Tracking reflectors movements

One of the most challenging problems of reflection tomography is to track correctly the movement of reflectors caused by changes in velocity. Usually the reflectors are parametrized independently from velocity and large reflectors movement can cause instability in the inversion process. One of the potential advantages of $\left(\tau,\xi\right)$tomography over $\left(z,x\right)$ tomography is that reflectors move less in the $\left(\tau,\xi\right)$ than in the $\left(z,x\right)$ domain, and that they move more consistently with the velocity function.

This reflector movement caused by velocity perturbations can be subdivided in a residual migration component and a residual mapping component. One of the advantages of $\left(\tau,\xi\right)$ tomography is that the residual mapping is automatically taken into account by the linearization introduced in equation (17). In contrast, $\left(z,x\right)$ tomography has an additional term to take into account both the residual mapping and migration effects. In the examples shown in this paper we used the following adaptation of the expression presented by Stork and Clayton 1991 to correct for the residual mapping in $\left(z,x\right)$ tomography:  

$$\Delta t = \Delta z \left( p_{z\Downarrow} + p_{z\Uparrow} \right),$$ (19)

where $\Delta z$ is the reflector vertical movement, while $p_{z\Downarrow}$and $p_{z\Uparrow}$are respectively the vertical ray parameters of the incident and reflected rays at the reflection point. To be consistent in the comparison between $\left(\tau,\xi\right)$ domain tomography and $\left(z,x\right)$ domain tomography, we computed $\Delta z$ as a residual mapping term along the vertical path. The residual migration term could be computed by performing a residual map migration of the zero-offset arrivals. This residual migration term could be added to both $\left(\tau,\xi\right)$ domain tomography and $\left(z,x\right)$ domain tomography. For $\left(\tau,\xi\right)$ tomography the expression linking the $\Delta \tau$caused by residual migration to the corresponding traveltime perturbations would be similar to equation (19); that is,

$$\Delta t = \Delta \tau \left( p_{\tau\Downarrow} + p_{\tau\Uparrow} \right).$$ (20)