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 |