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Journal articles on the topic 'Eikonal equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Hu, Jiangtao, Jianliang Qian, Jian Song, Min Ouyang, Junxing Cao, and Shingyu Leung. "Eulerian partial-differential-equation methods for complex-valued eikonals in attenuating media." GEOPHYSICS 86, no. 4 (2021): T179—T192. http://dx.doi.org/10.1190/geo2020-0659.1.

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Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distribut

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2

Fedorov, Arseny V., Christina A. Stepa, Anna V. Korolkova, Migran N. Gevorkyan, and Dmitry S. Kulyabov. "Methodological derivation of the eikonal equation." Discrete and Continuous Models and Applied Computational Science 31, no. 4 (2023): 399–418. http://dx.doi.org/10.22363/2658-4670-2023-31-4-399-418.

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Usually, when working with the eikonal equation, reference is made to its derivation in the monograph by Born and Wolf. The derivation of this equation was done rather carelessly. Understanding this derivation requires a certain number of implicit assumptions. For a better understanding of the eikonal approximation and for methodological purposes, the authors decided to repeat the derivation of the eikonal equation, explicating all possible assumptions. Methodically, the following algorithm for deriving the eikonal equation is proposed. The wave equation is derived from Maxwell’s equation. In

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3

Leung, Shingyu, Jiangtao Hu, and Jianliang Qian. "Liouville partial-differential-equation methods for computing 2D complex multivalued eikonals in attenuating media." GEOPHYSICS 87, no. 2 (2021): T71—T84. http://dx.doi.org/10.1190/geo2021-0113.1.

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We have developed a Liouville partial-differential-equation (PDE)-based method for computing complex-valued eikonals in real phase space in the multivalued sense in attenuating media with frequency-independent qualify factors, where the new method computes the real and imaginary parts of the complex-valued eikonal in two steps by solving Liouville equations in real phase space. Because the earth is composed of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy losses and phase distortions of wave propagation.

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4

Meher, Mehrollah, and Davood Rostamy. "Hybrid of differential quadrature and sub-gradients methods for solving the system of Eikonal equations." Nonlinear Engineering 10, no. 1 (2021): 436–49. http://dx.doi.org/10.1515/nleng-2021-0035.

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Abstract Many important natural phenomena of wave propagations are modeled by Eikonal equations and a variety of new methods are needed to solve them. The differential quadrature method (DQM) is an effective numerical method for solving the system of differential equations that can achieve accurate numerical results using fewer grid points and therefore requires relatively little computational effort. In this paper, we focus on the implementation of the non-smooth Eikonal optimization by using a hybrid of polynomial differential quadrature (PDQ) or Fourier differential quadrature (FDQ) method

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5

Liner, Christopher L. "Some analytic aspects of a 2.5-D wave equation." GEOPHYSICS 60, no. 5 (1995): 1536–40. http://dx.doi.org/10.1190/1.1443886.

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A recently proposed 2.5-D wave equation is transformed to the frequency domain where WKBJ analysis is applied. This analysis assumes a general c(x, z) velocity field. Eikonal and transport equations are derived and compared to their 2-D and in‐plane 3-D counterparts. The implied eikonal equation is shown to be identical to the 2-D and in‐plane 3-D eikonal. This assures traveltime accuracy of the 2.5-D wave equation for any velocity model. The implied transport equation is shown to be consistent with 3-D theory in a constant velocity region containing a point source.

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6

Gudynas, Gintautas. "Representation Formula for Solutions of Eikonal Type Equations." Nonlinear Analysis: Modelling and Control 6, no. 1 (2001): 57–65. http://dx.doi.org/10.15388/na.2001.6.1.15225.

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Equations of an eikonal type ones arise in many areas of applications, including optics, fluid mechanics, material sciences, and control theory. This article investigates the representation formula for semiconcave solutions of the boundary problem for eikonal type equation. The formula is given by forming envelope of some fundamental solutions of the equation. Using these fundamental solutions we obtain the representation formula for solutions of Cauchy problem for appropriate Hamilton – Jacobi equation.

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7

Bekar, Ali Can, Erdogan Madenci, Ehsan Haghighat, Umair bin Waheed, and Tariq Alkhalifah. "Solving the eikonal equation for compressional and shear waves in anisotropic media using peridynamic differential operator." Geophysical Journal International 229, no. 3 (2022): 1942–63. http://dx.doi.org/10.1093/gji/ggac037.

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SUMMARY Traveltimes of compressional (P) and shear (S) waves have proven essential in many earthquake and exploration seismology applications. An accurate and efficient traveltime computation for P and S waves is crucial for the success of these applications. However, solving the eikonal equation with a complex phase velocity field in anisotropic media is challenging. The eikonal equation is a first-order nonlinear hyperbolic partial differential equation. It represents the high-frequency asymptotic approximation of the wave equation. The fast marching and sweeping methods are commonly used du

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8

Sava, Paul, and Sergey Fomel. "3-D traveltime computation using Huygens wavefront tracing." GEOPHYSICS 66, no. 3 (2001): 883–89. http://dx.doi.org/10.1190/1.1444977.

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Traveltime computation is widely used in seismic modeling, imaging, and velocity analysis. The two most commonly used methods are ray tracing and numerical solutions to the eikonal equation. Eikonal solvers are fast and robust but are limited to computing only the first‐arrival traveltimes. Ray tracing can compute multiple arrivals but lacks the robustness of eikonal solvers. We propose a robust and complete method of traveltime computation. It is based on a system of partial differential equations, which is equivalent to the eikonal equation but formulated in the ray‐coordinates system. We us

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9

Bouin, Emeric, and Vincent Calvez. "A kinetic eikonal equation." Comptes Rendus Mathematique 350, no. 5-6 (2012): 243–48. http://dx.doi.org/10.1016/j.crma.2012.03.009.

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10

Li, Siwei, Alexander Vladimirsky, and Sergey Fomel. "First-break traveltime tomography with the double-square-root eikonal equation." GEOPHYSICS 78, no. 6 (2013): U89—U101. http://dx.doi.org/10.1190/geo2013-0058.1.

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First-break traveltime tomography is based on the eikonal equation. Because the eikonal equation is solved at fixed-shot positions and only receiver positions can move along the raypath, the adjoint-state tomography relies on inversion to resolve possible contradicting information between independent shots. The double-square-root (DSR) eikonal equation allows not only the receivers but also the shots to change position, and thus describes the prestack survey as a whole. Consequently, its linearized tomographic operator naturally handles all shots together, in contrast with the shotwise approac

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11

Schneider, William A. "Linearization of the P‐wave eikonal equation for weak vertical transverse isotropy." GEOPHYSICS 68, no. 3 (2003): 1075–82. http://dx.doi.org/10.1190/1.1581079.

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I develop approximate P‐wave and SV‐wave eikonal equations for weak vertical transversely isotropic (VTI) media and then use perturbation theory to solve the P‐wave eikonal equation. I proceed by expressing the exact VTI eikonal equations for P‐waves and SV‐waves in terms of the intuitive “Thomsen” anisotropy parameters and then linearizing the results with respect to these parameters. Next, I apply perturbation theory (with a heterogeneous, isotropic “reference” medium) to the P‐wave weak VTI eikonal equation. This results in a linear partial differential equation for the traveltime perturbat

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12

Vázquez, L., S. Jiménez, and A. B. Shvartsburg. "The wave equation: from eikonal to anti-eikonal approximation." Izvestiya Vysshikh Uchebnykh Zavedenii. Materialy Elektronnoi Tekhniki = Materials of Electronics Engineering 19, no. 2 (2016): 103–7. http://dx.doi.org/10.17073/1609-3577-2016-2-103-107.

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When the refractive index changes very slowly compared to the wave-length we may use the eikonal approximation to the wave equation. In the opposite case, when the refractive index highly variates over the distance of one wave-length, we have what can be termed as the anti-eikonal limit. This situation is addressed in this work. The anti- eikonal limit seems to be a relevant tool in the modelling and design of new optical media. Besides, it describes a basic universal behaviour, independent of the actual values of the refractive index and, thus, of the media, for the components of a wave with

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13

Vázquez, Luis, Salvador Jiménez, and Alexander B. Shvartsburg. "The wave equation: From eikonal to anti-eikonal approximation." Modern Electronic Materials 2, no. 2 (2016): 51–53. http://dx.doi.org/10.1016/j.moem.2016.11.001.

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14

Qian, Jianliang, and William W. Symes. "Finite‐difference quasi‐P traveltimes for anisotropic media." GEOPHYSICS 67, no. 1 (2002): 147–55. http://dx.doi.org/10.1190/1.1451438.

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The first‐arrival quasi‐P wave traveltime field in an anisotropic elastic solid solves a first‐order nonlinear partial differential equation, the q P eikonal equation. The difficulty in solving this eikonal equation by a finite‐difference method is that for anisotropic media the ray (group) velocity direction is not the same as the direction of the traveltime gradient, so that the traveltime gradient can no longer serve as an indicator of the group velocity direction in extrapolating the traveltime field. However, establishing an explicit relation between the ray velocity vector and the phase

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15

Wang, Guohui, Xuan Zhang, and Jin Cheng. "A Unified Shape-From-Shading Approach for 3D Surface Reconstruction Using Fast Eikonal Solvers." International Journal of Optics 2020 (May 29, 2020): 1–12. http://dx.doi.org/10.1155/2020/6156058.

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Object shape reconstruction from images has been an active topic in computer vision. Shape-from-shading (SFS) is an important approach for inferring 3D surface from a single shading image. In this paper, we present a unified SFS approach for surfaces of various reflectance properties using fast eikonal solvers. The whole approach consists of three main components: a unified SFS model, a unified eikonal-type partial differential image irradiance (PDII) equation, and fast eikonal solvers for the PDII equation. The first component is designed to address different reflectance properties including

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16

Kubera, P., and J. Felcman. "On a numerical flux for the pedestrian flow equations." Journal of Applied Mathematics, Statistics and Informatics 11, no. 2 (2015): 79–96. http://dx.doi.org/10.1515/jamsi-2015-0014.

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Abstract The pedestrian flow equations are formulated as the hyperbolic problem with a source term, completed by the eikonal equation yielding the desired direction of the pedestrian velocity. The operator splitting consisting of successive discretization of the eikonal equation, ordinary differential equation with the right hand side being the source term and the homogeneous hyperbolic system is proposed. The numerical flux of the Vijayasundaram type is proposed for the finite volume solution of the hyperbolic problem. The Vijayasundaram numerical flux, originally proposed for the hyperbolic

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17

Borovskikh, A. V. "Eikonal Equation for Anisotropic Media." Journal of Mathematical Sciences 197, no. 2 (2014): 248–89. http://dx.doi.org/10.1007/s10958-014-1714-5.

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18

Borovskikh, A. V. "The two-dimensional eikonal equation." Siberian Mathematical Journal 47, no. 5 (2006): 813–34. http://dx.doi.org/10.1007/s11202-006-0091-9.

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19

Ghanam, Ryad, and Gerard Thompson. "Symmetries of the Eikonal equation." Communications in Nonlinear Science and Numerical Simulation 60 (July 2018): 137–44. http://dx.doi.org/10.1016/j.cnsns.2018.01.007.

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20

Aldridge, David F. "Linearization of the eikonal equation." GEOPHYSICS 59, no. 10 (1994): 1631–32. http://dx.doi.org/10.1190/1.1443552.

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Seismic traveltime tomography is a nonlinear inverse problem wherein an unknown slowness model is inferred from the observed arrival times of seismic waves. Nonlinearity arises because the raypath connecting a given source and receiver depends on the slowness. Specifically, if L(s) designates a raypath through the slowness model s between two fixed endpoints, then the path integral for traveltime [Formula: see text] is a nonlinear functional of s because it does not, in general, satisfy the superposition condition (i.e., [Formula: see text] where [Formula: see text] and [Formula: see text] are

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21

Flores, Jesús, Ángel García, Mihaela Negreanu, Eduardo Salete, Francisco Ureña, and Antonio M. Vargas. "Numerical Solutions to Wave Propagation and Heat Transfer Non-Linear PDEs by Using a Meshless Method." Mathematics 10, no. 3 (2022): 332. http://dx.doi.org/10.3390/math10030332.

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The applications of the Eikonal and stationary heat transfer equations in broad fields of science and engineering are the motivation to present an implementation, not only valid for structured domains but also for completely irregular domains, of the meshless Generalized Finite Difference Method (GFDM). In this paper, the fully non-linear Eikonal equation and the stationary heat transfer equation with variable thermal conductivity and source term are solved in 2D. The explicit formulae for derivatives are developed and applied to the equations in order to obtain the numerical schemes to be use

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22

WERESZCZYŃSKI, A. "KNOTS, BRAIDS AND HEDGEHOGS FROM THE EIKONAL EQUATION." Modern Physics Letters A 20, no. 15 (2005): 1135–46. http://dx.doi.org/10.1142/s0217732305017330.

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The complex eikonal equation in the three space dimensions is considered. We show that apart from the recently found torus knots, this equation can also generate other topological configurations with a nontrivial value of the π2(S2) index: braided open strings as well as hedgehogs. In particular, cylindric strings, i.e. string solutions located on a cylinder with a constant radius are found. Moreover, solutions describing strings lying on an arbitrary surface topologically equivalent to cylinder are presented. We discuss them in the context of the eikonal knots. The physical importance of the

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23

Hao, Qi, and Tariq Alkhalifah. "An acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis." GEOPHYSICS 82, no. 1 (2017): C9—C20. http://dx.doi.org/10.1190/geo2016-0160.1.

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Seismic-wave attenuation is an important component of describing wave propagation. Certain regions, such as gas clouds inside the earth, exert highly localized attenuation. In fact, the anisotropic nature of the earth induces anisotropic attenuation because the quasi P-wave dispersion effect should be profound along the symmetry direction. We have developed a 2D acoustic eikonal equation governing the complex-valued traveltime of quasi P-waves in attenuating, transversely isotropic media with a vertical-symmetry axis (VTI). This equation is derived under the assumption that the complex-valued

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24

Qin, Fuhao. "On: “3-D traveltime compuation using the fast marching method” (J. A. Sethian and A. M. Popovici, GEOPHYSICS, 64, 516–523)." GEOPHYSICS 65, no. 2 (2000): 682. http://dx.doi.org/10.1190/1.1444764.

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The Sethian and Popovici paper “3-D traveltime computation using the fast marching method” that appeared in Geophysics, Vol. 64, 516–523, discussed a method to solve the eikonal equation for first arrival traveltimes which was called the “fast marching” method. The method, as the authors demonstrated, is very fast and stable. However, their method is very similar to the method discussed by F. Qin et al. (1992), entitled “Finite difference solution of the eikonal equation along expanding wavefronts,” Geophysics, Vol. 57, 478–487. F. Qin et al. first proposed the “expanding wavefront” method for

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25

Waheed, Umair bin. "A fast-marching eikonal solver for tilted transversely isotropic media." GEOPHYSICS 85, no. 6 (2020): S385—S393. http://dx.doi.org/10.1190/geo2019-0799.1.

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Fast and accurate traveltime computation for quasi-P waves in anisotropic media is an essential ingredient of many seismic processing and interpretation applications such as Kirchhoff modeling and migration, microseismic source localization, and traveltime tomography. Fast-sweeping methods are widely used for solving the anisotropic eikonal equation due to their flexibility in solving general equations compared to the fast-marching method. However, it has been observed that fast sweeping can be much less efficient than fast marching for models with curved characteristics and practical grid siz

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26

Gray, Samuel H., and William P. May. "Kirchhoff migration using eikonal equation traveltimes." GEOPHYSICS 59, no. 5 (1994): 810–17. http://dx.doi.org/10.1190/1.1443639.

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The use of ray shooting followed by interpolation of traveltimes onto a regular grid is a popular and robust method for computing diffraction curves for Kirchhoff migration. An alternative to this method is to compute the traveltimes by directly solving the eikonal equation on a regular grid, without computing raypaths. Solving the eikonal equation on such a grid simplifies the problem of interpolating times onto the migration grid, but this method is not well defined at points where two different branches of the traveltime field meet. Also, computational and data storage issues that are relat

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27

Zhang, Qingyu, Xiao Ma, and Yufeng Nie. "An iterative fast sweeping method for the eikonal equation in 2D anisotropic media on unstructured triangular meshes." GEOPHYSICS 86, no. 3 (2021): U49—U61. http://dx.doi.org/10.1190/geo2020-0187.1.

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Computation of traveltimes and raypaths is important for anisotropic tomography inversions. The eikonal-equation-based method outperforms traditional ray methods by producing more accurate results. However, most existing eikonal solvers are formulated on structured regular meshes, which are no longer accurate for models with the presence of irregular topography and subsurface interfaces. To solve the eikonal equation in vertically transversely isotropic (VTI) or tilted transversely isotropic (TTI) models with irregular geometry, we have formulated a new iterative fast sweeping method (FSM) on

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FRANKLIN, JONATHAN B., and JERRY M. HARRIS. "A HIGH-ORDER FAST MARCHING SCHEME FOR THE LINEARIZED EIKONAL EQUATION." Journal of Computational Acoustics 09, no. 03 (2001): 1095–109. http://dx.doi.org/10.1142/s0218396x01000784.

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We present a high-order upwind finite-difference scheme for solving a useful family of first-order partial differential equations, of which the linearized eikonal equation is a member. Fast solutions of the linearized eikonal equation have applications in traveltime tomography and residual migration algorithms. The technique, besides being both accurate and stable, escapes aperture limitations inherent in static marching schemes. We use a time-sequential evaluation method similar to Sethian's Fast Marching strategy to insure causal operator evaluation. We apply our technique to several complex

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29

Alexandrov, Dmitry, Umair bin Waheed, and Leo Eisner. "Microseismic Location Error Due to Eikonal Traveltime Calculation." Applied Sciences 11, no. 3 (2021): 982. http://dx.doi.org/10.3390/app11030982.

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The accuracy of computed traveltimes in a velocity model plays a crucial role in localization of microseismic events. The conventional approach usually utilizes robust fast sweeping or fast marching methods to solve the eikonal equation numerically with a finite-difference scheme. These methods introduce traveltime errors that strongly depend on the direction of wave propagation. Such error results in moveout changes of the computed traveltimes and introduces significant location bias. The issue can be addressed by using a finite-difference scheme to solve the factored eikonal equation. This e

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Filpo, Eduardo, Jessé Costa, and Jörg Schleicher. "Image-guided ray tracing and its applications." GEOPHYSICS 86, no. 3 (2021): U39—U47. http://dx.doi.org/10.1190/geo2020-0642.1.

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Eikonal solvers have important applications in seismic data processing and inversion, the so-called image-guided methods. To this day, in image-guided applications, the solution of the eikonal equation is implemented using partial-differential-equation solvers, such as fast-marching or fast-sweeping methods. We have found that alternatively, one can numerically integrate the dynamic Hamiltonian system defined by the image-guided eikonal equation and reconstruct the solution with image-guided rays. We evaluate interesting applications of image-guided ray tracing to seismic data processing, demo

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31

LIU SHEN-QUAN and LU QI-SHAO. "SOLVING EIKONAL EQUATION BY FIFE'S SCALING." Acta Physica Sinica 46, no. 8 (1997): 1457. http://dx.doi.org/10.7498/aps.46.1457.

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A. Peletier, Mark, and Marco Veneroni. "Stripe patterns and the Eikonal equation." Discrete & Continuous Dynamical Systems - S 5, no. 1 (2012): 183–89. http://dx.doi.org/10.3934/dcdss.2012.5.183.

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Hicks, R. Andrew, Ronald K. Perline, and Sarah G. Rody. "Anti-eikonal equation of an eigenmirror." Journal of the Optical Society of America A 37, no. 10 (2020): 1566. http://dx.doi.org/10.1364/josaa.401301.

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34

Wellner, M., and A. M. Pertsov. "Generalized eikonal equation in excitable media." Physical Review E 55, no. 6 (1997): 7656–61. http://dx.doi.org/10.1103/physreve.55.7656.

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35

Ekici, Cumali, and Çimdiker Muradiye. "A note on Berwald eikonal equation." Journal of Physics: Conference Series 766 (October 2016): 012029. http://dx.doi.org/10.1088/1742-6596/766/1/012029.

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36

Martı́nez-Niconoff, Gabriel, Patricia M. Vara, and Adrián Carbajal Domı́nguez. "Eikonal equation for partially coherent fields." Optics Communications 259, no. 2 (2006): 488–91. http://dx.doi.org/10.1016/j.optcom.2005.09.062.

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37

Cruz-Sampedro, J., and E. Skibsted. "Global solutions to the eikonal equation." Journal of Differential Equations 255, no. 12 (2013): 4337–77. http://dx.doi.org/10.1016/j.jde.2013.08.002.

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Zagatti, Sandro. "Maximal generalized solution of eikonal equation." Journal of Differential Equations 257, no. 1 (2014): 231–63. http://dx.doi.org/10.1016/j.jde.2014.04.001.

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39

Emel'yanova, I. S. "Jacobi method and the eikonal equation." Computational Mathematics and Modeling 9, no. 1 (1998): 55–63. http://dx.doi.org/10.1007/bf02404085.

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Waheed, Umair bin, and Tariq Alkhalifah. "A fast sweeping algorithm for accurate solution of the tilted transversely isotropic eikonal equation using factorization." GEOPHYSICS 82, no. 6 (2017): WB1—WB8. http://dx.doi.org/10.1190/geo2016-0712.1.

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Traveltime computation is essential for many seismic data processing applications and velocity analysis tools. High-resolution seismic imaging requires eikonal solvers to account for anisotropy whenever it significantly affects the seismic wave kinematics. Moreover, computation of auxiliary quantities, such as amplitude and take-off angle, relies on highly accurate traveltime solutions. However, the finite-difference-based eikonal solution for a point-source initial condition has upwind source singularity at the source position because the wavefront curvature is large near the source point. Th

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41

Zhou, Xiaole, Haiqiang Lan, Ling Chen, et al. "An iterative factored topography-dependent eikonal solver for anisotropic media." GEOPHYSICS 86, no. 5 (2021): U121—U134. http://dx.doi.org/10.1190/geo2020-0662.1.

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Accurate and efficient eikonal solvers for heterogeneous media play an important role in many areas of seismology, such as seismic tomography, migration, and earthquake localization. Incorporating seismic anisotropy and complex topography remain a computational challenge for finite-difference eikonal solvers. In recent years, the topography-dependent eikonal equation (TDEE) has been proposed as an effective way to calculate seismic traveltimes for isotropic and anisotropic media with irregular topography. However, the Lax-Friedrichs sweeping method used in previous studies to approximate the v

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42

Lehujeur, M., and S. Chevrot. "On the validity of the eikonal equation for surface-wave phase-velocity tomography." Geophysical Journal International 223, no. 2 (2020): 908–14. http://dx.doi.org/10.1093/gji/ggaa352.

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SUMMARY The phase velocity of surface waves can be directly determined from the amplitude and phase of the regional wavefield using the Helmholtz equation. However, the Helmholtz equation involves estimating the Laplacian of the amplitude field, a challenging operation to perform on noisy and sparsely sampled seismic data. For this reason, the amplitude information is often discarded. In that case, phase-velocity maps are reconstructed with the eikonal equation, which relates the local phase slowness to the gradient of the phase. Here, we derive analytical expression of the errors arising from

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Peletier, Mark A., and Marco Veneroni. "Stripe patterns and a projection-valued formulation of the eikonal equation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, no. 1965 (2012): 1730–39. http://dx.doi.org/10.1098/rsta.2011.0425.

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We describe recent work on striped patterns in a system of block copolymers. A by-product of the characterization of such patterns is a new formulation of the eikonal equation. In this formulation, the unknown is a field of projection matrices of the form P = e ⊗ e , where e is a unit vector field. We describe how this formulation is better adapted to the description of striped patterns than the classical eikonal equation, and illustrate this with examples.

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Hao, Qi, and Tariq Alkhalifah. "An acoustic eikonal equation for attenuating orthorhombic media." GEOPHYSICS 82, no. 4 (2017): WA67—WA81. http://dx.doi.org/10.1190/geo2016-0632.1.

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Attenuating orthorhombic models are often used to describe the azimuthal variation of the seismic wave velocity and attenuation in finely layered hydrocarbon reservoirs with vertical fractures. In addition to the P-wave related medium parameters, S-wave parameters are also present in the complex eikonal equation needed to describe the P-wave complex-valued traveltime in an attenuating orthorhombic medium, which increases the complexity of using the P-wave traveltime to invert for the medium parameters in practice. We have used the acoustic assumption to derive an acoustic eikonal equation that

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Sethian, James A., and A. Mihai Popovici. "3-D traveltime computation using the fast marching method." GEOPHYSICS 64, no. 2 (1999): 516–23. http://dx.doi.org/10.1190/1.1444558.

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We present a fast algorithm for solving the eikonal equation in three dimensions, based on the fast marching method. The algorithm is of the order O(N log N), where N is the total number of grid points in the computational domain. The algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain. The method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity. In addition, the method resolves any overturning propagati

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Alkhalifah, Tariq. "Prestack traveltime approximations." GEOPHYSICS 77, no. 3 (2012): U31—U37. http://dx.doi.org/10.1190/geo2011-0465.1.

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Many of the explicit prestack traveltime relations used in practice are based on homogeneous (or semi-homogenous, possibly effective) media approximations. This includes the multifocusing, based on the double square-root (DSR) equation, and the common reflection stack (CRS) approaches. Using the DSR equation, I constructed the associated eikonal form in the general source-receiver domain. Like its wave-equation counterpart, it suffers from a critical singularity for horizontally traveling waves. As a result, I recasted the eikonal in terms of the reflection angle, and thus, derived expansion b

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Cruz-Sampedro, Jaime. "Schrödinger-like operators and the eikonal equation." Communications on Pure and Applied Analysis 13, no. 2 (2013): 495–510. http://dx.doi.org/10.3934/cpaa.2014.13.495.

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Alimov, A. R. "Singularities of Solutions of the Eikonal Equation." Differential Equations 55, no. 10 (2019): 1311–16. http://dx.doi.org/10.1134/s0012266119100069.

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Nowack, Robert L. "Wavefronts and solutions of the eikonal equation." Geophysical Journal International 110, no. 1 (1992): 55–62. http://dx.doi.org/10.1111/j.1365-246x.1992.tb00712.x.

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Frittelli, Simonetta, Niky Kamran, and Ezra T. Newman. "The eikonal equation, envelopes and contact transformations." Classical and Quantum Gravity 20, no. 14 (2003): 3071–79. http://dx.doi.org/10.1088/0264-9381/20/14/309.

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