Higher-order schemes for 3D first-arrival traveltimes and amplitudes

Luo, Songting; Qian, Jianliang; Zhao, Hongkai

doi:10.1190/geo2010-0363.1

Cited by 32 publications

(41 citation statements)

References 59 publications

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“…ansatz for the solution of the Helmholtz equation in the high-frequency regime. [31][32][33][34] There, similarly to our approach, the eikonal equation is used to eliminate the 2 term in (4) yielding the travel time, and the amplitude is obtained by eliminating the term in (4), by solving the real-valued transport equation…”

Section: Introductionmentioning

confidence: 99%

“…To get an accurate solution for the amplitude in (7), the numerical approximation for (⃗ x) has to be very accurate. 33 Because the analytical (⃗ x) is nonsmooth at the point source, the numerical solution of Equation (6) is polluted with errors when it is computed using the aforementioned standard finite difference methods. 45 To overcome this, 32,33 use the factored version of the eikonal equation, which was originally suggested in the work of Pica.…”

Section: Introductionmentioning

confidence: 99%

“…33 Because the analytical (⃗ x) is nonsmooth at the point source, the numerical solution of Equation (6) is polluted with errors when it is computed using the aforementioned standard finite difference methods. 45 To overcome this, 32,33 use the factored version of the eikonal equation, which was originally suggested in the work of Pica. 46 The new equation is obtained by setting = 0 1 in (6), where the function 0 (⃗ x) is known and its derivatives are computed analytically.…”

Section: Introductionmentioning

confidence: 99%

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A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

Treister

Haber

2018

Numerical Linear Algebra App

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The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time of the wave and one for its amplitude. The travel time is obtained by a solution of the factored eikonal equation, and the amplitude is obtained by solving a complex-valued advection-diffusion-reaction equation. The reformulated equation is equivalent to the original Helmholtz equation, and the differences between the numerical solutions of these equations arise only from discretization errors. We develop an efficient multigrid solver for obtaining the amplitude given the travel time, which can be efficiently computed. This approach is advantageous because the amplitude is typically smooth in this case and, hence, more suitable for multigrid solvers than the standard Helmholtz discretization. We demonstrate that our second-order advection-diffusion-reaction discretization is more accurate than the standard second-order discretization at high wave numbers, as long as there are no reflections or caustics. Moreover, we show that using our approach, the problem can be solved more efficiently than using the common shifted Laplacian multigrid approach.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

Treister

Haber

2018

Numerical Linear Algebra App

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“…From the wave equation, the corresponding eikonal equation can be obtained. The eikonal approximation provides information about first arrivals [24], does not account for caustics [25] and requires a sufficiently well defined source [26] if amplitudes are of interest. Even so, eikonal models are widely used in many fields as approximation due to their simplicity [27], [28].…”

Section: Introductionmentioning

confidence: 99%

Acoustic wave and eikonal equations in a transformed metric space for various types of anisotropy

Noack

Clark

2017

Heliyon

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Acoustic waves propagating in anisotropic media are important for various applications. Even though these wave phenomena do not generally occur in nature, they can be used to approximate wave motion in various physical settings. We propose a method to derive wave equations for anisotropic wave propagation by adjusting the dispersion relation according to a selected type of anisotropy and transforming it into another metric space. The proposed method allows for the derivation of acoustic wave and eikonal equations for various types of anisotropy, and generalizes anisotropy by interpreting it as a change of the metric instead of a change of velocity with direction. The presented method reduces the scope of acoustic anisotropy to a selection of a velocity or slowness surface and a tensor that describes the transformation into a new metric space. Experiments are shown for spatially dependent ellipsoidal anisotropy in homogeneous and inhomogeneous media and sandstone, which shows vertical transverse isotropy. The results demonstrate the stability and simplicity of the solution process for certain types of anisotropy and the equivalency of the solutions.

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“…The fast sweeping method is an efficient iterative method for solving (1.1). It has been developed and used successfully for various hyperbolic problems (e.g., see [6,7,14,15,17,20,23,26,28,30,34,35,40,[43][44][45] and references therein for development, and [3,18,21,24,25,27,29,31] and references therein for applications to different problems). The key ingredients for the success of the FSM are the following:…”

Section: 4)mentioning

confidence: 99%

Convergence analysis of the fast sweeping method for static convex Hamilton–Jacobi equations

Luo

Zhao

2016

Res Math Sci

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In this work, we study the convergence of an efficient iterative method, the fast sweeping method (FSM), for numerically solving static convex Hamilton-Jacobi equations. First, we show the convergence of the FSM on arbitrary meshes. Then we illustrate that the combination of a contraction property of monotone upwind schemes with proper orderings can provide fast convergence for iterative methods. We show that this mechanism produces different behavior from that for elliptic problems as the mesh is refined. An equivalence between the local solver of the FSM and the Hopf formula under linear approximation is also proved. Numerical examples are presented to verify our analysis. Keywords: Fast sweeping method, Hamilton-Jacobi equation, Contraction property, Iterative method, Fast convergence, Error estimate Mathematics Subject Classification: 35L02, 65N06, 65N12 BackgroundEfficient and robust iterative methods are highly desirable for solving a variety of static hyperbolic partial differential equations (PDEs) numerically. The most important property for hyperbolic problems is that the information propagates along the characteristics. For linear hyperbolic problems, the characteristics are known á priori and do not intersect. For nonlinear hyperbolic problems, the characteristics are not known á priori and may intersect. Consequently the information that propagates along different characteristics has to compromise in certain ways when the characteristics intersect to render the desired weak solution. When discretizing hyperbolic PDEs, monotone upwind schemes are an important class of schemes that use stencils following the characteristics in the direction from which the information comes (e.g., see [39]). How to design an effective iterative method for hyperbolic problems also needs to fully utilize the above properties. We use the fast sweeping method (FSM) [43] as an example to show that the combination of monotone upwind schemes with Gauss-Seidel iterations and proper orderings can provide fast convergence. The convergence mechanism is different from that of iterative methods for elliptic problems, where relaxation is the underlying mechanism for convergence and the key point is how to deal with long-range interactions through short-range interactions

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Higher-order schemes for 3D first-arrival traveltimes and amplitudes

Cited by 32 publications

References 59 publications

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

Acoustic wave and eikonal equations in a transformed metric space for various types of anisotropy

Convergence analysis of the fast sweeping method for static convex Hamilton–Jacobi equations

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