Impulse response

To better understand the PS-AMO operator, I compute and analyze its impulse response. Figure  compares the AMO impulse responses obtained with the filters in equation (top) and equation (bottom). Both are obtained with a value of $\gamma=1$ and v_p=2.0 km/s, and are kinematically equivalent.

Figure  presents a similar comparison to Figure  for the case of converted waves. Here, we use $\gamma=1.2$ and v_p=2.0 km/s. Both impulse responses, Figures  and , also illustrate the differences in the dynamic behavior of the operator. The top panels for both figures show the impulse responses using the operator from equation , which is based on the known PS-DMO operator of Xu et al. (2001). In contrast, the bottom panels show the PS-AMO operator from equation , which is based on the new PS-DMO operator, presented in Chapter 2, equivalent to the Zhou et al. (1996) PP-DMO operator. The arrows in Figures  and  show that the operator from equation has stronger amplitudes for steeply dipping events than the operator from equation . The area marked by the oval in the bottom panel for both Figures  and  shows that impulse response for the PS-AMO operator is not center at zero inline and crossline midpoint location, as it is the case of the PP-AMO operator.

 

both_new_ant
Figure 2 PP-AMO impulse response comparison, filter in equation (top), and filter in equation (bottom), both with $\gamma=1$ and v_p=2.0km/s. The arrows mark the difference between both operators. The bottom figure presents stronger amplitudes for high dip values. This is an unfold 3-D cube representation of the results. The crossing solid lines represent position of the unfold planes.

 

both2_new_ant
Figure 3 PS-AMO impulse response comparison, filter in equation (top), and filter in equation (bottom), with $\gamma=1.2$ and v_p=2.0km/s. As in Figure  the new filter has more energy at high dip values. It is also possible to note the asymmetric behavior of the PS-AMO operator. This is an unfold 3-D cube representation of the results. The crossing solid lines represent position of the unfold planes.

Figure  shows two important characteristics of PS-AMO. First, the PS-AMO operator is asymmetric because of the difference between the downgoing and upgoing raypaths. Second, the PS-AMO operator varies with respect to traveltime, even for a constant velocity medium; this behavior is caused by all the non-linear dependencies of the PS-AMO operator with respect to traveltime, P velocity, and $\gamma$. The vertical variation of the lateral shift reflects that the lateral displacement between the CMP and CRP also varies with the traveltime. Both characteristics are intrinsic of converted-wave data.

 

amo2
Figure 4 PS-AMO impulses response variation with traveltime t. Observe the lateral displacement and the asymmetric behavior of the PS-AMO operator. This is characteristic of converted-wave operators. This is an unfold 3-D cube representation of the results. The crossing solid lines represent position of the unfold planes.