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# WEMVA objective function

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 after wavefield extrapolation to the migrated image using a generic differentiation function characterizing image imperfections
 (56)
where is the imaging operator applied to the extrapolated wavefield .In compact matrix form, we can write this relation as:
 (57)
The image is subject to optimization from which we derive the velocity updates.

Two examples of transformation functions are:

where u denotes an extrapolated wavefield and is a known target wavefield. A WEMVA method based on this criterion optimizes
 (58)
where stands for the target wavefield. For this method, we can use the acronym TIF standing for target image fitting (10; 88; 89; 93). where is a known operator. A WEMVA method based on this criterion optimizes
 (59)
If is a differential semblance operator, we can use the acronym DSO standing for differential semblance optimization (105; 97).

In general, such transformations belong to a family of affine functions that can be written as
 (60)
or in compact matrix form as
 (61)
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. With the definition in affine.z, we can write the objective function J as: Js &=& 12_,m,h || ||^2
&=& 12_,m,h || _z_z- _z_z||^2 ,

where s is the slowness function, and stand respectively for depth, and the midpoint and offset vectors. In compact matrix form, we can write the objective function as:
 (62)

In the Born approximation, the total wavefield is related to the background wavefield by the linear relation
 (63)
If we can replace the total wavefield in the objective function objective, we obtain
 (64)
objective.linear describes a linear optimization problem, where we obtain by minimizing the objective function
 (65)
where , and .The operator is constructed based on the Born approximation (56), and involves the pre-computed background wavefield through the background medium. A discussion on the implementation details for operator is presented in Appendix C. The convex optimization problem defined by the linearization in objective.linear can be solved using standard conjugate-gradient techniques.

Since, in most practical cases, the inversion problem is not well conditioned, we need to add constraints on the slowness model via a regularization operator. In these situations, we use the modified objective function
 (66)
Here, is a regularization operator, is a weighting operator on the data residual, and is a scalar parameter which balances the relative importance of the data residual () and of the model residual ().

Next: WEMVA operator Up: Wave-equation migration velocity analysis Previous: Recursive wavefield extrapolation
Stanford Exploration Project
11/4/2004