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Residual migration can be used to create image perturbations ().
In its simplest form, we can build as a difference between an
*improved* image () and the *reference* image ()
| |
(9) |

where is derived from as a function of the parameter ,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 ,
we can also write the discrete version
of the image perturbation as

| |
(10) |

which can also be written in differential form as
| |
(11) |

or, equivalently, using the chain rule, as
| |
(12) |

where 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 (),
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

| |
(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
starting from
the double square root (DSR)
equation written for prestack Stolt residual migration
Sava (2000):

where is given by the expression:
| |
(14) |

are the depth wavenumbers
and are the spatial wavenumbers for the sources
and receivers, respectively.
The derivative of with respect to is

| |
(15) |

therefore we can write
| |
(16) |

Finally, can be picked from the set of residually migrated images at
various values of the parameter Sava (2000).
One criterion that could be used to estimate 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:** Synthetic example
** Up:** Sava and Biondi: WEMVA
** Previous:** Wavefield scattering
Stanford Exploration Project

7/8/2003