Maslov theory for surface wave propagation on a laterally heterogeneous earth

Abstract

S U M M A R YT h e usual J W K B ray-theoretical description of Love a n d Rayleigh surface wave propagation on a smooth, laterally heterogeneous earth model breaks down in the vicinity of caustics, near the source and its antipode. In this paper we use Maslov theory t o obtain a representation of the wavefield that is valid everywhere, even in the presence of caustics. T h e surface wave trajectories lie o n a 3-D manifold in 4-D phase space (0, @, k e , k,), where 0 is the colatitude, @ is the longitude, and k , and k, are the covariant components of the wave vector. There are no caustics in phase space; it is only when the rays are projected onto configuration space (0, @), the mixed spaces (k,, @) and (0, k @ ) , or momentum space (k,, k,), that caustics occur. T h e essential strategy is t o employ a mixed-space o r momentum-space representation in the vicinity of configuration-space caustics, where the (0, @) representation fails. By this means we obtain a uniformly valid Green's tensor and an explicit asymptotic expression for the surface wave response to a moment tensor source.

Surface Wave Caustics

“…Tromp & Dahlen (1993) have used Maslov theory to obtain a more general representation of the wavefield that is valid everywhere, including in the vicinity of caustics. The Maslov representation specifically accounts for the interference between nearly simultaneously arriving waves such as R1 and R2 or R2 and R3, etc.…”

“…Surface wave ray theory (the JWKB approximation) is invalid in the vicinity of caustics, because the geometrical amplitude, which is inversely proportional to the square root of the ray-tube width, I d~l -' '~, diverges. Tromp & Dahlen (1993) have used Maslov theory to obtain a more general representation of the wavefield that is valid everywhere, including in the vicinity of caustics. The Maslov representation specifically accounts for the interference between nearly Snapshots of Wavefronl Passing Through Zeroth Antipodal Figure 15.…”

A global model of Love and Rayleigh surface wave dispersion and anisotropy, 25-250 s

“…The propagation of fundamental‐mode Love and Rayleigh waves can be described using ray theory on a sphere (e.g. Tromp & Dahlen 1992, 1993). The surface wave seismogram u (ω) is written where A (ω) and Φ(ω) are the amplitude and phase, respectively, of the wave as functions of angular frequency ω.…”

Imaging the Earth Using Time‐Domain Surface‐Wave Measurements: Evaluation and Correction of the Finite‐Frequency Phase Shift

“…In the vicinity (near field) of a caustic or semi‐caustic, the phase advance deviates from its far‐field value (Kästle et al., 2016; Pollitz, 2001; Schwab & Kausel, 1976; Tromp & Dahlen, 1993; Wielandt, 1980). The exact amount of the phase shift can be either estimated analytically, by adding higher‐order terms of Taylor expansions into the analysis (Herrmann, 1973; Wielandt, 1980), or computed numerically (Schwab & Kausel, 1976).…”