The first step of the procedure is shown in the upper panel of Figure 2. The rugosity of the true salt velocity model leads to a plane-wave rayfield containing a substantial number of triplications. To avoid this problem, we iteratively smooth the velocity model with stationary triangular operators. The velocity model shown in the upper panel is the roughest model where rays did not cross. The result of the second step of the procedure, calculating an initial rayfield, is presented in the middle panel. Here, the rays are calculated with a Huygens' wavefront tracing procedure Sava and Fomel (2001).
The third step of the procedure is to extrapolate a monochromatic wavefield on the initial rayfield generated in the step 2. This is done using a 5Hz wavefield and the RWE procedure employing the ray-coordinate 15 equation. The resulting wavefield, interpolated to Cartesian coordinates, is shown in the upper panel of Figure 3.
The wavefield above and to the left of the salt body is well behaved; however, beneath the top of the salt boundary it exhibits a cross-hatched pattern characteristic of the superposition of different triplication branch phases. This phenomenon is more evident in the sub-salt regions where the lens-like focusing effects of the salt are visible.
The fourth step is to compute the phase-rayfield from the monochromatic wavefield produced in step 3. Results for the 5Hz phase-rayfield are shown in the lower panel of Figure 2. The horizontal extent of the computation grid is less than that of the wavefield in the upper panel, because phase-rays can only be calculated at locations where the initial wavefield was computed. The effects of the rougher velocity model are evident in the shorter wavelength features of the coordinate system. For example, coordinates in the salt body canyons are delayed relative to neighboring points inside the salt body, leading to the sharper corners of the ray-coordinate system below the salt body. Also, the kink in the rayfield in the lower right corner of the lower panel is caused by side boundary reflections masquerading as wavefield triplications. The fifth step of the procedure, calculating a monochromatic wavefield on the updated ray-coordinate system, is illustrated in the lower panel of Figure (3). The resulting wavefield is full of triplications, and the illumination gaps (areas of low wavefield intensity) observed by Clapp (2003) are now plainly visible.
Broadband wavefields computed on the initial and updated ray-coordinate systems are presented in the upper panels of Figure 4. For interest, we include the velocity model mapped into both of these coordinate systems (lower panels). The horizontal lines represent surfaces of constant extrapolation time.
The upper panels show the initial plane-wave after propagation from the surface to depth through the respective velocity models. Results are presented at three travel-time steps spaced out at one second intervals. If the coordinate system were perfect, extrapolated wavefronts would be collinear with the three straight lines. This would indicate that the wavefront remains perfectly conformal with the ray-coordinate system. (Diffracted energy, though, moves sub-parallel to the wavefront and therefore does not form a straight line).
Differences in wavefront curvature between the upper two panels illustrate that the angles formed between the wavefront and extrapolation axes have, indeed, changed. In particular, they are now greater on the left side than on the right, which is not surprising because our goal is creating a coordinate system more conformal with the wavefront orientation. (Recall that a decreased extrapolation angle corresponds to increased operator fidelity.) This improved alignment, though, does come at a cost: the flanks of the canyon between 17 and 20km in midpoint are now steeper than in the original velocity model. This means that extrapolation across the salt flanks is now less robust because the salt flank angles are too severe. A result of this problem is observable in the less realistic extrapolated wavefield in the salt canyon between 2-3s in time and 17 and 20km in midpoint.
From this example, we point out that the ability to create a ray-coordinate system actually leads to an extra degree of freedom in the extrapolation process. As a result, the practitioner must resolve the trade-off between how well a wavefront conforms with the ray-coordinate system and how steep the structural dips of the velocity model are mapped as a result. Thus, one must ask the question: is it better to account for the steepest structural dips in the velocity model, but with lower extrapolation accuracy? or is it better to handle only shallower structural dips but with greater accuracy? Our conjecture is that the answer lies somewhere between these two end members and strongly depends on the particular velocity model in question.
The wavefields of Figure 4, interpolated to a Cartesian basis, are presented in Figure 5.
The upper and lower panels present the broadband extrapolation results computed using the initial and updated ray-coordinate systems, respectively. Snapshots of the wavefields at the first depth level is fairly similar; however, the second and third are markedly different. The wavefield in the lower panel at the fourth snapshot is more continuous across the breadth of the computational grid. However, beneath the salt canyon described above, the extrapolated wavefields in the upper panel seem to be more representative of the expected propagation results. This lack of global improvement is an example of the trade-off discussed above.