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Tariq Alkhalifah - One of the best experts on this subject based on the ideXlab platform.

  • solving the Eikonal Equation for compressional and shear waves in anisotropic media using peridynamic differential operator
    arXiv: Computational Physics, 2021
    Co-Authors: Ali Can Bekar, Umair Bin Waheed, Erdogan Madenci, Ehsan Haghighat, Tariq Alkhalifah
    Abstract:

    The traveltime of compressional (P) and shear (S) waves have proven essential in many applications of earthquake and exploration seismology. An accurate and efficient traveltime computation for P and S waves is crucial for the success of these applications. However, solutions to the Eikonal Equation with a complex phase velocity field in anisotropic media is challenging. The Eikonal Equation is a first-order, hyperbolic, nonlinear partial differential Equation (PDE) that represents the high-frequency asymptotic approximation of the wave Equation. The fast marching and sweeping methods are commonly used due to their efficiency in numercally solving Eikonal Equation. However, these methods suffer from numerical inaccuracy in anisotropic media with sharp heterogeneity, irregular surface topography and complex phase velocity fields. This study presents a new method to solving the Eikonal Equation by employing the peridynamic differential operator (PDDO). The PDDO provides the nonlocal form of the Eikonal Equation by introducing an internal length parameter (horizon) and a weight function with directional nonlocality. The operator is immune to discontinuities in the form sharp changes in field or model variables and invokes the direction of traveltime in a consistent manner. The weight function controls the degree of association among points within the horizon. Solutions are constructed in a consistent manner without upwind assumptions through simple discretization. The capability of this approach is demonstrated by considering different types of Eikonal Equations on complex velocity models in anisotropic media. The examples demonstrate its unconditional numerical stability and results compare well with the reference solutions.

  • an acoustic Eikonal Equation for attenuating orthorhombic media
    Geophysics, 2017
    Co-Authors: Qi Hao, Tariq Alkhalifah
    Abstract:

    ABSTRACTAttenuating 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 approximately governs the complex-valued traveltime of P-waves in an attenuating orthorhombic medium. For a homogeneous attenuating orthorhombic media, we solve the Eikonal Equation using a combination of the perturbation method and Shanks transform. For a horizontal attenuating orthorhombic layer, the real and imaginary parts of the complex-valued reflection traveltime have nonhyperbolic behaviors in terms ...

  • An acoustic Eikonal Equation for attenuating transversely isotropic media with a vertical symmetry axis
    GEOPHYSICS, 2017
    Co-Authors: Qi Hao, Tariq Alkhalifah
    Abstract:

    ABSTRACTSeismic-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 traveltime of quasi P-waves in attenuating VTI media are independent of the S-wave velocity parameter υS0 in Thomsen’s notation and the S-wave attenuation coefficient AS0 in Zhu and Tsvankin’s notation. We combine perturbation theory and Shanks transform to develop practical approximations to the acoustic attenuating Eikonal Equation, capable of admitting an analytical description of the attenuation in homoge...

  • application of perturbation theory to a p wave Eikonal Equation in orthorhombic media
    Geophysics, 2016
    Co-Authors: Alexey Stovas, Nabil Masmoudi, Tariq Alkhalifah
    Abstract:

    ABSTRACTThe P-wave Eikonal Equation for orthorhombic (ORT) anisotropic media is a highly nonlinear partial differential Equation requiring the solution of a sixth-order polynomial to obtain traveltimes, resulting in complex and time-consuming numerical solutions. To alleviate this complexity, we approximate the solution of this Equation by applying a multiparametric perturbation approach. We also investigated the sensitivity of traveltime surfaces in ORT media with respect to three anelliptic parameters. As a result, a simple and accurate P-wave traveltime approximation valid for ORT media was derived. Two different possible anelliptic parameterizations were compared. One of the parameterizations includes anelliptic parameters defined at zero offset: η1, η2, and ηxy. Another parameterization includes anelliptic parameters defined for all symmetry planes: η1, η2, and η3. The azimuthal behavior of sensitivity coefficients with different parameterizations was used to analyze the crosstalk between anelliptic ...

  • efficient traveltime solutions of the acoustic ti Eikonal Equation
    Journal of Computational Physics, 2015
    Co-Authors: Umair Bin Waheed, Tariq Alkhalifah, Hui Wang
    Abstract:

    Numerical solutions of the Eikonal (Hamilton-Jacobi) Equation for transversely isotropic (TI) media are essential for imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI Eikonal Equation, which requires solving a quartic polynomial for every grid point. Analytical solutions of the quartic polynomial yield numerically unstable formulations. Thus, it requires a numerical root finding algorithm, adding significantly to the computational load. Using perturbation theory we approximate, in a first order discretized form, the TI Eikonal Equation with a series of simpler Equations for the coefficients of a polynomial expansion of the Eikonal solution, in terms of the anellipticity anisotropy parameter. Such perturbation, applied to the discretized form of the Eikonal Equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for models with complex distribution of velocity and anisotropic anellipticity parameter, such as that for the complicated Marmousi model. The formulation allows for large cost reduction compared to using the direct TI Eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy in the proposed algorithm, without any addition to the computational cost.

Zhongjie Zhang - One of the best experts on this subject based on the ideXlab platform.

  • calculating ray paths for first arrival travel times using a topography dependent Eikonal Equation solver
    Bulletin of the Seismological Society of America, 2014
    Co-Authors: Zhongjie Zhang
    Abstract:

    Abstract Irregular surfaces cause a number of problems for seismic processing and interpretation. The problems lie in the proper treatment of topography in the first‐arrival travel‐time calculation and ray‐path tracing, both of which are subject to preconditions in ray‐based seismogram synthesis, seismic tomography, and seismic migration calculations. Two treatment schemes for irregular surfaces have been used previously: (1) model expansion with the irregular surface treated as an inner discontinuity, and (2) flattening of the irregular surface using a transformation between curvilinear and Cartesian coordinates, while maintaining it as a free surface. In the first approach, first‐arrival travel times can be calculated using an Eikonal Equation solver, and rays are traced backward from the receiver to the source along the direction of the gradient of the travel‐time field. In the second scheme, a topography‐dependent Eikonal Equation is used to calculate irregular‐surface first‐arrival travel times. We present a ray‐path tracing scheme for irregular surfaces, which applies a travel‐time field calculated using a topography‐dependent Eikonal Equation. The scheme is realized using travel‐time gradients in curvilinear coordinates. The validity of the scheme for tracing ray paths in the presence of irregular topography is illustrated by five models containing differing degrees of topographical complexity. Comparison of ray paths and first‐arrival travel times between the two irregular‐surface treatment schemes suggests the topography‐flattened scheme avoids the difficulties of both discretizing the irregular surface and the velocity selection of the infill medium in the model expansion scheme. For the treatment of the inner discontinuity by model expansion, there is a possibility that ray paths will be traced outside the real physical model. Using the topography‐dependent Eikonal Equation solver with our ray‐path tracing scheme should provide an efficient means of dealing with irregular surfaces, which could be applied in the fields of seismic tomography, seismic migration, and tomographic static correction.

  • a model expansion criterion for treating surface topography in ray path calculations using the Eikonal Equation
    Journal of Geophysics and Engineering, 2014
    Co-Authors: Zhongjie Zhang
    Abstract:

    Irregular surface topography has revolutionized how seismic traveltime is calculated and the data are processed. There are two main schemes for dealing with an irregular surface in the seismic first-arrival traveltime calculation: (1) expanding the model and (2) flattening the surface irregularities. In the first scheme, a notional infill medium is added above the surface to expand the physical space into a regular space, as required by the Eikonal Equation solver. Here, we evaluate the chosen propagation velocity in the infill medium through ray path tracking with the Eikonal Equation-solved traveltime field, and observe that the ray paths will be physically unrealistic for some values of this propagation velocity. The choice of a suitable propagation velocity in the infill medium is crucial for seismic processing of irregular topography. Our model expansion criterion for dealing with surface topography in the calculation of traveltime and ray paths using the Eikonal Equation highlights the importance of both the propagation velocity of the infill physical medium and the topography gradient.

Thomas Mieling - One of the best experts on this subject based on the ideXlab platform.

  • the response of laser interferometric gravitational wave detectors beyond the Eikonal Equation
    Classical and Quantum Gravity, 2021
    Co-Authors: Thomas Mieling
    Abstract:

    The response of Michelson interferometers to weak plane gravitational waves is computed at one order of accuracy beyond the Eikonal Equation. The modulation of the electromagnetic field amplitude and polarisation are taken into account by solving the transport Equations of geometrical optics with boundary conditions adapted to laser interferometry. Considering both DC and balanced homodyne readout schemes, explicit formulae for the interferometer output signals are derived. These signals comprise perturbations of the optical path length, frequency and amplitude, and are shown to be insensitive to polarisation perturbations.

Eldad Haber - One of the best experts on this subject based on the ideXlab platform.

  • a fast marching algorithm for the factored Eikonal Equation
    Journal of Computational Physics, 2016
    Co-Authors: Eran Treister, Eldad Haber
    Abstract:

    The Eikonal Equation is instrumental in many applications in several fields ranging from computer vision to geoscience. This Equation can be efficiently solved using the iterative Fast Sweeping (FS) methods and the direct Fast Marching (FM) methods. However, when used for a point source, the original Eikonal Equation is known to yield inaccurate numerical solutions, because of a singularity at the source. In this case, the factored Eikonal Equation is often preferred, and is known to yield a more accurate numerical solution. One application that requires the solution of the Eikonal Equation for point sources is travel time tomography. This inverse problem may be formulated using the Eikonal Equation as a forward problem. While this problem has been solved using FS in the past, the more recent choice for applying it involves FM methods because of the efficiency in which sensitivities can be obtained using them. However, while several FS methods are available for solving the factored Equation, the FM method is available only for the original Eikonal Equation.In this paper we develop a Fast Marching algorithm for the factored Eikonal Equation, using both first and second order finite-difference schemes. Our algorithm follows the same lines as the original FM algorithm and requires the same computational effort. In addition, we show how to obtain sensitivities using this FM method and apply travel time tomography, formulated as an inverse factored Eikonal Equation. Numerical results in two and three dimensions show that our algorithm solves the factored Eikonal Equation efficiently, and demonstrate the achieved accuracy for computing the travel time. We also demonstrate a recovery of a 2D and 3D heterogeneous medium by travel time tomography using the Eikonal Equation for forward modeling and inversion by Gauss-Newton.

Songting Luo - One of the best experts on this subject based on the ideXlab platform.

  • hybrid fast sweeping methods for anisotropic Eikonal Equation in two dimensional tilted transversely isotropic media
    Journal of Scientific Computing, 2020
    Co-Authors: Guangnan Huang, Songting Luo
    Abstract:

    We present hybrid fast sweeping methods for computing first-arrival traveltime of the qP, qSV and qSH waves in two-dimensional tilted transversely isotropic media, based on solving the anisotropic Eikonal Equation. A factorization approach is applied to resolve the source singularity near the point source, which leads to a factored anisotropic Eikonal Equation whose solutions can be computed with high accuracy. The proposed methods solve the factored Equation in a neighborhood of the point source with the size of the neighborhood independent of the mesh, and solve the original Equation outside the neighborhood. The methods enjoy all the appealing features, such as efficiency, accuracy and convergence, of the usual fast sweeping method. Furthermore, the “super-convergence” property of the first-order fast sweeping method, i.e., both its numerical solution and gradient are first-order accurate, allows us to design a second-order fast sweeping method based on a linear discontinuous Galerkin formulation. As a post-processing procedure of the first-order method, the second-order method reduces the local degrees of freedom from three to one in the linear discontinuous Galerkin formulation, which implies a simple local updating formula, hence an efficient second-order scheme. Numerical experiments are presented to demonstrate the proposed methods.

  • first arrival tomography with fast sweeping method solving the factored Eikonal Equation
    Exploration Geophysics, 2019
    Co-Authors: Songting Luo, Guangnan Huang, Tryggvason Ari, David C Nobes
    Abstract:

    ABSTRACTThis paper presents a first-arrival tomography incorporating a fast sweeping method (FSM) solving the factored Eikonal Equation (factored FSM). The traveltime calculation method plays a significant role in velocity inversion. However, for a point source condition, all finite-difference based Eikonal solvers suffer from the source singularity problem. Numerical error caused by source singularity will propagate from the source to all computational domains, and makes traveltimes inaccurate. A FSM solving the factored Eikonal Equation can deal with the source singularity problem very well. Therefore, a first-arrival tomography is developed by incorporating 2D and 3D factored FSMs to provide more accurate traveltimes in velocity inversion. For comparison, an open source package PStomo_eq is used to invert the same data set. It incorporates the traveltime calculation algorithms fdtime2d.c and fdtime3d.c. Traveltime accuracy tests show that factored FSM can generate more accurate traveltimes than FSM, fd...

  • fast sweeping method for the factored Eikonal Equation
    Journal of Computational Physics, 2009
    Co-Authors: Sergey Fomel, Songting Luo, Hongkai Zhao
    Abstract:

    We develop a fast sweeping method for the factored Eikonal Equation. By decomposing the solution of a general Eikonal Equation as the product of two factors: the first factor is the solution to a simple Eikonal Equation (such as distance) or a previously computed solution to an approximate Eikonal Equation. The second factor is a necessary modification/correction. Appropriate discretization and a fast sweeping strategy are designed for the Equation of the correction part. The key idea is to enforce the causality of the original Eikonal Equation during the Gauss-Seidel iterations. Using extensive numerical examples we demonstrate that (1) the convergence behavior of the fast sweeping method for the factored Eikonal Equation is the same as for the original Eikonal Equation, i.e., the number of iterations for the Gauss-Seidel iterations is independent of the mesh size, (2) the numerical solution from the factored Eikonal Equation is more accurate than the numerical solution directly computed from the original Eikonal Equation, especially for point sources.