Analysis of rotational invariants of the magnetotelluric impedance tensor

Abstract

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.