Examples

For the purposes of the comparison, let us consider the simple acoustic subsurface model shown in Figure a. Figure b displays the simulated seismic response of one shot record and shows the AVO effect clearly.

To obtain the angle-dependent reflectivity with conventional prestack imaging, I applied shot profile imaging with phase-shift extrapolation, acquired the offset-dependent reflectivity image (Figure a), and transformed it to an angle-dependent reflectivity image (Figure b) using the incidence angle at each depth location. Figure  shows an angle-dependent reflectivity image obtained using de Bruin's imaging condition and Figure , one obtained using plane wave synthesis imaging. All of them retrieve the general amplitude's increasing pattern with respect to the incidence angle.

To see the amplitude more clearly, I picked the amplitude of each reflectivity image at the reflector depth and plotted the amplitude as a function of the incidence angles (Figure ). The comparison shows that the reflectivity obtained by the plane wave synthesis is closer to the theoretical solution.

For a more realistic example, the Marmousi dataset was tested. Figure  shows the image cube obtained by performing thirty-one constant incidence-angle PWS imaging and each constant incidence-angle image is obtained by patching twenty-one images obtained by synthesizing plane waves at twenty-one equi-spaced depth levels. The information of angle-dependent reflectivity is contained in the prestack image cube. To examine the angle-dependent reflectivity, four locations are selected and shown with theoretical angle-dependent reflectivities in Figures ,  , , and . We can see that the amplitude variations with angle in the images are very close to those of the theoretical reflectivity.

 

modl-shot
Figure 7 Subsurface model and simulated seismic response. (a): Subsurface model of a horizontally layered acoustic medium, containing one reflector at depth z = 500 m. (b): Simulated seismic response from the acoustic model.

 

prf-img
Figure 8 Offset-dependent (a) and angle-dependent (b) reflectivity image obtained by profile imaging with the conventional imaging condition.

 

prf-db-img Figure 9 Angle-dependent reflectivity image obtained by profile imaging with de Bruin's imaging condition.

 

pws-img Figure 10 Angle-dependent reflectivity image obtained by the plane wave synthesis imaging.

 

ava Figure 11 Angle-dependent reflection coefficient as a function of the angle of incidence: (a) the theoretical result, (b) the result of profile imaging with conventional imaging condition, (c) the result of profile imaging with de Bruin's imaging condition, and (d) the result of plane wave synthesis imaging.

 

pws-all-marint-ava
Figure 12 Marmousi image obtained by PWS imaging. After angle-dependent reflectivity recovery, four locations, (a),(b),(c), and (d), were chosen to compare with theoretical angle-dependent reflectivity.

 

marm-ava-rfl-com-1 Figure 13 Angle-dependent reflectivity obtained by PWS imaging (left) and theoretical reflectivity (right) for location (a).

 

marm-ava-rfl-com-2 Figure 14 Angle-dependent reflectivity obtained by PWS imaging (left) and theoretical reflectivity (right) for location (b).

 

marm-ava-rfl-com-3 Figure 15 Angle-dependent reflectivity obtained by PWS imaging (left) and theoretical reflectivity (right) for location (c).

 

marm-ava-rfl-com-4 Figure 16 Angle-dependent reflectivity obtained by PWS imaging (left) and theoretical reflectivity (right) for location (d).