Pure states, polarized waves, and principal components in the spectra of multiple, geophysical time-series

Abstract

This paper outlines the connection between pure states, polarized waves, and principal components in the spectral representation of multichannel or multivariate geophysical time series. If a wave is a pure state or a polarized wave (e.g. a teleseismic P-wave), the multidimensional spectral representation can be reduced to a one-dimensional spectral representation. The objective of this paper is to develop estimators of the spectral representations of these pure states, which are based on a number of optimality criteria. If nothing is known about the noise in the data, then an estimator similar to the principal components of real multivariate analysis is most useful. If some knowledge of the spectral density matrix of the noise is available, then a number of other optimality criteria can be used. Minimization of the prediction error in the spectral representation of the pure states leads to estimators similar to those used to estimate factor-scores in factor-analytic methods. These estimators of the spectral representations of polarized waves should be extremely useful to the geophysical experimenter because they allow a more objective evaluation of the parameters of the wave, and because they allow the multichannel data to be filtered to extract polarized waves, without specific knowledge of the type of polarization (e.g. elliptical or linear, and orientation of polarization ellipse). Some practical examples using spectral representations of ULF (1–20 mHz) magnetic fields are given in order to illustrate the usefulness of these methods in interpreting multichannel data.