Amplitude versus offset

As important as the relative response between different reflections is the amplitude response of a single reflection as a function of the offset, or the illumination angle. The two P-to-P reflections of the previous subsection are separately analyzed here. To allow a direct comparison with the analytical curve for the PP plane-wave reflection coefficient, the wavefields simulated by each of the methods were mapped to the tau-p domain, which corresponds to the plane-wave decomposition of the original wavefields. Each figure used in this analysis contains five curves. The two continuous curves correspond to the real and imaginary parts of the theoretical P-to-P plane-wave response as a function of the horizontal slowness. The theoretical response includes the PP reflection coefficient and the two-way (angle-dependent) PP transmission coefficient. The other three curves come from the tau-p transform of the simulated wavefields, multiplied by a scaling factor that normalizes the amplitudes of the fifteenth horizontal slowness to the theoretical amplitude.

Figure  shows the curves associated with the water-bottom reflection. In the low horizontal slowness region the dual-operator curve (dashed-line) fits the theoretical curve almost perfectly, while the other two curves also represent a good approximation to the theoretical curve. It is also important to notice that the curve from the traditional finite-difference modeling (dashed-dotted-line) oscillates more than the other two curves. In the high horizontal slowness region, the traditional finite-difference method completely fails to approximate the theoretical curve while the other two methods still maintain a reasonably close fitting of the theoretical curve. One reason for the larger misfit here is the strong interference from the reflection of the second interface in the far offsets.

 

avoresp1
Figure 6 Amplitude versus horizontal slowness of the PP reflection from the water-bottom interface. These curves were extracted from the plane-wave decomposition of the wavefields generated by the three modeling schemes. The thick continuous line and the thin continuous line correspond, respectively, to the real and imaginary parts of the theoretical PP plane-wave reflection coefficient; the dotted-line is the amplitude response from the propagator-matrix scheme; the dashed-line is the amplitude response from the dual-operator scheme; the thin dot-dashed-line comes from the traditional finite-difference scheme.

As in the previous case, the curves for the second PP primary reflection (Figure ) approximate well the theoretical curve for low values of horizontal slowness. Again, the curve from the traditional finite-difference scheme is much more jagged than the curves from the other two schemes. For larger values of horizontal slowness, the dual-operator and the propagator-matrix curves are closer to the theoretical curve than the traditional finite-difference curve with better fit than in the water bottom case, mainly because the interference from other events is much weaker in this case. Particularly impressive is the extremely good fit obtained by the dual-operator method (dashed-line) for the whole range of horizontal slownesses, including the almost perfect prediction of the important zero-crossing point.

 

avoresp2
Figure 7 Amplitude versus horizontal slowness of the PP reflection from the interface between the second and third layer of the model. These curves were extracted from the plane-wave decomposition of the wavefields generated by the three modeling schemes. The thick continuous line and the thin continuous line correspond, respectively, to the real and imaginary parts of the theoretical PP plane-wave reflection coefficient; the dotted-line is the amplitude response from the propagator-matrix scheme; the dashed-line is the amplitude response from the dual-operator scheme; the thin dot-dashed-line comes from the traditional finite-difference scheme.