The Numerical Modeling of 3-D Elastic Wave Equation Using a High-Order,Staggered-Grid, Finite Difference Scheme

This article provides the application of the high-order, staggered-grid, finite-difference scheme to model elastic wave propagation in 3-D isotropic media. Here, we use second-order, temporal-and high-order spatial finite-difference formulations with a staggered grid for discretization of the 3-D elastic wave equations of motion. The set of absorbing boundary conditions based on paraxial approximations of 3-D elastic wave equations are applied to the numerical boundaries. The trial resuits for the salt model show that the numerical dispersion is decreased to a minimum extent, the accuracy high and diffracted waves abundant. It also shows that this method can be used for modeling wave propagation in complex media with the lateral variation of velocity.

Numerical simulation of seismic wavefields in TTI media using the rotated staggered-grid compact finite-difference scheme

Numerical simulation in transverse isotropic media with tilted symmetry axis(TTI) using the standard staggered-grid finite-difference scheme(SSG)results in errors caused by averaging or interpolation. In order to eliminate the errors, a method of rotated staggered-grid finite-difference scheme(RSG) is proposed. However, the RSG brings serious numerical dispersion. The compact staggered-grid finite-difference scheme(CSG) is an implicit difference scheme, which use fewer grid points to suppress dispersion more effectively than the SSG. This paper combines the CSG with the RSG to derive a rotated staggered-grid compact finite-difference scheme(RSGC). The numerical experiments indicate that the RSGC has weaker numerical dispersion and better accuracy than the RSG.

Source wavefield reconstruction based on an implicit staggered-grid finite-difference operator for seismic imaging

Reverse time migration and full waveform inversion involve the crosscorrelation of two wavefields,propagated in the forward-and reverse-time directions,respectively.As a result,the forward-propagated wavefield needs to be stored,and then accessed to compute the correlation with the backward-propagated wavefield.Boundary-value methods reconstruct the source wavefield using saved boundary wavefields and can significantly reduce the storage requirements.However,the existing boundary-value methods are based on the explicit finite-difference(FD)approximations of the spatial derivatives.Implicit FD methods exhibit greater accuracy and thus allow for a smaller operator length.We develop two(an accuracy-preserving and a memory-efficient)wavefield reconstruction schemes based on an implicit staggered-grid FD(SFD)operator.The former uses boundary wavefields at M layers of grid points and the spatial derivatives of wavefields at one layer of grid points to reconstruct the source wavefield for a(2M+2)th-order implicit SFD operator.The latter applies boundary wavefields at N layers of grid points,a linear combination of wavefields at M–N layers of grid points,and the spatial derivatives of wavefields at one layer of grid points to reconstruct the source wavefield(0≤N<M).The required memory of accuracy-preserving and memory-efficient schemes is(M+1)/M and(N+2)/M times,respectively,that of the explicit reconstruction scheme.Numerical results reveal that the accuracy-preserving scheme can achieve accurate reconstruction at the cost of storage.The memory-efficient scheme with N=2 can obtain plausible reconstructed wavefields and images,and the storage amount is 4/(M+1)of the accuracy-preserving scheme.

Optimized staggered-grid finite-difference operators using window functions

蹒跚格子有限差别(SGFD ) 方法广泛地在地震前面的建模被使用了。向前当模特儿的结果的精确直接影响随后的地震倒置和迁居的结果。数字分散是在这个方法的问题之一。由用窗口操作符代替有限差别的操作符的窗口函数方法罐头还原剂分散，由截断 pseudospectral 方法的空间卷绕旋转系列获得了。尽管窗户操作员在 low-wavenumber 领域有高精确，他们的精确在 high-wavenumber 领域很快减少。我们开发一个最少的广场优化方法提高窗口函数方法获得的操作符的精确。我们把 SGFD 问题转变成一个最少的广场问题并且反复地发现最好的答案。当起始的值和优化的域被错误阀值放，窗口操作符被选择。conjugate 坡度方法也被采用增加答案的稳定性。近似错误分析和数字模拟结果建议建议方法增加窗户功能操作员和减少的精确数字分散。

Research on modeling source effect of 3D CSAMT based on staggered-grid finite difference method

There is usually source effect in the field work of controlled-source audio-frequency magnetotelluric method.Source effect is a thorny problem during field working,data processing and interpretation.Therefore,it is very important for the results of field prospecting to model source effect and summarize its influence rules.Based on the previous research,the authors use 3D finite difference method to simulate the electromagnetic field and set different anomaly situation to study the source effect in near-field measurement,then conclude the influence rules of source effect.Simulations provide the reference for the actual field work and data processing to correct the influence of source effect,so the information of the underground will be more approaching to the real.

Calculation of cell face velocity of non-staggered grid system

In this paper,the cell face velocities in the discretization of the continuity equation,the momentum equation,and the scalar equation of a non-staggered grid system are calculated and discussed.Both the momentum interpolation and the linear interpolation are adopted to evaluate the coefficients in the discretized momentum and scalar equations.Their performances are compared.When the linear interpolation is used to calculate the coefficients,the mass residual term in the coefficients must be dropped to maintain the accuracy and convergence rate of the solution.

3D numerical simulation in acoustic-elastic coupled media with staggered-grid finite-difference method

Acoustic-elastic coupled media is often encountered in most marine explorations, and accurate simulation of acoustic-elastic coupled media is of great significance. At present, the study of acoustic-elastic coupled media still assumes that the solid of the acoustic-elastic coupled media is isotropic, but this assumption is not in accordance with the actual situation. In this paper, we derive the solid media of acoustic-elastic coupled media from isotropic media to anisotropic media, and propose an acoustic-elastic coupled medium based ontransverse isotropic media with vertical symmetric axes(VTI) to improve the accuracy of forward modeling. Based on the relationship between the Thomsen parameter and the coefficient matrix of the anisotropic elastic wave equation, we transform the Thomson parameter into a velocity model with anisotropic properties. We use a staggered grid finite difference method to simulate the propagation of a wavefield in a three-dimensional acoustic-elastic coupled media. We obtain the snapshots of the wave field when the solid of the acoustic-elastic coupled media is an isotropic medium and a VTI media. When the solid of the acoustic-elastic coupled media is considered VTI media, we can observe the qP wave and qS wave that cannot be observed in the isotropic medium from the wave field snapshot. We can also find that the seismic records obtained by the method we use are more realistic. The algorithm proposed in this paper is of great significance for high-precision ocean numerical simulation.

Modeling of low-frequency seismic waves in a shallow sea using the staggered grid difference method

Elastic waves in the seabed generated by low-frequency noise radiating from ships are known as ship seismic waves and can be used to detect and identify ships.To obtain the propagation characteristics of ship seismic waves,an algorithm for calculating seismic waves at the seafl oor is presented based on the staggered-grid fi nite diff erence method.The accuracy of the algorithm was tested by comparison with analytical solutions.Numerical simulation of seismic waves generated by a low-frequency point sound source in a typical shallow sea environment was carried out.Using various source frequencies and locations in the numerical simulation,we show that the seismic waves in the near fi eld are composed mostly of transmitted S-waves and interface waves while transmitted P-waves are weak near the seafl oor.However,in the far fi eld,the wave components of the seismic wave are mainly normal modes and interface waves,with the latter being relatively strong in the waveforms.As the source frequency decreases,the normal modes become smaller and the interface waves dominate the time series of the seismic waves.

Accuracy of the staggered-grid finite-difference method of the acoustic wave equation for marine seismic reflection modeling

Seismic wave modeling is a cornerstone of geophysical data acquisition, processing, and interpretation, for which finite-difference methods are often applied. In this paper, we extend the velocity- pressure formulation of the acoustic wave equation to marine seismic modeling using the staggered-grid finite-difference method. The scheme is developed using a fourth-order spatial and a second-order temporal operator. Then, we define a stability coefficient (SC) and calculate its maximum value under the stability condition. Based on the dispersion relationship, we conduct a detailed dispersion analysis for submarine sediments in terms of the phase and group velocity over a range of angles, stability coefficients, and orders. We also compare the numerical solution with the exact solution for a P-wave line source in a homogeneous submarine model. Additionally, the numerical results determined by a Marmousi2 model with a rugged seafloor indicate that this method is sufficient for modeling complex submarine structures.

A staggered-grid high-order finite-difference modeling for elastic wave field in arbitrary tilt anisotropic media

The paper presents a staggered-grid any even-order accurate finite-difference scheme for two-dimensional (2D), three-component (3C), first-order stress-velocity elastic wave equation and its stability condition in the arbitrary tilt anisotropic media; and derives a perfectly matched absorbing layer (PML) boundary condition and its stag- gered-grid any even-order accurate difference scheme in the 2D arbitrary tilt anisotropic media. The results of nu- merical modeling indicate that the modeling precision is high, the calculation efficiency is satisfactory and the absorbing boundary condition is better. The wave-front shapes of elastic waves are complex in the anisotropic media, and the velocity of qP wave is not always faster than that of qS wave. The wave-front triplication of qS wave and its events in both reflected domain and propagated domain, which are not commonly hyperbola, is a common phenomenon. When the symmetry axis is tilted in the TI media, the phenomenon of S-wave splitting is clearly observed in the snaps of three components and synthetic seismograms, and the events of all kinds of waves are asymmetric.

Stability of High-Order Staggered-Grid Schemes for 3D Elastic Wave Equation in Heterogeneous Media

In this paper, we firstly derive the stability conditions of high-order staggered-grid schemes for the three-dimensional (3D) elastic wave equation in heterogeneous media based on the energy method. Moreover, the plane wave analysis yields a sufficient and necessary stability condition by the von Neumann criterion in homogeneous case. Numerical computations for 3D wave simulation with point source excitation are given.

A scheme to treat the singularity in global seismic wavefield simulation using pseudospectral method with staggered grids

The pseudospectral method has been applied to the simulation of seismic wave propagation in 2-D global Earth model.When a whole Earth model is considered,the center of the Earth is included in the model and then singularity arises at the center of the Earth where r=0 since the 1/r term appears in the wave equations.In this paper,we extended the global seismic wavefield simulation algorithm for regular grid mesh to staggered grid configuration and developed a scheme to solve the numerical problems associated with the above singularity for a 2-D global Earth model defined on staggered grid using pseudospectral method.This scheme uses a coordinate transformation at the center of the model,in which the field variables at the center are calculated in Cartesian coordinates from the values on the grids around the center.It allows wave propagation through the center and hence the wavefield at the center can be stably calculated.Validity and accuracy of the scheme was tested by compared with the discrete wavenumber method.This scheme could also be suitable for other numerical methods or models parameterized in cylindrical or spherical coordinates when singularity arises at the center of the model.

Comparison between staggered grid finite difference method and stochastic method in simulating strong ground motions

Strong ground motion of an earthquake is simulated by using both staggered grid finite difference method (FDM) and stochastic method, respectively. The acceleration time histories obtained from the both ways and their response spectra are compared. The result demonstrates that the former is adequate to simulate the low-frequency seismic wave; the latter is adequate to simulate the high-frequency seismic wave. Moreover, the result obtained from FDM can better reflect basin effects.

Essential consistency of pressure Poisson equation method and projection method on staggered grids

A new pressure Poisson equation method with viscous terms is established on staggered grids.The derivations show that the newly established pressure equation has the identical equation form in the projection method.The results show that the two methods have the same velocity and pressure values except slight differences in the CPU time.

A Staggered Grid Method for Solving Incompressible Flow on Unstructured Meshes

A finite volume method based unstructured grid is presented to solve the two dimensional viscous and incompressible flow.The method is based on the pressure-correction concept and solved by using a semi-staggered grid technique.The computational procedure can handle cells of arbitrary shapes,although solutions presented in this paper were only involved with triangular and quadrilateral cells.The pressure or pressure-correction value was stored on the vertex of cells.The mass conservation equation was discretized on the dual cells surrounding the vertex of primary cells,while the velocity components and other scale variables were saved on the central of primary cells.Since the semi-staggered arrangement can’t guarantee a strong coupling relationship between pressure and velocity,thus a weak coupling relationship leads to the oscillations for pressure and velocity.In order to eliminate such an oscillation,a special interpolation scheme was used to construct the pressure-correction equation.Computational results of several viscous flow problems show good agreement with the analytical or numerical results in previous literature.This semi-staggered grid method can be applied to arbitrary shape elements,while it has the most efficiency for triangular cells.

SIMULATION SEISMIC WAVE PROPAGATION IN TOPOGRAPHIC STRUCTURES USING ASYMMETRIC STAGGERED GRIDS

A new 3D finite-difference(FD) method of spatially asymmetric staggered grids was presented to simulate elastic wave propagation in topographic structures.The method approximated the first-order elastic wave equations by irregular grids finite difference operator with second-order time precise and fourth-order spatial precise.Additional introduced finite difference formula solved the asymmetric problem arisen in non-uniform staggered grid scheme.The method had no interpolation between the fine and coarse grids.All grids were computed at the same spatial iteration.Complicated geometrical structures like rough submarine interface,fault and nonplanar interfaces were treated with fine irregular grids.Theoretical analysis and numerical simulations show that this method saves considerable memory and computing time,at the same time,has satisfactory stability and accuracy.

速度-应力声波方程混合交错网格有限差分数值模拟及逆时偏移

用常规交错网格有限差分法(C‐SFD)进行速度—应力声波方程数值模拟,虽然空间差分算子能达到2M阶差分精度(M表示空间差分算子中与差分中心点等距的坐标轴网格点的组数),但差分离散声波方程仅具有2阶差分精度,所以其模拟精度低、稳定性差。为此,联合利用坐标轴网格点和非坐标轴网格点构建空间差分算子近似一阶空间偏导数,建立了一种适用于速度—应力声波方程的混合交错网格有限差分法(M‐SFD),并基于时空域频散关系和泰勒级数展开建立差分系数求解方程组,导出了差分系数通解。M‐SFD给出的差分离散声波方程可达到4、6、8阶、甚至任意偶数阶差分精度。频散分析表明:在特定的Courant条件数取值(如Courant条件数r=0.3)条件下,C‐SFD通过调整M的取值,无法将数值频散误差控制在1%以内;M‐SFD通过调整M和N的取值(N表示空间差分算子中与差分中心点等距的非坐标轴网格点的组数),基本可以将频散误差控制在1‰以内,甚至可以控制在0.1‰以内。稳定性分析显示:M取值相同时,M‐SFD的稳定性强于C‐SFD。数值模拟实例表明:计算效率基本相同时,M‐SFD比C‐SFD能更有效地压制数值频散,模拟精度更高;M‐SFD还能采用比C‐SFD更大的时间采样间隔以获得更高的计算效率,且模拟精度更高。进一步将M‐SFD推广应用于逆时偏移,M‐SFD作为逆时偏移中的波场传播算子,能够有效消除由于数值频散造成的成像假象,从而提高深层的构造成像精度和分辨率。

无储能光-电微网下的抽油机井群间抽混合整数非线性优化方法

多数低渗透油藏丛式井因供液不足而采用间抽采油方式,光伏、风电等绿电引入油田生产后在减少碳排放方面发挥了重要作用。为解决人工制定的错峰开井和间抽制度存在的井群运行能耗高等问题,以各抽油机的启停状态和电源频率,以及各光伏机组和高压电网的出力为决策变量,用直流母线实现抽油机组倒发电电能互馈,根据井群生产需求充分考虑源荷端约束,以系统运行成本最低为目标,建立一种无储能环节光-电微网下的错峰开井和间抽运行的混合整数非线性模型,用Gurobi对该问题求解。案例分析结果表明,所建模型求解结果与高压电网供电下井群常开以及人为制定间抽制度比较,井群耗电量分别降低35.687%和15.219%,系统运行成本分别降低35.471%和23.884%,并且相较于后者井群日产量提高21.620%。研究将光伏引入抽油机间抽系统,建立调度模型,大幅降低井群运行能耗和运行成本,凸显了模型的有效性。

Finite-difference modeling with variable grid-size and adaptive time-step in porous media

Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However,the finite-difference forward-modeling method is usually implemented with global spatial grid-size and time-step; it consumes large amounts of computational cost when small-scaled oil/gas-bearing structures or large velocity-contrast exist underground. To overcome this handicap,combined with variable grid-size and time-step,this paper developed a staggered-grid finite-difference scheme for elastic wave modeling in porous media. Variable finite-difference coefficients and wavefield interpolation were used to realize the transition of wave propagation between regions of different grid-size. The accuracy and efficiency of the algorithm were shown by numerical examples. The proposed method is advanced with low computational cost in elastic wave simulation for heterogeneous oil/gas reservoirs.