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(script capital L signmZ) and script capital L signm det(Z) [where det(ℛeZ) -det(script capital L signmZ) = 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) = Zxx2 + Zxy2 + Zyx2 + Zyy2. Its real-valued invariants are ssq(ℛeZ) = ℛe2Zxx + ℛe2Zxy + ℛe2Zyx + ℛe2Zyy; ssq(script capital L signmZ) = script capital L signm2Zxx + script capital L signm2Zxy + script capital L signm2Zyx + script capital L signm2Zyy; and script capital L signm ssq(Z) = 2(ℛeZxxscript capital L signmZxx+ℛeZxy script capital L signmZxy+ℛeZyx script capital L signmZyX + ℛeZyy script capital L signmZyy) where ℛe ssq(Z) = ssq(ℛeZ) - ssq(script capital L signmZ), and ssq(ℛeZ) + ssq(ℛmZ) = ∥Z∥f2; 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, script capital L signmZ1 [where Z1 = (Zxy -Zyx)/2], ℛeZ2, script capital L signmZ2 (where the trace Zxx + Zyy = 2Z2) and the three determinant-based real-valued invariants, det(ℛeZ), det(script capital L signmZ) and script capital L signm 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.
|Number of pages||10|
|Journal||Geophysical Journal International|
|Publication status||Published - jan. 1 1997|
ASJC Scopus subject areas
- Geochemistry and Petrology