Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Some noise appears at the top of the least-squares
sections in Figures and .
These first few samples correspond to the
earliest *t _{0}* values (hyperbolas with the largest
amount of moveout). The noise
does not appear to be due to frequency-domain wraparound,
or aliasing,
and damping the least-squares method doesn't seem to help.
The noise is sufficiently strong, particularly away from
the single image plane shown in the figures, that it is
a cause for concern.
We hope to come up with a solution in the near future,
but weren't able to do so in time for this report.
This problem doesn't take away from the main result, as shown
in the figures, that the focusing of migration is less
affected by the limited aperture in the least-squares case.

The cost of this method is considerable. The conjugate
gradient algorithm runs for an average of about twenty
iterations on each frequency. This means multiplication
by the *L* or *L*^{H} matrix an average of forty times,
versus a single multiplication for the non least-squares case.
The aperture compensation provided by least-squares must
be considerable if this method is to be worth the extra
cost. If least-squares allows us to migrate using fewer
traces, however, then the system of equations will be
smaller, and the extra cost will not be so large.
Another way to reduce the cost is to consider a target
oriented scheme, where only those hyperbola shapes appropriate
for the target zone are used for the third axis of the
migration cube. This would again make the least-squares
problems smaller, and reduce the cost.

11/17/1997