Estimation of the Magnetotelluric Response Function: The Path from Robust Estimation to a Stable Maximum Likelihood Estimator

The robust statistical model of a Gaussian core contaminated by outlying data in use since the 1980s, and which underlies modern estimation of the magnetotelluric (MT) response function, is re-examined from first principles. The residuals from robust estimators applied to MT data are shown to be systematically long-tailed compared to a distribution based on the Gaussian and hence inconsistent with the robust model. Instead, MT data are pervasively described by the stable distribution family for which the Gaussian is an end member, but whose remaining distributions have algebraic rather than exponential tails. The validity of the stable model is rigorously demonstrated using a permutation test. A maximum likelihood estimator (MLE), including the use of a remote reference, that exploits the stable nature of MT data is formulated, and its two-stage implementation, in which stable parameters are first fit to the residuals, and then the MT responses are solved for, with iteration between them, is described. The MLE is inherently robust, but differs from a conventional robust estimator because it is based on a statistical model derived from the data rather than being ad hoc. Finally, the covariance matrices obtained from MT data are pervasively improper as a result of weak non-stationarity, and the Cram,r-Rao lower bound for the improper covariance matrix is derived, resulting in reliable second-order statistics for MT responses. The stable MLE was applied to an exemplar broadband data set from northwest Namibia. The stable MLE is shown to be consistent with the statistical model underlying linear regression and hence is unconditionally unbiased, in contrast to the robust model. The MLE is compared to conventional robust remote reference and two-stage estimators, establishing that the standard errors of the former are systematically smaller than for either of the latter, and that the standardized differences between them exhibit excursions that are both too frequent and too large to be described by Gaussian statistics. These excursions are more prevalent when the tail thickness parameter of the stable distribution is small, and are attributed to rising bias in the robust estimator that is also consistent with the Berry-Ess,en theorem that defines the rate of convergence to the central limit theorem value. An explanation for weak non-stationarity of MT data is proposed, and several extensions to the present work are described.