Uniformly valid body-wave ray theory

Abstract

We present a uniformly valid ray theory for body-wave propagation in laterally heterogeneous earth models. This is accomplished by implementing Maslov theory, which is a 3-D analogue of the widely used WKBJ seismogram method for spherically symmetric earth models. Away from caustics, complete seismic waveforms can be calculated by solving a system of 14 coupled first-order ordinary differential equations: four equations determine the ray geometry, eight additional equations determine the amplitude, and two further equations determine traveltime and attenuation. In the vicinity of a caustic, neighbouring rays cross, and asymptotic ray theory breaks down. Rather than considering the contribution to the wavefield of one single ray, our strategy is to express the wavefield in the vicinity of a caustic as a summation over neighbouring, non-Fermat rays based upon Maslov theory. Away from caustics, Maslov theory reduces to asymptotic ray theory. We present examples of the ray geometry in the 3-D model SKS12WM13, and demonstrate that small-scale triplications in the traveltime curve associated with large-scale heterogeneities in the lowermost mantle are ubiquitous. The theory is applicable to direct, turning and reflected waves, may to a limited extent be advanced to include head waves, but does not describe waves that are diffracted into the deep shadow.The determination of the geometric ray that connects a given source and receiver is based upon a 'shooting' method. Initial guesses for the take-off angles are determined based upon perturbation theory, which substantially reduces the number of iterations required to hit a receiver. Perturbation theory also provides predictions for arrival angles and amplitude anomalies. These predictions incorporate the effects of longwavelength topography on internal boundaries and the free surface, and may be used as a basis for tomographic inversions. Generally, predictions based upon perturbation theory agree very well with exact 3-D ray tracing. Just as traveltime measurements provide constraints on velocity, arrival angles constrain velocity gradients, and amplitude anomalies put constraints on second derivatives in velocity. In SKS12WM13, traveltime anomalies can be as much as 10 s, arrival-angle anomalies can be larger than +5", and 3-D amplitudes can differ by more than 100 per cent from PREM amplitudes.

Pwave amplitudes in a 3-D earth

“…The standard deviations listed in Table 4 do however provide a yardstick to judge the adequacy of global models to represent accurate amplitudes of velocity perturbations. We note that the variations we observe are significantly larger than earlier estimates of the effects of focusing of 15 per cent by Lay & Helmberger (1981) or 25 per cent by Liu & Tromp (1996). The actual tomographic models fail to predict small variations mainly because the predicted focusing is far too small.…”

Fréchet kernels for body-wave amplitudes

“…In the case of a traveltime perturbation δ T , it is evident that the ω→∞ approximation to the 3D integral coincides with the result obtained using geometrical ray perturbation theory. A number of investigators, including Thomson (1983), Nowack & Lutter (1988), Coates & Chapman (1990), Neele et al (1993) and Liu & Tromp (1996) have presented numerical recipes for computing first‐order ray‐theoretical amplitude perturbations; however, we are not aware of a satisfactory explicit general formula for δ(ln A ) with which the 1D asymptotic integral representation can be compared. For this reason, we present our own strictly ray‐theoretical amplitude analysis in this and the following section.…”

Constructing synthetics from deep earth tomographic models

“…They used the earlier approach (Chapman 1976) in their development of a 3-D slowness method. Because WKBJ can be directly derived from classical optic theory, it proves relatively easy to treat 3-D problems for smoothly varying media (Liu & Tromp 1996). Sharp boundaries cause di¤culties following this approach where GRT has some advantages.…”