Image transformation

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 $\it{f}$, followed by imaging:

$$\mathcal P_{z }= {\bf I}_z\left[\it{f}_z\left(\u_{z } \right) \right].$$ (12)

In compact matrix form, we can write this relation as:

$$\mathcal P= {\bf I}\it{f}\left(\u \right).$$ (13)

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

Examples of transformation functions are:

• $\it{f}(x)=x-t$ where t is a known target. A WEMVA method based on this criterion optimizes

  $$\mathcal P_{z }:= {\bf I}_z\left[ \u_{z }- \mathcal T_{z } \right],$$ (14)

  where $\mathcal T_{z }$ 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).
• $\it{f}(x)=Dx$ where D is a known operator. A WEMVA method based on this criterion optimizes

  $$\mathcal P_{z }:= {\bf I}_z\left[ {\bf D}_z\left[\u_{z } \right] \right].$$ (15)

  If ${\bf D}$ 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

$$\mathcal P_{z }:= {\bf I}_z\left[\AA_z\left[\u_{z } \right] - {\bf B}_z\left[\mathcal T_{z } \right] \right],$$ (16)

or in compact matrix form as  

$$\mathcal P:= {\bf I}\left(\AA \u - {\bf B}\mathcal T\right),$$ (17)

where the operators $\AA$ and ${\bf B}$ are known and take special forms depending on the optimization criterion we use. For example, $\AA={\bf 1}$ and ${\bf B}={\bf 1}$ for TIF, and $\AA={\bf D}$ and ${\bf B}={\bf 0}$ for DSO. ${\bf 1}$ stands for the identity operator, and ${\bf 0}$ stands for the null operator.