Simultaneous computation of seismic slowness paths and the traveltime field in anisotropic media

We present a numerical implementation of an alternative formulation of the geometrical, or ray, acoustics, where wavefronts rather than rays are the primary objects. Rays are recovered as a by-product of wavefront tracing. The alternative formulation of the geometrical acoustics is motivated, first, by the observation that wavefronts are often more stable than rays at long-range sound propagation, and, second, by a need for computationally efficient modeling of high-frequency acoustic fields in three-dimensionally inhomogeneous, moving or motionless fluids. Wavefronts are found as a finite-difference solution to a system of partial differential equations, which is equivalent to the eikonal equation and is a direct implementation of the intuitive Huygens' wavefront construction. The finite-difference algorithm is an extension of the approach originally developed in the framework of an open source Madagascar project. Benchmark problems, which admit exact, analytic solutions of the eikonal equation, are formulated and utilized to verify the finite-difference wavefront tracing algorithm. Huygens' wavefront tracing (HWT) is applied to modeling sound propagation in three-dimensionally inhomogeneous ocean and atmosphere.

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“…At M = 0, t = 0 at R = R 0 in Eq. (31), and, hence, Eq. (31) represents timefronts due to a point source at R = R 0 .…”

Section: Maxwell's "Fish-eye" Lensmentioning

confidence: 94%

“…Equation (31) shows that a radiated pulse returns periodically to every observation point with the same period T = πa/c 0 .…”

Section: Maxwell's "Fish-eye" Lensmentioning

confidence: 99%

Tracing Three-Dimensional Acoustic Wavefronts in Inhomogeneous, Moving Media

et al. 2014

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“…In this paper, we do not distinguish between the phase velocity and the group velocity in ray tracing throughout the study area between two vertical boreholes, because these two parallel boreholes are so close to each other. When considering the exact phase velocity, Wang (2013) suggests to replace the raypath concept with the concept of slowness paths, and to solve a system of equations, consisting of slowness equations and an explicit normal constraint that slowness vectors are perpendicular to wavefronts.…”

Section: Anisotropic Traveltime Tomographymentioning

confidence: 99%

Crosshole seismic tomography with cross-firing geometry

et al. 2016

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We have developed a case study of crosshole seismic tomography with a cross-firing geometry in which seismic sources were placed in two vertical boreholes alternatingly and receiver arrays were placed in another vertical borehole. There are two crosshole seismic data sets in a conventional sense. These two data sets are used jointly in seismic tomography. Because the local sediment is dominated by periodic, flat, thin layers, there is seismic anisotropy with different velocities in the vertical and horizontal directions. The vertical transverse isotropy anisotropic effect is taken into account in inversion processing, which consists of three stages in sequence. First, isotropic traveltime tomography is used for estimating the maximum horizontal velocity. Then, anisotropic traveltime tomography is used to invert for the anisotropic parameter, which is the normalized difference between the maximum horizontal velocity and the maximum vertical velocity. Finally, anisotropic waveform tomography is implemented to refine the maximum horizontal velocity. The cross-firing acquisition geometry significantly improves the ray coverage and results in a relatively even distribution of the ray density in the study area between two boreholes. Consequently, joint inversion of two crosshole seismic data sets improves the resolution and increases the reliability of the velocity model reconstructed by tomography.

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“…where vðx; z; ϕÞ is the phase velocity, and ϕ is the phase angle (Wang, 2013). This is the eikonal equation, read as ∇τ · ∇τ ¼ 1∕v 2 .…”

Section: Ray Tracing Based On Raypath Equationsmentioning

confidence: 99%

“…In isotropic media, the raypath and the group velocity vector coincide with direction normal to the wavefront (Julian and Gubblins, 1977;Pereyra et al, 1980;Červený, 2001;Rawlinson et al, 2007). In anisotropic media, ray tracing remains a challenging problem especially in terms of numerical calculation, although the fundamental theory of anisotropy is well-known (Crampin, 1981;Fryer and Frazer, 1984;Shearer and Chapman, 1988;Červený, 2001;Wang, 2013). The raypath and the group velocity vector are not perpendicular to the wavefront.…”

Section: Introductionmentioning

confidence: 99%

Seismic ray tracing in anisotropic media: A modified Newton algorithm for solving highly nonlinear systems

Wang

2014

GEOPHYSICS

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Seismic ray tracing with a path-bending method leads to a nonlinear system that has much stronger nonlinearity in anisotropic media than the counterpart in isotropic media. Any path perturbation causes changes to directional velocities, which depend not only upon the spatial position but also upon the local propagation direction in anisotropic media. To combat the high nonlinearity of the problem, the Newton-type iterative algorithm is modified by enforcing Fermat's minimum-time principle as a constraint for the solution update, instead of conventional error minimization in the nonlinear system. As the algebraic problem is incorporated with the physical principle, it is able to stabilize the solution for such a highly nonlinear problem as ray tracing in realistically complicated anisotropic media. With this modified algorithm, two ray-tracing schemes are presented. The first scheme involves newly derived raypath equations, which are approximate for anisotropic media but the minimum-time constraint will ensure that the solution steadily converges to the true solution. The second scheme is based on the minimal variation principle. It is more efficient than the first one as it solves a tridiagonal system and does not need to compute the Jacobian and its inverse in each iteration. Even in this second scheme, Fermat's minimum-time constraint is employed for the solution update, so as to guarantee a robust convergence of the iterative solution in anisotropic media.

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Simultaneous computation of seismic slowness paths and the traveltime field in anisotropic media

Cited by 7 publications

References 17 publications

Tracing Three-Dimensional Acoustic Wavefronts in Inhomogeneous, Moving Media

Tracing Three-Dimensional Acoustic Wavefronts in Inhomogeneous, Moving Media

Crosshole seismic tomography with cross-firing geometry

Seismic ray tracing in anisotropic media: A modified Newton algorithm for solving highly nonlinear systems

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