To test the steering methodology versus a more tradition isotropic smoothing approach we created a synthetic velocity model with four reflectors (Figure 3). To simulate picked traveltimes we shot rays through the correct model using a paraxial ray tracing code Rekdal and Biondi (1994) and found ray pairs where Snell's Law was obeyed at the reflectors.

overlay-vel-cor
Initial model with reflectors superimposed.
Figure 3 |

rays-vel0
Starting model and reflector position with
model rays superimposed.
Figure 4 |

We use as our starting guess a *v*(*z*) velocity
model calculated by averaging over *x* at each depth. We map migrated the
correct depth surfaces according to our initial guess at the velocity model
to obtain our first guess at their position
(Figure 4).
We then shot and matched rays using the initial model and
map-migrated reflector position.
The difference between the modeled reflection times and the correct reflection
times (Figure 5) was then used as input to
our tomography problem fitting goals (6).

Figure 5

We performed two non-linear iterations of
tomography using both the inverse Laplacian
and steering filters, oriented along the map migrated reflector position
(Figure 6), as our preconditioner.
As Figure 7
shows, the Laplacian model estimate has already produced
an unwanted isotropic
anomaly to explain the *w*-shaped pattern in the time differentials of reflector
3 and 4 (Figure 5). On the other hand, the steering
filter preconditioned result has introduced velocity perturbations
that follow our reflector geometry.

angles
Angle (from horizontal) of the steering filters.
Figure 6 |

To test whether or not the steering filter approach would eventually fall
into the same trap as the Laplacian preconditioned scheme we performed
two more non-linear iterations. Figure 8 shows the results
of all four non-linear iterations using the steering filters. Note how
the dome like shape is developing with successive iterations
and the low velocity
doublet (at 3 km depth and approximately at *x*=10 km) is diminishing.

Figure 7

7/5/1998