previous: Wavefield scattering

Differential image perturbation

Residual migration can be used to create image perturbations ($\Delta \mathcal R$). In its simplest form, we can build $\Delta \mathcal R$ as a difference between an improved image ($\mathcal R$) and the reference image ($\mathcal R_b$)
\begin{displaymath}
\Delta \mathcal R= \mathcal R- \mathcal R_b,\end{displaymath} (9)
where $\mathcal R$ is derived from $\mathcal R_b$ as a function of the parameter $\rho$,which is the ratio of the original and improved velocities. Of course, the improved velocity map is not known explicitly, but is described indirectly by the ratio map of the two velocities.

If we denote by 1 the velocity ratio that corresponds to the background velocity model and define $\Delta \rho=\rho-1$, we can also write the discrete version of the image perturbation as
\begin{displaymath}
\Delta \mathcal R\approx \frac{\mathcal R- \mathcal R_b}{\rho-1} \Delta \rho,\end{displaymath} (10)
which can also be written in differential form as
\begin{displaymath}
\Delta \mathcal R\approx \left. \frac{d \mathcal R}{d \rho} \right\vert _{\rho=1} \Delta \rho,\end{displaymath} (11)
or, equivalently, using the chain rule, as  
 \begin{displaymath}
\Delta \mathcal R\approx \left. \frac{d \mathcal R}{d\k_z}
 \frac{d \k_z}{d \rho} \right\vert _{\rho=1} \Delta \rho,\end{displaymath} (12)
where $\k_z$ is the depth wavenumber defined for PSRM.

Equation (12) offers the possibility to build the image perturbation directly. We achieve this by computing three elements: the derivative of the image with respect to the depth wavenumber, and two weighting functions, one for the derivative of the depth wavenumber with respect to the velocity ratio parameter ($\rho$), and the other one for the magnitude of the $\Delta \rho$ perturbation from the reference to the improved image.

Firstly, the image derivative in the Fourier domain, $\frac{d \mathcal R}{d\k_z}$, is straightforward to compute in the space domain as
\begin{displaymath}
\left. \frac{d \mathcal R}{d\k_z}\right\vert _{\rho=1} = -i z \mathcal R_b.\end{displaymath} (13)
The derivative image is the imaginary part of the migrated image scaled by depth.

Secondly, we can obtain the weight representing the derivative of the depth wavenumber with respect to the velocity ratio parameter $\left(\left. \frac{d \k_z}{d \rho} \right\vert _{\rho=1} \right)$ starting from the double square root (DSR) equation written for prestack Stolt residual migration Sava (2000):
\begin{eqnarray}
\k_z&=& {\k_z}_s+ {\k_z}_r
 \nonumber \\  \nonumber 
&=& \frac{...
 ...ac{1}{2}\sqrt{ \rho^2\mu^2 - \left\vert {\bf \k}_r \right\vert^2},\end{eqnarray}
where $\mu$ is given by the expression:
\begin{displaymath}
\mu^2 = \frac{ \left[4 \left({{\k_z}_o}\right)^2 + \left(\le...
 ...s \right\vert\right)^2 \right]}
 {16\left({{\k_z}_o}\right)^2},\end{displaymath} (14)
$\left({\k_z}_s,{\k_z}_r\right)$ are the depth wavenumbers and $\left(\left\vert {\bf \k}_s \right\vert,\left\vert {\bf \k}_r \right\vert\right)$ are the spatial wavenumbers for the sources and receivers, respectively.

The derivative of $\k_z$ with respect to $\rho$ is
\begin{displaymath}
\frac{d \k_z}{d \rho} 
= \rho\left(\frac{\mu^2}{4{\k_z}_s} + 
 \frac{\mu^2}{4{\k_z}_r} \right),\end{displaymath} (15)
therefore we can write
\begin{displaymath}
\left. \frac{d \k_z}{d \rho} \right\vert _{\rho=1}
 = \frac{...
 ...rac{\mu^2}{2\sqrt{\mu^2 -\left\vert {\bf \k}_r \right\vert^2}}.\end{displaymath} (16)

Finally, $\Delta \rho$ can be picked from the set of residually migrated images at various values of the parameter $\rho$ Sava (2000). One criterion that could be used to estimate $\Delta \rho$is the flatness of the angle-domain image gathers, which can be measured through derived parameters such as stack power, semblance or differential semblance.


next up previous print clean
Next: Synthetic example Up: Sava and Biondi: WEMVA Previous: Wavefield scattering
Stanford Exploration Project
7/8/2003