Theory and Observations – Body Waves: Ray Methods and Finite Frequency Effects

Virieux, Jean; Lambaré, Gilles

doi:10.1016/b978-044452748-6.00004-3

Cited by 7 publications

(6 citation statements)

References 179 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Eikonal equation arises in many domains such as geometrical optics (Benamou 2003), optimality problems with shortest/geodesic path calculation (Moser 1991;Kimmel & Sethian 1998), computer vision problems with shape-from-shading (Kimmel & Sethian 2001;Rouy & Tourin 1992), as well as in geophysics and more particularly in seismic imaging for traveltime-based tomographic methods (Virieux & Lambaré 2007;Vidale 1988;Podvin & Lecomte 1991;Hole & Zelt 1995;Le Meur 1994;Le Meur et al 1997;Billette & Lambaré 1998;Leung & Qian 2006;Taillandier et al 2009). It establishes a mathematical link between the travel time of an oscillatory phenomenon from a source to any point of a given medium, and the speed at which this phenomenon propagates inside the medium.…”

Section: Introductionmentioning

confidence: 99%

“…They are efficiently solved by integration along rays for given initial conditions following a Lagrangian approach (Courant & Hilbert 1966). A review of ray methods can be found in Virieux & Lambaré (2007). However, ray tracing suffers from inherent difficulties, such as non-uniform sampling of the medium, the complexity related to triplications, the existence of shadow zones and the difficulty to capture the ray arriving at a receiver from a given source.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An accurate discontinuous Galerkin method for solving point-source Eikonal equation in 2-D heterogeneous anisotropic media

Bouteiller¹,

Benjemaa

Métivier

et al. 2017

Geophysical Journal International

View full text Add to dashboard Cite

Accurate numerical computation of wave traveltimes in heterogeneous media is of major interest for a large range of applications in seismics, among which phase identification, data windowing, traveltime tomography and seismic imaging. A high level of precision is needed for traveltimes and their derivatives in applications which require quantities such as amplitude or takeoff angle. Even more challenging is the anisotropic case, where the general Eikonal equation is a quartic in the derivatives of traveltimes. Despite their efficiency on Cartesian meshes, finite-difference solvers are inappropriate when dealing with unstructured meshes and irregular topographies. Moreover, reaching high orders of accuracy generally requires wide stencils and high additional computational load. To go beyond these limitations, we propose a discontinuous-finite-element-based strategy which has the following advantages: (1) the Hamiltonian formalism is general enough for handling the full anisotropic Eikonal equations; (2) the scheme is suitable for any desired high-order formulation or mixing of orders (p-adaptivity); (3) the solver is explicit what-2 P. Le Bouteiller et al. ever Hamiltonian is used (no need to find the roots of the quartic); (4) the use of unstructured meshes provides the flexibility for handling complex boundary geometries such as topographies (h-adaptivity) and radiation boundary conditions for mimicking an infinite medium. The point-source factorization principles are extended to this discontinuous Galerkin formulation. Extensive tests in smooth analytical media demonstrate the high accuracy of the method. Simulations in strongly heterogeneous media illustrate the solver robustness to realistic Earth-sciences-oriented applications.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An accurate discontinuous Galerkin method for solving point-source Eikonal equation in 2-D heterogeneous anisotropic media

Bouteiller¹,

Benjemaa

Métivier

et al. 2017

Geophysical Journal International

View full text Add to dashboard Cite

show abstract

“…References are available in the review article (Virieux and Lambaré, 2007). The equation that we will use throughout this article is the two or three-dimensional Helmholtz equation for the pressure field.…”

Section: Ray Tracing Backgroundmentioning

confidence: 99%

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

Yedlin

Vorst

Virieux

2012

Journal of Applied Geophysics

View full text Add to dashboard Cite

“…The HJE has seen only sporadic use in the geosciences -except in the field of seismology, where its static or eikonal form has been found to be particularly useful. 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., Cervený, 1989Cervený, , 2005Cervený, , 2002Dellinger, 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: Geoscience Applications Of the Hjementioning

confidence: 99%

The Direction of Erosion

Stark

2021

Preprint

View full text Add to dashboard Cite

Abstract. The rate of erosion of a geomorphic surface depends on its local gradient and on the material fluxes over it. 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 an 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 by 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 of solving 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, 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. Analysis of the Hamiltonian shows that these rays carry boundary-condition information upstream, and that they are geodesics, meaning that erosion takes the path of least erosion time. This constitutes a definition of the variational principle governing landscape evolution. In contrast with previous studies of network self-organization, neither energy nor energy dissipation is invoked in this variational principle, only geometry.

show abstract

Theory and Observations – Body Waves: Ray Methods and Finite Frequency Effects

Cited by 7 publications

References 179 publications

An accurate discontinuous Galerkin method for solving point-source Eikonal equation in 2-D heterogeneous anisotropic media

An accurate discontinuous Galerkin method for solving point-source Eikonal equation in 2-D heterogeneous anisotropic media

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

The Direction of Erosion

Contact Info

Product

Resources

About