** Next:** Objective function
** Up:** Theory of wave-equation MVA
** Previous:** Wavefield perturbations

Migration velocity analysis is based on estimating the velocity
that optimizes certain properties of the migrated images.
In general, measuring such properties involves making
a transformation to the extrapolated wavefield by some function
, followed by imaging:
| |
(12) |

In compact matrix form, we can write this relation as:
| |
(13) |

The image is subject to optimization from
which we derive the velocity updates.
Examples of transformation functions are:

- where
*t* is a known target.
A WEMVA method based on this criterion optimizes
| |
(14) |

where stands for the target wavefield.
For this method, we can use the acronym TIF
standing for *target image fitting*
Biondi and Sava (1999); Sava and Fomel (2002).
- where
*D* is a known operator.
A WEMVA method based on this criterion optimizes
| |
(15) |

If is a differential semblance operator,
we can use the acronym DSO standing for
*differential semblance optimization*
Stolk and Symes (2002); Symes and Carazzone (1991).

In general, both examples presented above belong to a
family of affine functions that can be written as
| |
(16) |

or in compact matrix form as
| |
(17) |

where the operators and are known and
take special forms depending on the optimization criterion we
use. For example, and for TIF,
and and for DSO.
stands for the identity operator, and
stands for the null operator.

** Next:** Objective function
** Up:** Theory of wave-equation MVA
** Previous:** Wavefield perturbations
Stanford Exploration Project

11/11/2002