声波方程逆散射反演的近似方法

我们在文献[1]里研究了介质参考波速沿某个方向线性变化时的三维声散射理论,导出了声波方程逆散射反演问题解的解析表达式.考虑到应用时的实际条件,本文根据上述反演方法导出2.5维模型的声波方程逆散射反演的波速扰动计算公式,给出该方法在“高频”近似条件下的波速扰动反演计算公式,从而使我们提出的“参考波速线性变化时的声波方程逆散射反演”理论更接近实际应用条件.本文给出的这些反演公式仍然具有原方法的优点,即不但可以使Born近似的假定在大多数情况下能得以满足,而且可以利用快速Fourier变换来快速实现介质波速扰动的反演成象.     

声波方程逆散射反演的近似方法

我们在文献[1]里研究了介质参考波速沿某个方向线性变化时的三维声散射理论,导出了声波方程逆散射反演问题解的解析表达式.考虑到应用时的实际条件,本文根据上述反演方法导出2.5维模型的声波方程逆散射反演的波速扰动计算公式,给出该方法在“高频”近似条件下的波速扰动反演计算公式,从而使我们提出的“参考波速线性变化时的声波方程逆散射反演”理论更接近实际应用条件.本文给出的这些反演公式仍然具有原方法的优点,即不但可以使Born近似的假定在大多数情况下能得以满足,而且可以利用快速Fourier变换来快速实现介质波速扰动的反演成象.     

二阶Born近似及其在逆散射GRT反演中的应用

传统的地震逆散射广义Radon变换(GRT)保幅反演方法是建立在散射场一阶Born近似(单散射)的基础上,仅仅适用于弱扰动介质模型.本文从散射场积分方程出发,通过研究二次散射的特征,讨论和验证了基于局部二阶Born近似的GRT非线性保幅反演方法,将传统GRT线性保幅反演算子的适用范围扩展至非均匀强扰动介质.数值测试结果表明:在散射场近似模拟方面,二阶Born近似比一阶Born近似更为准确,二次散射效应主要集中在主散射点周围的局部区域内,超过这一范围,二次散射强度趋于稳定;在保幅反演方面,本文基于局部二阶Born近似的GRT非线性反演算法,明显优于传统的GRT线性反演算法,可以准确重构强扰动介质模型,而计算效率与线性反演方法相当.     

基于T-matrix的非线性参数估计方法

全波形反演可提供高精度的地下介质参数空间分布,但传统的全波形反演方法建立在Born近似的基础上,对初始模型具有一定的依赖性.为了摆脱Born近似的束缚,本文基于二维常密度声波方程,在De Wolf近似的前提下,借助传输矩阵(T-matrix)方法,深入研究了逆薄板传播算子(Inverse Thin-Slab Propagator,ITSP),实现了速度扰动的非线性估计.ITSP方法避免了Born级数方法在扰动较强、扰动区域较大时的发散性问题,且只经过一次扫描校正,计算效率较高.二维模拟数据分析验证了本文方法的可行性以及有效性.     

二阶Born近似有限频率走时敏感核

有限频率层析成像考虑了非均匀介质中波的散射、衍射、波前愈合等物理性质,使得其对速度异常体的分辨能力远大于射线层析成像.推导和计算有限频率敏感核是进行有限频率层析成像的关键,当前推导有限频率敏感核多借助一阶Born近似,但这只适用于弱散射介质的情况.本文基于二阶Born近似并利用傅里叶变换推导了三维均匀介质情况下有限频率敏感核的解析表达式,并将其推广到非均匀介质中得到了三维非均匀介质中有限频率敏感核.研究表明:当介质中速度扰动小于2%时,基于二阶Born近似的有限频率敏感核与基于一阶Born近似的有限频率敏感核差别很小,可近似认为相同;当介质中速度扰动大于5%时,基于二阶Born近似的有限频率敏感核与基于一阶Born近似的有限频率敏感核有较大不同,表明此时已不能忽略二次散射.     

散射波积分方程的Adomian分解解法

在背景模型基础上,求解模型扰动后的地震波散射场,这是目前地震反演中的一个关键步骤.本文将计算数学中求解非线性积分方程的Adomian分解方法,应用到求解标量波散射场的Lippmann-Schwinger积分方程和Ricatti积分方程中,分别得到了散射场的Born序列解和Rytov序列解.通过一维和二维数值算例说明:在满足一定的条件下,散射场的这两种序列解稳定收敛,与传统的Born和Rytov近似解相比,引入散射序列中的高阶项可以更精确地描述地震波散射场.     

水平层状介质中基于DTA的三维电磁波逆散射快速模拟算法

为提高水平层状介质中三维电磁波散射和逆散射数值模拟的效率,在对角张量近似(DTA)的基础上根据不同回代方式得到了求解积分方程的DTA1和DTA2两种近似. 这两种近似可以作为计算积分方程稳定型双共轭梯度快速Fourier变换(BCGS-FFT)算法的初始猜测值和预条件因子，从而形成效率更高的混合DTA-BCGS算法. 散射实例说明了DTA2的高精度和混合DTA-BCGS算法尤其是混合DTA2-BCGS算法的高效率. 由于DTA2近似程度更高，将DTA2与变型Born迭代反演方法(DBIM)相结合形成了一种对三维异常体进行重构的快速电磁波逆散射技术. 文中的逆散射实例说明所开发的逆散射技术对重构水平层状介质中的任意三维异常体是非常有效的.     

横向各向同性介质拟声波方程及其在逆时偏移中的应用

地下岩石的速度各向异性影响地震波的传播与成像.横向各向同性(TI)介质为最普遍的等效各向异性模型.引入TI介质拟声波方程可以避免复杂的弹性波方程求解以及各向异性介质波场分离,以满足对纵波成像的实际需要.本文从垂直横向各向同性(VTI)介质弹性波方程出发,推导出正应力表达的拟声波方程以及相应的纵波分量的表达式,进而分析从频散关系得到的拟声波方程的物理意义,而后将拟声波方程扩展到更一般的倾斜横向各向同性(TTI)介质中.波前快照与群速度平面的对比验证了拟声波方程可以很好地近似描述qP波的运动学特征.在此基础上,将拟声波方程应用在逆时偏移中并与其特例声波近似方程进行对比,讨论了计算效率、稳定性等实际问题.数值试验表明VTI介质情况下采用声波近似方程可以提高计算效率,而TTI介质qP-qSV波方程则在效率相当的情况下可以保证稳定性.SEG/HESS模型和逆冲模型逆时偏移试验验证了本文TI介质拟声波方程的实用性.     

地球物理学中的电磁场积分方程正演

本文对地球物理学中的电磁场积分方程正演进行了综述,重点分析和讨论了积分方程正演中的散射场近似求解方法.散射场近似解法可以在保证计算精度的前提下有效的提高计算效率,使积分方程正演突破了传统简单孤立异常体研究的限制,适用于大规模复杂三维电磁场快速正演.本文着重对近年来国内外学者提出的散射场近似求解方法,如扩展Born近似、高阶广义Born近似、准线性(QL)近似和准解析近似(QA)等进行了分析和讨论,指出了各种近似解法的优缺点和适用范围.并在前人工作的基础上总结了地球物理学中的电磁场积分方程正演的基本原理和关键问题及解决方法,包括并矢格林函数、散射场的近似求解方法以及全积分求解方法等.最后,本文提出了积分方程法发展趋势和实际工程应用的前景以及面临的困难和待解决的问题.     

最大熵时域逆散射层析成像的研究

用最大熵求解二维时域逆散射问题的层析成像算法反演井间、ＶＳＰ和ＳＲＰ中的二维介质波速分布。首先给出时域散射场与介质扰动的关系式，然后，应用最大熵剑桥算法从所有满足时域散射数据的模型中挑选图像熵最大的模型作为反演结果，并给出了该算法的实现步骤。数值试验的结果表明，本文算法用于反演含噪声，不完全的时域波场数据时，十分有效。     

AN EXTENSION OF THE BORN INVERSION METHOD TO A DEPTH DEPENDENT REFERENCE PROFILE*

An extension of the multidimensional Born inversion technique for acoustic waves is described. In earlier work, a perturbation in reference sound velocity was determined by assuming that the reference velocity was constant. In this extension, we allow the reference velocity to be a function of the depth variable z. The output of this method is a high-frequency bandlimited reflectivity function of the subsurface. The reflectivity function is an array of bandlimited singular functions scaled by the normal reflection strength. Each singular function is a Dirac delta function of a scalar argument which measures distance normal to a reflecting interface. Thus, the reflectivity function is an indicator map of subsurface reflectors equivalent to the map produced by migration. In addition to the assumption of small perturbation, the method requires that the reflection data reside in the high frequency regime in a well-defined sense. The method is based on the derivation of an integral equation for the perturbation in sound velocity from a known reference velocity. When the reference velocity is constant, the integral equation admits an analytic solution as a multifold integral of the reflection data. Further high frequency asymptotic analysis simplifies this integral considerably and leads to an extremely efficient numerical algorithm for computing the reflectivity function. The development of a computer code to implement this constant-reference-velocity solution is published elsewhere. For a reference velocity c(z) we can no longer invert the integral equation exactly. However, we can write down an asymptotic high-frequency approximation for the kernal of the integral equation and an asymptotic solution for the perturbation. The computer implementation of this result is designed along the same lines as the code for constant background velocity. In tests the total processing time for this algorithm with depth-dependent background velocity is usually considerably less than that required by a standard Kirchhoff migration algorithm. The method is implemented as a migration technique and compared with alternative migration algorithms on the flanks of the salt dome.     

Seismic inversion with generalized Radon transform based on local second-order approximation of scattered field in acoustic media

Sound velocity inversion problem based on scattering theory is formulated in terms of a nonlinear integral equation associated with scattered field. Because of its nonlinearity, in practice, linearization algorisms (Born/single scattering approximation) are widely used to obtain an approximate inversion solution. However, the linearized strategy is not congruent with seismic wave propagation mechanics in strong perturbation (heterogeneous) medium. In order to partially dispense with the weak perturbation assumption of the Born approximation, we present a new approach from the following two steps: firstly, to handle the forward scattering by taking into account the second-order Born approximation, which is related to generalized Radon transform (GRT) about quadratic scattering potential; then to derive a nonlinear quadratic inversion formula by resorting to inverse GRT. In our formulation, there is a significant quadratic term regarding scattering potential, and it can provide an amplitude correction for inversion results beyond standard linear inversion. The numerical experiments demonstrate that the linear single scattering inversion is only good in amplitude for relative velocity perturbation ( \( \delta_{c}/c_{0} \) ) of background media up to 10 %, and its inversion errors are unacceptable for the perturbation beyond 10 %. In contrast, the quadratic inversion can give more accurate amplitude-preserved recovery for the perturbation up to 40 %. Our inversion scheme is able to manage double scattering effects by estimating a transmission factor from an integral over a small area, and therefore, only a small portion of computational time is added to the original linear migration/inversion process.     

PARAMETRIZATION OF GRT INVERSION FOR ACOUSTIC AND P–P SCATTERING1

The generalized Radon transform (GRT) inversion contains an explicit relationship between seismic amplitude variations, the reflection angle and the physical parameters which can be used to describe the earth efficiently for inversion purposes. Using this relationship, we have derived parametrizations for acoustic and P–P scattering so that the variations in seismic amplitude with reflection angle for each parameter are sufficiently independent. These parametrizations show that small offset and large offset amplitudes are related to different physical parameters. In the case of acoustic scattering, the small-offset amplitudes are related to impedance variations while large-offset amplitudes are related to velocity variations. A similar result has been established for P–P scattering. The Born approximation (which is used to derive the GRT inversion) does not correctly predict the amplitude due to velocity variations at large offsets, and thus the inversion of velocity is not as satisfactory as the inversion of impedance.     

T-matrix approach to seismic forward modelling in the acoustic approximation

Forward seismic modelling in the acoustic approximation, for variable velocity but constant density, is dealt with. The wave equation and the boundary conditions are represented by a volume integral equation of the Lippmann-Schwinger (LS) or Fredholm type. A T-matrix (or transition operator) approach from quantum mechanical potential scattering theory is used to derive a family of linear and nonlinear approximations (cluster expansions), as well as an exact numerical solution of the LS equation. For models of 4D anomalies involving small or moderate contrasts, the Born approximation gives identical numerical results as the first-order t-matrix approximation, but the predictions of an exact T-matrix solution can be quite different (depending on spatial extention of the perturbations). For models of fluid-saturated cavities involving large or huge contrasts, the first-order t-matrix approximation is much more accurate than the Born approximation, although it does not lead to significantly more time-consuming computations. If the spatial extention of the perturbations is not too large, it is practical to use the exact T-matrix solution which allows for arbitrary contrasts and includes all the effects of multiple scattering.     

积分道与基于Born散射的一维声波方程速度反演

基于Born散射理论，推导了适用于常速和变速背景的一维速度反演公式。与经典的Bleistein逆散射反演公式相比，本文考虑了声波在一维有限空间中的传播，选取的边界条件更合理。改进后的公式也揭示了积分道(对反射系数的积分)与绝对速度的关系，更有实用价值。     

用变分玻恩迭代方法重建二维非均匀介质结构

提出了用于二维轴对称非均匀介质结构的反演和成像的一种新的反演迭代方法──变分玻恩迭代方法（ＶＢＩＭ）．首先利用玻恩近似将非线性积分方程线性化,然后应用变分方法导出用于反演的电场积分方程．正演数据则利用高效的数值模式匹配方法获得．数值结果表明,ＶＢＩＭ与ＢＩＭ相比,其收敛速度、成像质量等均得到较大的改善。     

电离层声重波扰动的高频无线电诊断

为了充分发挥现代数字式电离层探测仪在电离层结构与优动研究中的潜力,必须在传统的频高图真高换算的基础上,发展新的反演理论和算法.本文介绍了随时空缓变的各向异性电离层介质中无线电波包传播的广义射线方程组,讨论了计算波包射线几何路径以及计算波包参数,如频移、波矢、群时延等沿射线路径变化的传播正问题,并详细讨论了根据波包参量的测定值反演介质结构和扰动伏态的传播反问题.文中若干实验实例表明,这种反演理论和算法可用于数字式电离层探测仪记录的分析,它使高频无线电波探测技术成为研究全球电离层声重波一类扰动的有力工具。     

时间二阶积分波场的全波形反演

通过对波场的时间二阶积分运算以增强地震数据中的低频成分,提出了一种可有效减小对初始速度模型依赖性的地震数据全波形反演方法—时间二阶积分波场的全波形反演方法.根据散射理论中的散射波场传播方程,推导出时间二阶积分散射波场的传播方程,再利用一阶Born近似对时间二阶积分散射波场传播方程进行线性化.在时间二阶积分散射波场传播方程的基础上,利用散射波场反演地下散射源分布,再利用波场模拟的方法构建地下入射波场,然后根据时间二阶积分散射波场线性传播方程中散射波场与入射波场、速度扰动间的线性关系,应用类似偏移成像的公式得到速度扰动的估计,以此建立时间二阶积分波场的全波形迭代反演方法.最后把时间二阶积分波场的全波形反演结果作为常规全波形反演的初始模型可有效地减小地震波场全波形反演对初始模型的依赖性.应用于Marmousi模型的全频带合成数据和缺失4Hz以下频谱成分的缺低频合成数据验证所提出的全波形反演方法的正确性和有效性,数值试验显示缺失4Hz以下频谱成分数据的反演结果与全频带数据的反演结果没有明显差异.