Suppress numerical dispersion in reversetime migration of acoustic wave equation using optimal nearly analytic discrete method

Using staggered-grid finite difference method to solve seismic wave equation,large spatial grid and high dominant frequency of source cause numerical dispersion,staggeredgrid finite difference method,which can reduce the step spatial size and increase the order of difference,will multiply the calculation amount and reduce the efficiency of solving wave equation.The optimal nearly analytic discrete(ONAD)method can accurately solve the wave equation by using the combination of displacement and gradient of spatial nodes to approach the spatial partial derivative under rough grid and high-frequency condition.In this study,the ONAD method is introduced into the field of reverse-time migration(RTM)for performing forward-and reverse-time extrapolation of a two-dimensional acoustic equation,and the RTM based on ONAD method is realized via normalized cross-correlation imaging condition,effectively suppressed the numerical dispersion and improved the imaging accuracy.Using ONAD method to image the groove model and SEG/EAGE salt dome model by RTM,and comparing with the migration sections obtained by staggered-grid finite difference method with the same time order 2 nd and space order 4 th,results show that the RTM based on ONAD method can effectively suppress numerical dispersion caused by the high frequency components in source and shot records,and archive accurate imaging of complex geological structures especially the fine structure,and the migration sections of the measured data show that ONAD method has practical application value.

GENETIC ALGORITHM IN REDUCTION OF NUMERICAL DISPERSION OF 3-D ADI-FDTD METHOD

A new method to reduce the numerical dispersion of the three-dimensional Alternating Di-rection Implicit Finite-Difference Time-Domain (3-D ADI-FDTD) method is proposed. Firstly,the numerical formulations of the 3-D ADI-FDTD method are modified with the artificial anisotropy,and the new numerical dispersion relation is derived. Secondly,the relative permittivity tensor of the artificial anisotropy can be obtained by the Adaptive Genetic Algorithm (AGA). In order to demon-strate the accuracy and efficiency of this new method,a monopole antenna is simulated as an exam-ple. And the numerical results and the computational requirements of the proposed method are com-pared with those of the conventional ADI-FDTD method and the measured data. In addition the re-duction of the numerical dispersion is investigated as the objective function of the AGA. It is found that this new method is accurate and efficient by choosing proper objective function.

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.

Numerical modeling of seismic wavefields in transversely isotropic media with a compact staggered-grid finite difference scheme

To deal with the numerical dispersion problem, by combining the staggeredgrid technology with the compact finite difference scheme, we derive a compact staggered- grid finite difference scheme from the first-order velocity-stress wave equations for the transversely isotropic media. Comparing the principal truncation error terms of the compact staggered-grid finite difference scheme, the staggered-grid finite difference scheme, and the compact finite difference scheme, we analyze the approximation accuracy of these three schemes using Fourier analysis. Finally, seismic wave numerical simulation in transversely isotropic （VTI） media is performed using the three schemes. The results indicate that the compact staggered-grid finite difference scheme has the smallest truncation error, the highest accuracy, and the weakest numerical dispersion among the three schemes. In summary, the numerical modeling shows the validity of the compact staggered-grid finite difference scheme.

NUMERICAL ANALYSIS AND CONSTRUCTION OF LIMITER OF HIGH RESOLUTION DIFFERENCE SCHEME

In the paper, based on the theory of the remainder effects of difference schemes, some typical limiters are analysed and compared. For different limiters, the different strength of numerical dissipation and dispersion of schemes is the reason why the schemes show obvious different characteristics. After analysing and comparing the numerical dissipation and dispersion of various schemes, a new kind of limiter is proposed. The new scheme has high resolution in sharp discontinuities, and avoids the 'distortion' due to the stronger numerical dispersion in the relatively more smooth region. Numerical experiments show that the scheme has good properties.

Optimized staggered-grid finite-difference operators using window functions

The staggered-grid finite-difference （SGFD） method has been widely used in seismic forward modeling. The precision of the forward modeling results directly affects the results of the subsequent seismic inversion and migration. Numerical dispersion is one of the problems in this method. The window function method can reduce dispersion by replacing the finite-difference operators with window operators, obtained by truncating the spatial convolution series of the pseudospectral method. Although the window operators have high precision in the low-wavenumber domain, their precision decreases rapidly in the high-wavenumber domain. We develop a least squares optimization method to enhance the precision of operators obtained by the window function method. We transform the SGFD problem into a least squares problem and find the best solution iteratively. The window operator is chosen as the initial value and the optimized domain is set by the error threshold. The conjugate gradient method is also adopted to increase the stability of the solution. Approximation error analysis and numerical simulation results suggest that the proposed method increases the precision of the window function operators and decreases the numerical dispersion.

3D elastic waveform modeling with an optimized equivalent staggered-grid finite-difference method

Equivalent staggered-grid(ESG) as a new family of schemes has been utilized in seismic modeling,imaging,and inversion.Traditionally,the Taylor series expansion is often applied to calculate finite-difference(FD) coefficients on spatial derivatives,but the simulation results suffer serious numerical dispersion on a large frequency zone.We develop an optimized equivalent staggered-grid(OESG) FD method that can simultaneously suppress temporal and spatial dispersion for solving the second-order system of the 3 D elastic wave equation.On the one hand,we consider the coupling relations between wave speeds and spatial derivatives in the elastic wave equation and give three sets of FD coefficients with respect to the P-wave,S-wave,and converted-wave(C-wave) terms.On the other hand,a novel plane wave solution for the 3 D elastic wave equation is derived from the matrix decomposition method to construct the time-space dispersion relations.FD coefficients of the OESG method can be acquired by solving the new dispersion equations based on the Newton iteration method.Finally,we construct a new objective function to analyze P-wave,S-wave,and C-wave dispersion concerning frequencies.The dispersion analyses show that the presented method produces less modeling errors than the traditional ESG method.The synthetic examples demonstrate the effectiveness and superiority of the presented method.

A spherical higher-order finite-difference time-domain algorithm with perfectly matched layer

A higher-order finite-difference time-domain（HO-FDTD） in the spherical coordinate is presented in this paper. The stability and dispersion properties of the proposed scheme are investigated and an air-filled spherical resonator is modeled in order to demonstrate the advantage of this scheme over the finite-difference time-domain（FDTD） and the multiresolution time-domain（MRTD） schemes with respect to memory requirements and CPU time. Moreover, the Berenger＇s perfectly matched layer（PML） is derived for the spherical HO-FDTD grids, and the numerical results validate the efficiency of the PML.

An efficient locally one-dimensional finite-difference time-domain method based on the conformal scheme

An efficient conformal locally one-dimensional finite-difference time-domain（LOD-CFDTD） method is presented for solving two-dimensional（2D） electromagnetic（EM） scattering problems. The formulation for the 2D transverse-electric（TE） case is presented and its stability property and numerical dispersion relationship are theoretically investigated. It is shown that the introduction of irregular grids will not damage the numerical stability. Instead of the staircasing approximation, the conformal scheme is only employed to model the curve boundaries, whereas the standard Yee grids are used for the remaining regions. As the irregular grids account for a very small percentage of the total space grids, the conformal scheme has little effect on the numerical dispersion. Moreover, the proposed method, which requires fewer arithmetic operations than the alternating-direction-implicit（ADI） CFDTD method, leads to a further reduction of the CPU time. With the total-field/scattered-field（TF/SF） boundary and the perfectly matched layer（PML）, the radar cross section（RCS） of two2 D structures is calculated. The numerical examples verify the accuracy and efficiency of the proposed method.

A nearly analytic exponential time difference method for solving 2D seismic wave equations

In this paper, we propose a nearly analytic exponential time difference （NETD） method for solving the 2D acoustic and elastic wave equations. In this method, we use the nearly analytic discrete operator to approximate the high-order spatial differential operators and transform the seismic wave equations into semi-discrete ordinary differential equations （ODEs）. Then, the converted ODE system is solved by the exponential time difference （ETD） method. We investigate the properties of NETD in detail, including the stability condition for 1-D and 2-D cases, the theoretical and relative errors, the numerical dispersion relation for the 2-D acoustic case, and the computational efficiency. In order to further validate the method, we apply it to simulating acoustic/elastic wave propagation in mul- tilayer models which have strong contrasts and complex heterogeneous media, e.g., the SEG model and the Mar- mousi model. From our theoretical analyses and numerical results, the NETD can suppress numerical dispersion effectively by using the displacement and gradient to approximate the high-order spatial derivatives. In addition, because NETD is based on the structure of the Lie group method which preserves the quantitative properties of differential equations, it can achieve more accurate results than the classical methods.

Effects of supply air temperature and inlet location on particle dispersion in displacement ventilation rooms

The effects of supply temperature and vertical location of inlet air on particle dispersion in a displacement ventilated （DV） room were numerically modeled with validation by experimental data from the literature. The results indicate that the temperature and vertical location of inlet supply air did not greatly affect the air distribution in the upper parts of a DV room, but could significantly influence the airflow pattern in the lower parts of the room, thus affecting the indoor air quality with contaminant sources located at the lower level, such as particles from working activities in an office. The numerical results also show that the inlet location would slightly influence the relative ventilation efficiency for the same air supply volume, but particle concentration in the breathing zone would be slightly lower with a low horizontal wall slot than a rectangular diffuser. Comparison of the results for two different supply temperatures in a DV room shows that, although lower supply temperature means less incoming air volume, since the indoor flow is mainly driven by buoyancy, lower supply temperature air could more efficiently remove passive sources （such as particles released from work activities in an office）. However, in the breathing zone it gives higher concentration as compared to higher supply air temperature. To obtain good indoor air quality, low supply air temperature should be avoided because concentration in the breathing zone has a stronger and more direct impact on human health.

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.

The ONAD method for solving the SH-wave equation and simulation of the SH-wave propagation in the Earth's mantle

The optimal nearly-analytic discrete(ONAD) method is a new numerical method developed in recent years for solving the wave equation.Compared with other methods,such as popularly-used finite-difference methods,the ONAD method can effectively suppress the numerical dispersion when coarse grids are used.In this paper,the ONAD method is extended to solve the 2-dimensional SH-wave equation in the spherical coordinates.To investigate the accuracy and the efficiency of the ONAD method,we compare the numerical results calculated by the ONAD method and other methods for both the homogeneous model and the inhomogeneous IASP91 model.The comparisons indicate that the ONAD method for solving the SH-wave equation in the spherical coordinates has the advantages of less numerical dispersion,small memory requirement for computer codes,and fast calculation.As an application,we use the ONAD method to simulate the SH-wave propagating between the Earth's surface and the core-mantle boundary(CMB).Meanwhile,we investigate the SH-wave propagating in the mantle through analyzing the wave-field snapshots in different times and synthetic seismograms.

AN OPERATOR-SPLITTING ALGORITHM FOR ADVECTION-DIFFUSION-REACTION EQUATION

An operator-splitting algorithm for three-dimensional advection-diffusion-reaction equation is presented.The method of characteristics is adopted for the pure advection operator, the explicit difference scheme is used for diffusion,and a prediction-correction scheme is em- ployed for reaction.The condition for stability of the algorithm is analysed.Severall inear and nonlinear examples are illustrated to test the convergence and accuracy of the numerical proce- dure,and satisfactory agreements between computed and analytical solutions are achieved.Due to its simplicity,stability,and validity for both one-and two-dimensional problems,the success- ful algorithm can be used to numerical simulations of viscous fluid flows,the transport of pollu- tants and sedimentations in reservoirs,lakes,rivers,estuaries and other environments,cooling- problems in heat or nuclear power plants,etc.

A Nearly Analytical Discrete Method for Wave-Field Simulations in 2D Porous Media

The nearly analytic discrete method(NADM)is a perturbation method originally proposed by Yang et al.(2003)[26]for acoustic and elastic waves in elastic media.This method is based on a truncated Taylor series expansion and interpolation approximations and it can suppress effectively numerical dispersions caused by the discretizating the wave equations when too-coarse grids are used.In the present work,we apply the NADM to simulating acoustic and elastic wave propagations in 2D porous media.Our method enables wave propagation to be simulated in 2D porous isotropic and anisotropic media.Numerical experiments show that the error of the NADM for the porous case is less than those of the conventional finite-difference method(FDM)and the so-called Lax-Wendroff correction(LWC)schemes.The three-component seismic wave fields in the 2D porous isotropic medium are simulated and compared with those obtained by using the LWC method and exact solutions.Several characteristics of wave propagating in porous anisotropic media,computed by the NADM,are also reported in this study.Promising numerical results illustrate that the NADM provides a useful tool for large-scale porous problems and it can suppress effectively numerical dispersions.

Directly Simulation of Second Order Hyperbolic Systems in Second Order Form via the Regularization Concept

We present an efficient and robustmethod for stresswave propagation problems(second order hyperbolic systems)having discontinuities directly in their second order form.Due to the numerical dispersion around discontinuities and lack of the inherent dissipation in hyperbolic systems,proper simulation of such problems are challenging.The proposed idea is to denoise spurious oscillations by a post-processing stage from solutions obtained from higher-order grid-based methods(e.g.,high-order collocation or finite-difference schemes).The denoising is done so that the solutions remain higher-order(here,second order)around discontinuities and are still free from spurious oscillations.For this purpose,improved Tikhonov regularization approach is advised.This means to let data themselves select proper denoised solutions(since there is no pre-assumptions about regularized results).The improved approach can directly be done on uniform or non-uniform sampled data in a way that the regularized results maintenance continuous derivatives up to some desired order.It is shown how to improve the smoothing method so that it remains conservative and has local estimating feature.To confirm effectiveness of the proposed approach,finally,some one and two dimensional examples will be provided.It will be shown how both the numerical(artificial)dispersion and dissipation can be controlled around discontinuous solutions and stochastic-like results.