Uniformly valid body-wave ray theory

Xian Feng Liu, Jeroen Tromp

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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 long-wavelength 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.

Original languageEnglish (US)
Pages (from-to)461-491
Number of pages31
JournalGeophysical Journal International
Issue number2
StatePublished - 1996

All Science Journal Classification (ASJC) codes

  • Geophysics
  • Geochemistry and Petrology


  • Body waves
  • Perturbation methods
  • Synthetic seismograms
  • Tomography


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