The non-commutivity of shear wave splitting operators at low frequencies and implications for anisotropy tomography
The non-commutivity of shear wave splitting operators at low frequencies and implications for anisotropy tomography
复制标题
低频剪切波分裂算子的非交换性及其对各向异性层析成像的影响
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
M. Long
中科院分区：
文献类型：
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作者：
P. Silver;M. Long
总结
测量地震横波的分裂或双折射是表征上地幔方位各向异性的一种强有力的和流行的技术。由于密集宽带地震台阵提供的数据集越来越多，人们对开发剪切波分裂数据的层析反演技术以及将分裂测量结果与从表面波分析等其他约束条件获得的各向异性上地幔模型进行比较产生了兴趣。两种不同的理论方法已被开发用于预测表观剪切波分裂参数（快方向和延迟时间）的模型，包括多层各向异性的深度，这是有用的比较方位各向异性的表面波模型与剪切波分裂测量。这些方法在一个关键方面有所不同，即剪切波分裂算子是否可以被视为交换算子。在本文中，我们调查这种差异的理论来源，并表明，在频率相关的上地幔各向异性的研究，在剪切波分裂算子的非交换性的多层分裂的表达式中的结果的长期必须保留。与此相反，与视在快方向和延迟时间密切相关的分裂强度，在这些频率下是可以互换的。我们说明这些推论与正演模拟的例子，并讨论其影响层析反演的剪切波分裂测量，比较面波模型与剪切波分裂观测和联合反演的面波和剪切波分裂观测上地幔各向异性模型。
SUMMARY
Measurements of the splitting or birefringence of seismic shear waves constitute a powerful and popular technique for characterizing azimuthal anisotropy in the upper mantle. The increasing availability of data sets from dense broad-band seismic arrays has driven interest in the development of techniques for the tomographic inversion of shear wave splitting data and in comparing splitting measurements with anisotropic upper-mantle models obtained from other constraints, such as surface wave analysis. Two different theoretical approaches have been developed for predicting apparent shear wave splitting parameters (fast direction and delay time) for models that include multiple layers of anisotropy at depth, which is useful for comparing azimuthally anisotropic surface wave models with shear wave splitting measurements. These approaches differ in one key aspect, which is whether or not the shear wave splitting operator can be treated as commutative. In this paper, we investigate the theoretical source of this discrepancy, and show that at frequencies relevant to most studies of upper-mantle anisotropy, the term that results in the non-commutivity of the shear wave splitting operator in the expressions for multiple-layer splitting must be retained. In contrast, the quantity known as the splitting intensity, which is closely related to the apparent fast direction and delay time, does commute at these frequencies. We illustrate these inferences with forward modelling examples and discuss their implications for the tomographic inversion of shear wave splitting measurements, the comparison of surface wave models with shear wave splitting observations and the joint inversion of surface wave and shear wave splitting observations for upper-mantle anisotropic models.