BACKGROUND

Following the methodology of Clapp and Biondi (1999), I will begin by considering a regularized tomography problem. I will linearize around an initial slowness estimate and find a linear operator in the vertical traveltime domain $\bf T_{}$ that relates change in slowness $\bf \Delta s$with our change in traveltimes $\bf \Delta t$.We will write a set of fitting goals,

$$\begin{eqnarray} \bf \Delta t&\approx&\bf T_{} \bf \Delta s\nonumber \\ \bf 0&\approx&\epsilon \bf A\bf \Delta s,\end{eqnarray}$$ (1)

where $\bf A$ is our steering filter operator Clapp et al. (1997) and $\epsilon$ is a Lagrange multiplier. However, these fitting goals don't accurately describe what we really want. Our steering filters are based on our desired slowness rather than change of slowness. With this fact in mind, we can rewrite our second fitting goal as:

$$\begin{eqnarray} \bf 0&\approx&\epsilon \bf A\left( {\bf s_0} + \bf \Delta s\right) \\ -\epsilon \bf A{\bf s_0} &\approx&\epsilon \bf A\bf \Delta s.\end{eqnarray}$$ (2) (3)

Our second fitting goal can not be strictly defined as regularization but we can still do a preconditioning substitution Fomel et al. (1997), giving us a new set of fitting goals:

$$\begin{eqnarray} \bf \Delta t&\approx&\bf T_{} \bf A^{-1}\bf p\nonumber \\ - \epsilon \bf A{\bf s_0} &\approx&\epsilon \bf I\bf p .\end{eqnarray}$$ (4)

The trouble is how to estimate $\bf \Delta t$. Previously I have calculated semblance at various hyperbolic moveouts. I then picked the moveout corresponding to the maximum semblance. To calculate $\Delta t$ I converted my picked moveout parameter back to a depth error $\Delta z$, which is converted into a travel time. Following the methodology of Stork (1992),  

$$\Delta t =\cos(\alpha) \cos(\beta) v \Delta z ,$$ (5)

where, $\alpha$ is the local dip, $\beta$ is the opening angle at the reflection point, and v is the local velocity. This approach is effective in areas which are generally flat and have a sufficient offset coverage, but as shown in Biondi and Symes (2002) it runs into problems when these conditions aren't met.

A real data example of this problem can be seen in Figure 1. Note the ``hockey stick'' behavior seen at `A'. If we follow the procedure of Stork (1992) we get unreasonably large travel time errors. Clapp (2002) shows that if we use these points for back projection, we get too large of velocity changes, which can lead to instability of the tomography problem.

 

gathers
Figure 1 Every 5th gather to the left edge of a salt body. Note the coherent, ``hockey stick'' behavior within `A'.