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Table 1 Classification of the components of an arbitrary MT tensor \(\hat{X}\) from the standpoint of the DR validity

From: Validity of the dispersion relations in magnetotellurics. Part II: synthetic and field data

\(X_{ij}\) component Class 1 Class 2 Class 3
\(\hat{X}\) causality Causal Causal Non-causal
DR-II validity Valid Violated by the amount of a positive monotonic phase lag \(\theta_{n}\) (Eq. 8 in Part I) Violated by the amount of a negative monotonic phase lag \(- \theta_{m}\), or a non-monotonic phase lag \(\theta_{n} - \theta_{m}\) (Eq. 8 in Part I)
DR-I validity Valid Valid Invalid (unless \(\theta_{n} \equiv \theta_{m}\))
Characteristic features of \(\arg X_{ij}\) on log-period scale Rolls out of its regular range upwards and/or rolls in from below Rolls out of its regular range downwards and/or rolls in from above at least at some angle of \(\hat{X}\) rotation
Typical features of \(\ln \left| {X_{ij} } \right|\) May reveal a negative cusp if \(\hat{X}\) is rotated May reveal a cusp if \(\hat{X}\) is rotated
How the DR-II can applied Directly Using (Eq. 7 in Part I) with \(n\) unknowns Consider the DR application to the inverse tensor \(\hat{X}^{ - 1}\) or the corresponding inter-site tensors instead
How the DR-I can 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