Tomography

In order to use this methodology our tomography problem has to be set up in a similar fashion to that of fitting goals (3). This isn't necessarily straight forward. Our first problem is that tomography is a non-linear process. The standard approach in ray-based tomography is to linearize around an initial slowness model $\bf s_{0}$. Our linearized tomography operator $\bf T_{0}$ is formed by rays traced through the background slowness. We then write a linear relation between the change in slowness $\bf \Delta s$ and the change in travel-time $\bf \Delta t$.

When doing migration velocity analysis in the depth domain, we are not dealing with travel-times but instead move-out as a function of some parameter (offset or azimuth) Stork (1992). Biondi and Symes (2003) showed how for angle domain migration there is a link between travel-time error dt, local dip $\phi$, the local slowness s depth of the reflection z, the reflection angle $\theta$,and scaling $\gamma$ of the background slowness model. This relation can be written in terms of an operator $\bf D$which maps from $1.-\gamma$ to $\bf \Delta t$ and whose elements are

$$D(\theta,\phi,z,s)= \frac{z s \sin(\theta)^2} { \cos(\phi)*(\cos(\theta)^2 - \sin(\theta)^2)} .$$ (4)

For our regularization operator we can use a steering filter Clapp et al. (1997); Clapp (2001a) oriented along reflector dips. Our basic linearized fitting goals become

$$\begin{eqnarray} \bf 0&\approx&\bf r_{data}= {\bf D} {\bf \gamma} - \bf T_{0} \b... ... r_{model}=\epsilon {\bf A} (\bf s_{0} + \bf \Delta s) \nonumber .\end{eqnarray}$$ (5)

The added term in our regularization fitting goal $\bf A\bf s_{0}$is due to the fact that we want to smooth slowness not change in slowness. Clapp (2003a) and () showed that adding noise to $\bf r_{model}$ produced velocity models with what looked like thin layers that had little effect on image kinematics but noticeable effects on amplitudes.

We run into problems when we want to explore the effect of adding noise to $\bf r_{data}$.Our $\gamma$ values, and therefore our data fitting error exist in some irregular space (potentially consistent angle sampling, but irregular in space). This makes making an effective noise covariance operator difficult.