The Magnetotelluric Phase Tensor: A Critical Review

Abstract

The magnetotelluric (MT) phase tensor is a property of the MT impedance that is resistant to a common form of distortion due to unresolvable local structure. Review of the theory leads to a new geometrical description that cleanly separates information about directionality and dimensionality of regional conductivity structure. This information is widely used to justify two-dimensional (2D) interpretation, but the case is seldom made convincingly. In particular, errors are largely ignored and it is not understood that full data covariance is essential for accurate error bars. It is also common to use 2D impedance tensor decompositions when the phase tensor shows this model to be inconsistent with the data. A phase tensor-consistent impedance tensor decomposition is introduced. Because the phase tensor is a distortion-free 3D response, it should be used as data for 3D inversions. Until codes for this become more developed, comparison of predicted and observed phase tensors can ascertain whether 3D aspects of the data have been adequately fit by other inversions or modeling.

Appendices

Appendix 1

1.1 Computing phase tensor parameters

All standard SVD algorithms easily compute the magnitudes of the tangents of the principal phase angles, but SVDs are not unique and so extracting an ellipse axis direction θ and determining the correct quadrant for the phases is a problem. A practical way to compute θ follows directly from the geometrical development above. With ω measured in the right-hand sense from the x-axis of the measurement coordinates, find the angle ω ₀ such that either the vector

$$ {\mathbf{p}}(\omega_{0} )\, = \,{\varvec{\Upphi}}\,{\mathbf{c}}(\omega_{0} + \psi )\, $$

(46)

is parallel to the vector c(ω₀ − ψ) or that

$$ \left[ {\frac{\partial }{\partial \omega }|{\mathbf{p}}(\omega )|^{2} } \right]_{{\omega = \omega_{0} }} = 0 $$

(47)

The first condition is probably easier to code in lower-level languages such as C or FORTRAN and should be more stable in nearly degenerate cases. The second is coded easily in high level languages such as Matlab that have built-in routines to find the zero of a function. Then, θ = ω ₀ and the lengths of the semi-axes are

$$ |\Upphi_{a} | = \left| {{\varvec{\Upphi}}_{\text{ellipse}} {\mathbf{c}}(\theta + \psi )} \right| $$

(48)

$$ |\Upphi_{b} | = \left| {{\varvec{\Upphi}}_{\text{ellipse}} {\mathbf{c}}(\theta + \psi + 90^{\text{o}} )} \right| $$

(49)

where Φ _ellipse is computed using Eq. (15). The direction of circulation around the ellipse relative to the unit circle is determined by computing p(ω) for two slightly increasing values of ω and seeing which direction the vector p rotates. Finally, determining the signs of Φ _a and Φ _b from the circulation of the ellipse is summarized in Table 2 in Section A.3.

1.2 Estimating Uncertainties

The phase tensor is a nonlinear function of the impedance, and the phase tensor decomposition parameters are nonlinear functions of the phase tensor. The situation is made worse by the fact that the phase tensor and derived parameters such as normalized skew are ratios of random variables. This can lead to distributions with formally infinite second moments. In a rigorous sense, the variance is then undefined (see Chave 2012b). This does not mean, however, that the uncertainties are unbounded or even that they are difficult to estimate. Statisticians have invented what is commonly called the “delta method” that is applicable to such situations (see Freedman, http://www.stat.berkeley.edu/~census/ratest.pdf and Efron 1982, chapter 6). Operationally, it amounts to linear propagation of errors.

In the context of phase tensor parameters, the N-parameters by 4 complex matrix J of derivatives of the parameters with respect to the complex data are computed and then the N by N covariance of the parameters is given by

$$ {\varvec{\Upsigma}}_{\text{parameters}} = {\mathbf{J\,\Upsigma }}_{\text{obs}} {\mathbf{J}}^{T} $$

(50)

where Σ_obs is the 4 by 4 complex covariance of the impedance (see Efron 1982, p43, eq 6.22 for the real case). The error estimates of the parameters are the square root of the diagonal of Σ_parameters. For most parameters, computing the derivatives is only practical numerically. The real and imaginary parts of the impedance elements are separately perturbed up and down by a small amount, and changes in the computed parameters are divided by twice the magnitude of the perturbation. Care is needed to detect and correct for parameter quadrant jumps between the positive and negative perturbations. The complex derivative is

$$ {\mathbf{J}} = {\mathbf{J}}_{\text{real}} + i{\mathbf{J}}_{\text{imag}} $$

(51)

Monte Carlo simulations can be used to verify the delta method results and illustrate the problems. I concentrate here on ψ because Jones (2012b, p265) singled it out for poor statistical performance. A very large number of realizations (10⁴ is generally much too small; I use 10⁶) are generated by adding random noise to the real and imaginary impedance elements using eight independent normal distributions. This generates circularly symmetric Gaussian noise in each complex element. The distributions are scaled so that their standard deviations equal the standard errors estimated from the observations. The parameters are calculated for each realization and their means and standard deviations are computed from all the realizations. It is important to point out that this does not simulate the effect of covariance and can only be compared to delta method results with the off-diagonals of Σ_obs set to zero.

Table 1 compares ψ and its uncertainties using the Monte Carlo and delta methods. For the highly distorted impedance used in Sect. 5.3 and ignoring the off-diagonal covariance, the error estimates are essentially identical with no significant bias. However, including the full covariance in the delta method decreases the error estimate by more than a factor of three!

Figure 10a is a Quantile–Quantile (Q–Q) plot of the ψ Monte Carlo realizations versus a normal distribution. Both distributions are normalized so that their means are 0 and their standard deviations are 1. The plot is constructed by sorting the realizations of ψ by size. Then, the number of points with values in fixed intervals are counted and plotted against the number expected for the normal distribution. If the points lie on a straight line with slope 1, the distributions are identical. They are clearly extremely close. Strictly speaking, angles like ψ should be compared to a “wrapped” Gaussian. However, even with a million points, the probability of one point outside of 4.5 standard deviations (±5°) is too small for any such points to exist. With such a small total angle spread, the wrapped and standard Gaussian are indistinguishable.

Table 1 The phase tensor skew angle ψ (deg) and its 1 standard deviation uncertainty at two MT sites using Monte Carlo (MC) with 10⁶ realizations and Delta (Δ) methods

Fig. 10

a Q–Q plot of 10⁶ random realizations of phase tensor skew angle ψ generated using observed impedance and error estimates at 80 s for site pam606. The realizations have been normalized so that they have zero mean and unit standard deviation. They are plotted against a normal distribution with the same mean and standard deviation. The axes units are standard deviations. If the ψ distribution is normal, the points should all fall on the straight line. b Histogram of 10⁶ realizations of phase tensor normalized skew for the same impedance, but whose error estimates have been inflated as described in the text. The reason for the non-Gaussian secondary hump at ±180° and the justification for suppressing it are discussed in the text

The situation is more complicated for data with larger errors. To simulate this, the estimated errors of the off-diagonal impedance phases are increased to 1.5° (10 % of their apparent resistivity or 5 % of their magnitude). The error estimates of the diagonal element magnitudes are set equal to the error estimate of the off-diagonal in the same row on the premise that the noise is from the electric field. The Monte Carlo estimate of the uncertainty in Table 1b is now much larger than the delta method ignoring the off-diagonal covariance. Comparison with the full covariance is not shown because I know of no consistent way to alter the off-diagonal covariance. To see what has happened, look at the histogram of the ψ realizations in Fig. 10b. There are secondary outlier peaks near ± 180°. These are due to realizations that have pushed ψ out of quadrant and produced a distribution that is clearly not a wrapped Gaussian. The poor performance of the “High Noise 3D” error simulation reported by Jones (2012b), Table 6.5a) may be the same problem. As noted in Sect. 2.2.3 and Fig. 11b of Appendix section “Principal Phase Signs,” values of ψ near ± 180° have the same degree of three dimensionality as values near 0. Thus, for a moderately 3D impedance with ψ _mean about 10°, outliers near ±180° are much more likely than highly 3D values of ψ near ±90°. A rigorous study of how this quadrant wrapping should be “unwound” is needed, but it is reasonable to suppress the influence of these “antipodal” outliers by simply “trimming” (i.e., dropping) angles outside of ±90°. Doing so (see Table 1b) brings the “trimmed” Monte Carlo standard deviation into close agreement with the delta method. The bias is only 1 % of the error. To avoid spurious asymmetry in the ψ distribution, one can make the region of included data symmetric about the mean. This reduces the already negligible bias a bit and so is not worth the effort.

Fig. 11

Phase tensor ellipses with axes lengths and coordinate rotation identical to Fig. 1b. The normalized skew angle in (a) is the same as Fig. 1b, but the sign of the principal value Φ _a is negative and thus its principal phase is out of quadrant. Note that ellipse circulation is counter to that of the unit circle and that the ellipse starting point has moved to the opposite side of the origin. Note also that ψ is still measured from the original ellipse axis direction. This remains true when ψ is increased from 20 to 160° in (b)

The large impact of the covariance in Table 1a is a common, but not general, situation. Table 1c also shows results for a site that has a larger normalized skew and is thus more 3D, but has less distortion (α _x  = 23.8°; α _y  = − 22.7°). The effect of covariance is much smaller for this site. Inflating the errors at this second site by the same amount as in Table 1b leads to a ψ distribution with only 7 out of 10⁶ antipodal values with essentially no impact on the Monte Carlo mean or error estimate (see Table 1d).

In my experience, large effects of covariance and strong distortion go together, but should never be assumed absent. Rotation always introduces covariance, and rotation of the variances alone will not in general give correct variances in the rotated frame. A rotation of 45° is the worst case. For example, ignoring the off-diagonals of the covariance at site pam604 (which is almost unaffected by covariance in Table 1(c)), the uncertainty at 80 s after a rotation of 45° is 32 % too small for |Z _xy | and 34 % too large for |Z _yx |. Paraphrasing the title of Jones and Groom (1993): “rotate [ignoring covariance] at your peril.” It is conceivable that their conclusions were actually the result of ignoring covariance.

One should not expect delta method uncertainties to be accurate for angles when the standard errors are greater than about ± 20° (4.5 standard deviations = 90°). Since angles with larger errors are of little use, this is not a practical problem. However, impedance covariance should never be ignored, especially for strongly distorted data. Computation of skew angle errors using a Monte Carlo method is clearly problematic. Not only must one identify and compensate for quadrant-flipped tails, but one cannot easily incorporate error covariance. Both can lead to grossly incorrect error estimates.

1.3 Principal Phase Signs

The sgn function is +1 when its argument is positive and −1 when its argument is negative. It can be used to rewrite (14) as

$$ {\mathbf{p}}\,(\omega ) = \left[ {\begin{array}{*{20}c} {|\Upphi_{a} |} & 0 \\ 0 & {|\Upphi_{b} |} \\ \end{array} } \right]\,{\mathbf{S}}\,{\mathbf{R}}(\psi )\,{\mathbf{c}}(\omega ) $$

(52)

where

$$ {\mathbf{S}} = \left[ {\begin{array}{*{20}c} {\text{sgn} (\Upphi_{a} )} & 0 \\ 0 & {\text{sgn} (\Upphi_{b} )} \\ \end{array} } \right]\,\, $$

(53)

is a “reflection” matrix. When Φ _a and Φ _b have opposite signs, this reflection reverses the circulation about the ellipse. Additionally, when Φ _a  < 0 and Φ _b  > 0, the starting point for the ellipse circulation moves to the opposite side of the origin. These relationships are summarized in Table 2 and examples are shown in Fig. 11. Plotting only 270° of the unit circle and ellipse and starting ω at θ + ψ (the semi-axis in the θ direction) makes it easy to see these relationships. Note that ψ is always measured from the “unreflected” semi-axis direction θ.

Table 2 Signs of phase tensor principal values as a function of circulation directions for the unit circle and phase tensor ellipse and ω ₀, the starting angle of ellipse circulation

1.4 Site Locations and Data

The locations of the three MT sites used in this paper are shown in Fig. 4. They were collected in cooperation with Argentine colleagues. Site pam885 (31.724°S 58.627°W) used in Fig. 2 is in Entre Rios Province near the border between Argentina and Uruguay at the Uruguay River. It was collected using a Narod NIMS system sampling at 4 Hz. Site pam606 (31.531°S 68.839°W) is used in Figs. 8, 9, and 10 and in Table 1 is about 25 km west of the city of San Juan in a side valley of the San Juan River canyon in the Pre-Cordillera mountains of San Juan Province, Argentina. Site pam604 (31.509°S 68.999°W) also used in Table 1 is 15 km up the main canyon west of pam606. They were collected using LRMT systems (Phoenix clones of the Canadian Geological Survey LIMS) sampling at 5 s. All sites used lead–lead chloride electrodes separated by about 100 m. These data are available from a link at http://earthweb.ess.washington.edu/booker and will become available from the IRIS DMC (http://www.iris.edu/data/).

Appendix 2

The misfit tensor defined by Heise et al. (2007) is

$$ {\varvec{\Updelta}} = {\mathbf{I}} - \frac{1}{2}\left( {\hat{\varvec{\Upphi}}}^{ - 1} {\varvec{\Upphi}} + {\varvec{\Upphi}} {\hat{\varvec{\Upphi}}}^{ - 1} \right) $$

(54)

(where the “hat” ^ signifies the phase tensor predicted by the model) with the exception that the role of the observed and predicted phase tensors have been interchanged. Rewriting Δ using the rotationally invariant parameters of the phase tensor parameterization (16), we can show that (54) depends on the coordinate system in which it is computed.

Δ is the average of “right-handed” and “left-handed” relative misfits

$$ {\varvec{\Updelta}}_{\text{right}} = \left( {\hat{\varvec{\Upphi }}} - {\varvec{\Upphi}} \right){\varvec{\Upphi}}^{ - 1} = {\mathbf{I}} - {\varvec{\Upphi }}{\hat{\varvec{\Upphi }}}^{ - 1} $$

(55)

$$ {\varvec{\Updelta}}_{left} \,\,\,\, = {\hat{\varvec{\Upphi }}}^{ - 1} \left( {\hat{\varvec{\Upphi }}} - {\varvec{\Upphi}} \right)\,\,\, = {\mathbf{I}} - {\hat{\varvec{\Upphi }}}^{ - 1} {\varvec{\Upphi}} $$

(56)

It is now clear why interchanging the roles of the observed and model phase tensors is a good idea. The predicted phase tensor is not subject to random noise and makes a more stable quantity against which to compare the phase tensor principal values residual.

Choosing the coordinate system aligned with the predicted phase tensor, we can use the more compact notation

$$ {\hat{\varvec{\Uplambda }}} \equiv \left[ {\begin{array}{*{20}c} {\hat{\Upphi }_{a} } & 0 \\ 0 & {\hat{\Upphi }_{b} } \\ \end{array} } \right] $$

(57)

and (16) to write the predicted phase tensor

$$ {\hat{\varvec{\Upphi}}} = {\hat{\varvec{\Uplambda}}}{\mathbf{R}}_{\hat{\psi}} $$

(58)

The observed phase tensor ellipse axes do not necessarily align with the predicted ellipse. Defining δθ as the angle of the observed ellipse axes to the predicted ellipse coordinate system, we can again use (16) to write the observed phase tensor in the predicted phase tensor coordinates

$$ {\varvec{\Upphi}} = {\mathbf{R}}_{\delta \theta } \left( {{\mathbf{\Uplambda R}}_{\psi } } \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathbf{R}}_{\delta \theta }^{ - 1} = \left( {{\mathbf{R}}_{\delta \theta } {\mathbf{\Uplambda R}}_{\delta \theta }^{ - 1} } \right)\,{\kern 1pt} {\mathbf{R}}_{\psi } \equiv {\mathbf{L}}\,{\mathbf{R}}_{\psi } $$

(59)

where L is the observed principal value matrix rotated to the predicted ellipse frame. Finally, in the predicted ellipse coordinate frame, the right-handed misfit tensor (55) becomes

$$ {\varvec{\Updelta}}_{right} = {\mathbf{I}} - {\mathbf{L}}\,{\mathbf{R}}_{\delta \psi } {\hat{\varvec{\Uplambda }}}^{ - 1} \equiv {\mathbf{I}} - {\tilde{\mathbf{L}}}\,{\hat{\varvec{\Uplambda }}}^{{ - {\mathbf{1}}}} $$

(60)

where the skew angle residual$\delta \psi \equiv \psi - \hat{\psi }$and

$$ {\tilde{\mathbf{L}}} \equiv {\mathbf{LR}}_{\delta \psi } = {\mathbf{R}}_{\delta \theta } {\mathbf{\Uplambda R}}_{\delta \theta - \delta \psi } $$

(61)

In the same coordinate frame, the left-handed misfit tensor (38) is

$$ {\varvec{\Updelta}}_{left} = {\mathbf{I}} - {\mathbf{R}}_{{\hat{\psi }}}^{ - 1} {\hat{\varvec{\Uplambda }}}^{ - 1} {\mathbf{L}}\,{\mathbf{R}}_{\psi } $$

(62)

This can be simplified by rotating its coordinates by normalized skew angle $\hat{\psi }$ relative to the axes of the predicted ellipse and using the fact that ${\hat{\varvec{\Uplambda }}}$ is diagonal. The rotated misfit tensor is given the new name

$$ {\tilde{\mathbf{\Updelta }}}_{left} \equiv {\mathbf{R}}_{{\hat{\psi }}} {\varvec{\Updelta}}_{left} {\mathbf{R}}_{{\hat{\psi }}}^{ - 1} = {\mathbf{I}} - \,{\hat{\varvec{\Uplambda }}}^{ - 1} {\mathbf{L}}\,{\mathbf{R}}_{\delta \psi } = {\mathbf{I}} - \,{\hat{\varvec{\Uplambda }}}^{ - 1} {\tilde{\mathbf{L}}} = {\mathbf{I}} - \,{\tilde{\mathbf{L}}}^{{\mathbf{T}}} {\hat{\varvec{\Uplambda }}}^{ - 1} $$

(63)

The first thing to notice about (60) and (63) is that they differ only in the transpose of ${\tilde{\mathbf{L}}}$. Thus, Δ _right and Δ _left contain the same coordinate-invariant information. Averaging them confuses the situation because their difference is a consequence of the coordinate systems in which they are evaluated (which are rotated ψ relative to each other). If one wants to plot a phase tensor misfit tensor using relative residuals, it is better to choose one of the definitions (56), (57), (60) or (63), and not (54).

The second thing to notice is that, when the skew angle residual is small, the net rotation due to R _δψ will be negligible even when ψ itself cannot be ignored. Thus, these misfit tensors are only weakly dependent on the magnitude of ψ and hence on the parameter that is unambiguously 3D. Better and far simpler ways of addressing how much a 2D or 3D inverse or model is violating 3D aspects of the data are pseudosections and maps of the normalized skew residual δψ and the angles δθ between the predicted and observed phase tensor ellipses.

Finally, if both δψ and δθ are negligible, (60) and (63) reduce to the result discussed earlier:

$$ {\varvec{\Updelta}}_{\text{left}} = {\varvec{\Updelta}}_{\text{right}} = {\mathbf{I}} - {\varvec{\Uplambda}}\,{\hat{\varvec{\Uplambda }}}^{ - 1} = \left[ {\begin{array}{*{20}c} {{\hat{\Upphi }_{a} - \Upphi_{a} }}/{{\hat{\Upphi }_{a} }} & 0 \\ 0 & {{\hat{\Upphi }_{b} - \Upphi_{b} }}/{{\hat{\Upphi }_{b} }} \\ \end{array} } \right] $$

(64)

Appendix 3

Smith (1995) showed that any static distortion matrix can be parameterized by

$$ {\mathbf{D}} = \left[ {\begin{array}{*{20}c} a & c \\ b & d \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\cos \,(\alpha_{x} )} & { - \sin \,(\alpha_{y} )} \\ {\sin \,(\alpha_{x} )} & {\cos \,(\alpha_{y} )} \\ \end{array} } \right]\,\left[ {\begin{array}{*{20}c} {g_{x} } & 0 \\ 0 & {g_{y} } \\ \end{array} } \right] $$

(65)

where

$$ \tan \,(\alpha_{x} ) = \frac{b}{a}\quad \tan \,(\alpha_{y} ) = \frac{ - c}{d}\quad g_{x} = \sqrt {a^{2} + b^{2} } \quad g_{y} = \sqrt {c^{2} + d^{2} } $$

(66)

“Distortion angle” α _x is the rotation and the “gain” factor g _x multiplies the magnitude of the x-component of the regional electric field; α _y is the rotation and g _y is the gain of the y-component of the regional electric field. It is then easy to show that the distorted regional impedance in regional coordinates can be expressed

$$ \begin{aligned} {\mathbf{Z}}_{D} & = {\mathbf{DZ}} = \left[ {\begin{array}{*{20}c} 1 & { - \tan (\alpha_{y} )} \\ {\tan (\alpha_{x} )} & 1 \\ \end{array} } \right]\,\left[ {\begin{array}{*{20}c} {\cos (\alpha_{x} )\,g_{x} Z_{xx} } & {\cos (\alpha_{x} )\,g_{x} Z_{xy} } \\ {\cos (\alpha_{y} )\,g_{y} Z_{yx} } & {\cos (\alpha_{y} )g_{y} Z_{yy} } \\ \end{array} } \right] \\ & \equiv {\tilde{\mathbf{D}}} {\tilde{\mathbf{Z}}} \\ \end{aligned} $$

(67)

where ${\tilde{\mathbf{D}}}$ and ${\tilde{\mathbf{Z}}}$ are just rescaled versions of D and Z.

Multiplying (67) by ${\tilde{\mathbf{D}}}^{ - 1}$ gives

$$ \begin{aligned} {\tilde{\mathbf{D}}}^{ - 1} {\mathbf{Z}}_{D} & = \frac{1}{{\det \left( {{\tilde{\mathbf{D}}}} \right)}}\left[ {\begin{array}{*{20}c} 1 & {\tan (\alpha_{y} )} \\ { - \tan (\alpha_{x} )} & 1 \\ \end{array} } \right]\,\left[ {\begin{array}{*{20}c} {Z_{Dxx} } & {Z_{Dxy} } \\ {Z_{Dyx} } & {Z_{Dyy} } \\ \end{array} } \right] \\ & = \frac{1}{{\det \left( {{\tilde{\mathbf{D}}}} \right)}}\left[ {\begin{array}{*{20}c} {Z_{Dxx} + Z_{Dyx} \tan (\alpha_{y} )} & {Z_{Dxy} + Z_{Dyy} \tan (\alpha_{y} )} \\ { - Z_{Dxx} \tan (\alpha_{x} ) + Z_{Dyx} } & { - Z_{Dxy} \tan (\alpha_{x} ) + Z_{Dyy} } \\ \end{array} } \right] \\ & = \left[ {\begin{array}{*{20}c} {\cos (\alpha_{x} )\,g_{x} Z_{xx} } & {\cos (\alpha_{x} )\,g_{x} Z_{xy} } \\ {\cos (\alpha_{y} )\,g_{y} Z_{yx} } & {\cos (\alpha_{y} )g_{y} Z_{yy} } \\ \end{array} } \right] \\ \end{aligned} $$

(68)

The magnitude of the ratio of the elements in the first row of (68) gives

$$ \left| {\frac{{\cos (\,\alpha_{x} )\,g_{x} Z_{xx} }}{{\cos (\,\alpha_{x} )\,g_{x} Z_{xy} }}} \right| = \left| {\frac{{Z_{xx} }}{{Z_{xy} }}} \right| = Q_{x} = \left| {\frac{{Z_{Dxx} + Z_{Dyx} \tan (\alpha_{y} )}}{{Z_{Dxy} + Z_{Dyy} \tan (\alpha_{y} )}}} \right| $$

(69)

The magnitude of the ratio of the elements in the second row of (68) gives

$$ \left| {\frac{{\cos (\,\alpha_{y} )\,g_{y} Z_{yy} }}{{\cos (\,\alpha_{y} )\,g_{y} Z_{yx} }}} \right| = \left| {\frac{{Z_{yy} }}{{Z_{yx} }}} \right| = Q_{y} = \left| {\frac{{Z_{Dyy} - Z_{Dxy} \tan (\alpha_{x} )}}{{Z_{Dyy} - Z_{Dyx} \tan (\alpha_{x} )}}} \right| $$

(70)

Thus, satisfying constraints (42) is a simple matter of searching for the angles $\alpha_{y}^{\hbox{min} }$ and $\alpha_{x}^{\hbox{min} }$ that minimize the terms on the right of (69) and (70).