3D Fourier finite-difference migration by alternating-direction-implicit plus interpolation

Conventional two‐way splitting Fourier finite‐difference migration for 3D complex media yields azimuthal anisotropy where an additional phase correction is needed with much increase of computational cost. We incorporate the alternating‐direction‐implicit plus interpolation scheme into the conventional Fourier finite‐difference method to reduce azimuthal anisotropy. This scheme retains the high‐order remnants ignored by the two‐way splitting in the form of a wavefield interpolation in the wavenumber domain. The wavefield interpolation for each step of downward extrapolation is implemented between the wavefields before and after the conventional Fourier finite‐difference extrapolation. As the Fourier finite‐difference migration is implemented in the space and wavenumber dual space, the Fourier transforms between space and wavenumber domain that were needed for the alternating‐direction‐implicit plus interpolation in frequency domain (FD) migration are saved in Fourier finite‐difference migration. Since the azimuth anisotropy in Fourier finite‐difference is much less than that in FD, the application of the alternating‐direction‐implicit plus interpolation scheme in Fourier finite‐difference migration is superior to that in FD migration in handling complex media with large velocity contrasts and steep dips. Impulse responses show that the presented method reduces the azimuthal anisotropy at almost no extra cost.     

2.5D finite-difference solution of the acoustic wave equation

The finite‐difference method applied to the full 3D wave equation is a rather time‐consuming process. However, in the 2.5D case, we can take advantage of the medium symmetry. By taking the Fourier transform with respect to the out‐of‐plane direction (the symmetry axis), the 3D problem can be reduced to a repeated 2D problem. The third dimension is taken into account by a sum over the corresponding wave‐vector component. A criterion for where to end this theoretically infinite sum derives from the stability conditions of the finite‐difference schemes employed. In this way, the computation time of the finite‐difference calculations can be considerably reduced. The quality of the modelling results obtained with this 2.5D finite‐difference scheme is comparable to that obtained using a standard 3D finite‐difference scheme.     

一种优化的频率域三维声波有限差分模拟方法

为提高频率域有限差分（FD,finite-difference）正演模拟技术的计算精度和效率,基于旋转坐标系统的优化差分格式被广泛应用,但是只应用于正方形网格的情况.基于平均导数法（ADM）的优化差分格式,应用于正方形和长方形网格模拟.这些频率域有限差分算子,各自具有不同的差分格式和对应的优化系数求解表达式.本文基于三维声波方程发展了一种新的优化方法,只要给定FD模板形式,可直接构造频散方程,求取FD模板上各节点的优化系数.此方法的优点在于频率域FD算子的优化系数对应各个节点,可扩展优化其他格式.运用此优化方法,计算得到了不同空间采样间距比情况下27点和7点格式的优化系数.数值实验表明,优化27点格式与ADM 27点格式具有相同的精度,优化7点格式比经典的7点格式具有更小的数值频散.

     

Effective filtering of artifacts for implicit finite-difference paraxial wave equation migration1

Implicit finite-difference implementations of the paraxial wave equation are widely used in industrial prestack and post-stack migration programs for imaging and velocity analysis. This type of implementation gives rise to numerical artifacts which, in general, do not degrade image quality but which do impede effective velocity analysis. This paper reviews the artifacts generated by the paraxial approximation and a post-extrapolation, spatially varying filtering scheme is described which completely eliminates these artifacts. The method is illustrated with numerous examples.     

三维波动方程时空域混合网格有限差分数值模拟方法

常规高阶和时空域高阶有限差分方法广泛应用于三维标量波动方程的数值模拟,这两种差分方法仅利用笛卡尔坐标系中的坐标轴网格点构建三维Laplace差分算子,相应的差分离散波动方程本质上仅具有2阶差分精度,模拟精度低.本文将三维笛卡尔坐标系中非坐标轴网格点分为两类:坐标平面内的非坐标轴网格点和坐标平面外的非坐标轴网格点,系统推导出了两类非坐标轴网格点构建三维Laplace差分算子的方法,进而提出了一种利用坐标轴网格点和非坐标轴网格点共同构建三维Laplace差分算子的混合网格有限差分方法,并利用时空域频散关系和泰勒展开建立差分系数方程,推导出了差分系数的通解.相比常规高阶和时空域高阶差分格式的2阶差分精度,时空域混合网格差分离散波动方程理论上能够达到任意偶数阶差分精度,模拟精度显著提高,同时稳定性更强.频散分析表明:相比常规高阶和时空域高阶差分格式,在计算效率基本相同时,时空域混合网格差分格式能更有效地减小数值频散,减弱数值各向异性,模拟精度更高;在模拟精度基本相当时,混合网格差分格式能采用更大的时间采样间隔,计算效率更高.数值模拟实例进一步验证了混合网格差分格式在提高模拟精度和计算效率方面的...     

Pure quasi-P wave equation and numerical solution in 3D TTI media

Based on the pure quasi-P wave equation in transverse isotropic media with a vertical symmetry axis (VTI media), a quasi-P wave equation is obtained in transverse isotropic media with a tilted symmetry axis (TTI media). This is achieved using projection transformation, which rotates the direction vector in the coordinate system of observation toward the direction vector for the coordinate system in which the z-component is parallel to the symmetry axis of the TTI media. The equation has a simple form, is easily calculated, is not influenced by the pseudo-shear wave, and can be calculated reliably when δ is greater than ε. The finite difference method is used to solve the equation. In addition, a perfectly matched layer (PML) absorbing boundary condition is obtained for the equation. Theoretical analysis and numerical simulation results with forward modeling prove that the equation can accurately simulate a quasi-P wave in TTI medium.     

Numerical modeling of wave equation by a truncated high-order finite-difference method

Finite-difference methods with high-order accuracy have been utilized to improve the precision of numerical solution for partial differential equations.However, the computation cost generally increases linearly with increased order of accuracy.Upon examination of the finite-difference formulas for the first-order and second-order derivatives, and the staggered finite-difference formulas for the first-order derivative, we examine the variation of finite-difference coefficients with accuracy order and note th...     

2D finite-difference viscoelastic wave modelling including surface topography

We have pursued two-dimensional (2D) finite-difference (FD) modelling of seismic scattering from free-surface topography. Exact free-surface boundary conditions for the particle velocities have been derived for arbitrary 2D topographies. The boundary conditions are combined with a velocity–stress formulation of the full viscoelastic wave equations. A curved grid represents the physical medium and its upper boundary represents the free-surface topography. The wave equations are numerically discretized by an eighth-order FD method on a staggered grid in space, and a leap-frog technique and the Crank–Nicholson method in time.
In order to demonstrate the capabilities of the surface topography modelling technique, we simulate incident point sources with a sinusoidal topography in seismic media of increasing complexities. We present results using parameters typical of exploration surveys with topography and heterogeneous media. Topography on homogeneous media is shown to generate significant scattering. We show additional effects of layering in the medium, with and without randomization, using a von Kármán realization of apparent anisotropy. Synthetic snapshots and seismograms indicate that prominent surface topography can cause back-scattering, wave conversions and complex wave patterns which are usually discussed in terms of inter-crust heterogeneities.     

2D finite-difference elastic wave modelling including surface topography1

A 2D numerical finite-difference algorithm accounting for surface topography is presented. Higher-order, dispersion-bounded, cost-optimized finite-difference operators are used in the interior of the numerical grid, while non-reflecting absorbing boundary conditions are used along the edges. Transformation from a curved to a rectangular grid achieves the modelling of the surface topography. We use free-surface boundary conditions along the surface. In order to obtain complete modelling of the effects of wave propagation, it is important to account for the surface topography, otherwise near-surface effects, such as scattering, are not modelled adequately. Even if other properties of the medium, for instance randomization, can improve numerical simulations, inclusion of the surface topography makes them more realistic.     

旋转网格和常规网格混合的时空域声波有限差分正演

压制数值频散,提高正演模拟精度,一直是有限差分正演模拟研究的重要内容.基于时空域频散关系的有限差分法,比基于空间域频散关系的传统有限差分法,模拟精度更高.时空域声波方程数值模拟,普遍采用常规十字交叉型高阶有限差分格式.而在频率-空间域,普遍采用旋转网格和常规网格混合的有限差分格式,有效提高了模拟精度和计算效率.本文将频率-空间域混合网格有限差分的思想引入到时空域,提出了时空域混合网格2M+N型声波方程有限差分方法.首先推导出基于时空域频散关系的混合网格差分系数计算方法,然后进行频散分析、稳定性分析,并和传统高阶、时空域高阶有限差分法对比,结果表明：计算量相同时,新方法能有效压制数值频散,显著提高模拟精度;新方法相比传统2M阶有限差分法,稳定性增强,与时空域2M阶有限差分法稳定性基本相当.最后利用新方法进行均匀介质、层状介质、盐丘模型的数值模拟和盐丘模型的逆时偏移,模拟效果和成像质量进一步证实了该方法的有效性和普遍适用性.     

基于波动方程三维表面多次波预测方法研究

与传统的二维表面多次波预测算法相比,基于波动方程的全三维表面多次波预测方法无需对地下介质做简单近似,其更符合地震波在地下介质中传播的真实状况,是地震资料处理中解决多次波预测问题的强有力工具.本文从三维多次波预测的基本理论出发,给出了全三维多次波预测算法的预测矩阵表示、计算方法以及实现条件,采用GPU(图形处理器)加速全三维表面多次波预测,较传统的CPU串行计算,GPU并行预测表面多次波的计算效率约提高165倍.文中分别利用二维和三维表面多次波预测算法对理论模拟的含表面多次波的三维地震数据进行多次波预测计算,对比分析结果表明,相比于二维算法,文中所述的基于波动方程的全三维表面多次波预测效果明显改善,其计算精度更高,辅以合理有效的自适应相减算法,可获得高精度的地震勘探资料表面多次波压制数据.     

三维偏移距平面波有限差分叠前时间偏移

本文提出了中点-半偏移距域内的三维偏移距平面波(offset plane-wave)方程,并给出了其有限差分解法.偏移距平面波可通过对CMP道集进行平面波分解(倾斜叠加或线性Radon变换)生成,然而这样做会产生严重的噪音干扰.本文提出了局部倾斜叠加方法(local slant-stacking)来消除离散线性Radon变换引入的噪音.针对实际三维数据的不规则性(中点-偏移距域内方位角展布不均匀及偏移距采样不规则),本文还提出了与方位角无关的三维倾斜叠加方法(azimuth-independent 3D slant-stacking),解决了三维平面波分解中存在的问题.使用文中提出的平面波分解方法,可以得到高信噪比的偏移距平面波数据体.同时,三维偏移距平面波偏移可以输出偏移距射线参数域共成像点道集,基于此道集的剩余速度分析方法可以用来更新偏移速度场.偏移距平面波偏移具有很高的计算效率,相较Kirchhoff积分叠前时间偏移有较好的保幅特性,可作为水平地表三维叠前时间偏移的一个很好的解决方案.     

拟空间域弹性波方程交错网格有限差分逆时偏移

弹性波逆时偏移是一种高精度的复杂构造地震成像方法.然而,在传统的基于矩形网格离散化的逆时偏移中,介质界面通常会产生畸变.另外,因使用双程波动方程进行波场延拓,其产生的反射波会在成像过程中产生偏移假象.为解决这些问题,本文提出了一种拟空间域弹性波方程高阶交错网格有限差分格式,并给出了差分格式的稳定性条件,进而实现了高精度的拟空间域弹性波方程有限差分逆时偏移.模型实验表明,若在计算拟空间域采样间隔时引入速度界面信息,则拟空间域弹性波方程高阶交错网格有限差分逆时偏移能够避免常规弹性波方程逆时偏移中弯曲界面形态畸变问题;此外基于该方法进行波场延拓时可有效压制弯曲界面的假散射现象,并能有效压制层间反射波,因此可以减少剖面上的偏移假象,从而显著提高成像的质量.

     

Fast and accurate calculation of seismic wave travel time in 3D TTI media

The Tilted tilted transversely isotropic(TTI)media,a kind of anisotropic medium,widely exists within the earth.For faster calculation of travel times in the TTI...     

用共方位角偏移实现叠前二维资料三维化成像

波动方程偏移保持了波场动力学特征,依此本文应用共方位角叠前深度偏移技术来实现二维资料三维化处理.本方法能使共反射点偏移成像与叠前数据插值一步完成,较好地解决二维资料处理中的成像点不能准确归位、侧面反射波难处理、主测线与联络线不闭合等问题.     

基于精细积分法的三维弹性波数值模拟(英文)

波动方程有限差分法是地震数值模拟中的一种重要的方法,对理解和分析地震传播规律、分析地震属性和解释地震资料有着非常重要的意义。但是有限差分法由于其离散化的思想,产生了不稳定性。精细积分法在有限差分法的基础上,在时间域采用解析解的表达形式,在空间域保留任意差分格式,发展成为半解析的数值方法。本文结合并发展了以往学者的成果,推导了任意精细积分法的三维弹性波正演模拟计算公式,并对其稳定性进行了数值分析。在计算实例中,实现了精细积分法二维和三维弹性波模型的地震正演模拟,对计算结果的分析表明,精细积分法反射信号走时准确,稳定性好,弹性波场相较于声波波场,弹性波波场成分更为丰富,包含了更多波型成分(PP-和PS-反射波、透射波和绕射波),这对实际地震资料的解释和储层分析有重要的意义。实践证明,该方法可直接应用到弹性波的地质模型的数值模拟中。     

时间-空间高阶精度矩形交错网格隐式有限差分声波正演模拟

有限差分方法因其操作简单、计算消耗低而成为地震勘探领域中最为常用的数值模拟方法之一, 然而用离散的显式差分算子数值逼近地震波动方程中的连续导数容易导致数值频散, 并且基于正方形网格离散形式的有限差分方法对不同地质模型的适应性较低.针对一阶变密度声波方程的数值模拟, 本文发展了一种适用于矩形网格离散形式的时间高阶空间隐式有限差分格式, 可以有效压制时间和空间频散, 同时灵活的网格剖分增强了其应用的广泛性.基于本文矩形交错网格时间高阶空间隐式有限差分格式的时空域频散关系和变量替换的思想, 首先采用泰勒级数展开方法求解不同方向的非轴上时间差分系数及轴上空间差分系数, 使本文差分格式可以获得任意偶数阶时间和空间精度.为了进一步提高本文差分格式在更大波数区域的空间模拟精度, 我们采用线性优化方法来求取新的轴上空间差分系数用于一阶变密度声波方程的波场迭代求解中.频散、稳定性分析及数值模拟算例表明: 相比于传统十字形空间域隐式有限差分格式, 本文矩形交错网格时间高阶空间隐式有限差分格式在精度、稳定性和效率方面均具有优势.

     

3D finite-difference modeling algorithm and anomaly features of ZTEM

The Z-Axis tipper electromagnetic (ZTEM) technique is based on a frequency-domain airborne electromagnetic system that measures the natural magnetic field. A survey area was divided into several blocks by using the Maxwell’s equations, and the magnetic components at the center of each edge of the grid cell are evaluated by applying the staggered-grid finite-difference method. The tipper and its divergence are derived to complete the 3D ZTEM forward modeling algorithm. A synthetic model is then used to compare the responses with those of 2D finite-element forward modeling to verify the accuracy of the algorithm. ZTEM offers high horizontal resolution to both simple and complex distributions of conductivity. This work is the theoretical foundation for the interpretation of ZTEM data and the study of 3D ZTEM inversion.     

常Q衰减介质分数阶波动方程优化有限差分模拟

本文基于Kjartansson常Q模型理论,推导了常Q衰减介质中黏声波和黏弹性波的速度-应力方程,并采用基于二项式窗函数的优化交错网格有限差分方法进行了数值模拟,同时引入不分裂的复频移卷积完全匹配层(CPML)吸收边界条件,以消除边界反射.使用基于自适应时间步长记忆方法的中心差分近似时间分数阶导数,与常用的短时记忆方法相比,提高了波动方程的离散化精度和计算效率.通过对比均匀模型下声波的数值解与解析解,验证了算法的精确性,并进一步分析了不同品质因子下地震波的频散及衰减特征.对BP盐丘模型的数值模拟结果可以较好地反映本文数值方法对复杂介质的适应性及频散压制效果.