全球尺度地震波传播的保结构算法研究

Numerical simulation of seismic waves constitues an important part for seismological research and seismic exploration.There are many methods, such as finite difference method, finite element mthod, can deal with simulation of small-scale or regional seismic wave propagation effectively. But for large-scale, especially for global simulation of seismic waves propagtion, no effective method is avaliable to deal with the problem of invetable accumulated errors and numerical dispersion that encounted in the present non-high precision structure-preserving methods because of the issue of long-time computation, casuing much difficulty in tracing global-scale seismic wave propagation. Therefore, it is very critical to develop a reliable algorithms of high-precision global-scale numerical simulation of seismic wave propagation to address the said issue. In this research project, the issue of tracing long-time simulation of elastic wave propagation in spherical coordinates will be addressed effectively via a constructed structure-preserving algorithm of combining spectral element method with new force-gradient symplectic method. And the problems like the stability,precision and time-space structure-preserving properties in the spherical coordinates will be well studied. Also, a kind of dicontinuous opteror is employed to deal with the inhomogenous layer of the Earth's interior discontinuity. The algorithm will be developed for simulation of large-scale even global-scale seismic wave propagation with high-precision in order to lower accumulated errors and suppress numerical dispersion in the long-time simulation. The appealing characters of the new algorithm would make it effective to model the global-scale and long-time seismic wave propagation in three dimensional inhomogeneous medium with high-precision and would be another effective choice to long-time simulation of seismic wave propagation.

地震波数值模拟是地震勘探和地震学研究的重要部分。尽管现有的如有限差分、有限元等技术基本能满足小尺度问题的模拟需要,然而对于大尺度,尤其是全球尺度地震波传播的计算而言,由于涉及到长时程追踪,且前述方法大多是非高精度保结构方法,长时程计算时不可避免地产生高积累误差,导致数值频散严重,很难精确处理全球尺度地震波传播这一长时程追踪问题。因此,发展针对于全球尺度地震波传播数值模拟的高精度保结构算法成为关键。本项目针对球坐标下非均匀介质中弹性波方程拟构造谱元法结合新推的力梯度辛算法的离散方法,通过对其保结构性、精度及效率等问题的研究,发展适用于全球尺度地震波传播的高精度保结构数值模拟方法,使长时程模拟时的积累误差大幅降低,数值频散得以压制；并用间断算子处理地球内部的非均匀层间断面,显著提高模拟精度。保结构新方法的推出,将为高精度处理类似全球尺度三维非均匀介质中地震波传播的长时程计算问题提供新的途径。

研究表明离散型哈氏算法的体系结构与守恒律完整并行, 高度逼近于哈氏原型, 而且拥有理论上无限长时程的跟踪能力, 因此求解诸如地球自由振荡、全球尺度的地震波传播等的长时程传播的模拟问题时, 要求算法自身应是保辛的. 本项目的目旳是在哈氏体系中发展适用于大尺度长时程地震波传播数值模拟的保结构算法, 使长时程模拟时的积累误差大幅降低, 数值频散得以压制, 显著提高模拟精度.首先运用相位误差进行约束优化，提出了一种构造辛算法的可行途径——相位误差最小原理，推导了优化的二阶、三级三阶非力梯度显式辛算法，然后结合谱元法、有限元法、褶积微分等空间域离散算法，构造了时间-空间域的保辛结构算法，并将其运用于求解地球自由振荡等长时程模拟问题中，提高了计算效率并保证了精度，且得到了有益的结果，说明发展辛算法的有效性。我们将地幔柱相关结构和物性变化引入径向非均匀理论地球模型中，构造了新的基于S362M地球模型的大尺度横向非均匀地球模型。以2012年4月印度尼西亚苏门答腊西北海域8.6级大地震为震源机制解，进行了地球自由振荡及全球尺度的地震波传播的长时程模拟，将模拟结果同观测数据进行直观的比较分析，吻合较好。充分验证了优化的辛-谱元法求解地球自由振荡问题的可行性和有效性，同时验证了构建的大尺度横向非均匀地球模型的正确性。