Tomography review

Tomography for migration velocity analysis is a non-linear problem that we linearize around an initial slowness model. In this discussion, I will be talking about the specific case of ray based tomography but most of the discussion is valid for other tomographic operators. We can linearize the problem around an initial slowness model and obtain a linear relation $\bf T_{}$ between the change in travel times $\bf \Delta t$ and change in slowness $\bf \Delta s$:

$$\bf \Delta t\approx \bf T_{} \bf \Delta s.$$ (6)

Certain components of the velocity model are better determined than others. As a result we need to either choose an intelligent model parameterization Cox et al. (2003) or regularize the model. I prefer the regularizing approach so that I can easily incorporate a priori information. I impose a regularization operator that tends to smooth the velocity according to reflector dip Clapp (2001b). My resulting fitting goals then become

$$\begin{eqnarray} \bf \Delta t&\approx&\bf T_{} \bf \Delta s\nonumber \\ \bf 0&\approx&\bf A(\bf s_0 + \bf \Delta s) .\end{eqnarray}$$ (7)

where $\bf A$ is a steering filter. In this paper I will do downward continuation based migration Ristow and Ruhl (1994); Stoffa et al. (1990) and then constructing angle gathers Prucha et al. (1999); Sava and Fomel (2000). Biondi and Symes (2003) and Biondi and Tisserant (2004) showed that the travel time error is related to the reflector dip $\alpha$, local slowness $\bf s_0$, depth z₀, and the velocity model scaled by $\gamma$ through  

$${\bf \Delta t}_{\rm RMO}= \frac{1-\rho}{\cos\alpha} \frac{\sin^2 \gamma}{\left(\cos^2\alpha - \sin^2\gamma\right)} s_0 z_0.$$ (8)

One of the best ways to scan over $\gamma$ is by doing Stolt Residual Migration (SRM) Sava (1999a,b). In Clapp (2002b) I showed how we can obtain better ray-based tomography results by doing SRM scanning versus simple vertical move-out analysis.