泄漏模式频散曲线的正反演及其在地下结构成像中的应用

频散是波沿非均匀模型传播时的固有现象，并且波的频散特征与模型结构有关。频散特征通常由频散曲线来描述，频散曲线在浅地表勘探、地震学、声学无损检测和光学等领域应用广泛。在浅地表勘探和地震学中，当地下结构接近层状时，从地震记录或背景噪声中提取出的面波频散曲线是反演横波速度结构的有效手段。地震和背景噪声数据中，除面波外，泄漏波也表现出频散特征。在模式理论中，面波可以用简正模式的叠加来表示；相应地，泄漏波可以用泄漏模式的叠加来表示，两种模式的频散都能反映地下结构信息。与面波相比，一部分泄漏波由纵波主导，并且能量泄漏较弱，这部分波称为P导波。利用P导波的频散信息有望约束纵波速度，从而弥补面波频散反演的不足。但现有的频散曲线正反演大多聚焦于简正模式，对泄漏模式的频散特征和数值计算方法缺乏探讨。本文基于谱元法开发了简正和泄漏模式的半解析求解方法并研究了它们的频散特征和敏感性特征，在此基础上实现了P导波和面波频散曲线的联合反演，从而可以同时约束地下的纵、横波速度结构。

简正和泄漏模式对应无源波动方程的非平凡解，数学上可以用特征值问题来描述。传统方法通过构建频散方程进而搜索实数根的方式，可以求解简正模式，即面波频散。然而泄漏模式对应频散方程的复数根，其求解难度大大增加，传统的实数轴搜根法难以获得完备的解集。针对层状半空间模型，本文基于谱元法实现了简正和泄漏模式的高效、高精度正演。与简正模式相比，计算泄漏模式最大的难点在于如何处理波场在半空间中满足的边界条件。本文采用罗宾边界条件和半无限元方法描述半空间中的波场，从而导出了线性特征值问题，之后利用特征值分解就可以稳定有效地计算简正和泄漏模式的半解析解。该方法的主要优势是其在求解过程中无需任何先验信息，且运算效率高，因而对反演非常友好。本文通过多个数值实验证明了该方法的有效性，体现了其精度高和不漏根的特点。模式分析表明，泄漏模式可以分为两种类型，一部分模式的波场由纵波主导且衰减较弱，称为P导波模式；而其他模式主要受横波影响。多种不同类型的模型实验表明，基于谱元法的模式求解方法应用场景广泛，除了由多个均匀层组成的模型外，它还可以应用于垂向速度渐变模型。

基于频散曲线的反演方法被广泛应用于地下速度结构的估计，然而传统频散反演主要利用简正模式，一般仅能反演横波速度。要恢复纵波速度，应考虑P导波的频散。本文基于上述正演方法导出了泄漏模式的敏感性计算方法，对简正和泄漏模式敏感性的定量分析表明，P导波联合面波频散可以同时约束纵横波速度。此外，本文利用P导波模式对纵波速度的高敏感性实现了模式分离，并基于纵波势函数场的振荡特征实现了P导波的模式阶数判别，这使得我们可以用面波分析方法来研究和反演P导波频散曲线。分离后的P导波频散曲线能够与理论模型的频散谱完美匹配，这为多阶P导波频散曲线的反演奠定了基础。

最后本文提出了基于面波和P导波频散曲线的联合反演方法，能够同时反演模型的纵、横波速度。该联合反演可以与多种局部线性化方法结合，包括阻尼最小二乘反演和最小梯度支持正则化反演。在理论模型实验后，本文利用阻尼最小二乘反演对布置在浅海的海底地震仪观测数据进行了反演，建立了海底浅层速度模型。之后利用最小梯度支持正则化反演处理了2008年美国内华达州的地震数据，优化了该地区的地壳速度结构。基于理论和实测数据的反演结果表明，简正和泄漏模式的联合反演能够同时有效地约束纵、横波速度结构，从而得到比传统面波频散反演更全面、准确的模型。

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