模拟地震波传播的谱元-有限差分混合方法

Spectral element-finite difference hybrid methods for seismic wave simulation

  • 摘要: 为了充分发挥SEM和FDM各自在数值模拟复杂介质中地震波传播的优点,同时避免各自缺点,本文提出了模拟地震波传播的谱元-有限差分(SEM-FDM)混合方法.该混合方法在起伏地形附近采用谱元法计算地表附近的地震波传播;在远离起伏地形的区域之下,采用有限差分法计算地震波传播;通过构建数据交换区实现两种方法的耦合.结果表明,本文提出的谱元-有限差分混合方法能有效模拟任意非均匀介质中地震波传播,具有对复杂地形的刻画能力,能处理自由地表边界条件,且能适应模型内速度间断面.通过模型试算,并与谱元法进行对比,验证了该混合方法用于地震波模拟的有效性和高精度性.

     

    Abstract: Efficient methods for seismic wave simulations play important roles in seismic waveform inversion. Currently, the spectral element method (SEM) and finite difference method (FDM) are mostly frequently used for simulating seismic waveforms in waveform inversion due to their efficiency. Compared with other methods, such as finite element method and pseudo spectral method, the computational costs of the SEM and FDM are relatively low, which is greatly important for seismic wave modeling in large-scale models and waveform inversion. Each method, the SEM or FDM, has its advantages and disadvantages when simulating seismic waveforms in complex medium. For example, the SEM can simulate seismic waves in arbitrary heterogeneous medium with velocity discontinuities due to its high-accuracy. In addition, the free-surface boundary can be automatically satisfied. However, designing a mesh with good quality is generally time-consuming. In some cases, it is difficult to construct an appropriate mesh for the SEM. The FDM generally uses regular meshes for seismic wave modeling. Therefore, it is extremely convenient to design a mesh for the FDM. However, free-surface boundary condition cannot be automatically satisfied and special treatments are required to incorporate the free-surface boundary condition. Although some special treatments are proposed in the past decades, algorithms for the approximation of free-surface boundary condition commonly have very low accuracy. Thus, it is difficult to accurately simulate surface waves using the FDM. To retain the advantages and avoid the disadvantages of the SEM and FDM, we develop a hybrid method, which is call SEM-FDM. For this method, the propagation of seismic waves near the free-surface boundary is simulated by the SEM, and in the area far away from the free-surface boundary is modeled by the FDM. A layer for data exchanging is used for coupling the two methods. The SEM-FDM hybrid method is efficient for the simulation of seismic waves in arbitrary heterogeneous media. The hybrid method has the properties for handling undulating topography, free-surface boundary condition and velocity discontinuities. Based on a plane wave analysis, we derive the stability conditions of the hybrid method, and present the stability conditions for spatial accuracy with different orders. In order to demonstrate the validity of the SEM-FDM hybrid method in seismic wave modeling, we construct four models for numerical tests. The first model is used to show the accuracy and efficiency of the SEM-FDM hybrid method. By a comparison with analytical solutions and results from the FDM and SEM, the SEM-FDM is indeed very suitable for seismic wave modeling. The second and third models are used to display the ability of the SEM-FDM hybrid method for modeling seismic wave in complex medium. Although relatively sample grids are adopted, the SEM-FDM can still accurately model seismic waves. The last is a really geological model, which is used to exhibit the usefulness of the proposed method in real cases. By a comparison with the results from the SEM, we demonstrate that the SEM-FDM hybrid method is a useful tool in real cases. Although the areas near the free-surface area are discretized by irregular elements, the process for generate irregular elements are not difficult. In our future work, we will use Fourier series to approximate the free surface and automatically generate a mesh based on the Fourier series and control parameters.

     

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