3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing,source localization,reflection migration,seismic m...3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing,source localization,reflection migration,seismic monitoring and tomographic imaging.In recent years,many advanced methods have been developed to solve the 3D eikonal equation in heterogeneous media.However,there are still challenges for the stable and accurate calculation of first-arrival traveltimes in 3D strongly inhomogeneous media.In this paper,we propose an adaptive finite-difference(AFD)method to numerically solve the 3D eikonal equation.The novel method makes full use of the advantages of different local operators characterizing different seismic wave types to calculate factors and traveltimes,and then the most accurate factor and traveltime are adaptively selected for the convergent updating based on the Fermat principle.Combined with global fast sweeping describing seismic waves propagating along eight directions in 3D media,our novel method can achieve the robust calculation of first-arrival traveltimes with high precision at grid points either near source point or far away from source point even in a velocity model with large and sharp contrasts.Several numerical examples show the good performance of the AFD method,which will be beneficial to many scientific applications.展开更多
Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of ...Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.展开更多
PFMM(perspective fast marching method)是一种有效解决透视投影下从明暗恢复形状(SFS)问题的方法,但是适应条件受限,且对初始数据的精度较为敏感。本文通过对Eikonal方程系数的分析,提出了在透视投影下基于自适应Eikonal方程的PFMM,...PFMM(perspective fast marching method)是一种有效解决透视投影下从明暗恢复形状(SFS)问题的方法,但是适应条件受限,且对初始数据的精度较为敏感。本文通过对Eikonal方程系数的分析,提出了在透视投影下基于自适应Eikonal方程的PFMM,解决了PFMM对初始数据过于依赖的问题,是PFMM的推广。对合成图像的实验表明本文算法比PFMM精度更高,对透视投影下SFS问题可以得到比较好的结果。展开更多
The Northeastern Tibetan plateau records Caledonian Qilian orogeny and Cenozoic reactivation by continental collision between the Indian and Asian plates. In order to provide the constraint on the Qilian orogenic mech...The Northeastern Tibetan plateau records Caledonian Qilian orogeny and Cenozoic reactivation by continental collision between the Indian and Asian plates. In order to provide the constraint on the Qilian orogenic mechanism and the expansion of the plateau,wide-angle seismic data was acquired along a 430 km-long profile between Jingtai and Hezuo. There is strong height variation along the profile,which is dealt by topography flattening scheme in our crustal velocity structure reconstruction. We herein present the upper crustal P-wave velocity structure model resulting from the interpretation of first arrival dataset from topography-dependent eikonal traveltime tomography. With topography flattening scheme to process real topography along the profile,the evenness of ray coverage times of the image area（upper crust）is improved,which provides upper crustal velocity model comparable to the classic traveltime tomography（with model expansion scheme to process irregular surface）. The upper crustal velocity model shows zoning character which matcheswith the tectonic division of the Qaidam-Kunlun-West Qinling belt,the Central and Northern Qilian,and the Alax blocks along the profile. The resultant upper crustal P-wave velocity model is expected to provide important base for linkage between the mapped surface geology and deep structure or geodynamics in Northeastern Tibet.展开更多
The aim in this paper is to construct an affine transformation using the classical physics analogy between the fields of optics and mechanics. Since optics and mechanics both have symplectic structures, the concept of...The aim in this paper is to construct an affine transformation using the classical physics analogy between the fields of optics and mechanics. Since optics and mechanics both have symplectic structures, the concept of optics can be replaced by that of mechanics and vice versa. We list the four types of eikonal (generating functions). We also introduce a unitary operator for the affine transformation. Using the unitary operator, the kernel (propagator) is calculated and the wavization (quantization) of the Gabor function is discussed. The dynamic properties of the affine transformed Wigner function are also discussed.展开更多
The precise determination of earthquake location is the fundamental basis in seismological community,and is crucial for analyzing seismic activity and performing seismic tomography.First arrivals are generally used to...The precise determination of earthquake location is the fundamental basis in seismological community,and is crucial for analyzing seismic activity and performing seismic tomography.First arrivals are generally used to practically determine earthquake locations.However,first-arrival traveltimes are not sensitive to focal depths.Moreover,they cannot accurately constrain focal depths.To improve the accuracy,researchers have analyzed the depth phases of earthquake locations.The traveltimes of depth phases are sensitive to focal depths,and the joint inversion of depth phases and direct phases can be implemented to potentially obtain accurate earthquake locations.Generally,researchers can determine earthquake locations in layered models.Because layered models can only represent the first-order feature of subsurface structures,the advantages of joint inversion are not fully explored if layered models are used.To resolve the issue of current joint inversions,we use the traveltimes of three seismic phases to determine earthquake locations in heterogeneous models.The three seismic phases used in this study are the first P-,sPg-and PmP-waves.We calculate the traveltimes of the three seismic phases by solving an eikonal equation with an upwind difference scheme and use the traveltimes to determine earthquake locations.To verify the accuracy of the earthquake location method by the inversion of three seismic phases,we take the 2021 M_(S)6.4 Yangbi,Yunnan earthquake as an example and locate this earthquake using synthetic and real seismic data.Numerical tests demonstrate that the eikonal equation-based earthquake location method,which involves the inversion of multiple phase arrivals,can effectively improve earthquake location accuracy.展开更多
In order to describe charge exchange reactions at intermediate energies,we implemented as a first step the formulation of the normal eikonal approach.The calculated differential cross-sections based on this approach d...In order to describe charge exchange reactions at intermediate energies,we implemented as a first step the formulation of the normal eikonal approach.The calculated differential cross-sections based on this approach deviated significantly from the conventional DWBA calculations for CE reactions at 140 MeV/nucleon.Thereafter,improvements were made in the application of the eikonal approximation so as to keep a strict three-dimensional form factor.The results obtained with the improved eikonal approach are in good agreement with the DWBA calculations and with the experimental data.Since the improved eikonal approach can be formulated in a microscopic way,it is easy to apply to CE reactions at higher energies,where the phenomenological DWBA is a priori difficult to use due to the lack,in most cases,of the required phenomenological potentials.展开更多
A generalized refractive index in the form of optic eikonal is defined through com- paring frame definitions of left-handed and right-handed sets and indicates the sign of the refractive index covered by the quadratic...A generalized refractive index in the form of optic eikonal is defined through com- paring frame definitions of left-handed and right-handed sets and indicates the sign of the refractive index covered by the quadratic form of the eikonal equation. Fer- mat’s principle is generalized, and the general refractive law is derived directly. Under this definition, the comparison between Fermat’s principle and the least ac- tion principle is made through employing path integral and analogizing L. de Broglie’s theory.展开更多
We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind sche...We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.展开更多
We develop a second-order continuousfinite element method for solving the static Eikonal equation.It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system.Mor...We develop a second-order continuousfinite element method for solving the static Eikonal equation.It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system.More specifically,the homotopy method is utilized to decrease the viscosity coefficient gradually,while Newton’s method is applied to compute the solution for each viscosity coefficient.Newton’s method alone converges for just big enough viscosity coefficients on very coarse grids and for simple 1D examples,but the proposed method is much more robust and guarantees the convergence of the nonlinear solver for all viscosity coefficients and for all examples over all grids.Numerical experiments from 1D to 3D are presented to confirm the second-order convergence and the effectiveness of the proposed method on both structured or unstructured meshes.展开更多
This paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation with a large wave numbe...This paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation with a large wave number.Rigorous matrix analysis is conducted in investigations and the oscillation-free computational procedure is proven to be stable in an asymptotic sense.Simulated examples are given to illustrate the conclusion.展开更多
We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inho...We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.展开更多
基金The authors thank the funds supported by the China National Nuclear Corporation under Grants Nos.WUQNYC2101 and WUHTLM2101-04National Natural Science Foundation of China(42074132,42274154).
文摘3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing,source localization,reflection migration,seismic monitoring and tomographic imaging.In recent years,many advanced methods have been developed to solve the 3D eikonal equation in heterogeneous media.However,there are still challenges for the stable and accurate calculation of first-arrival traveltimes in 3D strongly inhomogeneous media.In this paper,we propose an adaptive finite-difference(AFD)method to numerically solve the 3D eikonal equation.The novel method makes full use of the advantages of different local operators characterizing different seismic wave types to calculate factors and traveltimes,and then the most accurate factor and traveltime are adaptively selected for the convergent updating based on the Fermat principle.Combined with global fast sweeping describing seismic waves propagating along eight directions in 3D media,our novel method can achieve the robust calculation of first-arrival traveltimes with high precision at grid points either near source point or far away from source point even in a velocity model with large and sharp contrasts.Several numerical examples show the good performance of the AFD method,which will be beneficial to many scientific applications.
文摘Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.
文摘PFMM(perspective fast marching method)是一种有效解决透视投影下从明暗恢复形状(SFS)问题的方法,但是适应条件受限,且对初始数据的精度较为敏感。本文通过对Eikonal方程系数的分析,提出了在透视投影下基于自适应Eikonal方程的PFMM,解决了PFMM对初始数据过于依赖的问题,是PFMM的推广。对合成图像的实验表明本文算法比PFMM精度更高,对透视投影下SFS问题可以得到比较好的结果。
基金financial support for this work by the Ministry of Science and Technology of China (2011CB808904)the Ministry of Land and Resources of China (SinoProbe-02-02 or 201011041,SinoProbe-03-02 or 201011047)the National Nature Science Foundation of China (41174075,41021063,41274090 and 41174043)
文摘The Northeastern Tibetan plateau records Caledonian Qilian orogeny and Cenozoic reactivation by continental collision between the Indian and Asian plates. In order to provide the constraint on the Qilian orogenic mechanism and the expansion of the plateau,wide-angle seismic data was acquired along a 430 km-long profile between Jingtai and Hezuo. There is strong height variation along the profile,which is dealt by topography flattening scheme in our crustal velocity structure reconstruction. We herein present the upper crustal P-wave velocity structure model resulting from the interpretation of first arrival dataset from topography-dependent eikonal traveltime tomography. With topography flattening scheme to process real topography along the profile,the evenness of ray coverage times of the image area（upper crust）is improved,which provides upper crustal velocity model comparable to the classic traveltime tomography（with model expansion scheme to process irregular surface）. The upper crustal velocity model shows zoning character which matcheswith the tectonic division of the Qaidam-Kunlun-West Qinling belt,the Central and Northern Qilian,and the Alax blocks along the profile. The resultant upper crustal P-wave velocity model is expected to provide important base for linkage between the mapped surface geology and deep structure or geodynamics in Northeastern Tibet.
文摘The aim in this paper is to construct an affine transformation using the classical physics analogy between the fields of optics and mechanics. Since optics and mechanics both have symplectic structures, the concept of optics can be replaced by that of mechanics and vice versa. We list the four types of eikonal (generating functions). We also introduce a unitary operator for the affine transformation. Using the unitary operator, the kernel (propagator) is calculated and the wavization (quantization) of the Gabor function is discussed. The dynamic properties of the affine transformed Wigner function are also discussed.
基金supported by the National Natural Science Foundation of China(Grant Nos.42174111 and42064004)the Beijing Natural Science Foundation(Grant No.8222033)the Ningxia Science and Technology Leading Talent Training Program(Grant No.2022GKLRLX04)。
文摘The precise determination of earthquake location is the fundamental basis in seismological community,and is crucial for analyzing seismic activity and performing seismic tomography.First arrivals are generally used to practically determine earthquake locations.However,first-arrival traveltimes are not sensitive to focal depths.Moreover,they cannot accurately constrain focal depths.To improve the accuracy,researchers have analyzed the depth phases of earthquake locations.The traveltimes of depth phases are sensitive to focal depths,and the joint inversion of depth phases and direct phases can be implemented to potentially obtain accurate earthquake locations.Generally,researchers can determine earthquake locations in layered models.Because layered models can only represent the first-order feature of subsurface structures,the advantages of joint inversion are not fully explored if layered models are used.To resolve the issue of current joint inversions,we use the traveltimes of three seismic phases to determine earthquake locations in heterogeneous models.The three seismic phases used in this study are the first P-,sPg-and PmP-waves.We calculate the traveltimes of the three seismic phases by solving an eikonal equation with an upwind difference scheme and use the traveltimes to determine earthquake locations.To verify the accuracy of the earthquake location method by the inversion of three seismic phases,we take the 2021 M_(S)6.4 Yangbi,Yunnan earthquake as an example and locate this earthquake using synthetic and real seismic data.Numerical tests demonstrate that the eikonal equation-based earthquake location method,which involves the inversion of multiple phase arrivals,can effectively improve earthquake location accuracy.
基金Supported by the the National Key R&D Program of China(2018YFA0404403)the National Natural Science Foundation of China(11535004,11775013,11875074,11875073)
文摘In order to describe charge exchange reactions at intermediate energies,we implemented as a first step the formulation of the normal eikonal approach.The calculated differential cross-sections based on this approach deviated significantly from the conventional DWBA calculations for CE reactions at 140 MeV/nucleon.Thereafter,improvements were made in the application of the eikonal approximation so as to keep a strict three-dimensional form factor.The results obtained with the improved eikonal approach are in good agreement with the DWBA calculations and with the experimental data.Since the improved eikonal approach can be formulated in a microscopic way,it is easy to apply to CE reactions at higher energies,where the phenomenological DWBA is a priori difficult to use due to the lack,in most cases,of the required phenomenological potentials.
基金the National Natural Science Foundation of China (Grant No. 60601028)
文摘A generalized refractive index in the form of optic eikonal is defined through com- paring frame definitions of left-handed and right-handed sets and indicates the sign of the refractive index covered by the quadratic form of the eikonal equation. Fer- mat’s principle is generalized, and the general refractive law is derived directly. Under this definition, the comparison between Fermat’s principle and the least ac- tion principle is made through employing path integral and analogizing L. de Broglie’s theory.
基金partially supported by ONR Grant N00014-02-1-0090ARO MURI Grant W911NF-07-1-0185NSF Grant DMS0811254
文摘We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.
基金supported by Natural Science Foundation of Jiangsu Province(Nos.KFR21026,PAF20042)National Natural Science Foundation of China(Nos.GBA20029,GCA20004)+2 种基金Science Challenge Project(No.TZ2018002)National Science and Technology Major Project(No.J2019-II-0007-0027)WH is supported by NSF DMS-1818769.
文摘We develop a second-order continuousfinite element method for solving the static Eikonal equation.It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system.More specifically,the homotopy method is utilized to decrease the viscosity coefficient gradually,while Newton’s method is applied to compute the solution for each viscosity coefficient.Newton’s method alone converges for just big enough viscosity coefficients on very coarse grids and for simple 1D examples,but the proposed method is much more robust and guarantees the convergence of the nonlinear solver for all viscosity coefficients and for all examples over all grids.Numerical experiments from 1D to 3D are presented to confirm the second-order convergence and the effectiveness of the proposed method on both structured or unstructured meshes.
文摘This paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation with a large wave number.Rigorous matrix analysis is conducted in investigations and the oscillation-free computational procedure is proven to be stable in an asymptotic sense.Simulated examples are given to illustrate the conclusion.
文摘We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.
