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| 模拟地震波传播的优化质量矩阵Legendre谱元法 | |
| 引用本文： | 刘少林, 杨顶辉, 孟雪莉, 汪文帅, 徐锡伟, 李小凡. 2022. 模拟地震波传播的优化质量矩阵Legendre谱元法. 地球物理学报, 65(12): 4802-4815, doi: 10.6038/cjg2022Q0145 |
| 作者姓名： | 刘少林 杨顶辉 孟雪莉 汪文帅 徐锡伟 李小凡 |
| 作者单位： | 应急管理部国家自然灾害防治研究院,北京 100085;中国科学院地质与地球物理研究所,岩石圈演化国家重点实验室,北京 100029;清华大学数学科学系,北京 100084;清华大学数学科学系,北京 100084;应急管理部国家自然灾害防治研究院,北京 100085;宁夏大学数学统计学院,银川 7500215中国地质大学地球物理与空间信息学院,武汉 430074;宁夏大学数学统计学院,银川 7500215中国地质大学地球物理与空间信息学院,武汉 430074;应急管理部国家自然灾害防治研究院,北京 100085;中国地质大学地球物理与空间信息学院,武汉 430074 |
| 基金项目： | 岩石圈演化国家重点实验室开放课题(SKL-K202101)；;国家自然科学基金项目(42174111,42064004和42030108)联合资助； |
| 摘 要： |  波动方程的数值求解是地震波正反演的重要环节，而数值算法的计算精度直接关系到地震波的模拟结果和成像质量.当前，谱元法由于同时具备有限元法的网格灵活性与谱方法的高精度性已被成功应用于不同尺度模型中的地震波模拟.然而，常见的Legendre谱元法在求解地震波运动方程时采用Gauss-Lobatto-Legendre（GLL）数值积分计算质量矩阵所包含的积分项，由于GLL数值求积无法对积分项精确估计，从而造成谱元法精度损失.针对谱元法精度上的不足，本文提出一种优化算法用于提升其精度.首先构造关于GLL数值求积积分权与质量矩阵对角线元素精确值的最小二乘目标函数，然后利用共轭梯度法求解目标函数得到优化权系数，该权系数能减小质量矩阵的离散误差最终提高谱元法的计算精度.通过数值频散分析、数值算例证实了本文给出的优化算法用于提升谱元法数值模拟精度的可行性和有效性. |
| 关 键 词： | 数值模拟 谱元法 质量矩阵 数值精度 |
| 收稿时间： | 2022-03-04 |
| 修稿时间： | 2022-05-25 |
A Legendre spectral element method with optimal mass matrix for seismic wave modeling |
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| LIU ShaoLin, YANG DingHui, MENG XueLi, WANG WenShuai, XU XiWei, LI XiaoFan. 2022. A Legendre spectral element method with optimal mass matrix for seismic wave modeling. Chinese Journal of Geophysics (in Chinese), 65(12): 4802-4815, doi: 10.6038/cjg2022Q0145 | |
| Authors: | LIU ShaoLin YANG DingHui MENG XueLi WANG WenShuai XU XiWei LI XiaoFan |
| Affiliation: | 1. National Institute of Natural Hazards, Ministry of Emergency Management of China, Beijing 100085, China; 2. State Key Laboratory of Lithospheric Evolution, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China; 3. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China; 4. School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China; 5. Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China |
| Abstract: | Numerical solving seismic wave equation of motion plays an important role in investigating wave phenomenon and imaging subsurface structure. The accuracy of a numerical method for forward modeling can eventually determinates the effectiveness of numerical result and the image quality. At present, the spectral element method (SEM) is successfully applied to seismic wave propagation in models with different scales. However, the accuracy of SEM is not fully explored because the integrations involved in the mass matrix of SEM are not accurately approximated by Gauss-Lobatto-Legendre (GLL) quadrature rule. The numerical errors originating from numerical quadrature rule reduce the accuracy of SEM. In this article, we propose a strategy to increase the accuracy of SEM by constructing an optimal mass matrix. The optimal mass matrix is developed in two steps. First, we design a least-square problem in terms of the GLL quadrature weights and the integrations on the diagonal of mass matrix. Second, we adopt the conjugate gradient method to solve the least-square problem and obtain the optimal quadrature weights. The optimal quadrature weights can reduce the errors associated with the mass matrix and eventually increase the accuracy of SEM. The validity of the SEM with optimized mass matrix is demonstrated by numerical dispersion analysis and numerical examples. |
| Keywords: | Numerical simulation Spectral element method Mass matrix Numerical accuracy |
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