The distortion tensor of magnetotellurics: a tutorial on some properties
The distortion tensor of magnetotellurics: a tutorial on some properties
复制标题
DOI：
10.1071/eg14093
复制
发表时间：
2016-06
影响因子：
0.9
通讯作者：
F. Lilley
中科院分区：
文献类型：
--
作者：
F. Lilley
介绍了一个2×2矩阵，它将地质变形的观测点的电场与不受变形影响的区域电场联系起来。对于线性代数的学生来说，这个矩阵提供了一个实用的例子，用它来演示特征值分析和奇异值分解的基本和重要步骤。结果的意义是可视的，因为这样的大地失真矩阵的特征向量和它们的特征值都具有明确的实际意义。当实特征向量存在时，失真矩阵的莫尔图就会显示出来，并告诉我们它们的大小和方向。奇异值分解(SVD)的结果也具有明确的实际意义。这些结果也可以显示在莫尔图上。虽然实特征向量可能存在，也可能不存在，但奇异值分解总是可能的。矩阵的两个奇异值的比率给出了一个条件数，这对量化失真很有用。强烈的失真会使矩阵接近所谓的“奇点”。密切相关的各向异性数也可能是有用的，因为它告诉我们，当2×2矩阵具有负行列式时，其值将大于1。用两种线性代数方法：特征值分析和奇异值分解分析了大地电磁的变形张量。莫尔图显示了结果，并强调了扭曲的重要特征。举例比较了传统的Groom-Bailey分解和奇异值分解。
A 2 × 2 matrix is introduced which relates the electric field at an observing site where geological distortion applies to the regional electric field, which is unaffected by the distortion. For the student of linear algebra this matrix provides a practical example with which to demonstrate the basic and important procedures of eigenvalue analysis and singular value decomposition. The significance of the results can be visualised because the eigenvectors of such a telluric distortion matrix have a clear practical meaning, as do their eigenvalues. A Mohr diagram for the distortion matrix displays when real eigenvectors exist, and tells their magnitudes and directions. The results of singular value decomposition (SVD) also have a clear practical meaning. These results too can be displayed on a Mohr diagram. Whereas real eigenvectors may or may not exist, SVD is always possible. The ratio of the two singular values of the matrix gives a condition number, useful to quantify distortion. Strong distortion causes the matrix to approach the condition known as ‘singularity’. A closely-related anisotropy number may also be useful, as it tells when a 2 × 2 matrix has a negative determinant by then having a value greater than unity. The magnetotelluric distortion tensor is analysed by two methods of linear algebra: eigenvalue analysis and singular value decomposition. Mohr diagrams display the results, and emphasise important characteristics of the distortion. Examples compare traditional Groom-Bailey decomposition with singular value decomposition.