Building the steering filters

For constructing the steering filters I used the same methodology as described in Chapter . I first calculated the dip along the nine non-salt reflectors and then interpolated to the entire model space. Figure 7 shows the interpolated dip field and the result of applying $\frac{1}{\bf A \bf A'}$ to random noise. As you can see the preconditioning operator tends to create low frequency changes fairly flat in the left portion of the model and more ``U'' shaped changes to the right of the salt.

 

amp-vel0
Figure 7 The left panel is the dip field used for the first iteration of tomography. The right panel shows the result of applying $\frac{1}{\bf A \bf A'}$ to random noise.

The salt and salt boundary added additional complications. The sediment velocity abutting the salt dome is significantly different from the salt velocity. Using fitting goals () would create a smooth transition from the sediment layers to the salt velocity instead of the sharp contrast that often occurs when salt intrudes into sediments of significantly lower velocity. As a result it was necessary to make a slight modification to the tomography fitting goals.

I introduced a new diagonal operator $\bf V$, which is large where we want model smoothness and becomes smaller as we approach the salt boundary. Adding this operator to the model styling goal and then applying the same preconditioning trick we end up with

$$\begin{eqnarray} \bf \Delta t&\approx&\left({\bf T_{\tau,ref}}-{\bf T_{\tau,ray}... ...onumber \\ - \epsilon \bf A{\bf s_0} &\approx&\epsilon \bf I\bf p.\end{eqnarray}$$ (1)

The salt itself was an additional complication. Normally salt velocity is fairly constant. Reflector continuity disappears (and therefore can not be used reliably by the tomography operator) as we get close to the salt boundary and in lower portions of the model. Very few of the raypaths used in tomography pass through the salt structure. As a result the model fitting goal would be dominated by the model styling goal and instead of a constant velocity function we would get unrealistic smooth variation in the salt velocity. To counter this problem, I followed the common practice of not allowing the salt velocity to vary.