Image perturbation by residual migration

Residual migration can also be used to create an image perturbation. In its simplest form, we can build it as a difference between an improved image ($\mathcal R$) and the reference image ($\mathcal R_o$)

$$\Delta \mathcal R= \mathcal R- \mathcal R_o,$$ (9)

where $\mathcal R$ is derived from $\mathcal R_o$ as a function of the parameter $\gamma$,which is the ratio of the original and improved velocities Sava (2000). Of course, the improved velocity map is unknown explicitly, but it is described indirectly by the ratio map of the two velocities.

If we define $\Delta \gamma=\gamma-1$, we can also write the discrete version of the image perturbation as

$$\Delta \mathcal R\approx \frac{\mathcal R- \mathcal R_o}{\gamma-1} \Delta \gamma,$$ (10)

equation which can be written in differential form as

$$\Delta \mathcal R\approx \left. \frac{d \mathcal R}{d \gamma} \right\vert _{\gamma=1} \Delta \gamma,$$ (11)

or, equivalently, using the chain rule, as  

$$\fbox {$ \displaystyle \Delta \mathcal R\approx \left. \fra... ...frac{d \k_z}{d \gamma} \right\vert _{\gamma=1} \Delta \gamma,$}$$ (12)

where $\k_z$ is the depth wavenumber defined for PSRM.

The Equations (7) and (12) are very similar, which comes at no surprise since they effectively represent the same thing: the perturbation of the image given a perturbation of the slowness field, or equivalently, a perturbation of the $\gamma$ (ratio) field. We will use Equation (12) to create the image perturbation, which we will then backproject in the slowness space using the adjoint of the WEMVA equation (7).

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 ($\gamma$), and the other one for the magnitude of the $\Delta \gamma$ 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

$$\left. \frac{d \mathcal R}{d\k_z}\right\vert _{\gamma=1} = -i z \mathcal R_o.$$ (13)

The derivative image is nothing but the imaginary part of the migrated image, scaled by depth.

Secondly, we can obtain the weighting representing the derivative of the depth wavenumber with respect to the velocity ratio parameter, $\left. \frac{d \k_z}{d \gamma} \right\vert _{\gamma=1}$, starting from the double square root (DSR) equation written for prestack Stolt residual migration Sava (2000):

$$\begin{eqnarray} \k_z&=& {\k_z}^s+ {\k_z}^r \nonumber \\ \nonumber &=& \frac{... ...{1}{2}\sqrt{ \gamma^2\mu^2 - \left\vert {\bf \k}_r \right\vert^2},\end{eqnarray}$$

where $\mu$ is given by the expression:

$$\mu^2 = \frac{ \left[4 \left({{\k_z}^o}\right)^2 + \left(\le... ...s \right\vert\right)^2 \right]} {16\left({{\k_z}^o}\right)^2}.$$ (14)

The derivative of $\k_z$ with respect to $\gamma$ is

$$\frac{d \k_z}{d \gamma} = \gamma\left(\frac{\mu^2}{4{\k_z}^s} + \frac{\mu^2}{4{\k_z}^r} \right),$$ (15)

therefore

$$\left. \frac{d \k_z}{d \gamma} \right\vert _{\gamma=1} = \f... ...rac{\mu^2}{2\sqrt{\mu^2 -\left\vert {\bf \k}_r \right\vert^2}}.$$ (16)

Finally, $\Delta \gamma$ can be picked from the set of residually migrated images at various values of the parameter $\gamma$ Sava (2000). The main criterion that should be used is the flatness of the angle-domain image gathers, although in principle other derived parameters, such as stack power or semblance, can be used as well.