On the statistics of magnetotelluric rotational invariants

Abstract

The statistical properties of the Swift skew, the phase-sensitive skew and the WAL invariants I1-I7 and Q are examined through analytic derivation of their probability density functionsand/or simulation based on a Gaussian model for the magnetotelluric response tensor. The WAL invariants I1-I2 are shown to be distributed as a folded Gaussian, and are statisticallywell behaved in the sense that all of their moments are defined. The probability densityfunctions for Swift skew, phase-sensitive skew and the WAL invariants I3-I4,I7 and Q arederived analytically or by simulation, and are shown to have no moments of order 2 or more. Since their support is semi-infinite or infinite, they cannot be represented trigonometrically, andhence are inconsistent with a Mohr circle interpretation. By contrast, the WAL invariants I5-I6 are supported on [-1, 1], and are inferred to have a beta distribution based on analysis andsimulation. Estimation of rotational invariants from data is described using two approaches: asthe ratio of magnetotelluric responses that are themselves averages, and as averages of sectionby-section estimates of the invariant. Confidence intervals on the former utilize either Fieller'stheorem,which is preferred because it is capable of yielding semi-infinite or infinite confidenceintervals, or the less accurate delta method. Because section-by-section averages of most of therotational invariants are drawn from distributions with infinite variance, the classical centrallimit theorem does not pertain. Instead, their averaging is accomplished using the median inplace of the mean for location and an order statistic model to bound the confidence interval ofthe median. An example using real data demonstrates that the ratio of averages approach hasserious systematic bias issues that render the result physically inconsistent, while the averageof ratios result is a smooth, physically interpretable function of period, and is the preferredapproach. ©The Author 2013.