Example 3 - GRWE Green's Function Generation

The final test uses GRWE to generate estimates of Green's functions. The test velocity model is a slice of the SEG-EAGE salt model (see figure ). The contrast between the salt body and sediment velocities cause the wavefield to exhibit complicated propagation including significant wavefield triplication and multipathing. The upper left panel shows the velocity model with an overlain coordinate mesh generated by the differential method discussed in Shragge (2006a). The mesh is a ray-based coordinate system because the first and last extrapolation steps are formed by the 0.04 s and 2.25 s travel-time isochrons from a first-arrival Eikonal equation solution. The velocity model in the GRWE domain is illustrated in the upper right panel.

The lower right panel shows the impulse response test in the GRWE domain. The 7 impulsive waves conform fairly well to the travel-time steps, except where they enter the salt body to the lower left of the image. The migration results mapped back to Cartesian space are shown in the lower left panel. The wavefield to the left of the shot point is fairly complicated and the energy in the salt body and the corresponding upward refracted (and perhaps reflected?) wavefields are strongly present.

 

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Figure 5 Example of GRWE generated Green's Functions on structured non-orthogonal mesh for a slice through the SEG-EAGE Salt data set velocity model. Top left: Salt model in physical space with an overlain ray-based mesh. Top right: Velocity model in the GRWE domain. Bottom right: Wavefield propagated in ray-coordinates through velocity model shown in the upper right. Bottom left: Wavefield in bottom right interpolated back to Cartesian space.

Figure  presents a comparison test between GRWE and Cartesian extrapolation. Beneath and to the right of the shot point the wavefields are fairly similar except for a phase-change. However, they are significantly different to the left because Cartesian-based extrapolation cannot propagate energy laterally with the same accuracy and upward at all. Hence, this energy is absent from the wavefield in the lower panel.

 

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Figure 6 Results of a comparison test between generalized Riemannian wavefield extrapolation (top panel) and Cartesian-based extrapolation (bottom panel).