Guojie Song

Optimally Accurate Nearly Analytic Discrete Scheme for Wave-Field Simulation in 3D Anisotropic Media

We present a new numerical method for elastic wave modeling in 3D isotropic and anisotropic media, which is called the 3D optimal nearly analytic dis- crete method (ONADM) in this article. This work is an extension of the 2D ONADM (Yang et al., 2004) that models acoustic and elastic waves propagating in 2D media. The formulation is derived by using a multivariable truncated Taylor series expansion and high-order interpolation approximations. Our 3D ONADM enables wave propa- gation to be simulated in three dimensions through isotropic and anisotropic models. Promising numerical results show that the error of the ONADM for the 3D case is less than those of the conventional finite-difference (FD) method and the so-called Lax-Wendroff correction (LWC) schemes, measured quantitatively by the root- mean-square deviation from analytical solution. The seismic wave fields in the 3D isotropic and anisotropic media are simulated and compared with those obtained by using the fourth-order LWC method and exact solutions for the acoustic wave case. We show that, compared with the conventional FD method and the LWC schemes, the ONADM for the 3D case can reduce significantly the storage space and compu- tational costs. Numerical experiments illustrate that the ONADM provides a useful tool for the 3D large-scale isotropic and anisotropic problems and it can suppress effectively numerical dispersions caused by discretizing the 3D wave equations when too-coarse grids are used, which is the same as the 2D ONADM. Numerical modeling also implies that simultaneously using both the wave displacement and its gradients to approximate the high-order derivatives is important for both decreasing the nu- merical dispersion caused by the discretization of wave equations and compensating the important wave-field information included in wave-displacement gradients.

An improved nearly analytical discrete method: an efficient tool to simulate the seismic response of 2-D porous structures

The nearly analytic discrete method (NADM) for acoustic and elastic waves in porous elastic media is a perturbation method recently proposed by Yang et al (2006a Commun. Comput. Phys. 1 528–47). This method uses the truncated Taylor series expansion to approximate the time derivatives and the local high-order interpolation to approximate the spatial high-order derivatives by simultaneously using the displacements and its gradients and the velocity. As a result, it can suppress effectively numerical dispersions caused by the discretizing the wave equations when too-coarse grids are used. In this paper, we present an improved nearly-analytic discrete method (INADM) for the porous case. We compare numerically the error of the INADM with those of the original NADM and the so-called Lax–Wendroff correction (LWC) schemes for 1-D and 2-D cases, and give the wave-field modelling in 2-D porous isotropic and anisotropic media. We show that, compared with the original NADM, the INADM for the 2-D case can reduce significantly the storage space and increase time accuracy, while the space accuracy remains the same as that of the original one. Numerical experiments show that the error of the INADM for the porous case is less than those of the NADM and the fourth-order LWC scheme. The three-component seismic wave-fields in the 2-D porous isotropic medium are compared with those obtained by using the NADM, the LWC method, and exact solutions. Several characteristics of waves propagating in porous anisotropic media, computed by the INADM, are also reported in this study. Promising numerical results illustrate that the INADM provides a useful tool for large-scale porous problems and it can effectively suppress numerical dispersions.

An Explicit Method Based on the Implicit Runge–Kutta Algorithm for Solving Wave Equations

Abstract A new explicit differentiator series method based on the implicit Runge–Kutta method, called the IRK-DSM in brief, is developed for solving wave equations. To develop the new algorithm, we first transform the wave equation, usually described as a partial differential equation (PDE), into a system of first-order ordinary differential equations (ODEs) with respect to time t . Then we use a truncated differentiator series method of the implicit Runge–Kutta method to solve the semidiscrete ordinary differential equations, while the high-order spatial derivatives included in the ODEs are approximated by the local interpolation method. We analyze the theoretical properties of the IRK-DSM, including the stability criteria for solving the 1D and 2D acoustic-wave equations, numerical dispersion, discretizing error, and computational efficiency when using the IRK-DSM to model acoustic-wave fields. For comparison, we also present the stability criteria and numerical dispersion of the so-called Lax–Wendroff correction (LWC) methods with the fourth-order and eighth-order accuracies for the 1D case. Promising numerical results show that the IRK-DSM provides a useful tool for large-scale practical problems because it can effectively suppress numerical dispersions and source-noises caused by discretizing the acoustic- and elastic-wave equations when too-coarse grids are used or the models have a large velocity contrast between adjacent layers. Theoretical analysis and numerical modeling also demonstrate that the IRK-DSM, through combining both the implicit Runge–Kutta scheme with good stability condition and the approximate differentiator series method, is a robust wave-field modeling method.

Simulation of acoustic wavefields in heterogeneous media: A robust method for automatic suppression of numerical dispersion

Numerical dispersion limits the application of numerical simulation methods for solving the acoustic wave equation in largescalecomputation.ThenearlyanalyticdiscretemethodNADM and its improved version for suppressing numerical dispersion weredevelopedrecently.Thisnewmethodisarefinementoftwo previousmethodsandfurtherincreasestheabilityofsuppressing numerical dispersion for modeling acoustic wave propagation in 2D heterogeneous media, which uses acoustic wave displacement, particle velocity, and their gradients to restructure the acoustic wavefield via the truncated Taylor expansion and the high-order interpolation approximate method. For the method proposed,weinvestigateitsimplementationandcompareitwith the higher-order Lax-Wendroff correction LWC scheme, the original nearly analytic discrete method NADM and its improved version with regard to numerical dispersion, computational costs, and computer storage requirements. The numerical dispersionrelationsprovidedbytherefinedalgorithmfor1Dand 2D cases are analyzed, as well as the numerical results obtained bythismethodagainsttheexactsolutionforthe2Dacousticcase. Numerical results show that the refined method gives no visible numericaldispersionforverylargespatialgridincrements.Itcan simulatehigh-frequencywavepropagationforagivengridinterval and automatically suppress the numerical dispersion when the acoustic wave equation is discretized, when too few samples per wavelength are used, or when models have large velocity contrasts. Unlike the high-order LWC methods, our present methodcansavesubstantialcomputationalcostsandmemoryrequirements because very large grid increments can be used. The refined method can be used for the simulation of large-scale wavefields.

A Low-Dispersive Symplectic Partitioned Runge-Kutta Method for Solving Seismic-Wave Equations: I. Scheme and Theoretical Analysis

Abstract In this article, a series of nearly analytic symplectic partitioned Runge–Kutta (NSPRK) methods for the 3D seismic‐wave equation are developed. First, the Hamiltonian formulations for acoustic and elastic‐wave equations are presented, and then the spatial derivatives are discretized by a nearly analytic discrete operator to obtain a semidiscrete Hamiltonian system. The second‐, third‐, and fourth‐order symplectic partitioned Runge–Kutta schemes are then applied as the time integrator. A semianalytic procedure to facilitate the analysis of stability conditions and numerical dispersion relations is presented, and subsequent theoretical analysis shows that the NSPRK schemes preserve the wave velocity better than conventional symplectic schemes, especially on coarser grids. The numerical solutions computed by NSPRK schemes are compared with analytic solutions for the 3D acoustic and elastic cases. We implemented the NSPRK and some conventional schemes for a 3D acoustic wave propagation simulation in a parallel computer and compared their computational efficiencies. To generate comparable results, the NSPRK schemes require much less computer memory, central processing unit time, and communication time, which substantially accelerates the computation speed. The final simulation in the two‐layer acoustic model shows that the NSPRK schemes can suppress numerical dispersion and preserve the waveforms better than conventional symplectic schemes.

A time-space domain stereo finite difference method for 3D scalar wave propagation

The time-space domain finite difference methods reduce numerical dispersion effectively by minimizing the error in the joint time-space domain. However, their interpolating coefficients are related with the Courant numbers, leading to significantly extra time costs for loading the coefficients consecutively according to velocity in heterogeneous models. In the present study, we develop a time-space domain stereo finite difference (TSSFD) method for 3D scalar wave equation. The method propagates both the displacements and their gradients simultaneously to keep more information of the wavefields, and minimizes the maximum phase velocity error directly using constant interpolation coefficients for different Courant numbers. We obtain the optimal constant coefficients by combining the truncated Taylor series approximation and the time-space domain optimization, and adjust the coefficients to improve the stability condition. Subsequent investigation shows that the TSSFD can suppress numerical dispersion effectively with high computational efficiency. The maximum phase velocity error of the TSSFD is just 3.09% even with only 2 sampling points per minimum wavelength when the Courant number is 0.4. Numerical experiments show that to generate wavefields with no visible numerical dispersion, the computational efficiency of the TSSFD is 576.9%, 193.5%, 699.0%, and 191.6% of those of the 4th-order and 8th-order Lax-Wendroff correction (LWC) method, the 4th-order staggered grid method (SG), and the 8th-order optimal finite difference method (OFD), respectively. Meanwhile, the TSSFD is compatible to the unsplit convolutional perfectly matched layer (CPML) boundary condition for absorbing artificial boundaries. The efficiency and capability to handle complex velocity models make it an attractive tool in imaging methods such as acoustic reverse time migration (RTM). A time-space domain stereo finite difference method for 3D scalar wave equation is proposed.The method can suppress numerical dispersion effectively with high computational efficiency.It minimizes the maximum phase velocity error directly for different time steps.It uses constant interpolation coefficients to reduce the computational time cost.The stability condition is improved by adjusting the interpolation coefficients.