C. J. Thomson

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Seismic waveform variations seen via a vector narrow-angle one-way propagator for three-dimensional anisotropic inhomogeneous media.

EGS - AGU - EUG Joint Assembly (2003)

Doug Angus C. J. Thomson R. G. Pratt

Propagator

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Modeling RTM extended image gathers

C. J. Thomson James Hobro Robin Fletcher P. W. Kitchenside

We show examples of modeling subsurface space-time domain extended image gathers and their blurring functions for a synthetic-survey reverse-time migration using the SIGSBEE test model. We choose two image points and extended patches, in well- and moderately illuminated places. There is blurring over angle as well as in space, and we find it to be insignificant (around 5 degrees) and significant (around 10 degrees) in the well- and moderately illuminated places respectively. Our modeling is purely acoustic and only in two space dimensions. However, it demonstrates numerically how an interface reflection function, built of plane-wave reflection coefficients, is approximated by an extended image gather when we have a good migration velocity model. The theory of this blurred relationship is well established, yet our numerical experience is limited. When the incident and back propagated wavefields are even moderately complicated it seems our ability to model the EIG with precision given the reflectivity and blurring functions is notably degraded. The causes of this could be approximations made in the computations, such as integrand sampling and apertures. The fact that it is possible to get precise agreement in the well-illuminated location is encouraging and it indicates we still have things to learn from numerical modeling elsewhere. Presentation Date: Wednesday, September 27, 2017 Start Time: 4:45 PM Location: 371A Presentation Type: ORAL

10.1190/segam2017-17655625.1

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Corrections for grazing rays in 2-D seismic modelling

Geophysical Journal International (1989)

C. J. Thomson

SUMMARY The ray method of calculating body-wave seismograms fails for a ray that just grazes a reflector, because no provision exists for interference between the primary and secondary waves when they are travelling almost parallel to each other. Such interference is important both near the grazing point and near the grazing ray after it leaves the reflector. It gives rise to narrow, frequency-dependent transition regions within which the individual geometrical waves (incident, reflected and diffracted) cannot be separately identified and across which the overall signal strength changes rapidly, though continuously, with position. The diffraction is basically a local phenomenon. Therefore, it is possible to solve the wave equation first in the transition region surrounding the grazing point and then to obtain, by matching, the solution in the neighbourhood of the grazing ray at points far from the reflector. The purpose of this paper is to describe this ‘boundary-layer’ method for an acoustic problem in an introductory and self-contained way and also to present some apparently new observations and results. The latter are the extension to transmitted waves, an alternative description of the Fresnel field and more details concerning the geometrical limit of the reflection (in particular how the reflected and diffracted wavefronts are related at large distances). Though there are many intermediate steps, the far field is the region of main interest in seismology and it is natural to express the results there in ray coordinates. The diffraction formulas obtained are then easier to visualize and to apply than one might expect. Numerical examples are given to indicate the potential importance of the diffraction, by comparing the boundary-layer results with those of ray theory and its extension Maslov theory (which also fails at grazing incidence). The former gives no field in the shadow, of course, and may either over- or underestimate the field at the shadow edge, depending on whether the backscattered or transmitted wave is observed. The latter always overestimates the field in the shadow.

10.1111/j.1365-246x.1989.tb06005.x

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Citations (11)

Coherent-state analysis of the seismic head-wave problem: an overcomplete representation and its relationship to rays and beams

AGUFM (2003)

C. J. Thomson

SUMMARY The coherent-state transform (CST) is a Gaussian-windowed Fourier transform and compared with the usual plane wave expansion of seismic wavefields it provides an overcomplete basis of partial waves. These overcomplete partial waves have associated rays and the relationship of these rays to those of a physical wave front is explained here by appealing to a familiar analytic example. The exact CST of the standard Fourier plane wave summation for a point-source primary wave can be interpreted as a sum of damped plane waves forming a focussed or beam-like wavefield. This primary wave CST can also be approximated as a bundle of complex rays in a position-slowness space with higher dimension than that usually considered, a consequence of the overcompleteness. The higher-dimensional ray spreading controls the coherent-state (CS) amplitude. This complex-ray bundle in turn can be approximated as a paraxial Gaussian beam carried by a real ray associated with the physical wave front. The exact CST of the standard plane wave summation for a point-source reflection shows how the complex-ray and paraxial-beam approximations generalize to interfaces. Around a critical angle the standard plane wave summation has an asymptotic form involving the Weber function and this function also necessarily arises in the complex-ray and paraxial-beam approximations for the reflection CST. The inverse CST giving the point-source wavefield is a sum of coherent states and those contributing rays or paraxial beams that are incident around the critical angle should be given a reflection coefficient involving the Weber function. CS beams that are incident away from the critical angle collect a peripheral branch-point diffraction. In general, both types of critical-ray/branch-point signal should be included if the integrated CS response is to correctly describe the head wave. An advantage of the CST is that it combines with ray theory for gradually varying media to give a convenient solution to the caustic and pseudocaustic problems. It may be said to unify the Maslov and standard Gaussian-beam methods of seismic modelling, in part by avoiding the ray-centered coordinates conventionally employed in the latter. The general idea of such overlapping partial wave expansions, their flexibility and the smoothing they impart may have other benefits in seismic analysis and processing.

Slowness

Reflection

Geometrical acoustics

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Accuracy & Computational Considerations for Wide--Angle One--way Seismic Propagators and Multiple Scattering by Invariant Embedding

AGU Fall Meeting Abstracts (2004)

C. J. Thomson

Propagator

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Geometrical theory of shear-wave splitting: corrections to ray theory for interference in isotropic/anisotropic transitions

Geophysical Journal International (1992)

C. J. Thomson J. M. Kendall W. S. Guest

An S-wavefront from an isotropic region is expected to separate into two fronts when it passes into a gradually more anisotropic region. Standard ray expansions may be used to continue the waves in the anisotropic region when these two S-wavefronts have separated sufficiently. However, just inside the anisotropic region the two S-waves interfere with an effect that is stronger than the usual ω-1 corrections of the ray method. A waveform distortion can occur and this should be considered when modelling S-waves in, e.g., subduction zones with regions of isotropy grading into regions of anisotropy. The interference is studied here by local analysis of an integral equation obtained by the Green's function method. It is found that if the elasticity and its first two derivatives are continuous at the isotropy/anisotropy border, then zeroth-order ray theory may still be used to continue the incident wave into the anisotropic region. The incident displacement is simply resolved into two definite directions at the point where the anisotropy begins. These two directions are the limits of the unique eigenvectors on the anisotropic rays as the point of isotropy (onset of splitting) is approached. If the nth derivative of the elasticity is discontinuous at the isotropy/anisotropy border, then the scattering integral which describes the interference makes a correction to ray theory which is O(ω−1/n+1) in magnitude. Hence, the interference effect is stronger when the emergence of anisotropy is more gradual. Although the corrections are given by simple expressions, it is not reasonable to specify numerical velocity models up to such high-order derivatives. For a smooth interpolation scheme, such as cubic splines, it is more practical to monitor the splitting rays obtained by ray tracing and to use the best-fitting ‘equivalent’ high-order discontinuity. This will lead to an estimate of the importance of the correction terms. An example is given for a subduction zone model involving olivine alignment in the mantle-wedge above the slab.

Shear waves

10.1111/j.1365-246x.1992.tb00862.x

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Citations (27)

Using inverse transmission matrices for Marchenko redatuming in highly complex media

Ivan Vasconcelos Kees Wapenaar Joost van der Neut C. J. Thomson Matteo Ravasi

Summary The goal of Marchenko redatuming is to reconstruct, from single-sided reflection data, wavefields at virtual subsurface locations containing transmitted and reflected primaries and internal multiples, while relying on limited or no knowledge of discontinuties in subsurface properties. Here, we address the limitations of the current Marchenko scheme in retrieving waves in highly heterogeneous media, such as subsalt or sub-basalt. We focus on the initial focusing function that plays a key role in the iterative scheme, and propose an alternative focusing function that uses an estimate of the inverse transmission operator from a reference model that contains sharp contrasts (e.g., salt boundaries). Using a physics-driven estimate of the inverse transmission operator, we demonstrate that the new approach retrieves improved subsurface wave-fields, including enhanced amplitudes and internal multiples, in a subsalt environment.

Multiple

Reflection

Operator (biology)

10.1190/segam2015-5896301.1

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Citations (20)

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

Geophysical Journal International (1992)

J. M. Kendall W. S. Guest C. J. Thomson

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 times 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.

Bipolar coordinates

Prolate spheroidal coordinates

Parabolic coordinates

Reciprocity

Action-angle coordinates

Gaussian curvature

10.1111/j.1365-246x.1992.tb00863.x

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Citations (46)

Modelling converted seismic waveforms in isotropic and anisotropic 1-D gradients: discontinuous versus continuous gradient representations

Studia Geophysica et Geodaetica (2012)

Doug Angus C. J. Thomson