Theory and Observations: Body Waves, Ray Methods, and Finite-Frequency Effects

Virieux, Jean; Lambaré, Gilles

doi:10.1016/b978-0-444-53802-4.00004-x

Cited by 4 publications

(1 citation statement)

References 320 publications

(264 reference statements)

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“…The eikonal equation is a good approximation for seismic wave propagation in the so-called "high-frequency limit" at which seismic wavelengths are very small compared to the scale of wave propagation (e.g. Červený, 1989, 2005Dellinger, 1997;Mensch and Farra, 1999;Rawlinson et al, 2008;Slawinski, 2014;Virieux and Lambaré, 2007;Woodhouse and Deuss, 2007). From this approximation arises the convenient fiction of seismic rays, which are both the characteristics of the HJE and solutions of Hamilton's equations.…”

Section: A1 Geoscience Applications Of the Hjementioning

confidence: 99%

The direction of landscape erosion

Stark¹,

Stark

2022

Earth Surf. Dynam.

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Abstract. The rate of erosion of a landscape depends largely on local gradient and material fluxes. Since both quantities are functions of the shape of the catchment surface, this dependence constitutes a mathematical straitjacket, in the sense that – subject to simplifying assumptions about the erosion process, and absent variations in external forcing and erodibility – the rate of change of surface geometry is solely a function of surface geometry. Here we demonstrate how to use this geometric self-constraint to convert a gradient-dependent erosion model into its equivalent Hamiltonian, and explore the implications of having a Hamiltonian description of the erosion process. To achieve this conversion, we recognize that the rate of erosion defines the velocity of surface motion in its orthogonal direction, and we express this rate in its reciprocal form as the surface-normal slowness. By rewriting surface tilt in terms of normal slowness components and deploying a substitution developed in geometric mechanics, we extract what is known as the fundamental metric function of the model phase space; its square is the Hamiltonian. Such a Hamiltonian provides several new ways to solve for the evolution of an erosion surface: here we use it to derive Hamilton's ray-tracing equations, which describe both the velocity of a surface point and the rate of change of the surface-normal slowness at that point. In this context, gradient-dependent erosion involves two distinct directions: (i) the surface-normal direction, which points subvertically downwards, and (ii) the erosion ray direction, which points upstream at a generally small angle to horizontal with a sign controlled by the scaling of erosion with slope. If the model erosion rate scales faster than linearly with gradient, the rays point obliquely upwards, but if erosion scales sublinearly with gradient, the rays point obliquely downwards. This dependence of erosional anisotropy on gradient scaling explains why, as previous studies have shown, model knickpoints behave in two distinct ways depending on the gradient exponent. Analysis of the Hamiltonian shows that the erosion rays carry boundary-condition information upstream, and that they are geodesics, meaning that surface evolution takes the path of least erosion time. Correspondingly, the time it takes for external changes to propagate into and change a landscape is set by the velocity of these rays. The Hamiltonian also reveals that gradient-dependent erosion surfaces have a critical tilt, given by a simple function of the gradient scaling exponent, at which ray-propagation behaviour changes. Channel profiles generated from the non-dimensionalized Hamiltonian have a shape entirely determined by the scaling exponents and by a dimensionless erosion rate expressed as the surface tilt at the downstream boundary.

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Section: A1 Geoscience Applications Of the Hjementioning

confidence: 99%

The direction of landscape erosion

Stark¹,

Stark

2022

Earth Surf. Dynam.

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Uniform asymptotic conversion of Helmholtz data from 3D to 2D

Yedlin

Vorst

Virieux

2012

Journal of Applied Geophysics

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Computation of ray-Born seismograms using isochrons

Sarajærvi

Keers

2018

GEOPHYSICS

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Seismic modeling in heterogeneous media is accomplished by using either approximate or fully numerical methods. A popular approximate method is ray-Born modeling, which requires the computation of 3D integrals. We have developed an integration technique for accurate and, under certain circumstances, efficient evaluation of the ray-Born integrals in the time domain. The 3D integrals are split into several 2D integrals, each of which gives the wavefield at a certain time, so that the waveform at each time step is computed independently of all other times. We compute seismograms for 3D heterogeneous acoustic media using this technique and compare these seismograms with seismograms computed using two other modeling methods: frequency-domain ray-Born modeling and finite-difference modeling of the acoustic wave equation. Our method can also be applied to elastic ray-Born modeling. Velocity models with smooth scatterers and the SEG/EAGE overthrust model are used for comparison. The ray-Born seismograms computed using the time- and frequency-domain ray-Born modeling methods are identical, as expected. The comparison between the ray-Born modeling and the finite-difference-modeling method indicates that the waveforms are similar for both types of velocity models. We evaluate the discrepancies in terms of multiple scattering and multipathing.

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Theory and Observations: Body Waves, Ray Methods, and Finite-Frequency Effects

Cited by 4 publications

References 320 publications

The direction of landscape erosion

The direction of landscape erosion

Uniform asymptotic conversion of Helmholtz data from 3D to 2D

Computation of ray-Born seismograms using isochrons

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