Linearized image perturbations

A linearized image perturbation is computed using a prestack residual migration operator ($\Kop$) using a relation like

$$\Delta \RR\approx \left. \Kop^{'} \right\vert _{\rho=1} \lb \widetilde{\RR}\rb \Delta \rho \;.$$ (107)

The operator $\Kop$ depends on the scalar parameter $\rho$which is a ratio of the velocity to which we residually migrate and the background velocity (storm). The background image corresponds to $\rho=1$.

Using the chain rule of differentiation, we can write  

$$\Delta \RR\approx \left. \frac{d \Kop}{d \kzz} \frac{d \kzz... ...} \right\vert _{\rho=1} \lb \widetilde{\RR}\rb \Delta \rho \;,$$ (108)

where $\kzz$ is the depth wavenumber defined for prestack Stolt residual migration.

dR offers the possibility to build the image perturbation directly, 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, $\frac{d \Kop }{d \kzz}$, is straightforward to compute in the space domain as

$$\left. \frac{d \Kop}{d\kzz}\right\vert _{\rho=1} \lb \widetilde{\RR}\rb= -i z \widetilde{\RR}\;.$$ (109)

The derivative image is represented by 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, $\left. \frac{d \kzz}{d \rho} \right\vert _{\rho=1}$, starting from the double square root (DSR) equation written for prestack Stolt residual migration (87):

&=& _s+ _r
&=& 12 ^2 ^2 - |k_s|^2 + 12 ^2 ^2 - |k_r|^2 ,

where $\mu$ is given by the expression:

$$\mu^2 = \frac{ \lb 4 \lp{{\kzz}_0}\rp^2 + \lp {\bf k}_r- {\b... ...p^2 + \lp {\bf k}_r+ {\bf k}_s\rp^2 \rb } {16{{\kzz}_0}^2} \;.$$ (110)

The derivative of $\kzz$ with respect to $\rho$ is

$$\frac{d \kzz}{d \rho} = \rho \lp \frac{\mu^2}{4{\kzz}_s} + \frac{\mu^2}{4{\kzz}_r} \rp \;,$$ (111)

therefore, at $\rho=1$, we can write:

$$\left. \frac{d\kzz}{d \rho} \right\vert _{\rho=1} = \frac{\... ...t^2}} + \frac{\mu^2}{2\sqrt{\mu^2 -\vert{\bf k}_r\vert^2}} \;.$$ (112)

For common-azimuth data, the double square root (DSR) equation written for prestack Stolt residual migration (87) is:

$$\left\{ \bea{l} {\kzz}_x= {{\kzz}_x}_s + {{\kzz}_x}_r \\ \kzz = \sqrt{ {\kzz}_x^2 - {k_m}_y^2} \;, \eea \right .$$ (113)

where ${{\kzz}_x}_s$ and ${{\kzz}_x}_r$ are given by the expressions

_x_s &=& 12 ^2 _c^2 - k_m_x-k_h_x^2
_x_r &=& 12 ^2 _c^2 - k_m_x+k_h_x^2 ,

and $\mu_c$ is given by the expression:

$$\mu_c^2=\frac{ \lb {{\kzz}_0}^2 + {k_m}_y^2 + {k_h}_x^2 \rb ... ...}^2 + {k_m}_y^2 + {k_m}_x^2 \rb } {{{\kzz}_0}^2+{k_m}_y^2} \;.$$ (114)

The derivative of $\kzz$ with respect to $\rho$ is

d d &=& d d _x d _xd
&=& _x _c^24_x_s + _c^24_x_r .

At $\rho=1$, we can write: