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Table 1 Classification of the impedance tensor components from the standpoint of the DR validity

From: Validity of the dispersion relations in magnetotellurics: Part I—theory

\(Z_{ij}\)

Class 1

Class 2

Class 3

\(\hat{Z}\) causality

Causal

Causal

Non-causal

DR-II validity

Valid

Violated by the amount of a positive monotonic phase lag \(\theta_{n}\) (Eq. 8)

Violated by the amount of a negative monotonic phase lag \(- \theta_{m}\), or a non-monotonic phase lag \(\theta_{n} - \theta_{m}\) (Eq. 8)

DR-I validity

Valid

Valid

Invalid (unless \(\theta_{n} \equiv \theta_{m}\))

Characteristic features of \(\arg \left[ {Z_{ij} } \right]\) on log-period scale

Rolls out of its quadrant upwards and/or rolls in from below

Rolls out of its quadrant downwards and/or rolls in from above at least at some angle of \(\hat{Z}\) rotation

Typical features of \(\ln \left| {Z_{ij} } \right|\)

May reveal a negative cusp if \(\hat{Z}\) is rotated

May reveal a cusp if \(\hat{Z}\) is rotated

How the DR-II could be applied

Directly

By using Eq. (7) with \(n\) unknowns

Consider the DR application to \(\hat{Y} = \hat{Z}^{ - 1}\) or the corresponding inter-site tensors instead

How the DR-I could be applied

Directly

Directly

  1. Class 1—refers to MP components of a causal tensor, class 2—to non-MP components of a causal tensor, and class 3—to all components of a non-causal tensor