Impulse response

To better understand the PS-AMO operator, I compute and analyze its impulse response. Figure 2 compares the AMO impulse responses obtained with the filters in equation 10 (top) and equation 11 (bottom). Both are obtained with a value of $\gamma=1$ and v_p=2.0 km/s, and are kinematically equivalent.

Figure 3 presents a similar comparison to Figure 2 for the case of converted waves. Here, we use $\gamma=1.2$ and v_p=2.0 km/s. Both impulse responses, Figures 2 and 3, 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 10, which is based on the known PS-DMO operator of (). In contrast, the bottom panels show the PS-AMO operator from equation 11, which is based on the new PS-DMO operator, presented in Chapter 2, equivalent to the () PP-DMO operator. The arrows in Figures 2 and 3 show that the operator from equation 11 has stronger amplitudes for steeply dipping events than the operator from equation 10. The area marked by the oval in the bottom panel for both Figures 2 and 3 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.

 

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Figure 2 PP-AMO impulse response comparison, filter in equation 10 (top), and filter in equation 11 (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.

 

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Figure 3 PS-AMO impulse response comparison, filter in equation 10 (top), and filter in equation 11 (bottom), with $\gamma=1.2$ and v_p=2.0km/s. As in Figure 2 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 4 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.

 

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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.