Analysis of rotational invariants of the magnetotelluric impedance tensor
Analysis of rotational invariants of the magnetotelluric impedance tensor
复制标题
大地电磁阻抗张量旋转不变量分析
DOI：
10.1111/j.1365-246x.1997.tb00942.x
复制
发表时间：
1997
影响因子：
2.8
通讯作者：
M. Menvielle
中科院分区：
文献类型：
--
作者：
L. Szarka;M. Menvielle
总结
大地电磁阻抗张量Z的旋转不变量可以作为最紧凑的三维解释参数，因为它们不依赖于感应场的方向，并且它们可以在三维物体上具有各种形态特征。本文首次对它们的完整体系进行了评述。它表明，复杂的Z-有八个实值独立元素-有七个独立的旋转不变量。复行列式det（Z）包含三个独立的实值不变量：det（ReZ），det（ImZ）和Imdet（Z）[其中det（ReZ）-det（ImZ）=Re det（Z）]，而不是通常从其复特征假设的两个。对于Z的元素的平方和也是如此，ssq（Z）= Z2 xx + Z2 xy + Z2 yx + Z2 yy。其实值不变量是ssq（ReZ）= Re 2 Zxx + Re 2 Zxy + Re 2 Zyx + Re 2 Zyy; ssq（ImZ）= Im 2 Zxx + Im 2 Zxy + Im 2 Zyx + Im 2 Zyy;以及Im ssq（Z）= 2（ReZxx ImZxx+ ReZxy ImZxy+ ReZyx ImZyx+ ReZyy ImZyy），其中Re ssq（Z）= ssq（ReZ）-ssq（ImZ），以及ssq（ReZ）+ ssq（ImZ）=|| Z|| 2F：这里||Z|| f是Z的Frobenius范数。七个独立旋转不变量的集合可以以许多不同的方式选择。在经典的大地电磁集合中，建议使用ReZ_1，ImZ_1 [其中Z_1 =（Zxy-Zyx）/2]，ReZ_2，ImZ_2（其中道Z_x + Zyy= 2 Z_2）和三个基于行列式的实值不变量det（ReZ），det（ImZ）和Im det（Z）。如果迹、行列式和ssq（Z）被接受为基本标量函数（我们称它们为不变量的数学选择），则可以选择八个不同的独立不变量集。旋转不变量的几何意义说明使用两种不同的图形表示：复平面椭圆和莫尔圆。为了电磁成像的目的，建议应使用从真实的张量ReZ导出的那些参数中的一些，因为在现实时期，薄片状3-D模型的模型几何形状在例如ReZ 1、det（ReZ）和ssq（ReZ）中比在任何其他不变量中更好地反映。
SUMMARY
The rotational invariants of the magnetotelluric impedance tensor Z may serve as the most compact 3-D interpretational parameters, since they do not depend on the direction of the inducing field, and they may have various morphological characteristics over 3-D bodies. Their complete system is reviewed for the first time in this paper. It is demonstrated that the complex Z—having eight real-valued independent elements—has seven independent rotational invariants. The complex determinant det(Z) contains three independent real-valued invariants: det(ReZ), det(ImZ) and Imdet(Z) [where det(ReZ)–det(ImZ) =Re det(Z)] and not two, as is usually assumed from its complex character. The same is true for the sum of the squares of the elements of Z, ssq(Z) = Z2xx+ Z2xy+ Z2yx+ Z2yy. Its real-valued invariants are ssq ReZ) = Re2Zxx+ Re2Zxy+ Re2Zyx+ Re2Zyy; ssq(ImZ) = Im2Zxx+ Im2Zxy+ Im2Zyx+ Im2Zyy; and Im ssq(Z) = 2(ReZxx ImZxx+ ReZxy ImZxy+ ReZyx ImZyx+ ReZyy ImZyy) where Re ssq(Z) = ssq(ReZ)–ssq(ImZ), and ssq(ReZ) + ssq(ImZ) =||Z||2f; here ||Z||f is the Frobenius norm of Z. The sets of seven independent rotational invariants can be selected in many different ways. In the classical magnetotelluric set, ReZ1, ImZ1 [where Z1= (Zxy– Zyx)/2], ReZ2, ImZ2 (where the trace Zxx+ Zyy= 2Z2) and the three determinant-based real-valued invariants, det(ReZ), det(ImZ) and Im det(Z), are suggested for use. If the trace, the determinant and ssq(Z) are accepted as basic scalar functions (we call them the mathematical selection of invariants), eight different sets of independent invariants can be selected. The geometrical meaning of rotational invariants is illustrated using two different graphic representations: complex-plane ellipses and Mohr circles. For electromagnetic imaging purposes it is suggested that some of those parameters that are derived from the real tensor ReZ should be used, since at realistic periods the model geometry of thin-sheet-like 3-D models is much better reflected in, for example, ReZ1, det(ReZ) and ssq(ReZ) than in any other invariants.