Impulse response

The spreading surfaces for the theoretical impulse response of the common-azimuth migration operator (Appendix F) are shown in Figures  and   for the single-mode case, and the converted-mode case, respectively. Figure  shows the PP spreading surface in midpoint-offset coordinates for an event with a total traveltime of 0.300 s, a P-velocity of 2500 m/s, and inline-offset of 200 m. Figure  shows the equivalent PS spreading surface using an S-velocity of 2000 m/s.

 

pptheory
Figure 3 Single-mode (PP) spreading surface for an event with total travel-time of 0.300 s, using constant P-velocity=2500 m, and inline surface-offset=200 m.

 

pstheory
Figure 4 Converted-mode (PS) spreading surface for an event with total travel-time of 0.300 s, using constant P-velocity=2500 m, constant S-velocity=2000 m, and inline surface-offset=200 m.

Figures  and  represents the impulse response for PS common-azimuth migration downward-continuation operator. Figure  shows the result using the same velocity for the propagation of the downgoing and upgoing wavefields, which is equivalent to the single-mode common-azimuth operator. Figure  shows the result with a different propagation velocities for each of the two wavefields, which represents the converted-wave common-azimuth operator.

Figures  and  are a four-dimensional representation of the prestack image with dimensions ($z, {\rm CMP}_X, {\rm CMP}_Y,H_X$). For both figures, the top panel shows the inline-midpoint and crossline-midpoint cube that corresponds to zero inline subsurface-offset. And, the bottom panel shows the inline-midpoint and inline subsurface-offset cube that corresponds to zero crossline-midpoint.

The depth slices in Figures  and  exhibits interesting characteristics of the common-azimuth operator. For the single-mode case, Figure , the depth slice that corresponds to the inline-midpoint and crossline-midpoint cube (top panel) shows a circle, whereas the depth slice for inline-midpoint and inline subsurface-offset sections (bottom panel) shows a rectangle. For the converted-mode case, Figure , the depth slice in the inline-midpoint and crossline-midpoint cube (top panel) has similar characteristic as the single-mode case. However, the depth slice for the inline-midpoint and inline subsurface-offset sections (bottom panel) displays a rectangle that has been sheared and rotated. This deformation is the result of using two different propagation velocities for the downgoing and upgoing wavefields.

 

pp_imp_resp
Figure 5 4-D prestack PP impulse response. Top panel, ${\rm CMP}_X$ and ${\rm CMP}_Y$ sections for H_X=0. Bottom panel, ${\rm CMP}_X$ and H_X sections for ${\rm CMP}_Y$=0. The spike is located at zero surface-offset.

 

ps_imp_resp
Figure 6 4-D prestack PS impulse response. Top panel, ${\rm CMP}_X$ and ${\rm CMP}_Y$ sections for H_X=0. Bottom panel, ${\rm CMP}_X$ and H_X sections for ${\rm CMP}_Y$=0. The spike is located at zero surface-offset.

To validate the implementation of the PS-CAM operator, I compare the PS impulse response with the spreading surfaces, which represent the theoretical solution. Figure  shows several sections for the PS impulse response with the theoretical solution superimposed (blue dotted curve). This figure shows two panels side-by-side, each panel consists of five sections. The left panel shows inline-midpoint sections for zero crossline-midpoint, and the right panel displays crossline-midpoint sections for zero inline-midpoint. All the sections correspond to zero subsurface-offset. Each row represents an impulse response that corresponds to an spike located at different surface-offset locations. From top to bottom, the surface-offsets are: -250 m, -150 m, 0 m, 150 m, and 250 m, respectively. Notice that the center panel, that corresponds to zero offset, the impulse response is completely symmetric, as it is expected. The asymmetry characteristic for the PS-CAM operator is observed along the inline direction for different offset values, as it can be seen along the inline sections (left panel) in Figure . However, the crossline-midpoint location is symmetric (right panel in Figure ).

 

ps-ir-z-mx-my-hx
Figure 7 Impulse response for the PS common-azimuth downward-continuation operator. The superimposed blue-dotted curves represent the spreading surfaces. The left panel represents the inline-midpoint section for zero crossline-midpoint, the right panel represents the crossline-midpoint section for zero inline-midpoint, all the sections correspond to zero subsurface-offset. Each row represents an impulse response for an spike located at different surface-offset locations. From top to bottom, surface-offset: -250 m, -150 m, 0 m, 150 m , and 250 m.

Since the prestack image is four-dimensional, I use another comparison to truly validate the PS-CAM implementation. The second test corresponds to an spike located at a fixed surface-offset of 200 m, I take five inline-midpoint sections that correspond to different crossline-midpoint locations, and vice versa. Figure  shows these sections, the left panel represents the inline-midpoint sections for five different crossline-midpoint locations. The right panel displays the crossline-midpoint sections for five different inline-midpoint locations, all the sections correspond to zero subsurface-offset. Both, the inline-midpoint and crossline-midpoint locations are, from top to bottom, -250 m, -150 m, 0 m, 150 m, 250 m, respectively. As in the first comparison, the theoretical solution is superimposed on the PS impulse response as the blue dotted curve, in this case the spreading surface in Figure  shows the theoretical solution.

 

ps-ir-z-mx-my-200
Figure 8 Theoretical solution for the impulse response superimposed with the impulse response on Figure . This figure represents a fixed surface-offset of 200 m, the left panel represents the inline-midpoint direction and the right panel the crossline-midpoint direction, all of them corresponds to zero subsurface-offset. From top to bottom I present sections at different inline and crossline position: -250 m, -150 m, 0 m, 150 m, 250 m.