退化的Fourier偏移算子及其在复杂断块成像中的应用

波动方程宽角抛物逼近得到的通常是非常系数的单程波传播算子，其系数是速度横向变化的函数，因此需要利用有限差分(FD)进行数值实施. 通过对Lippmann Schwinger单程波动积分方程的退化核逼近,本文研究了一类宽角退化算子的偏移成像. 这种退化偏移算子只用快速Fourier变换进行波场延拓，将常规的Fourier分裂步地震偏移方法(SSF)推广适应强速度横向变化介质和大角度传播波场. 退化的Fourier偏移算子通过在两个分裂步项之间作波数域线性插值来实现波场延拓，每延拓一层需要比常规的SSF地震偏移方法多一次快速Fourier变换(FFT). 通过SEG/EAGE盐丘模型和实际地震资料的应用表明，退化Fourier偏移算子能很好地对盐下的陡倾角断层和实际地震剖面上的复杂小断块和大断裂地质构造成像.     

地震逆散射波场和算子的谱分解

本文对地震逆散射的研究，旨在于为抑制层间多次波和地震波场多重散射对一次反射干扰效应提供理论依据.这对薄互层地层滤波的高频恢复、保幅弹性反演、衍射地震勘探及海洋地震勘探中的干扰消除皆具重要意义.本文基于上下行波分解及弹性波互易定理，导出横向变速介质条件下线性预测算子的表达式和反射数据的广义谱分解方程. 文中先由上覆地层广义反射透射矩阵的元素定义线性预测算子，并将其表示成一系列单程波算子的线性组合,之后将横向变速介质条件下线性预测方程表达为反射数据与线性预测算子及其逆的乘积. 对该方程的求解可获得上覆地层的线性预测算子，从而可借以求出相应的反射透射算子. 本文先将水平层状介质条件下垂直入射的一维线性预测方程推广到斜入射的情况，以此为参照，导出横向非均匀介质条件下反射数据的地震逆散射广义谱分解方程.文中也揭示了单程波地震逆散射算子、反射透射算子的性态.本文还针对水平层状介质条件，给出斜入射的数值结果.     

三维非均匀介质中真振幅地震偏移算子研究

利用三维非均匀介质中的波动方程,进行振幅保真波场偏移算子分解,得到用于真振幅偏移的单程波方程. 经过数学推理,导出裂步Fourier法真振幅偏移和Fourier有限差分法真振幅偏移的算子方程,并给出其具体的实现过程.     

地震偏移的最优可分近似算法实现

针对地震偏移算法中单程波算子的特征函数近似，采用最优可分表示法将该特征函数展开为空间变量(g)和水平波数(k)的可分表达式，以此可分近似表达式为基础，运用正反Fourier变换重新构造单程波算子. 为了克服特征函数最优可分近似计算在奇点及其领域产生的数值振荡，引入等价黏性技巧以增强算法的数值计算稳定性，并采用分频最优可分及对空间变量(g)线性插值的方法，不仅提高了计算精度，也节约计算机时. 文中具体研究了单程波算子的脉冲响应和二维叠前深度偏移. 结果表明，在不甚大步长情况下，本文构造的算子具有适应横向强变速的能力，运用Marmousi模型验证了本文方法适合复杂构造的成像.     

界面起伏条件下反射／透射算子+单程波方程的地震波模拟方法

本文导出了一种由单程波方程利用反射／透射算子的可分表示方法模拟复杂介质中一次反射地震波的数值算法. 文中利用算子可分表示理论将反射／透射算子分解成适合于双域（空间域和波数域）运算的表达形式，使得本文得到的地震波数值模拟算法可适应于一定程度横向非均匀介质和界面起伏情况，在入射角小于45°时能够准确模拟振幅随入射角(AVA)的关系. 就模拟一次反射地震波而言，与前人研究的双程波动方程伪谱法地震波模拟相比，本文算法具有足够高的模拟精度，且计算效率成倍地提高.     

粘弹性介质单程波法非零偏移距地震数值模拟与偏移

地震资料分辨率降低,得不到深层介质的精确信息实际上是由于大地吸收效应的影响.同时与双程波动方程相比单程波动方程避免了多次波的干扰并且计算效率高、占用内存少.本文首先基于开尔芬粘弹性介质模型将品质因子与单程波分步傅立叶法波场延拓算子相结合,实现了粘弹性介质波场延拓,从而将单程波弹性介质波场延拓推广到了粘弹性介质.然后在定位原理,数学检波器原理以及等时叠加原理的基础之上实现了粘弹性介质非零偏移距叠前正演模拟.最后将数值模拟得到的正演记录进行弹性偏移和粘弹性偏移并进行对比分析.通过数值算例可以看出,粘弹性介质叠前正演深层的反射波能量减弱,同相轴变粗,频带变窄,主频减小,分辨率降低;粘弹性偏移不但实现了振幅的恢复,而且同时偏移剖面的垂向空间分辨率也得到了提高.     

混合域单程波传播算子及其在偏移成像中的应用

以地震波的单程波传播方程为基础,利用算子近似展开的方法推导出了当前波动方程叠前深度偏移方法研究中广泛使用的裂步Fourier、Fourier有限差分和广义屏传播算子的一般形式及其近似式．讨论了它们间差异、相互关系以及他们的特点,最后给出了基于裂步Fourier、Fourier有限差分和广义屏传播算子的偏移成像方法时Marmousi模型的偏移成像结果,以说明它们间的优劣与计算效率．     

复杂介质中平面波模拟的单程波方法

地震勘探中平面波分解技术在地震资料处理与反演领域均得以广泛应用,通过正演算法模拟平面波记录有助于理解和检验平面波分解的效果和精度.本文给出一种用以模拟复杂介质中平面波的数值算法,考虑了振幅随入射角度的变化.与前人研究的平面波模拟算法相比,该算法基于波动方程的单程波解法,不但适用于横向非均匀介质,在地层倾角水平条件下还能够准确模拟振幅随入射角关系.通过Marmousi模型的计算实例,证明了本文方法的可行性.     

波动方程数值模拟的三种方法及对比

波动方程数值模拟方法是研究地震波场传播的一种重要手段,本文采用交错网格高阶有限差分方法分别对双程声波方程和双程弹性波方程进行了波场数值模拟,并且根据定位原理采用傅立叶有限差分算子进行了单程波方程数值模拟,在分析定位原理的基础上,对其计算过程稍作修改,将延拓到地面的波场直接由每个检波点接收,无需横向叠加过程,得到了单程声波方程共炮记录.基于不同波动方程的数值模拟结果表明,双程波方程结果包含直达波、多次波等干扰波,信噪比低;单程波数值模拟结果只包含了介质分界面的一次反射波,信噪比高,但对于大角度入射波误差较大,并且对于同一个地质模型而言,双程弹性波方程计算速度最慢,双程声波方程次之,单程声波方程计算速度最快.因此对于复杂地质模型,三种模拟方法可以取长补短,综合应用.     

三维复杂构造中地震波模拟的单程波方法

复杂构造中单程波与双程波方法模拟结果的比较表明,就地震勘探中主要关心的一次反射波而言,单程波算法已具有足够的精度. 使用单程波方程将极大地减少数值计算的计算量,同时对介质的几何和物理参数建模也降低了要求. 单程波算法可视为深度偏移的“逆运算”,这样可以很好地借用已知的深度偏移方法及其程序系统. 基于计算效率和计算精度的双重考虑,本文在介质速度结构较复杂时采用显式短算子波场延拓方法,而在介质速度结构相对简单时采用分裂步相移法. 反射系数的计算中考虑了其随入射角的变化.     

正交各向异性介质中qP波入射的二阶近似反射系数与透射系数

地震波在各向异性介质中以一个准P波(qP)和两个准S波(qS1和qS2)的形式传播.研究三种波的相速度、群速度以及偏振方向等传播性质能够为各向异性介质中的正反演问题提供有效支撑.具有比横向各向同性(TI)介质更一般对称性的正交各向异性介质通常需要9个独立参数对其进行描述,这使得对传播特征的计算更为复杂.当两个准S波速度相近时具有耦合性,从而令慢度的计算产生奇异性.因此,奇异点(慢度面的鞍点和交叉点)附近的反射与透射(R/T)系数的求解不稳定,会导致波场振幅不准确.本文首次通过结合耦合S波射线理论和基于迭代的各向异性相速度与偏振矢量的高阶近似解,得到了适用于正交各向异性介质以qP波入射所产生的二阶R/T系数的计算方法.与基于一阶近似的结果相比,基于二阶近似的方法提高了qP波R/T系数的精度,能得到一阶耦合近似无法表达的准确的qP-qS转换波的R/T系数解,且方法适用于较强的各向异性介质.     

基于照明补偿的单程波最小二乘偏移

最小二乘偏移是一种基于反射地震数据与地下反射率间线性关系而建立起来的地震数据线性反演方法,相比常规偏移成像具有更好的保幅性能.本文提出了一种基于照明补偿的单程波最小二乘偏移方法,首先利用单程波方程的稳定Born近似广义屏波场传播算子构建反射地震数据与地下反射率间的线性算子,然后再应用线性最优化方法求解最小二乘偏移所对应的线性反问题.在迭代求解最优化问题的过程中,以地震波场的地下照明强度作为迭代反演的预条件算子加快迭代的收敛速度.单程波传播过程中考虑了速度分界面产生的透射效应,并用单极震源代替常规偏移中的偶极震源.把本文提出的方法应用于层状理论模型和Marmosi模型地震数据的数值试验中均取得了理想的结果.     

Synthetic seismograms in heterogeneous media by one-return approximation

When reverberations between heterogeneities or resonance scattering can be neglected but accumulated effects of forward scattering are strong, the Born approximation is not valid but the De Wolf approximation can be applied in such cases. In this paper, renormalized MFSB (multiple-forescattering single-backscattering) equations and the dual-domain expression for scalar, acoustic and elastic waves are derived by a unified approach. Two versions of the one-return method (using MFSB approximation) are given: One is the wide-angle dual-domain formulation (thin-slab approximation); the other is the screen approximation. In the screen approximation, which involves a small-angle approximation for the wave-medium interaction, it can be seen clearly that the forward scattered, or transmitted waves are mainly controlled by velocity perturbations; while the backscattered or reflected waves, by impedance perturbations. The validity of the method and the wide-angle capability of the dual-domain implementation are demonstrated by numerical examples. Reflection coefficients of a plane interface derived from numerical simulations by the wide-angle method match the theoretical curves well up to critical angles. For the reflections of a low-velocity slab, the agreement between theory and synthetics only starts to deteriorate for angles greater than 70°. The accuracy of the wide-angle version of the method could be further improved by optimizing the wave-number filtering for the forward propagation and shrinking the step length along the propagation direction.     

波动方程的高阶广义屏叠前深度偏移

不同于常规广义屏传播算子的推导中使用散射理论，本文利用单平方根算子的渐近展开，推导出了单程波方程广义屏传播算子的高阶表达式.高阶广义屏传播算子不仅可提高常规广义屏传播算子的计算精度，而且还能改善广义屏传播算子对速度强横向变化介质的适应性.把高阶广义屏传播算子应用于波动方程叠前深度偏移，可得到比常规广义屏传播算子更好的效果.高阶广义屏传播算子的阶数越高，计算精度越高，但计算量也越多.以SEG EAGE二维盐丘模型数据的波动方程叠前深度偏移为例，二阶广义屏传播算子相对于常规（一阶）广义屏传播算子增加了30%的计算量.高阶广义屏传播算子是常规广义屏传播算子理论的发展和完善.     

时-空局域化地震波传播方法:Dreamlet叠前深度偏移

提出了一种在时间和空间上完全局域化的波场分解和传播算法─dreamlet偏移方法.Dreamlet是一种脉冲-小波束形式的波场分解原子,它利用多维局部分解变换,把时空域波场映射到局部时间-频率-空间-波数相空间,并用局部相空间的传播算子(dreamlet算子)沿深度延拓.本文利用多维局部余弦变换实现dreamlet算法,分解后的波场系数和传播算子不仅有很好的稀疏性,且均为实数,也即波的传播和成像过程完全在实数域实现.文中推导了局部余弦基dreamlet波场分解和传播算子理论公式并将其应用于叠前深度偏移.在dreamlet相空间波的传播过程为稀疏矩阵相乘,而且延拓后的地表数据波场的有效时间长度随深度的增加不断减小,从而可以减少需要传播的波场系数.二维SEG/EAGE盐丘和SIGSBEE模型算例验证了理论推导的正确性,成像结果显示该方法在横向速度变化剧烈情况下有很好的精度.     

Lateral velocity variation related correction in asymptotic true‐amplitude one‐way propagators

It was mathematically proved that the asymptotic true‐amplitude one‐way wave equation could provide the same amplitude as the full‐wave equation in heterogeneous lossless media in the sense of high‐frequency asymptotics. Much work has been done on the vertical velocity variation related amplitude correction term but the lateral velocity variation related term has not received much attention, even being excluded in some asymptotic true‐amplitude one‐way propagator formulations. Here we analyse the effects of different amplitude correction terms in the asymptotic true‐amplitude one‐way propagator, especially the effect related to the lateral velocity variation, by comparing the wavefield amplitude from the one‐way propagator with that from full‐wave modelling. We derive a dual‐domain wide‐angle screen type asymptotic true‐amplitude one‐way propagator and evaluate two implementations of the amplitude correction. Numerical examples show that the lateral velocity variation related correction term can play a significant role in the asymptotic true‐amplitude one‐way propagator. Optimization of the expansion coefficients in the asymptotic true‐amplitude one‐way propagator can improve both the amplitude and phase accuracy for wide‐angle waves.     

Analytical wavefront curvature correction to plane‐wave reflection coefficients for a weak‐contrast interface

Most amplitude versus offset (AVO) analysis and inversion techniques are based on the Zoeppritz equations for plane‐wave reflection coefficients or their approximations. Real seismic surveys use localized sources that produce spherical waves, rather than plane waves. In the far‐field, the AVO response for a spherical wave reflected from a plane interface can be well approximated by a plane‐wave response. However this approximation breaks down in the vicinity of the critical angle. Conventional AVO analysis ignores this problem and always utilizes the plane‐wave response. This approach is sufficiently accurate as long as the angles of incidence are much smaller than the critical angle. Such moderate angles are more than sufficient for the standard estimation of the AVO intercept and gradient. However, when independent estimation of the formation density is required, it may be important to use large incidence angles close to the critical angle, where spherical wave effects become important. For the amplitude of a spherical wave reflected from a plane fluid‐fluid interface, an analytical approximation is known, which provides a correction to the plane‐wave reflection coefficients for all angles. For the amplitude of a spherical wave reflected from a solid/solid interface, we propose a formula that combines this analytical approximation with the linearized plane‐wave AVO equation. The proposed approximation shows reasonable agreement with numerical simulations for a range of frequencies. Using this solution, we constructed a two‐layer three‐parameter least‐squares inversion algorithm. Application of this algorithm to synthetic data for a single plane interface shows an improvement compared to the use of plane‐wave reflection coefficients.     

Wave Propagation, Scattering and Imaging Using Dual-domain One-way and One-return Propagators

— Dual-domain one-way propagators implement wave propagation in heterogeneous media in mixed domains (space-wavenumber domains). One-way propagators neglect wave reverberations between heterogeneities but correctly handle the forward multiple-scattering including focusing/defocusing, diffraction, refraction and interference of waves. The algorithm shuttles between space-domain and wavenumber-domain using FFT, and the operations in the two domains are self-adaptive to the complexity of the media. The method makes the best use of the operations in each domain, resulting in efficient and accurate propagators. Due to recent progress, new versions of dual-domain methods overcame some limitations of the classical dual-domain methods (phase-screen or split-step Fourier methods) and can propagate large-angle waves quite accurately in media with strong velocity contrasts. These methods can deliver superior image quality (high resolution/high fidelity) for complex subsurface structures. One-way and one-return (De Wolf approximation) propagators can be also applied to wave-field modeling and simulations for some geophysical problems. In the article, a historical review and theoretical analysis of the Born, Rytov, and De Wolf approximations are given. A review on classical phase-screen or split-step Fourier methods is also given, followed by a summary and analysis of the new dual-domain propagators. The applications of the new propagators to seismic imaging and modeling are reviewed with several examples. For seismic imaging, the advantages and limitations of the traditional Kirchhoff migration and time-space domain finite-difference migration, when applied to 3-D complicated structures, are first analyzed. Then the special features, and applications of the new dual-domain methods are presented. Three versions of GSP (generalized screen propagators), the hybrid pseudo-screen, the wide-angle Padé-screen, and the higher-order generalized screen propagators are discussed. Recent progress also makes it possible to use the dual-domain propagators for modeling elastic reflections for complex structures and long-range propagations of crustal guided waves. Examples of 2-D and 3-D imaging and modeling using GSP methods are given.     

Convergence analyses of different modeling schemes for generalized Lippmann–Schwinger integral equation in piecewise heterogeneous media

For wave propagation simulation in piecewise heterogeneous media, Gaussian-elimination-based full-waveform solutions to the generalized Lippmann–Schwinger integral equation (GLSIE) are highly accurate, but involved with extremely time-consuming computations because of the very large size of the resulting boundary–volume integral equation matrix to be inverted. Several flexible approximations to the GLSIE are scaled in an iterative way to adapt numerical solutions to the smoothness of heterogeneous media in terms of incident wavelengths, with a great saving of computing time and memory. Among various typical iterative schemes to the GLSIE matrix, the generalized minimal residual method (GMRES) is an efficient approach to reduce the computational intensity to some degree. The most efficient approximation can be obtained using a Born series, as an alternative iterative solution, to both the boundary-scattering and volume-scattering waves, leading to the Born-series approximation (BSA) scheme and the improved Born-series approximation (IBSA) scheme. These iteration schemes are validated by dimensionless frequency responses to a heterogeneous semicircular alluvial valley, and then applied to a heterogeneous multilayered model by calculating synthetic seismograms to evaluate approximation accuracies. Numerical experiments, compared with the full-waveform numerical solution, indicate that the convergence rates of these methods decrease gradually with increasing velocity perturbations. The comparison also shows that the BSA scheme has a faster convergence than the GMRES method for velocity perturbations less than 10 percent, but converges slowly and even hardly achieves convergence for velocity perturbations greater than 15 percent. The IBSA scheme gives a superior performance over the other methods, with the least iterations to achieve the necessary convergence.