基于Chebyshev走时逼近的三维多次反射射线计算

doi:10.13278/j.cnki.jjuese.20170301

吉林大学学报(地球科学版) ›› 2018, Vol. 48 ›› Issue (3): 890-899.doi: 10.13278/j.cnki.jjuese.20170301

基于Chebyshev走时逼近的三维多次反射射线计算

孙建国, 苗贺

吉林大学地球探测科学与技术学院, 长春 130026

收稿日期:2017-11-14 出版日期:2018-05-26 发布日期:2018-05-26
作者简介:孙建国(1956-),男,教授,博士生导师,博士,主要从事地下波动理论与成像技术、计算地球物理、岩石物理、科学计算方法与技术、反射地震资料处理、钻孔电磁探测理论、地球物理中的天线问题、可视化技术及其在地球物理场数值模拟与观测数据解释中的应用等方面的教学和研究工作,Tel:0431-88502537,E-mail:sun_jg@jlu.edu.cn
基金资助:
国家自然科学基金项目（41274120）

Computation of Three Dimensional Multi-Reflection Rays Based on Traveltimes Numerical Approximation Using Chebyshev Polynomials

Sun Jianguo, Miao He

College of GeoExploration Science and Technology, Jilin University, Changchun 130026, China

Received:2017-11-14 Online:2018-05-26 Published:2018-05-26
Supported by:
Supported by National Natural Science Foundation of China (41274120)

摘要/Abstract

摘要： 在利用有限差分等基于网格的数值分析方法解地震波走时所满足的程函方程时，由于速度模型的网格化离散等原因，会使走时在各网格节点之间不具有计算射线路径所要求的光滑性，即走时在邻近网格节点之间不具有连续的一阶导数。因此，直接利用网格节点走时计算射线路径会使最终的射线路径不光滑。为解决这个问题，已有研究者提出了基于B样条插值的逆向梯度方案（法）。然而，在速度发生突变时，B样条逆向梯度法所计算出的射线路径会具有较大的误差。针对这个问题，首先采用适合于解最小零偏差逼近及最佳平方逼近问题的Chebyshev多项式取代B样条对来自于分区多级计算方案的网格节点走时进行最佳逼近，得到在最小平方意义下的最优走时公式；然后采用与B样条逆向梯度法类似的计算过程得到光滑的射线路径。数值实验表明，利用Chebyshev多项式逼近走时可以得到具有很高精度的多次反射射线路径，在多次波偏移成像研究中具有潜在的价值。

关键词: 多次反射, 射线走时, 射线路径, 走时逼近, Chebyshev多项式

Abstract: When using the grid methods, such as the finite difference method for solving the eikonal equation satisfied by seismic traveltimes, the traveltimes obtained at the adjacent grid points may not have the smoothness needed for computing ray trajectories. This is because of the discretization of the velocity model by grids. To solve this problem, some researchers proposed a backward gradient scheme based on a B-spline interpolation. However, the ray trajectories calculated by the B-spline interpolation scheme will have large errors when the velocity changes abruptly in some model region. Aiming at this problem, instead of the B-splines, we first utilized the Chebyshev polynomials, which are optimal for solving the zero deviation and the least square problem, to obtain the formulas that can optimally approximate the grid point traveltimes resulted from the finite different solution of the eikonal equation. Then, we used a computation procedure which is similar to that used in the B-spline scheme for obtaining smooth ray trajectories. The numerical test showed that the use of Chebyshev polynomials for approximating traveltimes led to smooth ray trajectories of multiply reflected waves with high accuracy. It should be investigated further in the future work.

Key words: multiple reflections, ray traveltimes, ray trajectories, traveltimes approximation, Chebyshev polynomials

中图分类号:

P631.4

孙建国, 苗贺. 基于Chebyshev走时逼近的三维多次反射射线计算[J]. 吉林大学学报(地球科学版), 2018, 48(3): 890-899.

Sun Jianguo, Miao He. Computation of Three Dimensional Multi-Reflection Rays Based on Traveltimes Numerical Approximation Using Chebyshev Polynomials[J]. Journal of Jilin University(Earth Science Edition), 2018, 48(3): 890-899.

参考文献

[1] 孙建国. Kirchhoff型偏移理论的研究历史、研究现状与发展趋势展望:与光学绕射理论的类比、若干新结果、新认识以及若干有待于解决的问题[J]. 吉林大学学报(地球科学版),2012,42(5):1521-1552. Sun Jianguo. The History, the State of the Art and the Future Trend of The Research of Kirchhoff-Type Migration Theory:A Comparison with Optical Diffraction Theory, Some New Understanding and Some Problems to be Solved[J]. Journal of Jilin University (Earth Science Edition), 2012, 42(5):1521-1552.
[2] 孙建国. 高频渐近散射理论及其在地球物理场数值模拟与反演成像中的应用:研究历史与研究现状概述以及若干新进展[J]. 吉林大学学报(地球科学版),2016,46(4):1231-1259. Sun Jianguo. High-Frequency Asymptotic Scattering Theories and Their Applications in Numerical Modeling and Imaging of Geophysical Fields:An Overview of the Research History and the State-of-the-Art, and Some New Developments[J]. Journal of Jilin University (Earth Science Edition), 2016, 46(4):1231-1259.
[3] Vidale J E. Finite-Difference Calculations of Travel-Times[J]. Bulletin of the Seismological Society of America, 1988, 78:2062-2076.
[4] Vidale J E. Finite-Difference Calculations of Travel-times in Three Dimensions[J]. Geophysics, 1990, 55:521-526.
[5] Rawlinson N, Sambridge M. Multiple Reflection and Transmission Phases in Complex Layered Media Using a Multistage Fast Marching Method[J]. Geophysics, 2004, 69(5):1338-1350.
[6] de Kool M, Rawlinson N, Sambridge M. A Practical Grid-Based Method for Tracking Multiple Refraction and Reflection Phases in Three-Dimensional Heterogeneous Media[J]. Geophysical Journal International, 2006, 167(1):253-270.
[7] 孙章庆,孙建国,王雪秋,等. 三维复杂山地多级次群推进迎风混合法多波型走时计算[J]. 地球物理学报,2017,60(5):1861-1873. Sun Zhangqing, Sun Jianguo, Wang Xueqiu, et al. Computation of Multiples Seismic Traveltime in Mountainous Areas with Complex 3D Conditions Using Multistage Group Marching up Wind Hybrid Method[J]. Chinese Journal of Geophysics, 2017, 60(5):1861-1873.
[8] 唐小平,白超英. 最短路径算法下三维层状介质中多次波追踪[J]. 地球物理学报,2009,52(10):2635-2643. Tang Xiaoping, Bai Chaoying. Multiple Ray Tracing Within 3-D Layered Media with the Shortest Path Method[J]. Chinese Journal of Geophysics, 2009, 52(10):2635-2643.
[9] Sun Jianguo, Sun Zhangqing, Han Fuxing. A Finite Difference Scheme for Solving the Eikonal Equation Including Surface Topography[J]. Geophysics, 2011, 76(4):T53-T63.
[10] 石秀林,孙建国,孙辉,等. 基于波前构建法的时间域深度偏移:Delta波包途径[J]. 吉林大学学报(地球科学版),2016,46(6):1847-1854. Shi Xiulin, Sun Jianguo, Sun Hui, et al. Depth Migration in Time Domain Using Wavefront Construction:Delta Packet Approach[J]. Journal of Jilin University (Earth Science Edition), 2016, 46(6):1847-1854.
[11] 石秀林,孙建国,孙辉,等.基于delta波包叠加的时间域深度偏移[J].地球物理学报,2016,59(7):2641-2649. Shi Xiulin, Sun Jianguo, Sun Hui, et al. The Time Domain Depth Migration by the Summation of Delta Packets[J]. Chinese Journal of Geophysics, 2016, 59(7), 2641-2649.
[12] 孙建国,何洋. 基于波前构建的射线追踪:一种Java实现[J].吉林大学学报(地球科学版),2007, 37(4):814-820. Sun Jianguo, He Yang. Ray-Tracing Based on Wavefront Construction:A Java Implementation[J]. Journal of Jilin University (Earth Science Edition), 2007, 37(4):814-820.
[13] 韩复兴, 孙建国, 王坤. 速度模型光滑分析[J]. 吉林大学学报(地球科学版),2011, 41(5):1610-1616. Han Fuxing, Sun Jianguo, Wang Kun. Analysis of Velocity Model Smoothing[J]. Journal of Jilin University (Earth Science Edition), 2011, 41(5):1610-1616.
[14] 韩复兴,孙建国,孙章庆. 波前构建法研究现状[J]. 地球物理学进展,2011,26(3):1045-1051. Han Fuxing, Sun Jianguo, Sun Zhangqing. Research Status of the Wavefront Construction Method[J]. Progress in Geophysics, 2011, 26(3):1045-1051.
[15] 张东,张婷婷,乔友锋,等.三维旅行时场B样条插值射线追踪方法[J]. 石油地球物理勘探,2013,48(4):559-566. Zhang Dong, Zhang Tingting, Qiao Youfeng, et al. A Ray Tracing Method Based on B-Spline Traveltime Interpolation[J]. Oil and Gas Prospecting, 2013, 48(4):559-566.
[16] 张婷婷,张东,邱达. 三维层状介质中基于走时梯度的多次波射线追踪[J]. 石油地球物理勘探,2014,49(6):1097-1105. Zhang Tingting, Zhang Dong, Qiu Da. Multiple Ray Tracing in 3D Layered Media Using the Traveltime Gradient Method[J]. Oil and Gas Prospecting, 2014,49(6):1097-1105.
[17] 张婷婷,邱达,张东. 一种改进的三维旅行时梯度射线追踪方法[J]. 石油地球物理勘探,2016,51(5):916-923. Zhang Tingting, Qiu Da, Zhang Dong. A Modified 3D Traveltime Gradient Ray Tracing Method[J]. Oil and Gas Prospecting, 2016,51(5):916-923.
[18] Press W H, Teukolsky S A, Vettering W T, et al. Numerical Recipies in FORTRAN[M]. Cambridge:Cambridge University Press, 1992.
[19] 蒋迅,王淑红. 切比雪夫和切比雪夫多项式的故事[J]. 科学, 2016, 68(4):54-58. Jiang Xun, Wang Shuhong. The Story of Chebyshev and Chebyshev Polynomials[J]. Science, 2016, 68(4):54-58.

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基于Chebyshev走时逼近的三维多次反射射线计算

Computation of Three Dimensional Multi-Reflection Rays Based on Traveltimes Numerical Approximation Using Chebyshev Polynomials

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