Kirchhoff migration in constant velocity

For Kirchhoff migration, I used Claerbout's kirchfast() subroutine (Claerbout, 1992b). The kirchfast() subroutine is the simplest zero-offset Kirchhoff migration program I could find, but it produces many artifacts when is applied to a sparsely sampled data because of the nearest neighbor interpolation. Figure (a) shows a simple model with a syncline image. The Kirchhoff modeling for the model [Figure (a)] is shown in Figure (b), and the migration result is shown in Figure (c). The image in Figure (c) is very blurred compared to the original image. We can suppress this artifact by the least-squares imaging; the result is shown in Figure (d). The image in Figure (d) is very close to the original image, without any artifacts. This result was obtained after only 5 iterations of the conjugate gradient algorithm.

For more realistic synthetic model, I used Claerbout's synthetic model, which has a very complicated geological model, as shown in Figure (a). Figure (b) shows the synthetic data generated by the subroutine kirchfast(), and Figure (c) the result of migration. The image after migration is smoother than the original image. This suggests applying an $\vert\omega\vert$ filter to recover the high frequency. The image after the least-squares inversion, shown in Figure (d), contains more high frequency than the image after migration. The frequency recovery in the image after migration and inversion is shown in Figure . We can see that the spectrum of the image obtained by the least-squares imaging is very close to the spectrum of the original image. Figure  illustrates the same experiment as Figure , except that the sampling interval is two times that of the sampling interval of the model used in Figure . The image after migration in Figure (c) shows artifacts because of the change in the sampling interval. But the image obtained by the least-squares imaging, Figure (d), is still clear.

 

Kirmiginv
Figure 1 Least-squares Kirchhoff imaging: (a) Synthetic image model containing a syncline reflector, (b) Kirchhoff modeling, (c) Kirchhoff migration for (b), (d) Least-squares Kirchhoff imaging for (b) (after 5 iterations of the conjugate gradient algorithm).

 

kirinv3
Figure 2 Kirchhoff inversion: (a) Synthetic image model with a syncline (nz=128 and nx=200), (b) Kirchhoff modeling, (c) Kirchhoff migration for (b), (d) Least-squares Kirchhoff inversion for (b) (after 10 iterations of the conjugate gradient algorithm).

 

kirspec3 Figure 3 Spectrum recovery after migration and inversion. From the top: The spectrum of the original image, of the inversed image, of the migration image times frequency, and of the migration image.

 

kirinv2
Figure 4 Kirchhoff inversion: (a) Synthetic image model with a syncline (nz=128 and nx=100), (b) Kirchhoff modeling, (c) Kirchhoff migration for (b), (d) Least-squares Kirchhoff inversion for (b) (after 10 iterations of the conjugate gradient algorithm).

 

kirspec2 Figure 5 Spectrum recovery after migration and inversion. From the top: The spectrum of the original image, of the inversed image, of the migration image times frequency, and of the migration image.