SYNTHETIC TESTS

The impulse response of the migration algorithm is shown in Figure . To test the effect of a limited aperture, we generated a simple synthetic zero-offset section containing a single hyperbolic event, as shown in Figure . A conventional Kirchhoff migration, using all the traces, is shown in Figure , and a migration where the aperture is limited to the nearest 15 traces (the section contains 75 traces) is shown in Figure . The resolution in the limited aperture migration is not as good. The same two plots are shown for the least-squares case in Figures  and . The limited aperture result has improved resolution compared to the non least-squares case.

 

impulse
Figure 3 Impulse response of migration algorithm. This is a single slice from the 3D cube, taken along the t=t₀ plane.

 

model
Figure 4 Synthetic dataset used to test the migration scheme. Single point scatterer, constant velocity of 3 km/sec. 75 traces with a trace spacing of 25 meters.

 

stack
Figure 5 Conventional Kirchhoff migration of synthetic dataset. Single slice from 3D cube at t=t₀.

 

stacklim
Figure 6 Conventional Kirchhoff migration of synthetic dataset, but migration aperture has been limited to 15 of 75 traces.

 

lsmig
Figure 7 Least-squares Kirchhoff migration. Gives a better focus because effect of aperture has been removed.

 

lsmiglim
Figure 8 Least-squares Kirchhoff migration, aperture limited to 15 of 75 traces.

Some noise appears at the top of the least-squares sections in Figures  and . These first few samples correspond to the earliest t₀ 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.