基于优化数值积分的谱元法模拟地震波传播

Spectral-element method based on optimal numerical integration for seismic waveform modeling

复杂介质的高精度地震波数值模拟一直是地球物理研究领域的难点,研发高精度、高效率的数值算法对地震波正反演研究至关重要。谱元法现已被成功应用于各种尺度模型的地震波数值模拟。然而,传统谱元法采用的Gauss-Lobatto-Legendre数值求积算法无法精确计算质量矩阵和刚度矩阵包含的多项式积分,以至于难以充分展现谱元法的高精度优点。针对传统谱元法数值积分精度不足问题,文中提出一种优化积分算法。即首先构造关于数值求积与精确积分的最小二乘目标函数,然后利用共轭梯度优化方法求取优化的数值积分权系数,从而提升数值积分精度,并最终提高谱元法模拟地震波的数值精度。通过理论分析和模型试算,证实基于优化数值积分的谱元法在减少数值频散、提升计算精度等方面具有优势。

High-accuracy seismic waveform modeling for complex media is a difficult issue in the geophysics community,and developing a high-accuracy and efficient numerical algorithm is crucial to the research on the forward modeling and inversion of seismic waveforms.At present,the spectral-element method(SEM)has been successfully applied to seismic wave simulation by models on different scales. However,the Gauss-Lobatto-Legendre(GLL)numerical integration algorithm used by the conventional SEM is not able to accurately calculate the polynomial integration involved in the mass and stiffness matrices,which thus decreases the accuracy of SEM.Here we propose an optimal numerical integration algorithm to solve the abovementioned problem.We first construct the leastsquare formation of the objective functions for numerical integration and exact integration.After that,we utilize the conjugate gradient method to solve the weight coefficient of optimal numerical integration,which increases the accuracy of the numerical integration and thereby improves the numerical accuracy of SEM.Theoretical analyses and numerical examples verify that the spectral-element method based on optimal numerical integration performs better in suppressing numerical dispersion and increasing calculation accuracy.