Above is the image generated by Salvo for a simple impulse response. The Li filter was used which very well corrects for approximations in the algorithms development. Although the f-x migration is only accurate up to 65 degrees, the Li filter compensates "perfectly" in a constant velocity media. The phase errors along the diagonals have been corrected as can be seen by the circular response shown on the right.

Absorbing Boundary Conditions

         
    Fig. 1: No Absorbing BC           Fig. 2: Clayton & Engquist                 Fig. 3: Xu's ABC

The impulse response problem is very useful in testing absorbing boundary conditions (ABCs). In Fig. 1, the image created with Salvo without ABCs is shown. In this prestack run, it should be noted that not only do we see the primary reflection but we also see a stronger secondary reflection from the boundary. This secondary reflection is due to a cross-correlation between the source wave reflection from the boundary and the primary arrival of the receiver wave (Additionally, the cross-correlation between the primary arrival of the source wave and the receiver wave reflection off the boundary thus accounting for the strength of the signal.)

In Fig. 2, the impulse response is shown with Clayton and Enquist's ABCs. The primary reflection has been removed but the secondary reflection is still present and is weaker in strength. Although the reflected waves are greatly reduced, they still correlate with the primary arrival waves and produce an image.

In Fig. 3, the impulse response using the ABCs of Liyang Xu are shown. These boundary conditions are similar to Clayton and Enquist's but use optimized coefficients to further reduce the reflected energy.

Curtis C. Ober

Last modified: October 9, 1998