Ray perturbation theory and the Born approximation

Coates, R.; Chapman, Chris

doi:10.1111/j.1365-246x.1990.tb00692.x

Cited by 43 publications

(39 citation statements)

References 26 publications

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“…This implies that we must also consider lower terms in w in this case. A detailed study of Born scattering in the forward direction was presented by Coates & Chapman (1990a). It was shown there that, to leading order, the results produced by the Born approximation agree with ray theory results for the perturbed medium.…”

Section: A(x)supporting

confidence: 55%

“…where T(R, I) = R(I)R-'(R, To proceed further analytically we need to be able to relate the Hessian matrix to the geometrical spreading functions of the ray, Such a relationship is known for rays propagating along adjacent sections of a single ray path (Coates & Chapman 1990a). We now show that this relationship may be extended to include specularly reflected waves from curved interfaces.…”

Section: Appendix a : Asymptotic R A Y T H E O R Y A N D F O R W A R mentioning

confidence: 96%

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Generalized Born scattering of elastic waves in 3-D media

Coates

Chapman

2007

Geophysical Journal International

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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.

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Section: A(x)supporting

confidence: 55%

Section: Appendix a : Asymptotic R A Y T H E O R Y A N D F O R W A R mentioning

confidence: 96%

Generalized Born scattering of elastic waves in 3-D media

Coates

Chapman

2007

Geophysical Journal International

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“…The arrival-angle predictions can be used as a basis for tomographic inversions. The ray perturbation theory presented here is similar to that discussed by Farra & Madariaga (1987) and Coates & Chapman (1990).…”

Section: Estimating Take-off a N D Arrival Anglesmentioning

confidence: 63%

Uniformly valid body-wave ray theory

Liu

Tromp

1996

Geophysical Journal International

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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.

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“…For realistic problems, inhomogeneous reference models are necessary and lead to the distorted Born approximation (Chapman, 2004). For remedying this, Coates and Chapman (1990) and Chapman and Coates (1994) proposed a generalization of the Born approximation, which takes into account approximate Green function. We obtain then the WKBJ þ Born (Clayton and Stolt, 1981;Ikelle et al, 1988) or ray þ Born approximations (Bleistein, 1984(Bleistein, , 1987Beylkin, 1985;Beydoun and Mendes, 1989;Beylkin and Burridge, 1990;Cohen et al, 1986;Jin et al, 1992;Thierry et al, 1999a,b;Operto et al, 2000;Xu et al, 2001 where A (r, x, s) ¼ A (s, x) is the product of the amplitude of asymptotic Green functions for the rays x !…”

Section: Ray þ Born/rytov Formulationmentioning

confidence: 99%