Phase-ray examples

Examples of traced phase-rays are presented in this section using a salt body velocity field as a didactic model. The background velocity of the model, shown in figure e, is a typical Gulf of Mexico v(z) velocity gradient. The superposed salt body is characterized by higher wave speeds (4700 m/s) and a fairly rugose bottom of salt interface.

 

G5paper
Figure 2 Five phase-rays traced through monochromatic wavefields. Wavefields were generated by a split-step Fourier operator in Cartesian coordinates for a shot point at 11700 m using the velocity model illustrated in e). a) 5 Hz wavefield; b) 10 Hz wavefield; c) 15 Hz wavefield; d) 20 Hz wavefield; and e) smoothed salt body model velocity field.

Figure presents five phase-rays computed from four different monochromatic wavefields. The wavefields were generated for a shot point located at 11700 m using a split-step Fourier operator Stoffa et al. (1990) in a Cartesian coordinate system. Each ray begins at the same point in all panels. The rays to the extreme left and right in each panel show little variability in their spatial location; however, the three remaining rays are attracted to regions of greater wavefield amplitude and their spatial locations vary with a range up to 2000 m. Accordingly, because rays originate at the same spot, observed phase-ray movement is caused by changes in wavefield solution and indicates frequency-dependent behavior.

Phase-rays computed according to equations (9) may be traced in reverse, from observation to source point, by using a negative step interval. Figure illustrates this situation with the same model as in figure .

 

G5backpaper
Figure 3 200 phase-rays traced backwards through monochromatic wavefields for velocity model shown in e). Initial ray locations are points at regular intervals on a semicircular arc of radius 5000 m. a) 5 Hz wavefield; b) 10 Hz wavefield; c) 15 Hz wavefield; d) 20 Hz wavefield; and e) smoothed salt body model velocity field.

Initial ray locations are points at regular intervals on a semicircular arc of radius 5000 m. Calculated phase-rays do not overlap and the ray-field is caustic-free. Phase-ray density, though, is frequency-dependent, with significant coverage gaps of variable size appearing in all four panels. This suggests that an additional condition is required to ensure that, when needed, ray density is more uniform. One solution is to shoot a new ray between two successive rays wherever intra-ray distance exceeds some threshold value.

In summary, these results illustrate a number of advantageous characteristics of phase-rays: i) phase-ray ray-fields are triplication-free; ii) ray tracing from areas of low wavefield amplitude (e.g. shadow zones) to the source point is possible; and iii) sufficient phase-ray density may be ensured by an additional shooting of phase-rays wherever intra-ray spacing is too large. These three traits provide the main impetus for using phase-ray coordinates as a generalized coordinate system for wavefield extrapolation Sava and Fomel (2003).