Multi-scale seismic wave numerical simulation can take into account large-scale macro effects and small-scale fine structure, thereby effectively saving the amount of calculation, and the local model usually needs to be refined in the implementation process. The domain reduction method based on the finite element method can flexibly deal with the cross-scale problems of local complex structures and undulating surfaces, and is one of the common means to achieve multi-scale simulation. During processing, the transition region needs to be interpolated into the outer domain (coarse mesh) wave field to achieve upsampling to accommodate the wave field of the inner domain (fine mesh) (Figure 1). However, commonly used wavefield interpolation methods have low precision and typically only achieve a small upsampling magnification (M).
Figure 1 Schematic diagram of two regions of interest for seismic wave field simulations: Ω and Ω+ represent the inner domain (thin mesh) and outer domain (coarse mesh), respectively; the transition area Γ includes Γe and Γb, where subscripts b and e represent the boundary domain and the outer domain, respectively
Zhang Lei, postdoctoral fellow of the Key Laboratory of the Institute of Geology and Geophysics, Chinese Academy of Sciences, and Researcher Zhang Jinhai, co-supervisor, proposed a finite element wave field sampling method based on Fourier interpolation (Figure 2). Since Fourier interpolation requires a uniform distribution of spatial sampling points, this method requires uniform sampling in the transition region. However, due to the characteristics of the domain reduction method, this uniform sampling involves only one layer of elements (that is, two layer nodes) between the coarse mesh and the fine mesh. Numerical test results show that the method is better than linear interpolation or cubic interpolation in accuracy, and even if the upsampling magnification of 40 times (M=40), its relative error is less than 2%, compared with the relative error of linear interpolation and cubic interpolation, which is as high as 90% and 41%, respectively.
Fig. 2 Schematic diagram of the Fourier interpolation method for cross-scale seismic wave field simulation: (a) the coarse mesh used in the first step of the domain reduction method has a transition region at the boundary of the region of interest, and the displacement time history of the two nodes Γe and Γb is obtained in the coarse mesh simulation for subsequent calculations; (b) the wave field is smoothed through the Hanning window, and the node displacement of the elements along the transition area needs to be interpolated to the original M times (M=3 here;(c) The wave field after the Fourier transform; (d) the spectrum of the split wave field, Zero values are inserted at the Nyquist frequency; (e) inverse Fourier transform is used to achieve wave field sampling
In order to extend this method to the case of non-uniform meshes (such as undulating surfaces and complex subsurface structures), they first regularized the wave fields on the non-uniform mesh using cubic interpolation in the target area, and used reasonable interpolation to avoid spatial discontinuities in the wave fields near the undulating surface, and then performed the Fourier interpolation method. The results of numerical experiments show that the proposed method can simultaneously adapt to the three-dimensional models such as undulating surface, multi-layer non-uniform media and background random disturbance (Figure 3), and significantly improve the efficiency of cross-scale seismic wave field simulation while maintaining the accuracy of numerical simulation.
Figure 3 Snapshot of vertical displacement in different 3D models simulated: (a) body wave at t=4.065 s; (b) surface wave at t=6.900 s; (c) maximum displacement distribution map in all time intervals; (d) Comparison of vertical displacement waveform obtained by this method with theoretical results, the receiving point is the blue point along the surface (north-south) in Figure A
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