Global SH-wave propagation in a 2D whole Moon model using the parallel hybrid PSM/FDM method

We present numerical modeling of SH-wave propagation for the recently proposed whole Moon model and try to improve our understanding of lunar seismic wave propagation. We use a hybrid PSM/FDM method on staggered grids to solve the wave equations and implement the calculation on a parallel PC cluster to improve the computing efficiency. Features of global SH-wave propagation are firstly discussed for a 100-km shallow and900-km deep moonquakes, respectively. Effects of frequency range and lateral variation of crust thickness are then investigated with various models. Our synthetic waveforms are finally compared with observed Apollo data to show the features of wave propagation that were produced by our model and those not reproduced by our models. Our numerical modeling show that the low-velocity upper crust plays significant role in the development of reverberating wave trains. Increasing frequency enhances the strength and duration of the reverberations.Surface multiples dominate wavefields for shallow event.Core–mantle reflections can be clearly identified for deep event at low frequency. The layered whole Moon model and the low-velocity upper crust produce the reverberating wave trains following each phases consistent with observation. However, more realistic Moon model should be considered in order to explain the strong and slow decay scattering between various phases shown on observation data.

A Weighted Runge-Kutta Method with Weak Numerical Dispersion for Solving Wave Equations

In this paper, we propose a weighted Runge-Kutta (WRK) method to solvethe 2D acoustic and elastic wave equations. This method successfully suppresses thenumerical dispersion resulted from discretizing the wave equations. In this method,the partial differential wave equation is first transformed into a system of ordinarydifferential equations (ODEs), then a third-order Runge-Kutta method is proposedto solve the ODEs. Like the conventional third-order RK scheme, this new methodincludes three stages. By introducing a weight to estimate the displacement and itsgradients in every stage, we obtain a weighted RK (WRK) method. In this paper, weinvestigate the theoretical properties of the WRK method, including the stability criteria, numerical error, and the numerical dispersion in solving the 1D and 2D scalarwave equations. We also compare it against other methods such as the high-ordercompact or so-called Lax-Wendroff correction (LWC) and the staggered-grid schemes.To validate the efficiency and accuracy of the method, we simulate wave fields in the2D homogeneous transversely isotropic and heterogeneous isotropic media. We conclude that the WRK method can effectively suppress numerical dispersions and sourcenoises caused in using coarse grids and can further improve the original RK methodin terms of the numerical dispersion and stability condition.

A Strong Stability-Preserving Predictor-Corrector Method for the Simulation of Elastic Wave Propagation in Anisotropic Media

In this paper,we propose a strong stability-preserving predictor-corrector(SSPC)method based on an implicit Runge-Kutta method to solve the acoustic-and elastic-wave equations.We first transform the wave equations into a system of ordinary differential equations(ODEs)and apply the local extrapolation method to discretize the spatial high-order derivatives,resulting in a system of semi-discrete ODEs.Then we use the SSPC method based on an implicit Runge-Kutta method to solve the semi-discrete ODEs and introduce a weighting parameter into the SSPC method.On top of such a structure,we develop a robust numerical algorithm to effectively suppress the numerical dispersion,which is usually caused by the discretization of wave equations when coarse grids are used or geological models have large velocity contrasts between adjacent layers.Meanwhile,we investigate the performance of the SSPC method including numerical errors and convergence rate,numerical dispersion,and stability criteria with different choices of the weighting parameter to solve 1-D and 2-D acoustic-and elastic-wave equations.When the SSPC is applied to seismic simulations,the computational efficiency is also investigated by comparing the SSPC,the fourth-order Lax-Wendroff correction(LWC)method,and the staggered-grid(SG)finite differencemethod.Comparisons of synthetic waveforms computed by the SSPC and analytic solutions for acoustic and elastic models are given to illustrate the accuracy and the validity of the SSPCmethod.Furthermore,several numerical experiments are conducted for the geological models including a 2-D homogeneous transversely isotropic(TI)medium,a two-layer elastic model,and the 2-D SEG/EAGE salt model.The results show that the SSPC can be used as a practical tool for large-scale seismic simulation because of its effectiveness in suppressing numerical dispersion even in the situations such as coarse grids,strong interfaces,or high frequencies.