Using the chain rule of differentiation, we can write
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 (), and the other one for the magnitude of the perturbation from the reference to the improved image.
Firstly, the image derivative in the Fourier domain, , is straightforward to compute in the space domain as
Secondly, we can obtain the weighting representing the derivative of the depth wavenumber with respect to the velocity ratio parameter, , 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 is given by the expression:
The derivative of with respect to is
_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 is given by the expression:
d d _x
&=& _x _c^24_x_s + _c^24_x_r .
At , we can write: