An Elasticity Theorem for Heterogeneous Media, with an Example of Body Wave Dispersion in the Earth

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.

Section: G = a ( P~r S I N I S I N < J ) -' /~E X P I W T -M -$A8 mentioning

confidence: 96%

Uniformly valid body-wave ray theory

Liu

Tromp

1996

“…The solution to the transport equation may be written as (e.g. Richards 1971) where J(x,, xI) models the geometrical spreading of a pencil of paraxial rays emanating from a point source at x1 and observed at x2. To find the excess pressure Green's function at any point on the ray, we need to first find its value at one point before applying equation (17).…”

Section: P ( X )mentioning

confidence: 99%

“…where J ( x , x,) = a p 2 ( x l ) Ixxll' for x near x1 (e.g. Richards 1971). The zeroth-order approximation to the excess pressure P(")(x, x1 : w ) , due to a point source with an arbitrary source spectrum F ( w ) , may now be obtained from the Green's function,…”

Section: P ( X )mentioning

confidence: 99%

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Generalized Born scattering of elastic waves in 3-D media

Coates

Chapman

2007

S U M M A R Y It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansatze.In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution.The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach.The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed.

“…The reciprocity of point-source geometrical spreading functions is at the heart of the issue and Coates & Chapman (1990) refer to Richards (1971, section 3.5) for the proof. The discussion in Richards (1971) applies to multilayered media, but anisotropy and the relevance of symmetry in the geometrical spreading equations are not considered. Here, the approximate Green's function is found by matching zeroth-order ray theory with the exact point-source solution for a homogenous anisotropic medium obtained by Lighthill (1960), Buchwald (1959) and Burridge (1967).…”

Section: Introductionmentioning

confidence: 98%

Ray-theory Green's function reciprocity and ray-centred coordinates in anisotropic media

Kendall

Guest

Thomson

1992

S U M M A R Y Lighthill and others have expressed the ray-theory limit of Green's function for a point source in a homogeneous anisotropic medium in terms of the slowness-surface Gaussian curvature. Using this form we are able to match with ray theory for inhomogeneous media so that the final solution does not depend on arbitrarily chosen 'ray coordinates' or 'ray parameters' (e.g. take-off angles at the source). The reciprocity property is clearly displayed by this 'ray-coordinate-free' solution. The matching can be performed straightforwardly using global Cartesian coordinates.However, the 'ray-centred' coordinate system (not to be confused with 'ray coordinates') is useful in analysing diffraction problems because it involves 2 X 2 matrices not 3 x 3 matrices. We explore ray-centred coordinates in anisotropic media and show how the usual six characteristic equations for three dimensions can be reduced to four, which in turn can be derived from a new Hamiltonian. The corresponding form of the ray-theory Green's function is obtained. This form is applied in a companion paper.