Conventional Sources Least-squares Migration and Deblurring

Figure a shows the 2D prestack Kirchhoff migration image (color scale boosted to show deep structures) for a conventional acquisition geometry of 320 individual shots with 320 receivers per shot. To reduce the artifacts, a non-stationary preconditioner (also denoted as a deblurring filter in Aoki and Schuster (2009)) is applied to the Kirchhoff migration image to give the result shown in Figure b. It is referred to as the deblurred image.

Comparison of the deblurred image and non-deblurred images shows that the deblurred image has a more balanced reflectivity amplitude, which means that amplitude weakening due to geometric spreading is compensated. The migration artifacts are also suppressed in the deblurred image. However, the deblurring filter also introduces some high-frequency noise into the deblurred image because it only approximates the inverse Hessian (see Appendix A for details). In the end, the filter is effective for deblurring Kirchhoff migration images, but it comes with the price of adding high-frequency noise. A more effective deblurring filter (Yu et al., 2006) can be used but comes with added computation cost.

Figure 2.3: (a) Kirchhoff migration image for conventional sources data; (b) KM image after deblurring (deblurred image); (c) Least-squares migration image after 30 CG iterations; (d) Preconditioned least-squares migration image after 30 DCG iterations.

To summarize the overall effect of the deblurring filter, Figure  depicts the convergence curves for both standard (CG) and deblurred LSM (it is referred to as DCG). Here, the CG result after one iteration is equivalent to the Kirchhoff migration image, and the first iteration result of DCG represents the deblurred image. I can see that the deblurring filter reduces the data residual by 52%, in spite of the high-frequency noise it introduced. It is not used after a few iterations and allows the least-squares migration to reduce the remaining noise.

Figure 2.4: Normalized data residual plotted against iteration number. The line with stars indicates the convergence of the conjugate gradient method and the line with squares shows the convergence when the deblurring filter is used as a preconditioner.

Figure c shows the conventional sources least-squares migration image after 30 CG iterations, which is almost identical to the original model. It demonstrates that least-squares migration can sometimes produce images of higher quality and resolution compared to Kirchhoff migration (Nemeth et al., 1999), if the migration velocity is a somewhat accurate rendering of the actual smoothed velocity.