Two‐point ray tracing and controlled initial‐value ray tracing in 3‐D heterogeneous block structures

Bulant, Petr

doi:10.1190/1.1885765

Cited by 15 publications

(19 citation statements)

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“…The proposed algorithm of recursive subdivision of triangles, based on algebraic criteria to decide whether the solution curve traverses the triangle or its interior, bears some resemblance to algorithms developed by Bulant (1996Bulant ( , 1999. However, those algorithms are aimed at obtaining a triangulation of the entire parameter plane, so that each triangle has vertices pertaining to the same ray history (sequence of model layers visited).…”

Section: Discussionmentioning

confidence: 98%

“…Hence, we aim at obtaining the solution curve in polygonal representation. For the same reason, triangles are more suitable than rectangles (Vinje et al, 1996;Bulant, 1996Bulant, , 1999Moser and Pajchel, 1997). As a criterion to certify if the solution curve F(γ ) = 0 intersects an elementary triangle, this paper proposes to use Lipschitz continuity In the application on point-to-curve raytracing, the assumption of Lipschitz continuity of the ray emergence point with respect to its initial parameters is valid for smooth media.…”

Section: Implicit Curve Tracing -Global Approachmentioning

confidence: 98%

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Point-to-curve raytracing by algebraic rasterization

Moser

2006

Stud Geophys Geod

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Boundary-value raytracing problems can be concatenated to a smooth one-parameter family of problems, that can be solved by continuation. This has been the purpose of point-to-curve raytracing. A global approach, based on algorithms taken from computer graphics (algebraic rasterization of implicit curves), has several advantages. Subject to relatively mild assumptions -Lipschitz continuity of the emergence point as function of initial parameters -all solution branches are found, there are no problems with initialization, bifurcation, or closed loop solutions. The algebraic rasterization benefits to boundary value raytracing problems in a wide range of applications: shot-to-profile shooting, VSP raytracing, normal raytracing, and more. The algorithm is sufficiently robust to continue even beyond points where the Lipschitz continuity does not apply, such as faults.

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Section: Discussionmentioning

confidence: 98%

Section: Implicit Curve Tracing -Global Approachmentioning

confidence: 98%

Point-to-curve raytracing by algebraic rasterization

Moser

2006

Stud Geophys Geod

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“…The average travel times in seconds closely correspond to the average lengths in kilometres in this model. The travel times are calculated in a grid of 121 Â 241 points covering the model box by means of interpolation within ray cells (BULANT, 1997(BULANT, , 1999a(BULANT, , b, c, 2000BULANT and KLIMESˇ, 1999). Figure 3 shows the rays shot from the point source shifted to 1:85 km from the left-hand corner.…”

Section: Numerical Examplementioning

confidence: 99%

Lyapunov Exponents for 2-D Ray Tracing Without Interfaces

Klimeš¹

2002

Pure and Applied Geophysics

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The Lyapunov exponents quantify the exponential divergence of rays asymptotically, along infinitely long rays. The Lyapunov exponent for a finite 2-D ray and the average Lyapunov exponents for a set of finite 2-D rays and for a 2-D velocity model are introduced. The equations for the estimation of the average Lyapunov exponents in a given smooth 2-D velocity model without interfaces are proposed and illustrated by a numerical example. The equations allow the average exponential divergence of rays and exponential growth of the number of travel-time branches in the velocity model to be estimated prior to ray tracing.

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“…Further, once the rays and traveltimes have been computed, interpolation to a uniform grid is required, which can introduce error. The problem of finding raypaths between a source and/or receiver location and a subsurface gridpoint can be solved by two-point ray tracing, for which there are two main approaches: shooting (Bulant, 1999) and bending (Um and Thurber, 1987). Another family of ray calculation technique is rayfield propagation, in which the whole wavefield for a given mode [P-or S-wave in isotropic media; qP-, qSV-, or qSH-wave in transversely isotropic (TI) media] is propagated rather than individual rays (Vinje et al, 1993).…”

Section: Introductionmentioning

confidence: 99%

Traveltime calculation and prestack depth migration in tilted transversely isotropic media

2004

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The principal objective of our work is to develop a technique for prestack depth migration in tilted transversely isotropic (TTI) media in which the axis of symmetry is not vertical and may be spatially varying. Such models are required to image seismic data in geologically complex regions such as the Canadian Foothills. We have developed a 2D Kirchhoff integral-based migration algorithm in which the traveltime computation comprises the major task. Among the existing traveltime computation algorithms such as ray tracing with interpolation, ray bending, and eikonal solvers, a direct or a brute force approach of traveltime computation is generally highly robust. We have modified a direct method of first-arrival P-wave traveltime computation in 2D media that accounts for TTI. The algorithm requires that the group velocity be computed at each gridpoint, using either an analytic solution or by an approximate Fourier series expansion. The P-wave traveltime contours computed for complex geologic models show the pronounced effects from TTI media. Our results, using laboratory P-wave data collected over a physical model of an anisotropic thrust sheet, reveal that a 2D Kirchhoff migration based on our traveltime algorithm (TTI model) images the structure beneath the thrust sheet very well. The vertical transversely isotropic (assuming a vertical axis of symmetry) or isotropic imaging introduces false anticlinal structures. We compare our results with those obtained by a recursive-extrapolation method and find that our approach images the underside of one of the thrust sheets better.

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Two‐point ray tracing and controlled initial‐value ray tracing in 3‐D heterogeneous block structures

Cited by 15 publications

References 1 publication

Point-to-curve raytracing by algebraic rasterization

Point-to-curve raytracing by algebraic rasterization

Lyapunov Exponents for 2-D Ray Tracing Without Interfaces

Traveltime calculation and prestack depth migration in tilted transversely isotropic media

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