Lebedev网格改进差分系数TTI介质正演模拟方法研究

本文采用一种新的交错网格-Lebedev网格(LG)进行TTI介质的正演模拟研究,避免了Virieux标准交错网格(SSG)算法在处理TTI、单斜等各向异性介质时波场插值引入的数值误差,提高了模拟精度.在方法实现过程中,本文针对有限差分正演模拟面临的网格频散与边界反射两个关键性问题分别做了优化,并通过模型试算验证了它们的有效性与可行性:(1)结合最小二乘思想推导出新的频散改进差分系数(DIC),该系数比Taylor系数更能有效地压制粗网格引起的数值频散,可以节约内存,提高计算效率;(2)将分裂的多轴完全匹配层(M-PML)吸收边界条件引入到LG算法中,解决了传统PML边界条件在某些各向异性介质中的不稳定现象并且具有较好的边界吸收效果.     

横向各向同性介质地震波场数值模拟研究

地震波场数值模拟是理解地震波在地下介质中的传播特点,帮助解释观测数据的有效手段,而提高计算精度和运算效率是所有波场数值模拟方法研究所追求的目标.有限差分技术是求解波动方程计算效率最高、应用最为广泛的方法之一.但传统的有限差分技术计算过程中的数值频散问题影响了该技术的计算精度与计算效率.本文通过交错网格高阶有限差分技术与通量校正传输方法（Flux|corrected transport method,FCT）相结合, 对横向各向同性介质（Transverse isotropic medium,TI）一阶速度|应力弹性波动方程组进行了数值求解研究.波场快照数值模拟结果表明,本文研究的数值模拟方法与波动方程二阶有限差分方法、交错网格四阶有限差分方法相比,在压制网格数值频散方面有明显的优势,计算精度提高,而且可以利用较大的空间步长,提高计算效率.     

Lebedev网格改进差分系数TTI介质正演模拟方法研究

本文采用一种新的交错网格-Lebedev网格（LG）进行TTI介质的正演模拟研究,避免了Virieux标准交错网格（SSG）算法在处理TTI、单斜等各向异性介质时波场插值引入的数值误差,提高了模拟精度.在方法实现过程中,本文针对有限差分正演模拟面临的网格频散与边界反射两个关键性问题分别做了优化,并通过模型试算验证了它们的有效性与可行性：（1）结合最小二乘思想推导出新的频散改进差分系数（DIC）,该系数比Taylor系数更能有效地压制粗网格引起的数值频散,可以节约内存,提高计算效率;（2）将分裂的多轴完全匹配层（M-PML）吸收边界条件引入到LG算法中,解决了传统PML边界条件在某些各向异性介质中的不稳定现象并且具有较好的边界吸收效果.     

二维三分量TTI 介质Lebedev 网格正演模拟

鉴于三维各向异性介质(TTI、单斜等)正演模拟在计算量与内存上的巨大消耗以及标准交错网格机制波场插值带来的数值频散,本文采用二维三分量Lebedev交错网格有限差分方法对TTI介质进行波场模拟,利用二维介质便可得到3个相互垂直分量的弹性波场,并利用余弦相似度将其与完全三维正演波场进行对比,分析了该方法的模拟精度。对比测试结果表明,本文方法避免了插值误差,能够精确反映地震波在二维观测平面内的运动学特征,并且平面内质点的偏振速度、振幅能量与三维结果具有较高的相似度,而模拟占用的计算机资源却只相当于三维模拟中的一个二维剖面,是一种高效、准确的各向异性介质数值模拟方法。     

TTI介质的交错网格伪P波正演方法

研究了三维弱各向异性近似下,利用伪P波(伪纵波)模拟弹性波场P分量在倾斜对称轴的横向各向同性(TTI)介质中的传播过程,并对比了分别基于弹性Hooke定律、弹性波投影和运动学色散方程所建立的三种二阶差分伪P波方程的正演特点.目前这些伪P波方程数值计算主要采用规则网格差分,但是规则网格在TTI模拟中有低效率、低精度以及不稳定的缺点.为了提高计算的精度,本文构建出相应方程的交错网格有限差分格式.通过对比伪P波方程在三维TTI介质中不同的数值模拟的表达形式,本文认为基于色散方程所建立的伪P波方程在模拟弹性波中P波传播的过程中具有最小的噪声.本文分析不同的各向同性对称轴空间角度的频散特征,并引入适当的横波速度维持计算的稳定.二维模型算例表明,本文提出的交错网格正演算法可以得到稳定光滑的伪P波正演波场.使用本文交错网格算法对二维BP TTI模型的逆时偏移也具有较稳定的偏移结果.     

一种基于全局优化的交错网格有限差分法

在地震波场数值模拟中, 交错网格有限差分技术得到了广泛的应用, 但是在弹性模量变化较大时, 通常会因插值而导致模拟误差增大. 旋转交错网格可以很好地克服这个缺点, 因而适合于各向异性介质正演模拟. 但是对于同样大小的网格单元, 旋转交错网格需要的步长比常规交错网格要大, 这会使梯度和散度算子的误差增大因而更易产生空间数值频散. 针对这些问题, 本文提出了旋转交错网格与紧致有限差分相结合的方法, 并基于模拟退火算法进行全局优化, 压制数值频散, 拓宽波数范围. 数值模拟结果表明, 此方法可以有效地压制数值频散, 且具有较高的模拟精度.      

TTI介质弹性波FCT有限差分数值模拟

TTI(Tilted Transversely Isotropic)各向异性是对地下岩石中广泛存在的规则发育的裂缝和层理的一种有效的弹性近似,基于TTI介质的地震波数值模拟技术是分析地震波在复杂各向异性介质中的传播机理的有效工具.同时,高精度的数值模拟算法也能为后续的逆时偏移技术提供重要的技术支撑.由于TTI介质中地震波方程的弹性参数众多且变化复杂,常规有限差分技术在解决TTI介质正演模拟问题时往往会产生严重的数值频散现象,降低了数值模拟精度.通量校正传输(FluxCorrected Transport,FCT)技术能够有效地压制由空间离散产生的数值频散.本文将FCT技术用于TTI介质中弹性波方程的交错网格高阶精度差分正演,在数值模拟过程中通过对波场进行漫射和反漫射校正实现了空间网格频散的压制.模型模拟结果表明,与常规有限差分算法相比,本文算法能够有效的压制大网格条件下的数值频散,提高模拟精度.     

黏弹TTI介质中旋转交错网格高阶有限差分数值模拟

以Carcione黏弹各向异性理论为基础,给出了适用于黏弹性具有任意倾斜对称轴横向各向同性介质(黏弹TTI介质)的二维三分量一阶速度-应力方程,采用旋转交错网格任意偶数阶精度有限差分格式求解该方程,并推导出了二维黏弹TTI介质完全匹配层(PML)吸收边界条件公式和相应的旋转交错网格任意偶数阶精度有限差分格式,实现了该类介质的地震波场数值模拟.数值模拟结果表明:该方法模拟精度高,边界吸收效果好,可以得到高精度的波场快照和合成记录;并且波场快照和合成记录能较好地反映地下介质的各向异性特征和黏弹性特征.     

波场模拟中的数值频散分析与校正策略

波动方程有限差分法正演模拟,对认识地震波传播规律、进行地震属性研究、地震资料地质解释、储层评价等,均具有重要的理论和实际意义.但有限差分法本身固有存在着数值频散问题,数值频散在正演模拟中是一种严重的干扰,会降低波场模拟的精度与分辨率.针对TI介质波场模拟的交错网格有限差分方法,本文从空间网格离散、时间网格离散和算子近似等三个方面对其产生的数值频散进行了分析,并结合其他学者的研究成果给出了TI介质波场模拟中压制数值频散的方法与策略：在已知介质频散关系时,对差分算子可实施算子校正;通过提高差分方程的阶数来提高波场模拟精度;采用流体力学中守恒式方程的通量校正传输方法来压制波场模拟中的数值频散;在实际正演模拟时,采用交错网格高阶有限差分方程,不仅在空间上采用高阶差分,而且在时间上也要采用高阶差分,否则只在单一方向上（空间或时间）提高方程的阶数对压制数值频散也不会取得理想的效果.     

一种全局优化的隐式交错网格有限差分算法及其在弹性波数值模拟中的应用

在数值模拟中,隐式有限差分具有较高的精度和稳定性.然而,传统隐式有限差分算法大多由于需要求解大型矩阵方程而存在计算效率偏低的局限性.本文针对一阶速度-应力弹性波方程,构建了一种优化隐式交错网格有限差分格式,然后将改进格式由时间-空间域转换为时间-波数域,利用二范数原理建立目标函数,再利用模拟退火法求取优化系数.通过对均匀模型以及复杂介质模型进行一阶速度-应力弹性波方程数值模拟所得单炮记录、波场快照分析表明:这种优化隐式交错网格差分算法与传统的几种显式和隐式交错网格有限差分算法相比不但降低了计算量,而且能有效的压制网格频散,使弹性波数值模拟的精度得到有效的提高.     

Numerical simulation of seismic wave in elastic and viscoelastic TTI media*

Based on the two-dimensional (2D) three-component first-order velocity-stress equation, the high order staggered mesh finite difference numerical simulation method was used to simulate the elastic and viscoelastic tilted transversely isotropic (TTI) media. The perfect matched layer (PML) absorption boundary condition was selected to eliminate the boundary effect. The results show that: ① Under the condition of fixed elastic parameters of elastic TTI medium, when the polarization angle and azimuth are 60° and 45° respectively, the degree of shear wave splitting is significantly greater than the angle of 0°; ② The influence of viscoelasticity on TTI medium is mainly reflected in the amplitude. If the quality factor decreases, the attenuation of the seismic wave amplitude increases, causing the waveform to become wider and distorted. If the quality factor increases, the viscoelastic medium becomes closer to elastic medium; ③ For TTI medium with different polarization angle and azimuth angle in the upper and lower layers, the shear wave can multiple splits at the interface of medium. The symmetry of seismograms is affected by the polarization angle and azimuth angle of TTI medium; ④ Viscoelasticity has a great influence on reflected wave, transmitted wave and converted wave in the low-velocity model. When the viscoelasticity is strong, the weaker waves may not be shown.     

线性粘弹介质中地震波场数值模拟

本文将以往两种粘弹介质中地震波模拟方法的优点结合起来，以模型理论和积分本构方程为基础，从理论上分析了模型对地震波场的影响；采用交错网格有限差分法对粘弹介质中的地震波进行数值模拟.数值计算结果表明该方法不仅便于计算，同时也便于从力学的角度来分析地震波的传播.数值计算结果与理论分析一致，说明这种方法可以更为有效地模拟粘弹介质中地震波的传播.     

Weak‐anisotropy approximation for P‐wave reflection coefficient at the boundary between two tilted transversely isotropic media

Existing and commonly used in industry nowadays, closed‐form approximations for a P‐wave reflection coefficient in transversely isotropic media are restricted to cases of a vertical and a horizontal transverse isotropy. However, field observations confirm the widespread presence of rock beds and fracture sets tilted with respect to a reflection boundary. These situations can be described by means of the transverse isotropy with an arbitrary orientation of the symmetry axis, known as tilted transversely isotropic media. In order to study the influence of the anisotropy parameters and the orientation of the symmetry axis on P‐wave reflection amplitudes, a linearised 3D P‐wave reflection coefficient at a planar weak‐contrast interface separating two weakly anisotropic tilted tranversely isotropic half‐spaces is derived. The approximation is a function of the incidence phase angle, the anisotropy parameters, and symmetry axes tilt and azimuth angles in both media above and below the interface. The expression takes the form of the well‐known amplitude‐versus‐offset “Shuey‐type” equation and confirms that the influence of the tilt and the azimuth of the symmetry axis on the P‐wave reflection coefficient even for a weakly anisotropic medium is strong and cannot be neglected. There are no assumptions made on the symmetry‐axis orientation angles in both half‐spaces above and below the interface. The proposed approximation can be used for inversion for the model parameters, including the orientation of the symmetry axes. Obtained amplitude‐versus‐offset attributes converge to well‐known approximations for vertical and horizontal transverse isotropic media derived by Rüger in corresponding limits. Comparison with numerical solution demonstrates good accuracy.     

不分裂卷积完全匹配层与旋转交错网格有限差分在孔隙弹性介质模拟中的应用

完全匹配层吸收边界在地震波模拟中已广泛使用,但常用的场分裂格式完全匹配层吸收边界(SPML)和传统的不分裂完全匹配层吸收边界(NPML)对极低频入射波或大角度入射波的边界吸收效果不好.一种无需分裂和显式卷积计算的完全匹配层吸收边界(CPML)不仅能够解决常规PML吸收边界的不足,而且具有存储量小、计算效率高、易于编程实现的特点.本文将这种完全匹配层(CPML)吸收边界引入到孔隙弹性介质速度-应力格式的旋转交错网格有限差分算法中,对完全匹配层吸收边界参数进行数值分析,得到一组优化的参数.孔隙弹性介质数值模拟结果表明这种不分裂卷积完全匹配层的吸收效果优于常规完全匹配层.     

Image-domain wavefield tomography for tilted transversely isotropic media

Transversely isotropic models with a tilted symmetry axis have become standard for imaging beneath dipping shale formations and in active tectonic areas. Here, we develop a methodology of wave-equation-based image-domain tomography for acoustic tilted transversely isotropic media. We obtain the gradients of the objective function using an integral wave-equation operator based on a separable dispersion relation that takes the symmetry-axis tilt into account. In contrast to the more conventional differential solutions, the integral operator produces only the P-wavefield without shear-wave artefacts, which facilitates both imaging and velocity analysis. The model is parameterized by the P-wave zero-dip normal-moveout velocity, the Thomsen parameter δ, anellipticity coefficient η and the symmetry-axis tilt θ. Assuming that the symmetry axis is orthogonal to reflectors, we study the influence of parameter errors on energy focusing in extended (space-lag) common-image gathers. Distortions in the anellipticity coefficient η introduce weak linear defocusing regardless of reflector dip, whereas δ influences both the energy focusing and depth scale of the migrated section. These results, which are consistent with the properties of the P-wave time-domain reflection moveout in tilted transversely isotropic media, provide important insights for implementation of velocity model-building in the image-domain. Then the algorithm is tested on a modified anticline section of the BP 2007 benchmark model.     

TTI介质隐式有限差分平面波偏移

文章研究了TTI介质隐式有限差分(IFD)波场外推算子和TTI介质平面波偏移.与各向同性和VTI介质相比,TTI介质频散关系要复杂的多,很难得到频散关系的显性表达,也不能再只用对称的偶函数表示,需要加入一个奇函数.文中设计TTI介质IFD波场外推算子并用非线性优化方法求解算子系数.将各向同性介质下平面波偏移理论与TTl...     

THE APPLICATION OF PML BOUNDARY CODITIONS IN THE SIMULATION OF SEISMIC WAVE BY THE HIGH-ORDER STAGGERED-GRID FINITE DIFFERENCES AT THE TRANSVERSELY ISOTROPIC MEDIA

In the realm of the numerical simulation, finite difference method and finite element method are more intuitive and effective than other simulation methods. In the process of simulating seismic wave propagation, the finite differences method is widely used because of its high computational efficiency and the advantage of the algorithm is more efficient. With the demand of precision, more and more researchers have proposed more effective methods of finite differences, such as the high-order staggered-grid finite differences method, which can restore the actual process of wave propagation on the premise of ensuring accuracy and improving the efficiency of operation. In the past numerical simulation of seismic wave field, different models of isotropic medium are mostly used, but it is difficult to reflect the true layer situation. With the research demand of natural seismology and seismic exploration, the research on anisotropic media is more and more extensive. Transversely isotropic(TI)media can well simulate the seismic wave propagation in the formation medium, such as gas-bearing sandstone, mudstone, shale et al., the character of TI media is reflected by introducing the Thomsen parameters to reflect its weak anisotropy of vertical direction by using Thomson parameter. Therefore, studying the process of seismic wave propagation in TI media can restore the true information of the formation to the greatest extent, and provide a more reliable simulation basis for the numerical simulation of seismic wave propagation. In the geodynamic simulation and the numerical simulation of the seismic wave field, under the limited influence of the calculation area, if no boundary conditions are added, a strong artificial boundary reflection will be generated, which greatly reduces the validity of the simulation. In order to minimize the influence of model boundaries on the reflection of seismic waves, it is often necessary to introduce absorbing boundary conditions. At present, there are three types of absorption boundary conditions: one-way wave absorption boundary, attenuation absorption boundary, and perfectly matched layer(PML)absorption boundary. In terms of numerical simulation of seismic waves, the boundary absorption effect of PML is stronger than the first two, which is currently the most commonly used method, and it also represents the cutting-edge development direction of absorption boundary technology. The perfectly matched layer absorbing boundary is effectively applied to eliminating the reflective waves from model boundaries, but for transversely isotropic medium, the effect of the absorbing is not very well. For this reason, the elastic dynamic wave equations in transversely isotropic media are derived, and we describe a second-order accurate time, tenth-order accurate space, formulation of the Madariaga-Virieux staggered-grid finite difference methods with the perfectly matched layer(PML)are given. In addition, we have established vertical transversely isotropic(VTI)media and arbitrary inclined tilted transversely isotropic(TTI)media models, using a uniform half-space velocity model and a two-layer velocity model, respectively. By combining the actual geoscience background, we set the corresponding parameters and simulation conditions in order to make our model more research-oriented. When setting model parameters, different PML thickness, incident angle, source frequency and velocity layer models were transformed to verify the inhibition of boundary reflection effect by PML absorption boundary layer. The implementations of this simulation show that the formula is correct and for the transversely isotropic(TI)media of any angular symmetry axis, when the thickness of the PML layer reaches a certain value, the seismic wave reflection effect generated by the artificial boundary can be well suppressed, and the absorption effect of PML is not subject to changes in incident angle and wave frequency. Therefore, the results of our study indicate that our research method can be used to simulate the propagation process of seismic waves in the transversely isotropic(TI)media without being affected by the reflected waves at the model boundary to restore the actual formation information and more valuable geological research.     

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.