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.