The Magnetotelluric Phase Tensor A Critical Revie

Authors: John R Booker

Publish Date: 2013/05/23

Volume: 35, Issue: 1, Pages: 7-40

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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 twodimensional 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 tensorconsistent impedance tensor decomposition is introduced Because the phase tensor is a distortionfree 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 modelingI thank my Argentine colleagues Alicia Favetto and Cristina Pomoposiello for many stimulating discussions I also thank them and their field technician Gabriel Giordanengo and my graduate students Aurora Burd and Jeremy Smith for helping me collect MT sites used as examples Friendly arguments with Alan Jones were responsible for much of what is in this paper Support for this research was provided by US National Science Foundation NSF Grants EAR9909390 EAR0310113 and EAR0739116 and US Department of Energy Office of Basic Energy Sciences grant DEFG0399ER14976 MT data in Argentina were collected with equipment from the EMSOC Facility supported by NSF Grants EAR9616421 and EAR0236538 The research in Argentina also received support from the Agencia Nacional de Promocion Cientifica y Tecnologica PICT 2005 No 38253The 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//wwwstatberkeleyedu/~census/ratestpdf and Efron 1982 chapter 6 Operationally it amounts to linear propagation of errorsMonte 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 offdiagonals of Σobs set to zeroTable 1 compares ψ and its uncertainties using the Monte Carlo and delta methods For the highly distorted impedance used in Sect 53 and ignoring the offdiagonal 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

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