List of Figures

1. (a) Time-shifted shot gathers, (b) blended supergather created by blending $S$ time-shifted shot gathers, (c) migration images after migrating the supergather for each shot position with SNR approximately $\frac{1}{\sqrt{S-1}}$ , (d) final image after summing $S$ migration images. The final SNR is $\frac{\sqrt{S}}{\sqrt{S-1}}$ .
2. 2D SEG/EAGE salt model (reflectivity).
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.
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.
5. Kirchhoff migration images obtained from the following clusters of supergathers, (a) thirty-two 10-shot supergathers, (b) sixteen 20-shot supergathers, (c) eight 40-shots supergathers, (d) four 80-shot supergathers, (e) two 160-shot supergathers and (f) one 320-shot supergather. Here, all shot gathers consisted of 320 traces, and each supergather in a cluster was formed from a unique set of shot gathers.
6. The predicted and measured signal-to-noise ratios of iterative stacking method are plotted against iteration number as dashed and solid lines. The measurements have been normalized by the 1st iteration result.
7. Stacked images for iterative stacking after (a) 1 iteration; (b) 5 iterations; (c) 10 iterations; (d) 20 iterations.
8. Same as Figure 2.5 except the supergather clusters consisted of (a) one 40-shot supergather, (b) one 80-shot supergather and (c) one 160-shot supergather.
9. Least-squares migration images of a 320-shot supergathers after (a) 10, (b) 30, (c) 60 iterations with static encoding or (d) 10, (e) 30, (f) 60 iterations with dynamic encoding.
10. The solid line with squares shows the measured SNR for images of one 320-shot supergather with static encoding; the solid line with stars shows the results with dynamic encoding. Here the measured SNR is normalized by the first iteration result. The dashed line indicates the prediction from equation 2.11.
11. The diagram of plane wave encoding (reproduced from Zhang et al. (2005)), where the time shift is linear function to the source location $x$ and the slope is the ray parameter $p$ .
12. (a) Modified Marmousi2 model and (b) the smooth migration velocity model. The migration velocity is smoothed by a triangle smoothing filter with a window length of 100 m to get rid of the fine scale structures.
13. The conventional shot-domain RTM image for the Marmousi2 model.
14. A plane-wave gather with p=22.2 $\mu$ s/m for the Marmousi2 model.
15. The plane-wave RTM image of the Marmousi2 model with only one angle ($p=0$ ); (b) The plane-wave LSRTM image of the marmousi2 Model with only one angle ($p=0$ ) after 30 iterations; and (c) The plane-wave LSRTM image of the Marmousi2 model with only one angle per iteration. The angle is dynamically changed at every iteration.
16. (a) The plane-wave RTM image of the Marmousi2 model and (b) plane-wave LSRTM image of the Marmousi2 model after 30 iterations. All the 31 plane-wave gathers are used.
17. The common image gathers extracted from the plane-wave RTM image.
18. The common image gathers extracted from the plane-wave LSRTM image after 30 iterations.
19. The convergence curves for LSRTM with the stacked image and the prestack image. It is clear that the convergence of LSRTM is improved when more unknowns are incorporated into the inversion and the migration velocity contains 5% error.
20. (a) The plane-wave RTM image of the Marmousi2 model, (b) image obtained by plane-wave LSRTM with the stacked image after 30 iterations, and (c) image obtained by plane-wave LSRTM with the prestack image after 30 iterations. All the 31 plane-wave gathers are migrated with 5% velocity error.
21. The common image gathers extracted from the plane-wave RTM image obtained with 5% velocity error.
22. The common image gathers extracted from the plane-wave LSRTM image after 30 iterations when the migration velocity contains 5% error.
23. A plane-wave gather with zero surface shooting angle (p=0 $\mu$ s/m).
24. The migration velocity model for the field data test, obtained by full waveform inversion.
25. The migration images obtained by: (a) conventional shot-domain reverse time migration, (b) plane-wave reverse time migration, (c) plane-wave least-squares reverse time migration and (d) plane-wave LSRTM with dynamic encoding. The blue and red boxes indicate the areas for zoom view.
26. The zoom views of the red boxes: (a) conventional shot-domain RTM, (b) the plane-wave RTM, (c) the plane-wave LSRTM and (d) the plane-wave LSRTM images with dynamic encoding.
27. The zoom views of the blue boxes: (a) conventional shot-domain RTM, (b) the plane-wave RTM, (c) the plane-wave LSRTM and (d) the plane-wave LSRTM images with dynamic encoding.
28. The misfit vs iteration number curve for plane-wave LSRTM shows fast and stable convergence even when the velocity is not completely accurate.
29. The common image gathers extracted from the plane-wave RTM image of the field data.
30. The common image gathers extracted from the plane-wave LSRTM image after 30 iterations for the field data test.
31. (a) A velocity model with a horizontal reflector and a vertical reflector. The yellow arrows indicate the ray path for a prism wave from the source at the star to the receiver at the triangle; (b) the wave path of the prism wave with a 20-Hz Ricker wavelet; and (c) the trace recorded at the triangle. The two arrivals in the red window are the reflections from the horizontal reflector and the prism wave in panel (b).
32. (a) A two-layer velocity model. The star and triangle indicate the source and receiver locations. The yellow arrow is the ray path for the direct wave and the red arrows show the ray path for the reflected wave. (b) The trace recorded at the triangle. It is simulated with a 20-Hz Ricker wavelet.
33. (a) The homogeneous velocity ($2~km/s$ ) with a horizontal reflector embedded ($2.5~km/s$ ); (b) the migration image of the data within the red window in Figure 4.1(a) with the velocity model in panel (a).
34. Diagrams of the ray paths illuminating the process of prism wave migration: (a) source and receiver wavefields correlate at the correct image point. Panels (b) and (c) show the ray paths to two image points that are above and below the right location. The black vertical curve plots part of the prism wave migration kernel. The circles along the curve show the locations of trial image points.
35. (a) The migration kernel of the prism wave corresponding to the term in equation 4.9 in the case the vertical reflector is on the left side. (b) The outline of the migration kernel in panel (a) according to the geometric interpretation. The star and triangle indicate the source and receiver locations respectively.
36. (a) The ray path for the prism wave with a vertical reflector on the right side; (b) the migration kernel of the prism wave corresponding to the term in equation 4.10 in the case the vertical reflector is on the right side; and (c) the outline of the migration kernel in panel (b) according to the geometric interpretation. The star and triangle indicate the source and receiver location respectively.
37. A shot gather with the source at $x=4.6~km$ . The shot gather contains the direct wave, the reflection off the horizontal reflector, and the diffraction from the top of the vertical reflector. The yellow arrow points out the prism wave.
38. (a) The RTM image obtained with a homogeneous velocity model. The vertical reflector is not illuminated. (b) The RTM image of the prism waves with homogeneous velocity and the reflectivity image in panel (a). The vertical reflector is well imaged.
39. (a) A velocity model with a salt body on the left side; (b) the smooth migration velocity model without the salt body.
40. A shot gather with the source at $x=4~km$ . The yellow arrows point out the prism waves.
41. (a) The velocity model with subhorizontal reflectors embedded; (b) the RTM image obtained with the velocity model in panel (a). The irregular salt boundary is well imaged.
42. (a) The RTM image obtained with the smooth migration velocity model. Along the salt boundary, only a few diffractors are visible. (b) The RTM image of the prism waves with the same velocity model. The irregular salt boundary is well imaged.
43. (a) The RTM image obtained with the smooth migration velocity model after dip filtering to keep subhorizontal reflectors only; (b) the RTM image of the prism waves.
44. (a) The RTM image of the prism waves after dip filtering for subvertical reflectors only; (b) the sum of two partial images: one from conventional RTM and one from migration of the prism waves.
45. Steps for computing the deblurring filter. Step (a) Define smooth velocity model with point scatterers denoted as circles in (b). Generate multisource data in (c), migrate the multisource data and get an image shown in (d). Step (e), in each sub-section, compute a local filter according to $\textbf {[m}_{mig\_ref}\textbf {]}_{i}*\textbf {f}_{i}=\textbf {[m}_{ref}\textbf {]}_{i}$ and combine all the local filters into the deblurring filter $\textbf {F}$ .