S U M M A R YIn this study, we propose a new numerical method, named as Traction Image method, to accurately and efficiently implement the traction-free boundary conditions in finite difference simulation in the presence of surface topography. In this algorithm, the computational domain is discretized by boundary-conforming grids, in which the irregular surface is transformed into a 'flat' surface in computational space. Thus, the artefact of staircase approximation to arbitrarily irregular surface can be avoided. Such boundary-conforming gridding is equivalent to a curvilinear coordinate system, in which the first-order partial differential velocity-stress equations are numerically updated by an optimized high-order non-staggered finite difference scheme, that is, DRP/opt MacCormack scheme. To satisfy the free surface boundary conditions, we extend the Stress Image method for planar surface to Traction Image method for arbitrarily irregular surface by antisymmetrically setting the values of normal traction on the grid points above the free surface. This Traction Image method can be efficiently implemented. To validate this new method, we perform numerical tests to several complex models by comparing our results with those computed by other independent accurate methods. Although some of the testing examples have extremely sloped topography, all tested results show an excellent agreement between our results and those from the reference solutions, confirming the validity of our method for modelling seismic waves in the heterogeneous media with arbitrary shape topography. Numerical tests also demonstrate the efficiency of this method. We find about 10 grid points per shortest wavelength is enough to maintain the global accuracy of the simulation. Although the current study is for 2-D P-SV problem, it can be easily extended to 3-D problem.
SUMMARY Surface topography has been considered a difficult task for seismic wave numerical modelling by the finite‐difference method (FDM) because the most popular staggered finite‐difference scheme requires a rectilinear grid. Even though there are numerous collocated grid schemes in other computational fields that could be used to solve the first‐order hyperbolic equations, the lack of a stable free‐surface boundary condition implementation for curvilinear grids also obstructs the adoption of curvilinear grids in seismic wave FDM modelling. In this study, we use generalized curvilinear grids that can fit the surface topography to discretize the computational domain and describe the implementation of a collocated grid finite‐difference scheme, a higher order MacCormack scheme, to solve the first‐order hyperbolic velocity‐stress equations on the curvilinear grid. To achieve a sufficiently accurate and stable free‐surface boundary condition implementation on the curvilinear grids, we propose the traction image method that antisymmetrically images the traction components instead of the stress components to the ghost points above the free surface. Since the velocity derivatives at the free surface are provided by the free‐surface condition, we use a compact scheme to compute the velocity derivatives near the free surface and avoid the use of velocity values on the ghost points. Numerical tests verify that using the curvilinear grid, the collocated finite‐difference scheme and the traction image technique can simulate seismic wave propagation in the presence of surface topography with sufficient accuracy.
S U M M A R YTo simulate seismic wave propagation in the spherical Earth, the Earth's curvature has to be taken into account. This can be done by solving the seismic wave equation in spherical coordinates by numerical methods. In this paper, we use an optimized, collocated-grid finitedifference scheme to solve the anisotropic velocity-stress equation in spherical coordinates. To increase the efficiency of the finite-difference algorithm, we use a non-uniform grid to discretize the computational domain. The grid varies continuously with smaller spacing in low velocity layers and thin layer regions and with larger spacing otherwise. We use stressimage setting to implement the free surface boundary condition on the stress components. To implement the free surface boundary condition on the velocity components, we use a compact scheme near the surface. If strong velocity gradient exists near the surface, a lower-order scheme is used to calculate velocity difference to stabilize the calculation. The computational domain is surrounded by complex-frequency shifted perfectly matched layers implemented through auxiliary differential equations (ADE CFS-PML) in a local Cartesian coordinate. We compare the simulation results with the results from the normal mode method in the isotropic and anisotropic models and verify the accuracy of the finite-difference method.
Differences between 3-D numerical predictions of earthquake ground motion in the Mygdonian basin near Thessaloniki, Greece, led us to define four canonical stringent models derived from the complex realistic 3-D model of the Mygdonian basin. Sediments atop an elastic bedrock are modelled in the 1D-sharp and 1D-smooth models using three homogeneous layers and smooth velocity distribution, respectively. The 2D-sharp and 2D-smooth models are extensions of the 1-D models to an asymmetric sedimentary valley. In all cases, 3-D wavefields include strongly dispersive surface waves in the sediments. We compared simulations by the Fourier pseudo-spectral method (FPSM), the Legendre spectral-element method (SEM) and two formulations of the finite-difference method (FDM-S and FDM-C) up to 4Hz. The accuracy of individual solutions and level of agreement between solutions vary with type of seismic waves and depend on the smoothness of the velocity model. The level of accuracy is high for the body waves in all solutions. However, it strongly depends on the discrete representation of the material interfaces (at which material parameters change discontinuously) for the surface waves in the sharp models. An improper discrete representation of the interfaces can cause inaccurate numerical modelling of surface waves. For all the numerical methods considered, except SEM with mesh of elements following the interfaces, a proper implementation of interfaces requires definition of an effective medium consistent with the interface boundary conditions. An orthorhombic effective medium is shown to significantly improve accuracy and preserve the computational efficiency of modelling. The conclusions drawn from the analysis of the results of the canonical cases greatly help to explain differences between numerical predictions of ground motion in realistic models of the Mygdonian basin. We recommend that any numerical method and code that is intended for numerical prediction of earthquake ground motion should be verified through stringent models that would make it possible to test the most important aspects of accuracy.
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