频率域全波形反演是当今地球物理领域的研究热点和难点之一，许多计算数学家与地球物理学家为此做出过不少杰出的工作。本文将近似解析离散化 (NAD)方法引入到频率域波形反演中的正演计算过程。首先利用四阶 NAD 方法离散频率域波动方程，详细研究了 NAD 类方法的离散过程和完美匹配层 (PML) 吸收边界条件的实现办法，进而得到一个大型稀疏线性代数方程组。然后较为细致地分析了这个线性方程组的结构特点，并构造了一类不精确旋转分块预处理子加速 Krylov 迭代方法对其进行求解，进一步推导了相应预处理矩阵的特征值性质结论并且通过数值试验比较了本文所提出的预处理方法与另外两种传统预处理方法的计算效率。数值结果显示出文中方法的优势，在计算速度方面提升数十倍。同时针对几种地质介质模型进行了波场模拟以及数值频散分析，并将四阶 NAD 方法与另外两种经典数值方法进行了比较。数值实验结果均表明频率域 NAD 方法具有较强的压制数值频散的能力。在四阶 NAD 方法基础上，本文分别通过两种途径改进数值格式来提高频率域正演计算过程的效率。第一种思路是采用更高精度的六阶 NAD 格式。首先详细介绍了频率域波动方程的离散方法和相应线性代数方程组系数矩阵的分块结构特点以及具体的元素组成情况，然后进行了频率域波场模拟并与四阶 NAD 方法进行了比较。数值结果表明这种六阶 NAD 方法具有更强的压制数值频散的能力，从而可以节省大致 25% 的正演计算时间。第二种思路是通过优化数值格式的系数以改进四阶 NAD 方法，使得离散得到的线性方程组系数矩阵条件数降低，从而更加容易求解，以节省计算时间。波场模拟结果表明，改进 NAD 方法可以在保持压制数值频散能力不受损失的前提下，节省大约 10% 的正演计算时间。最后，本文给出了基于 NAD 方法的频率域全波形反演过程算法。首先介绍了频率域反演的基本原理与方法，然后对反演过程中的若干重要环节做了深入研究。选取不同规模的双层介质模型和复杂的 Marmousi 模型进行反演，均得到了很好的反演结果，验证了本文所提出方法的有效性。
Full waveform inversion in the frequency domain is one of the hot and difficult spotsin geophysical researches. Many computational mathematicians and geophysicists havemade a lot of outstanding work. This dissertation proposes to use nearly-analytic discrete(NAD) methods for forwarding modelling in frequency-domain full waveform inversionprocesses. At first, the frequency-domain wave equations are discretized by forth-orderNAD method. The NAD discrete processes and the achievements of Perfectly MatchedLayer (PML) absorbing boundary condition are introduced in detail to obtain a large s-parse linear algebraic system. The structure properties of this linear system are analysedin detail and solved by a class of inexact rotated block triangular preconditioned Krylovsubspace methods. The eigen-properties of the corresponding preconditioned matricesare further developed and numerical experiments are implemented to compare the effi-ciencies of such preconditioned iteration methods and other two conventional precondi-tioned methods. The numerical results show the advantages of the methods introduces inthis dissertation, to be more specific, speeding up for dozens of times. Then, the wave-field simulations and dispersion analyses are performed in various media to compare theefficiencies of forth-order NAD method and other two classical numerical schemes. Nu-merical results indicate that frequency domain NAD methods have stronger ability tosuppress numerical dispersion.Based on the forth-order NAD method, this dissertation develops numerical schemesto accelerate frequency-domain forward modelling according to two approaches, respec-tively. One idea is to use more accurate sixth-order NAD method. At first, the construc-tion of numerical stencils is introduced in detail for discretizing frequency-domain waveequations to obtain a linear system and the block structure and expressions of elements ofits coefficient matrix are introduced in detail. Then, wave-field simulations are performedto compare the efficiency of sixth-order and forth-order NAD methods. Numerical resultsindicate that sixth-order NAD method is more able to suppress numerical dispersion andcan save approximately 25% computing time in forward modelling. The other idea isto develop forth-order NAD method according to optimize the coefficients of numericalstencils to make the condition numbers of the linear system after discretion decrease.Thus, it is easier to solve such linear system and save computing time. According towave-field simulation, developed NAD methods can save approximately 10% computingtime under the premise of keeping the ability to suppress numerical dispersion.Finally, this dissertation presents frequency-domain full waveform inversion algo-rithm based on NAD methods. The fundamental theory and methods of frequency-domain inversion are introduced and some details of inversion process are described.Then, the inversion processes are performed in two-layer media of different scales andmore complicated Marmousi medium. The accurate inversion results are obtained to il-lustrate the effectiveness of proposed methods
