According to Table 1, an intrinsic DR-II violation in any MT transfer function is always associated with some “unexpected” positive or negative 180° phase flips (i.e., sign reversals), which naturally suggests looking for such irregularities in the synthetic models where some of the MT field components reverse their initial direction with frequency.

### 1-D model with anisotropy

In isotropic 1-D media the off-diagonal impedance components \(Z_{xy}\) and \(Z_{yx}\) are causal and minimum phase (Weidelt 1972; see also Berdichevsky and Pokhotelov 1997; Weidelt 2003 for the case of polarizable layers), while the diagonal components \(Z_{xx}\) and \(Z_{yy}\) are identically equal to zero. However, a completely different situation arises if some of the layers in a 1-D model are considered to be anisotropic: in this case, \(Z_{xx}\) and \(Z_{yy} = - Z_{xx}\) become generally non-zero and may show numerous sign reversals (Marti 2014).

The simplest model we found to reveal a non-MP magnetotelluric response is given in Fig. 1. It consists of two anisotropic layers with the same values of principal resistivities \(\rho_{x'} = 1\; \varOmega {\text{m}}\), \(\rho_{y'} = 100\; \varOmega {\text{m}}\) and \(\rho_{z'} = 100 \; \varOmega {\text{m}}\), but with different strike angles \(\alpha_{\text{strike}}\) (azimuth between the \(x\)-axis and the corresponding principal direction \(x'\) of the resistivity tensor): − 45° for the upper layer, and + 45° for the lower one.

The forward problem was solved in the MT1DAnisoFwd software (author D. Alekseev) based on the numerical algorithm developed by Kovacikova and Pek (2002). The \(Z_{yy}\) response calculated for the upper layer’s thickness \(h\) = 0.1 km, 1 km and 10 km is shown in Fig. 2 (note using Basokur’s frequency normalization \(\hat{Z}^{n} = \hat{Z}/\sqrt {i\omega \mu_{0} }\), discussed in Part I, for the lower curves). The depicted MT data clearly shows that \(Z_{yy}\) represents a causal but non-MP function (class 2), with valid DR-I and distinctly violated DR-II. This violation is associated with the reversal of the corresponding component of the induced electric current, which takes place at the periods when the relative influence of the basement layer on the measured response outweigh that of the upper layer.

Since in a two-layered medium the impedance phase values represent functions of \(h\sqrt \omega\) (Kovacikova and Pek 2002), a tenfold increase of \(h\) naturally “shifts” the whole \(Z_{yy}\) phase curve along with all its irregularities by 2 decades of period rightwards. As a result, the DR-II violation becomes practically observable only if the upper layer’s thickness corresponds to the measured frequency range. For \(h > 1\) km and \(T < 1\) s the influence of the basement layer on \(\hat{Z}\) is negligible and the DR-II in \(Z_{yy}\) is effectively valid. Similarly, for \(h < 0.1\) km and \(T > 10\) s, the subsurface layer ceases to affect the frequency behavior of \(\hat{Z}\), and the DR-II in \(Z_{yy}\) turns out to be valid as well, albeit after the corresponding correction for the initial sign of the measured response. This means that MP or non-MP behavior is essentially a local (in frequency) rather than global attribute of a transfer function, and the same component of \(\hat{Z}\) may be relegated to different classes of the DR validity, depending on the particular period range. For example, in the considered 1-D model with \(h\) = 1 km (Fig. 2, middle column) \(Z_{yy}\) is effectively MP (class 1) within the AMT period range (from 10^{−4} s to 1 s), but turns out to be non-MP (class 2) when considered at MT periods (from 3 · 10^{−3} s to 10^{3} s).

The DR-II violation in \(Z_{yy}\) defined as

$$\tilde{\varphi }_{yy} = \arg Z_{yy} - {\text{DR}}\left[ {\ln \left| {Z_{yy} } \right|} \right],$$

(3)

and calculated for various thicknesses \(h\) of the upper layer is shown in Fig. 3 (dotted lines). All curves have the same smooth shape, which can be easily identified (cf. Fig. 9a in Part 1) as that of the elementary phase lag \(\theta_{1}\), given by

$$\theta_{1} \left( \omega \right) = 2\arctan \frac{{\left| {\varOmega_{1} } \right|}}{\omega }.$$

(4)

This implies that, when considered on the complex \(\varOmega\)-plane, \(Z_{yy} \left( \varOmega \right)\) has a single zero right on the negative half-axis of imaginary frequencies, and its location \(\varOmega_{1} = - iz_{1}\) may be steadily defined by fitting the obtained data with the corresponding theoretic curves (Fig. 3, solid lines). The resulting estimates of \(z_{1}\) (namely 2.8 · 10^{1} rad/s for \(h\) = 0.1 km, 3.1 · 10^{−1} rad/s for \(h\) = 1 km and 2.9 · 10^{−3} rad/s for \(h\) = 10 km) characterize the central frequencies of the DR-II violation in \(Z_{yy}\) for the given \(h\) values and expectedly differ from each other by about two orders of magnitude.

### 2-D model with topography

As follows from numerous published results (see Marti 2014 for a review), incorporation of strongly anisotropic objects into a 2-D model may lead to PROQ not only in diagonal, but also in off-diagonal impedance components as well. However, all such examples apparently belong to the same class 2 of the DR validity as the \(Z_{yy}\) component in the above 1-D model, with the DR-II violated by the amount of an elementary phase lag \(\theta_{n}\) and the DR-I being universally valid. Emergence of a non-causal (class 3) impedance component requires strong perturbations of horizontal magnetic field (see Part I), which could be associated with the phenomenon of excessive concentration of the induced currents in highly conductive elongated structures, usually referred to as “current channeling” (Simpson and Bahr 2005; Berdichevsky and Dmitriev 2008).

Since the rigorous proof of the DR validity in MT impedance on the surface of isotropic 2D Earth exists only for H-polarized models (Weidelt and Kaikkonen 1994), the efforts of many researchers within the last decades were directed to numerical investigations of the MT curves in the vicinity of highly conductive E-polarized 2-D bodies (Berdichevsky and Dmitriev 2008). To the best of our knowledge, none of the proposed models have revealed any observable DR violation. However, in some synthetic examples with huge resistivity contrast (Parker 2010; Selway et al. 2012) the impedance phase curve was found out to be slightly leaving its quadrant downwards and then immediately returning back to “normal” values from the same side (i.e., without any phase flips). Though such behavior does not yet imply the DR violation (see Zorin and Alekseev 2018), it clearly indicates the existence of strong magnetic field perturbations, which cannot be encountered on the surface of H-polarized media. A possible reason why these perturbations do not lead to the breakdown of causality is that in flat models considered by the above authors the induced currents are flowing exclusively below the observation sites. As a result, the horizontal component of the secondary magnetic field produced by a 2-D current channel of any intensity has the same sign as the primary magnetic field, and hence cannot force the observed response to reverse its polarity. In this situation we may yet expect the total magnetic field to be reversed at an observation site located below the conductive body, e.g., due to steep 2-D topography, as that shown in Fig. 4.

To check this assumption, we assembled the given model of an “elevated 2-D conductor” on a 220 × 80 finite-element grid with minimum mesh size of 2.5 × 5 m, and applied to it the NWMT2D code (author A. Kaminsky). The obtained MT response reveals dramatic violations of DR-I and DR-II in the considered transfer functions (Fig. 5), thus showing that both \(\hat{Z}\) and \(\hat{W}\) observed near the cliff are non-causal.

As follows from Table 1, one of the possible ways to assess quality of non-causal (class 3) transfer functions using the DR technique is to consider the corresponding inverse tensors: \(\hat{Y}\) instead of \(\hat{Z}\), \(\hat{Y}^{n} = \hat{Y}\sqrt {i\omega \mu_{0} }\) instead of \(\hat{Z}^{n} = \hat{Z}/\sqrt {i\omega \mu_{0} }\), and so on. Nonsquare tipper matrix \(\hat{W}\) for this purpose might be replaced by the pseudoinverse matrix \(\hat{W}^{ + }\) defined by the following equation:

$$\varvec{H} = \left[ {\begin{array}{*{20}c} {H_{x} } \\ {H_{y} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {W^{ + }_{xz} } \\ {W^{ + }_{yz} } \\ \end{array} } \right]H_{z} = \hat{W}^{ + } H_{z} .$$

(5)

The spectral components of \(Y_{yx}\) and \(W_{yz}^{ + }\) are plotted in Fig. 6. Though these functions represent exactly the same data as contained in \(Z_{xy} = 1/Y_{yx}\) and \(W_{zy} = 1/W_{yz}^{ + }\), respectively, they both belong to the (lower) class 2 of the DR validity and hence are both causal.

The obtained results also reveal an interesting aspect of the DR violations in non-causal 2-D MT tensors, which at first glance may even seem to be self-contradictory. Indeed, on the one side, we see that the impedance \(\hat{Z}\) observed near the cliff is non-causal (Fig. 5); hence, all its components formally belong to the class 3 of the DR validity and both dispersion relations in them are meant to be violated. But at the same time, as shown by Weidelt and Kaikkonen (1994), in any isotropic 2-D model with arbitrary topography, the impedance tensor component corresponding to H-polarization of the medium must be minimum phase.

To solve this apparent contradiction we note that, for the model in hand, Eq. (13 in Part I) is simplified as follows:

$$\left[ {\begin{array}{*{20}c} 0 \\ {Z_{yx} } \\ \end{array} \begin{array}{*{20}c} {Z_{xy} } \\ 0 \\ \end{array} } \right] = \frac{1}{{\det \hat{h}}}\left[ {\begin{array}{*{20}c} 0 \\ {h_{yy} e_{yx} } \\ \end{array} \begin{array}{*{20}c} {h_{xx} e_{xy} } \\ 0 \\ \end{array} } \right],$$

(6)

where the H-polarization magnetic response function \(h_{xx}\) is MP, while the E-polarization magnetic response function \(h_{yy}\) is not and hence have at least one lower half-plane zero. As a consequence, each of these zeros is translated from \(h_{yy}\) right into \(\det \hat{h} = h_{xx} h_{yy}\), thus becoming a non-causal singularity of the whole tensor \(\hat{Z}\). However, \(h_{yy}\) also appears as a multiplier within the \(Z_{yx}\) component (Eq. 6), so that all its poles and zeros are mutually canceled out, and the transverse impedance \(Z_{yx}\) turns out to be MP in the given coordinate system.

To have a closer look at this phenomenon we shall make use of the rotation procedure. Figure 7 shows the apparent resistivity \(\rho_{xy}\), impedance phase \(\arg Z_{xy}\), DR-II violation curve \(\tilde{\varphi }_{xy} = \arg Z_{xy} - {\text{DR}}\left[ {\ln \left| {Z_{xy} } \right|} \right]\) and corresponding best-fit locations of the lower half-plane poles and zeros of the \(Z_{xy}\) component rotated 0°, 15°, 30°, 45°, 60°, 75° and 90° clockwise, respectively. As expected, the locus of two thuswise discovered poles (viz., \(\varOmega_{1} \approx - 8.6i\) rad/s and \(\varOmega_{2} \approx - 91i\) rad/s) does not show any dependence on the rotation angle \(\alpha\), and the shape of the associated negative part of the total DR-II violation \(\tilde{\varphi }_{xy}\) remains the same for all values of \(\alpha\). On the other hand, gradual rotation of the coordinate system forces the zeros of \(Z_{xy} \left( \varOmega \right)\) to move along the complex frequency plane, and, as could be anticipated from the emerging negative cusp of \(\lg \rho_{xy}\), at \(\alpha\) higher than ~ 30° a pair of such zeros approaches the real-frequency axis from above. At \(\alpha \approx\) 39° these zeros cross the \(\omega\)-axis at \(\omega_{1,2} \approx\) ± 20 rad/s (± 3 Hz), turning for a moment the cusp of \(\lg \rho_{xy}\) into an infinite spike and introducing an additional positive 360° phase lag into the total DR-II violation curve. At \(\alpha\)\(\approx\) 56° the zeros meet at the imaginary frequency axis, split up, and start moving from each other in the opposite directions. Finally, at \(\alpha =\) 90°, i.e., in the proper H-polarization mode, both zeros coincide with the corresponding poles, the positive and negative phase flips fully compensate each other, and the dispersion relations in \(Z_{xy}\) become valid (slight mismatch of the compared curves at the very ends of the available frequency range is attributed to the inherent properties of the DR calculation—see Part I for details).

The above example shows that a component of a non-causal MT tensor may indeed be minimum phase—for that to happen, its lower half-plane poles have to be exactly overlapped by the corresponding zeros. However, such MP property is not the same as that in a causal tensor, since an infinitely small rotation of the coordinate system in any direction would lead to its failure. As a result, it seems reasonable to leave the general classification given in Table 1 intact and consider the MP components of a non-causal tensor as those belonging to the third class of the DR validity with the dispersion relations being violated by the amount of zero (i.e., \(\theta_{n} \equiv \theta_{m}\)).

### 2-D model with bathymetry

While placing an MT station below a 2-D conductor requires rather exotic geological conditions on the land surface, this is evidently a common practice for the seafloor observations, which implies that non-causal impedance tensors and tippers be encountered in marine magnetotellurics on a regular basis. Indeed, the unique behavior of the offshore MT data in the coastal regions, which is “seldom, if ever, encountered on land” (Constable et al. 2009), was revealed decades ago and successfully simulated with the help of simple E-polarized 2-D models (e.g., Vanyan and Palshin 1990, White and Heinson 1994). Characterized by numerous negative phase flips, this behavior is reported to be a plain indicator of the DR-II violation in the offshore impedances (Alekseev et al. 2009, Kapinos and Brasse 2011; Kaufman et al. 2014). Surprisingly, none of the authors tried to check the validity of the DR-I in such data since this would have marked a decisive end to the hot discussion of the unsupported statement of Yee and Paulson (1988) about the general causality of \(\hat{Z}\) in real MT data (see Berdichevsky and Dmitriev 2008) by providing a simple and geologically reasonable counterexample. In this section, we fill this gap by pointing out the non-causal behavior of \(\hat{Z}\) in a simple E-polarized model of the coast effect.

The model consists of a conductive (0.3 Ωm) semi-infinite 2-D “ocean” located on top of a more resistive (100 Ωm) “continent”, as shown in Fig. 8 (bottom). The origin of the horizontal axis \(y\) is chosen at the place where the ocean depth reaches half (0.5 km) of its maximum (1 km) value, which happens to be a convenient reference point for estimating the spatial limits of the coast effect influence on MT data (Worzewski et al. 2012). The forward MT responses were calculated using the NWMT2D code at 101 observation sites equally spaced along the 50-km survey line, with an increased period density of 15 per decade, necessary for accurate representation of the anomalous curves.

The map given in the middle part of Fig. 8 shows the absolute value of the DR-II violation in \(Z_{xy}\) defined as

$$\left| {\tilde{\varphi }} \right|_{xy} = \arccos \left[ {\cos \left[ {\arg Z_{xy} - {\text{DR}}\left[ {\ln \left| {Z_{xy} } \right|} \right]} \right]} \right].$$

(7)

This parameter is restricted to lie within the interval \(\left[ {0,\pi } \right]\) and thus is more convenient for examination of the MT responses with particularly complex frequency behavior, than the simple sign-variable parameter \(\tilde{\varphi }_{xy} = \arg Z_{xy} - {\text{DR}}\left[ {\ln \left| {Z_{xy} } \right|} \right]\) used above (cf. Figs. 3, 7). As seen from the map, the DR-II is prominently violated at all underwater stations from the very coastline and up to those located at about \(y \approx\) 30 km. At the same time, the frequency behavior of this violation varies greatly with distance, involving almost all periods within the considered range of 1 s–10,000 s near the foot of the continental slope, but becoming sharply localized around the periods of 100 s–200 s at the remote offshore sites. To further investigate this phenomenon we have examined each survey site separately (some examples are depicted at the top part of Fig. 8) and fitted all the anomalous curves with the help of elementary phase lags described in Part I.

The absolute values of the obtained complex frequencies at which \(Z_{xy} \left( \varOmega \right)\) has a pole (or a pair of poles) are plotted with correspondingly labeled hatched lines over the DR-II violation map in Fig. 8 (middle). These curves are, in fact, describing how the non-causal poles of \(Z_{xy} \left( \varOmega \right)\) change their locations in the complex frequency plane when we travel along the survey line (at any particular station their locations are securely fixed, like those of the zeros shown in Fig. 7). Now, going towards the coastline from the open water, we can see that at \(y \approx\) 30 km, a conjugate pair of poles crosses the real frequency axis at \(\omega \approx\) ± 0.035 rad/s (\(T \approx\) ± 180 s) from above, resulting in the breakdown of \(\hat{Z}\) causality. At \(y \approx\) 19 km, the poles reach the imaginary axis, split up, and start moving from each other, making the period range with the out-of-quadrant impedance phase wider. Around \(y \approx\) 10 km, one of these poles finds yet another pole to combine with and cross the real frequency axis again at \(\omega \approx\) ± 5.2 rad/s (\(T \approx\) ± 1.2 s), while the other one stays alone up to the very coastline, where it finally gets out of the available period range towards higher frequencies.

Non-causal behavior of \(Z_{xy}\) component at the MT stations marked with red color in Fig. 8(bottom) is also confirmed by Fig. 9, which shows the map of normalized DR-I violation parameter for \(Z_{xy}\) defined as

$$\widetilde{\text{Im}}^{n} \left[ {Z_{xy}^{n} } \right] = \frac{{{\text{Im}}Z_{xy}^{n} - {\text{DR}}\left[ {\text{Re} Z_{xy}^{n} } \right]}}{{\left| {Z_{xy}^{n} } \right|}}.$$

(8)

As follows from the figure, at periods corresponding to the non-causal poles of \(Z_{xy}^{n} \left( \varOmega \right)\), the DR-I fails to correctly “predict” the actual sign of \({\text{Im}}Z_{xy}^{n}\), so that the inaccuracy of its application comes up to ~ 200% of the total \(Z_{xy}^{n}\) amplitude, which is in good agreement with the general expectations of the DR-I violation in this type of functions (see Appendix 2 in Part I).

Figure 10 shows the \(\rho_{xy}\) values along the survey line. As should be expected, the regions where the complex plane poles of \(Z_{xy} \left( \varOmega \right)\) approach the \(\omega\)-axis are characterized by emerging positive cusps on the resistivity curves, which turn into infinite spikes at the points where the poles are purely real, i.e., where the corresponding magnetic field component totally vanish. According to the papers (Key and Constable 2011; Worzewski et al. 2012), in simplistic 2-D models of the coast effect, one such point is always present at the distance \(r \approx 0.09\rho_{l} \left( {h/\rho_{w} } \right)\) from the middle of the continental slope and the period \(\tau \approx 0.19\rho_{l} \left( {h/\rho_{w} } \right)^{2}\), where \(\rho_{w}\) and \(\rho_{l}\) stand for the water and land resistivities, respectively, and \(h\) is the ocean depth in kilometers. Substituting \(\rho_{w}\) = 0.3 Ωm, \(\rho_{l}\) = 100 Ωm, and \(h\) = 1 km into the above equations, we get the values very close to those describing the apparent resistivity peak observed at \(y \approx\) 30 km. Though this particular singularity of \(\rho_{xy}\) is not unique for the present model (we see another one at about \(y \approx\) 10 km, \(T \approx\) 1 s, and probably, there are even more of a kind at higher frequencies), it appears to be of most practical importance as the one drawing a clear boundary between the MT sites with causal and non-causal behavior of the impedance tensor at long periods.

In conclusion, it should be noted that the transverse (H-polarization) impedance value \(Z_{yx}\) obtained for the given periods and model parameters demonstrates MP behavior at all MT stations along the survey line. This means that the same mechanism of “compensating zeros” encountered in the above model of an elevated 2-D conductor (Fig. 7) applies to the H-polarized seafloor models as well. Furthermore, the observed MP behavior of the horizontal electric field components ensures causality of the admittance tensor \(\hat{Y} = 1/\hat{Z}\) along the whole survey line, thus solving the issue of the DR application for quality assessment of the MT data heavily distorted by the 2-D coast affect.

### 3-D model with a curved conductor

For completeness we shall consider one of the curved conductive 3-D structures known for producing MT responses with PROQ (e.g., Weckmann et al. 2003; Thiel et al. 2009; Ichihara and Mogi 2009; Kaufman et al. 2014). The most simple and, hence, geologically reasonable among them is the L-form model of Ichihara and Mogi (2009). As shown by these authors, anomalous behavior of \(\hat{Z}\) over the L-shaped conductor is associated with local “reversed channeling” (sign reversal) of the electric current observed in a minor region of the model, while the horizontal magnetic field “varies slightly in all area”. This result suggests that in the L-form model one can encounter the impedance components only of the first two classes of the DR validity, while the emergence of a non-causal impedance tensor on the flat Earth’s surface requires more exotic shape of the conductor, such as the “S” shape proposed by Aleksanova and Blinova (Kaufman et al. 2014). Indeed, in the S-form model the reversed current is observed in a very large region and happens to be so intense that we should expect the associated secondary magnetic field to dominate the total magnetic field within this region, which may even cause its sign reversals along with the breakdown of \(\hat{Z}\) causality.

To observe the MT responses produced by both L-shaped and S-shaped conductors at once we have constructed a combination of these two models shown in Fig. 11. The resulting “SL-shaped” conductor (0.5 Ωm) is located 50 m below the flat Earth’s surface, has constant thickness of 400 m and extends infinitely along both directions of the \(y\)-axis. The background medium is a resistive (500 Ωm) homogeneous half-space.

The forward problem was solved for 57 periods on a rectangular grid of 627,200 (112 × 112 × 50) meshes using the ModEM code of Egbert and Kelbert (2012). To reduce excessive time consumption of the corresponding calculations, they were performed on the supercomputing complex Lomonosov at Moscow State University (Sadovnichy et al. 2013).

Figure 12a, b shows the maps of maximum absolute values of the DR-II violation (Eq. 7) in the off-diagonal impedance components \(Z_{xy}\) and \(Z_{yx}\) within the period range from 10^{−3} s to 10^{3} s. The same maps obtained for the coordinate system rotated 45° clockwise are shown in Fig. 12c, d. The validity of the DR-II in \(\hat{Z}\) components depends on the employed coordinate system, and this dependence also considerably varies from site to site. Particularly, we have discovered that for each point over the L-shaped part of the conductor (e.g., in the site A), there exist some rotation angles at which both \(Z_{xy}\) and \(Z_{yx}\) are minimum phase. In more heavily distorted area over the central part of the conductor (e.g., in the sites B and C) at least one of these components happens to be non-MP, irrespective of the rotation angle. Trying to delineate this area, we found out that a reasonable approximation could be obtained by highlighting the region of the DR-II violation in the Berdichevsky impedance \(Z_{brd}\) (Fig. 12e), which is basically an arithmetic mean of the principal impedance components with an intrinsic property of being a rotational invariant (Berdichevsky and Dmitriev 2008):

$$Z_{\text{brd}} = \frac{{Z_{xy} - Z_{yx} }}{2}.$$

(9)

The DR-II violation map for another important rotational invariant—the magnetic tensor determinant \(\det \hat{M}\)—is given in Fig. 12f. As pointed out in Part I, both \(\hat{Z}\) and \(\hat{W}\) are causal at all MT sites with MP behavior of \(\det \hat{M}\) obtained with the base site placed in an area with regular field behavior (i.e., over the 1-D background or, simply, far enough from strong magnetic anomalies). Consequently, the given DR-II violation map for \(\det \hat{M}\) implies that all components of \(\hat{Z}\) and \(\hat{W}\) observed at the sites A and B should belong to either of the first two classes of the DR validity, while at the site C, we should encounter only those of the class 3, which is fully confirmed by Fig. 13.

For being able to apply the DR technique for quality assessment of non-causal MT data, it is necessary to use some other transfer functions, which contain the same information but are causal. In accordance with Table 1, for this purpose one may try to consider the corresponding inverse or inter-site transfer functions from the given response space. The first approach, proved useful in the above 2-D examples, is not applicable to the present model, since all of the MT field components observed over the middle part of the SL-shaped conductor are greatly distorted. However, if the MT measurements are synchronously taken at some remote station with regular field behavior, it is always possible to express local MT transfer functions via the corresponding causal inter-site transfer functions. For instance, using the data from the base site shown in Fig. 12, the pair of tensors \(\hat{Z}\) and \(\hat{W}\) at sites A, B and C can be equivalently represented by causal tensors \(\hat{M}\), \(\hat{Q} = \hat{Z}\hat{M}\) and \(\hat{S} = \hat{W}\hat{M}\) (see Eqs. 12–17 in Part I), whose components belong either to class 1 or class 2 of the DR validity (Fig. 14).