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 $\it{f}$characterizing image imperfections

$$\PP\atzo= \it{f}{\Mop_z\lb \UU\atzo \rb} \;,$$ (56)

where $\Mop$ is the imaging operator applied to the extrapolated wavefield $\UU$.In compact matrix form, we can write this relation as:

$$\PP = \it{f}\lp \Mop{\UU} \rp \;.$$ (57)

The image $\PP$ is subject to optimization from which we derive the velocity updates.

Two examples of transformation functions are:

$\it{f}(u)=u-\bar{u}$ where u denotes an extrapolated wavefield and $\bar{u}$ is a known target wavefield. A WEMVA method based on this criterion optimizes

$$\PP\atzo= \Mop_z\lb \UU\atzo \rb - \Mop_z\lb {\TT}\atzo \rb \;,$$ (58)

where ${\TT}\atzo$ stands for the target wavefield. For this method, we can use the acronym TIF standing for target image fitting (10; 88; 89; 93). $\it{f}(u)=\Dop\lb u \rb$ where $\Dop$ is a known operator. A WEMVA method based on this criterion optimizes

$$\PP\atzo= \Dop_z\lb \Mop_z\lb \UU\atzo \rb \rb.$$ (59)

If $\Dop$ 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  

$$\PP\atzo= \Aop_z\lb \Mop_z\lb \UU\atzo \rb \rb - \Bop_z\lb \Mop_z\lb {\TT}\atzo \rb \rb,$$ (60)

or in compact matrix form as  

$$\PP = \Aop \Mop \UU - \Bop \Mop \TT,$$ (61)

where the operators $\Aop$ and $\Bop$ are known and take special forms depending on the optimization criterion we use. For example, $\Aop=\Iop$ and $\Bop=\Iop$ for TIF, and $\Aop=\Dop$ and $\Bop=\Zop$ for DSO. $\Iop$ stands for the identity operator, and $\Zop$ 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 $\zz,{\bf m},{\bf h}$ stand respectively for depth, and the midpoint and offset vectors. In compact matrix form, we can write the objective function as:  

$$J\lp s \rp= \frac{1}{2} \Vert \Aop \Mop \UU - \Bop \Mop \TT \Vert^2 \;.$$ (62)

In the Born approximation, the total wavefield $\UU$ is related to the background wavefield $\widetilde{\UU}$by the linear relation

$$\UU \approx \widetilde{\UU}+ \Gop \Delta s\;.$$ (63)

If we can replace the total wavefield in the objective function objective, we obtain  

$$J\lp s \rp = \frac{1}{2} \Vert \Aop \Mop \widetilde{\UU}- \Bop \Mop \TT + \Aop \Mop \Gop \Delta s\Vert^2 \;.$$ (64)

objective.linear describes a linear optimization problem, where we obtain $\Delta s$ by minimizing the objective function  

$$J \lp \Delta s\rp = \Vert \Delta \RR- \Lop \Delta s\Vert ^2 \;,$$ (65)

where $\Delta \RR=-\lp \Aop \Mop \widetilde{\UU}- \Bop \Mop \TT \rp$, and $\Lop=\Aop \Mop \Gop$.The operator $\Lop$ 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 $\Lop$ 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  

$$J \lp \Delta s\rp = \Vert \Wop \lp \Delta \RR- \Lop \Delta s\rp \Vert^2 + \epsilon^2 \Vert \Rop \Delta s\Vert^2 \;.$$ (66)

Here, $\Rop$ is a regularization operator, $\Wop$ is a weighting operator on the data residual, and $\epsilon$ is a scalar parameter which balances the relative importance of the data residual ($\Wop \lp \Delta \RR- \Lop \Delta s\rp$) and of the model residual ($\Rop \Delta s$).