ALBERT

Description: 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 4 Hz. 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.

Description: In this study, we present a new method for simulating the 3-D dynamic rupture process occurring on a non-planar fault. The method is based on the curved-grid finite-difference method (CG-FDM) proposed by Zhang & Chen and Zhang et al. to simulate the propagation of seismic waves in media with arbitrary irregular surface topography. While keeping the advantages of conventional FDM, that is computational efficiency and easy implementation, the CG-FDM also is flexible in modelling the complex fault model by using general curvilinear grids, and thus is able to model the rupture dynamics of a fault with complex geometry, such as oblique dipping fault, non-planar fault, fault with step-over, fault branching, even if irregular topography exists. The accuracy and robustness of this new method have been validated by comparing with the previous results of Day et al. , and benchmarks for rupture dynamics simulations. Finally, two simulations of rupture dynamics with complex fault geometry, that is a non-planar fault and a fault rupturing a free surface with topography, are presented. A very interesting phenomenon was observed that topography can weaken the tendency for supershear transition to occur when rupture breaks out at a free surface. Undoubtedly, this new method provides an effective, at least an alternative, tool to simulate the rupture dynamics of a complex non-planar fault, and can be applied to model the rupture dynamics of a real earthquake with complex geometry.

Description: Perfectly matched layer (PML) is an efficient absorbing technique for numerical wave simulations. Since it appeared, various improvements have been made. The complex frequency-shifted PML (CFS-PML) improves the absorbing performance for near-grazing incident waves and evanescent waves. The auxiliary differential equation (ADE) formulation of the PML provides a convenient unsplit-field PML implementation that can be directly used with high order time marching schemes. The multi-axial PML (MPML) stabilizes the PML on anisotropic media. However, these improvements were generally developed for Cartesian grids. In this paper, we extend the ADE CFS-PML to general curvilinear (non-orthogonal) grids for elastic wave modelling. Unlike the common implementations to absorb the waves in the computational space, we apply the damping along the perpendicular direction of the PML layer in the local Cartesian coordinates. Further, we relate the perpendicular and parallel components of the gradient operator in the local Cartesian coordinates to the derivatives in the curvilinear coordinates, to avoid mapping the wavefield to the local Cartesian coordinates. It is thus easy to be incorporated with numerical schemes on curvilinear grids. We derive the PML equations for the interior region and for the free surface separately because the free surface boundary condition modifies the elastic wave equations. We show that the elastic wave modelling on curvilinear grids exhibits anisotropic effects in the computational space, which may lead to unstable simulations. To stabilize the simulation, we adapt the MPML strategy to also absorb the wavefield along the two parallel directions of the PML. We illustrate the stability of this ADE CFS-MPML for finite-difference elastic wave simulations on curvilinear grids by two numerical experiments.

Description: A discontinuous grid finite-difference (FD) method with non-uniform time step Runge–Kutta scheme on curvilinear collocated-grid is developed for seismic wave simulation. We introduce two transition zones: a spatial transition zone and a temporal transition zone, to exchange wavefield across the spatial and temporal discontinuous interfaces. A Gaussian filter is applied to suppress artificial numerical noise caused by down-sampling the wavefield from the finer grid to the coarser grid. We adapt the non-uniform time step Runge–Kutta scheme to a discontinuous grid FD method for further increasing the computational efficiency without losing the accuracy of time marching through the whole simulation region. When the topography is included in the modelling, we carry out the discontinuous grid method on a curvilinear collocated-grid to obtain a sufficiently accurate free-surface boundary condition implementation. Numerical tests show that the proposed method can sufficiently accurately simulate the seismic wave propagation on such grids and significantly reduce the computational resources consumption with respect to regular grids.