Wavefront construction using waverays

LOMAX'S WAVERAYS

Two basic ideas characterize Lomax's method: (1) it does a wavelength-dependent velocity smoothing and (2) it uses Huygen's principle to track the motion of narrow-band wavefronts at a number of center frequencies. Narrow-band wavefronts are defined as surfaces (lines in 2-D) of constant phase or traveltime in a narrow-band ``wavefield''. As these narrow-band wavefronts propagate with time they define a wavepath, which is frequency dependent. The wavelength-dependent smoothing of the velocity is done by averaging with a Gaussian weighting curve. The smoothing is done along the wavefronts. The final result; i.e, the broadband wavefield, is constructed by summing the results of independent narrow-band wavefields at many center frequencies.

Three important advantages of using Lomax's waveray over conventional ray tracing methods are:

• It increases the stability of wavepaths compared to the paths produced by high frequency methods, due to the wavelength-dependent smoothing.
• It provides waverays with a sensitivity that produces frequency dependent scattering as a function of the ratio of wavelength to the characteristic size of the scattering region. Figure 3 illustrates this point.
• It is capable of handling large to small inhomogeneity sizes, since in the former case it is similar to ray theory and in the latter it responds to a smooth, averaged velocity structure.

Lomax (1994) approximates the narrow-band wavefronts at any time by a plane wavefront (see Figure 1). This approximation requires that the radius of curvature of the wavefront be large relative to a wavelength. A better approximation to the wavefronts could probably be obtained using a parabolic approximation. For the sake of computational time, plane wavefronts are used.

Before considering the details of the waveray technique, two points need to be emphasized. First, Lomax (1994) points out, ``it is the wavelength dependent smoothing that makes the waveray method a broadband wave propagation technique, and distinguishes it from the high frequency ray methods.'' Second, there are no equations that give the waveray method a theoretical basis. Its support comes from the fact that it reproduces high-frequency ray propagation and produces a good approximation of broadband wave phenomena.

lomax1 Figure 1. Waveray wavepath and wavelength-dependent velocity smoothing at point ${\bf\vec{x}}_{\nu}$ Adapted from Lomax (1994).

Wavefront construction using waverays