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Journal articles on the topic 'One-Way equations'

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Published: 1 February 2025

Last updated: 31 July 2025

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1

Bschorr, Oskar, and Hans-Joachim Raida. "Factorized One-Way Wave Equations." Acoustics 3, no. 4 (2021): 717–22. http://dx.doi.org/10.3390/acoustics3040045.

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The method used to factorize the longitudinal wave equation has been known for many decades. Using this knowledge, the classical 2nd-order partial differential Equation (PDE) established by Cauchy has been split into two 1st-order PDEs, in alignment with D’Alemberts’s theory, to create forward- and backward-traveling wave results. Therefore, the Cauchy equation has to be regarded as a two-way wave equation, whose inherent directional ambiguity leads to irregular phantom effects in the numerical finite element (FE) and finite difference (FD) calculations. For seismic applications, a huge number

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2

Halpern, Laurence, and Lloyd N. Trefethen. "Wide‐angle one‐way wave equations." Journal of the Acoustical Society of America 84, no. 4 (1988): 1397–404. http://dx.doi.org/10.1121/1.396586.

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3

Lee, Myung W., and Sang Y. Suh. "Optimization of one‐way wave equations." GEOPHYSICS 50, no. 10 (1985): 1634–37. http://dx.doi.org/10.1190/1.1441853.

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The theory of wave extrapolation is based on the square‐root equation or one‐way equation. The full wave equation represents waves which propagate in both directions. On the contrary, the square‐root equation represents waves propagating in one direction only.

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4

Towne, Aaron, and Tim Colonius. "One-way spatial integration of hyperbolic equations." Journal of Computational Physics 300 (November 2015): 844–61. http://dx.doi.org/10.1016/j.jcp.2015.08.015.

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5

Bschorr, Oskar, and Hans-Joachim Raida. "One-Way Wave Equation Derived from Impedance Theorem." Acoustics 2, no. 1 (2020): 164–71. http://dx.doi.org/10.3390/acoustics2010012.

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The wave equations for longitudinal and transverse waves being used in seismic calculations are based on the classical force/moment balance. Mathematically, these equations are 2nd order partial differential equations (PDE) and contain two solutions with a forward and a backward propagating wave, therefore also called “Two-way wave equation”. In order to solve this inherent ambiguity many auxiliary equations were developed being summarized under “One-way wave equation”. In this article the impedance theorem is interpreted as a wave equation with a unique solution. This 1st order PDE is mathema

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6

Sheinman, Izhak, and Yeoshua Frostig. "Constitutive Equations of Composite Laminated One-Way Panels." AIAA Journal 38, no. 4 (2000): 735–37. http://dx.doi.org/10.2514/2.1025.

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7

Sheinman, Izhak, and Yeoshua Frostig. "Constitutive equations of composite laminated one-way panels." AIAA Journal 38 (January 2000): 735–37. http://dx.doi.org/10.2514/3.14474.

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8

You, Jiachun, Ru-Shan Wu, and Xuewei Liu. "One-way true-amplitude migration using matrix decomposition." GEOPHYSICS 83, no. 5 (2018): S387—S398. http://dx.doi.org/10.1190/geo2017-0625.1.

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To meet the requirement of true-amplitude migration and address the shortcomings of the classic one-way wave equations on the dynamic imaging, one-way true-amplitude wave equations were developed. Migration methods, based on the Taylor or other series approximation theory, are introduced to solve the one-way true-amplitude wave equations. This leads to the main weakness of one-way true-amplitude migration for imaging the complex or strong velocity — contrast media — the limited imaging angles. To deal with this issue, we apply a matrix decomposition method to accurately calculate the square-ro

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9

Bérczes, Attila, J. Ködmön, and Attila Pethő. "A one-way function based on norm form equations." Periodica Mathematica Hungarica 49, no. 1 (2004): 1–13. http://dx.doi.org/10.1023/b:mahu.0000040535.45427.38.

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10

Chen, Jing-bo, and Shu-yuan Du. "Multisymplectic Structures and Discretizations for One-way Wave Equations." Letters in Mathematical Physics 79, no. 2 (2006): 213–20. http://dx.doi.org/10.1007/s11005-006-0119-x.

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11

Пачев, У. М., А. Х. Кодзоков, А. Г. Езаова, А. А. Токбаева, and З. Х. Гучаева. "On One Way to Solve Linear Equations Over a Euclidean Ring." Вестник КРАУНЦ. Физико-математические науки 46, no. 1 (2024): 9–21. http://dx.doi.org/10.26117/2079-6641-2024-46-1-9-21.

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Линейным уравнениям, т.е. уравнениям первой степени, а также системам из таких уравнений уделяется большое внимание как в алгебре, так в теории чисел. Наибольший интерес представляет случай таких уравнений с целыми коэффициентами и при этом их нужно решать в целых числах. Такие уравнения с указанными условиями называют линейными диофантовыми уравнениями. Еще Эйлер рассматривал способы решения линейных диофантовых уравнений с двумя неизвестными, причем один из этих способов был основан на применении алгоритма Евклида. Другой способ решения таких уравнений, основанный на цепных дробях, применялс

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12

Wapenaar, C. P. A. "Representation of seismic sources in the one‐way wave equations." GEOPHYSICS 55, no. 6 (1990): 786–90. http://dx.doi.org/10.1190/1.1442892.

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One‐way extrapolation of downgoing and upgoing acoustic waves plays an essential role in the current practice of seismic migration (Berkhout, 1985; Stolt and Benson, 1986; Claerbout, 1985; Gardner, 1985). Generally, one‐way wave equations are derived for the source‐free situation. Sources are then represented as boundary conditions for the one‐way extrapolation problem. This approach is valid provided the source representation is done with utmost care. For instance, it is not correct to represent a monopole source by a spatial delta function and to use this as input data for a standard one‐way

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13

Hakimov, Abdusalom, Baxiyor Hayitovich Ungarov, and Maftuna Abdinazarova. "The roots of some algebraic equations one way to determine." ACADEMICIA: An International Multidisciplinary Research Journal 11, no. 9 (2021): 255–59. http://dx.doi.org/10.5958/2249-7137.2021.01904.2.

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14

Towne, Aaron, and Tim Colonius. "Efficient jet noise models using the one-way Euler equations." Journal of the Acoustical Society of America 136, no. 4 (2014): 2081. http://dx.doi.org/10.1121/1.4899470.

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15

Wetton, Brian T. R., and Gary H. Brooke. "One‐way wave equations for seismoacoustic propagation in elastic waveguides." Journal of the Acoustical Society of America 87, no. 2 (1990): 624–32. http://dx.doi.org/10.1121/1.398931.

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16

Dong, Liangguo, Zhongyi Fan, Hongzhi Wang, Benxin Chi, and Yuzhu Liu. "Correlation-based reflection waveform inversion by one-way wave equations." Geophysical Prospecting 66, no. 8 (2018): 1503–20. http://dx.doi.org/10.1111/1365-2478.12668.

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17

Porter, Michael B., Carlo M. Ferla, and Finn B. Jensen. "The problem of energy conservation in one‐way wave equations." Journal of the Acoustical Society of America 86, S1 (1989): S54. http://dx.doi.org/10.1121/1.2027554.

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18

Kamel, A. H. "Time‐domain behavior of wide‐angle one‐way wave equations." GEOPHYSICS 56, no. 3 (1991): 382–84. http://dx.doi.org/10.1190/1.1443054.

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The constant‐coefficient inhomogeneous wave equation reads [Formula: see text], Eq. (1) where t is the time; x, z are Cartesian coordinates; c is the sound speed; and δ(.) is the Dirac delta source function located at the origin. The solution to the wave equation could be synthesized in terms of plane waves traveling in all directions. In several applications it is desirable to replace equation (1) by a one‐way wave equation, an equation that allows wave processes in a 180‐degree range of angles only. This idea has become a standard tool in geophysics (Berkhout, 1981; Claerbout, 1985). A “wide

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19

Zhu, Min, and Aaron Towne. "Recursive one-way Navier-Stokes equations with PSE-like cost." Journal of Computational Physics 473 (January 2023): 111744. http://dx.doi.org/10.1016/j.jcp.2022.111744.

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20

Vivas, Flor A., and Reynam C. Pestana. "True-amplitude one-way wave equation migration in the mixed domain." GEOPHYSICS 75, no. 5 (2010): S199—S209. http://dx.doi.org/10.1190/1.3478574.

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One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called “true-amplitude one-way wave equations,” provide amplitudes that

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21

Yusupov, A. K., Kh M. Muselemov, and R. I. Vishtalov. "Calculation models of slabs with one-way connections." Herald of Dagestan State Technical University. Technical Sciences 52, no. 1 (2025): 227–41. https://doi.org/10.21822/2073-6185-2025-52-1-227-241.

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Objective. The aim of the study is to determine the performance characteristics of plate structures with one-sided connections. The paper considers issues that allow constructing calculation models of slabs on discrete supports and solid arrays with one-sided connections, and provides algorithms that allow analyzing the performance of plate structures (two-dimensional structures in plan).Method. Analytical and graphical dependencies between slab displacements and their support reactions are constructed. Properties of generalized functions are used to describe nonlinearities that occur during s

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22

Bernstein, Alexander, Richard Rand, and Robert Meller. "The Dynamics of One Way Coupling in a System of Nonlinear Mathieu Equations." Open Mechanical Engineering Journal 12, no. 1 (2018): 108–23. http://dx.doi.org/10.2174/1874155x01812010108.

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Background: This paper extends earlier research on the dynamics of two coupled Mathieu equations by introducing nonlinear terms and focusing on the effect of one-way coupling. The studied system of n equations models the motion of a train of n particle bunches in a circular particle accelerator. Objective: The goal is to determine (a) the system parameters which correspond to bounded motion, and (b) the resulting amplitudes of vibration for parameters in (a). Method: We start the investigation by examining two coupled equations and then generalize the results to any number of coupled equations

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23

Sun, Yan Jun, Chang Ming Liu, Hai Yu Li, and Zhe Yuan. "One-Way Function Construction Based on the MQ Problem and Logic Function." Applied Mechanics and Materials 220-223 (November 2012): 2360–63. http://dx.doi.org/10.4028/www.scientific.net/amm.220-223.2360.

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Multivariate quadratic based public-key cryptography called MQ problem which based on calculation of a secure cryptography of multivariate equations and MQ cryptography security is based on the difficulty of the solution of multivariate equations. But computer and mathematician scientists put a lot of effort and a long time to research MQ cryptography and they have proved that MQ cryptography is NP complete problem. Therefore, before the P problem Equal to the NP problem we do not figure out selected multivariate equations by random in polynomial time. So we can use this feature to construct t

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24

Charara, Marwan, and Albert Tarantola. "Boundary conditions and the source term for one‐way acoustic depth extrapolation." GEOPHYSICS 61, no. 1 (1996): 244–52. http://dx.doi.org/10.1190/1.1443945.

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The one‐way acoustic wave equations can be derived, using eigenvalue decomposition of the two‐way wave equation in the Fourier domain, in such a manner that the source term and the free‐surface boundary condition are explicitly introduced. The proposed form of the one‐way wave equations is well adapted to seismic reflection modeling because it allows a pressure field recorded at the surface to be extrapolated directly in depth. A numerical example illustrates the appropriate implementation of the source term and the free‐surface boundary conditions. A comparison with a two‐way modeling shows a

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25

Godin, Oleg A. "Three‐dimensional, energy‐conserving, and reciprocal one‐way acoustic wave equations." Journal of the Acoustical Society of America 102, no. 5 (1997): 3149. http://dx.doi.org/10.1121/1.420705.

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26

Godin, Oleg A. "On reciprocity and energy conservation for one‐way acoustic wave equations." Journal of the Acoustical Society of America 100, no. 4 (1996): 2835. http://dx.doi.org/10.1121/1.416693.

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27

Trefethen, Lloyd N., and Laurence Halpern. "Well-posedness of one-way wave equations and absorbing boundary conditions." Mathematics of Computation 47, no. 176 (1986): 421. http://dx.doi.org/10.1090/s0025-5718-1986-0856695-2.

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28

Fishman, Louis. "Derivation of one‐way wave equations for vector wave propagation problems." Journal of the Acoustical Society of America 89, no. 4B (1991): 1895. http://dx.doi.org/10.1121/1.2029420.

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29

Berjamin, Harold. "On the accuracy of one-way approximate models for nonlinear waves in soft solids." Journal of the Acoustical Society of America 153, no. 3 (2023): 1924–32. http://dx.doi.org/10.1121/10.0017681.

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Simple strain-rate viscoelasticity models of isotropic soft solid are introduced. The constitutive equations account for finite strain, incompressibility, material frame-indifference, nonlinear elasticity, and viscous dissipation. A nonlinear viscous wave equation for the shear strain is obtained exactly and corresponding one-way Burgers-type equations are derived by making standard approximations. Analysis of the travelling wave solutions shows that these partial differential equations produce distinct solutions, and deviations are exacerbated when wave amplitudes are not arbitrarily small. I

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30

Yusupov, A. K., Kh M. Muselemov, and R. I. Vishtalov. "Calculation models of beams with one-way connections." Herald of Dagestan State Technical University. Technical Sciences 51, no. 4 (2025): 236–48. https://doi.org/10.21822/2073-6185-2024-51-4-236-248.

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Objective. The article considers the operational features of beam structures with one-way ties. Method. By studying the operational features of beam structure supports, analytical and graphical dependencies between beam displacements and their support reactions are derived. The properties of generalized functions are used to describe the nonlinearities that occur during structural deformations. Differential equations for transverse bending of beams are presented, taking into account nonlinearities and methods for their solution. Result. Design and calculation schemes have been developed that t

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31

Angus, D. A. "True amplitude corrections for a narrow-angle one-way elastic wave equation." GEOPHYSICS 72, no. 2 (2007): T19—T26. http://dx.doi.org/10.1190/1.2430694.

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Wavefield extrapolators using one-way wave equations are computationally efficient methods for accurate traveltime modeling in laterally heterogeneous media, and have been used extensively in many seismic forward modeling and migration problems. However, most leading-order, one-way wave equations do not simulate waveform amplitudes accurately and this is primarily because energy flux is not accounted for correctly. I review the derivation of a leading-order, narrow-angle, one-way elastic wave equation for 3D media. I derive correction terms that enable energy-flux normalization and introduce a

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32

Dyussembayeva, L., Zh. Keulimzhayeva, M. Serimbetov, and M. Tilepiyev. "ONE WAY TO DERIVE THE FORMULA FOR THE INVERSE MATRIX." Znanstvena misel journal, no. 94 (September 30, 2024): 41–44. https://doi.org/10.5281/zenodo.13860776.

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This paper discusses an alternative method for the appearance of an inverse matrix. The work used the “reverse” method, that is Assuming knowledge of the final result, innovative approaches were found to find the inverse matrix. A special structure of the “discharged” molecular system was used, which makes it possible to break down a large part of the source system into a smaller method system.

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33

ZHU, JIANXIN, and YA YAN LU. "VALIDITY OF ONE-WAY MODELS IN THE WEAK RANGE DEPENDENCE LIMIT." Journal of Computational Acoustics 12, no. 01 (2004): 55–66. http://dx.doi.org/10.1142/s0218396x0400216x.

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Numerical solutions of the Helmholtz equation and the one-way Helmholtz equation are compared in the weak range dependence limit, where the overall range distance is increased while the range dependence is weakened. It is observed that the difference between the solutions of these two equations persists in this limit. The one-way Helmholtz equation involves a square root operator and it can be further approximated by various one-way models used in underwater acoustics. An operator marching method based on the Dirichlet-to-Neumann map and a local orthogonal transform is used to solve the Helmho

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34

Khoma, H., Viktor Chornyi, and Svitlana Khoma-Mohylska. "About one way of construction of t-periodic solutions to hyperbolic type equations." Visnyk of Zaporizhzhya National University. Physical and Mathematical Sciences, no. 1 (2018): 153–60. http://dx.doi.org/10.26661/2413-6549-2018-1-15.

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35

Zhang, Yu, Guanquan Zhang, and Norman Bleistein. "True amplitude wave equation migration arising from true amplitude one-way wave equations." Inverse Problems 19, no. 5 (2003): 1113–38. http://dx.doi.org/10.1088/0266-5611/19/5/307.

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36

Cheng, Yingda, Ching-Shan Chou, Fengyan Li, and Yulong Xing. "$L^2$ stable discontinuous Galerkin methods for one-dimensional two-way wave equations." Mathematics of Computation 86, no. 303 (2016): 121–55. http://dx.doi.org/10.1090/mcom/3090.

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37

McCoy, John J., and L. Neil Frazer. "Pseudodifferential operators, operator orderings, marching algorithms and path integrals for one-way equations." Wave Motion 9, no. 5 (1987): 413–27. http://dx.doi.org/10.1016/0165-2125(87)90030-8.

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38

Zhang, Yu, Guanquan Zhang, and Norman Bleistein. "Theory of true-amplitude one-way wave equations and true-amplitude common-shot migration." GEOPHYSICS 70, no. 4 (2005): E1—E10. http://dx.doi.org/10.1190/1.1988182.

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One-way wave operators are powerful tools for forward modeling and migration. Here, we describe a recently developed true-amplitude implementation of modified one-way operators and present some numerical examples. By “true-amplitude” one-way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct. That is, ray theory applied to these equations yields the upward- and downward-traveling eikonal equations of the full wave equation, and the amplitude satisfies the transport equation of the full wave equation. The solutions of these equations are used in

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39

Amazonas, Daniela, Rafael Aleixo, Gabriela Melo, Jörg Schleicher, Amélia Novais, and Jessé C. Costa. "Including lateral velocity variations into true-amplitude common-shot wave-equation migration." GEOPHYSICS 75, no. 5 (2010): S175—S186. http://dx.doi.org/10.1190/1.3481469.

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In heterogeneous media, standard one-way wave equations describe only the kinematic part of one-way wave propagation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical techniques. We present an approach to circumvent these problems by implementing approximate solutions based on the one-dimensional analyt

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40

Sun, Weijia, and Li-Yun Fu. "Compensation for transmission losses based on one-way propagators in the mixed domain." GEOPHYSICS 77, no. 3 (2012): S65—S72. http://dx.doi.org/10.1190/geo2011-0460.1.

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Most true-amplitude migration algorithms based on one-way wave equations involve corrections of geometric spreading and seismic [Formula: see text] attenuation. However, few papers discuss the compensation of transmission losses (CTL) based on one-way wave equations. Here, we present a method to compensate for transmission losses using one-way wave propagators for a 2D case. The scheme is derived from the Lippmann-Schwinger integral equation. The CTL scheme is composed of a transmission term and a phase-shift term. The transmission term compensates amplitudes while the wave propagates through

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41

Javanmardi, Nasrin, and Parviz Ghadimi. "Hydroelastic analysis of surface-piercing propeller through one-way and two-way coupling approaches." Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment 233, no. 3 (2018): 844–56. http://dx.doi.org/10.1177/1475090218791617.

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Semi-submerged operation of surface-piercing propellers can cause severe structural stress and deflection due to the high-rate variation of the hydrodynamic loads on each blade in one cycle of revolution. Therefore, a proper hydroelastic analysis regarding these propellers seems imperative. Accordingly, in this article, hydrodynamic performance of a surface-piercing propeller is studied by considering the structural flexibility. The one-way and two-way coupled unsteady Reynolds-averaged Navier–Stokes equations (finite volume method used for the fluid analysis) and direct finite element method

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42

Naghadeh, Diako Hariri, and Mohamad Ali Riahi. "One-way wave-equation migration in log-polar coordinates." GEOPHYSICS 78, no. 2 (2013): S59—S67. http://dx.doi.org/10.1190/geo2012-0229.1.

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We obtained acoustic wave and wavefield extrapolation equations in log-polar coordinates (LPCs) and tried to enhance the imaging. To achieve this goal, it was necessary to decrease the angle between the wavefield extrapolation axis and wave propagation direction in the one-way wave-equation migration (WEM). If we were unable to carry it out, more reflection wave energy would be lost in the migration process. It was concluded that the wavefield extrapolation operator in LPCs at low frequencies has a large wavelike region, and at high frequencies, it can mute the evanescent energy. In these coor

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43

Austin, Joe Dan, and H. J. Vollrath. "Representing, Solving, and Using Algebraic Equations." Mathematics Teacher 82, no. 8 (1989): 608–12. http://dx.doi.org/10.5951/mt.82.8.0608.

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Students of beginning algebra are quickly expected to solve linear equations. The solution procedures are generally abstract, involving the manipulation of numbers and algebraic symbols. Many students, even after completing a year of algebra, do not understand variables, equations, and solving equations (cf. Carpenter et al. [1982]). One way to help students learn to solve equations is to use physical objects, diagrams, and then symbols to represent equations. (Bruner [1964, 1967] calls such representations enactive (concrete), iconic (pictorial), and symbolic.) Although solving equations symb

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44

Dolgaya, A. A., H. R. Zainulabidova, A. S. Mozzhukhin, et al. "About one way to integrate the equations of seismic vibrations of seismically isolated structures." Herald of Dagestan State Technical University. Technical Sciences 51, no. 2 (2024): 190–96. http://dx.doi.org/10.21822/2073-6185-2024-51-2-190-196.

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Objective. The purpose of the study is to develop a new method of analyzing seismic vibrations of seismically isolated structures.Method. The structure is modeled by a system with one degree of freedom with a nonlinear stiffness characteristic. In the absence of damping an analytical solution of oscillation equations on the phase plane is obtained. In this case the connection between speeds and displacements is accurate, and the law of uniformly variable motion is used to calculate displacements.Result. To take into account the attenuation, the law of change in mechanical energy at each integr

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45

Pelykh, V., and Y. Taistra. "On the null one-way solution to Maxwell equations in the Kerr space-time." Mathematical Modeling and Computing 5, no. 2 (2018): 201–6. http://dx.doi.org/10.23939/mmc2018.02.201.

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46

Zhong, Jize, and Zili Xu. "An energy method for flutter analysis of wing using one-way fluid structure coupling." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 231, no. 14 (2016): 2560–69. http://dx.doi.org/10.1177/0954410016667146.

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In this paper, an energy method for flutter analysis of wing using one-way fluid structure coupling was developed. To consider the effect of wing vibration, Reynolds-averaged Navier–Stokes equations based on the arbitrary Lagrangian Eulerian coordinates were employed to model the flow. The flow mesh was updated using a fast dynamic mesh technology proposed by our research group. The pressure was calculated by solving the Reynolds-averaged Navier–Stokes equations through the SIMPLE algorithm with the updated flow mesh. The aerodynamic force for the wing was computed using the pressure on the wi

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47

Zhu, Rui, Da-duo Chen, and Shi-wei Wu. "Unsteady Flow and Vibration Analysis of the Horizontal-Axis Wind Turbine Blade under the Fluid-Structure Interaction." Shock and Vibration 2019 (March 31, 2019): 1–12. http://dx.doi.org/10.1155/2019/3050694.

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A 1.5 MW horizontal-axis wind turbine blade and fluid field model are established to study the difference in the unsteady flow field and structural vibration of the wind turbine blade under one- and two-way fluid-structure interactions. The governing equations in ﬂuid field and the motion equations in structural were developed, and the corresponding equations were discretized with the Galerkin method. Based on ANSYS CFX fluid dynamics and mechanical structural dynamics calculation software, the effects of couplings on the aerodynamic and vibration characteristics of the blade are compared and

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48

Gustavson, Richard, and Sarah Rosen. "A Reduction Algorithm for Volterra Integral Equations." PUMP Journal of Undergraduate Research 6 (May 30, 2023): 172–91. http://dx.doi.org/10.46787/pump.v6i0.3631.

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An integral equation is a way to encapsulate the relationships between a function and its integrals. We develop a systematic way of describing Volterra integral equations – specifically an algorithm that reduces any separable Volterra integral equation into an equivalent one in operator-linear form, i.e., one that only contains iterated integrals. This serves to standardize the presentation of such integral equations so as to only consider those containing iterated integrals. We use the algebraic object of the integral operator, the twisted Rota-Baxter identity, and vertex-edge decorated roote

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Bleistein, Norman, Yu Zhang, and Guanquan Zhang. "Analysis of the neighborhood of a smooth caustic for true-amplitude one-way wave equations." Wave Motion 43, no. 4 (2006): 323–38. http://dx.doi.org/10.1016/j.wavemoti.2006.01.003.

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Golikov, Pavel, and Alexey Stovas. "Traveltime parameters in tilted transversely isotropic media." GEOPHYSICS 77, no. 6 (2012): C43—C55. http://dx.doi.org/10.1190/geo2011-0457.1.

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Traveltime parameters define the coefficients of the Taylor series for traveltime or traveltime squared as a function of offset. These parameters provide an efficient tool for analyzing the effect of the medium parameters for short- and long-offset reflection moveouts. We derive the exact equations for one-way and two-way traveltime parameters in a homogeneous transversely isotropic medium with the tilted symmetry axis (TTI). It is shown that most of the one-way traveltime parameters in TTI differ from the two-way traveltime parameters, and we observe strong dependence of all traveltime parame

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