Modulated Oscillations in Three Dimensions

TL;DR

This paper develops a comprehensive method for analyzing the three-dimensional polarization and oscillation characteristics of nonstationary signals, introducing new measures and formulas for instantaneous frequency, bandwidth, and ellipse parameters.

Contribution

It introduces a novel framework for fully characterizing 3D oscillations using the analytic operator, including new expressions for ellipse parameters and generalized instantaneous measures.

Findings

The method accurately describes nonstationary trivariate signals.

The trivariate instantaneous bandwidth has five distinct contributions.

Application to seismic data demonstrates practical utility.

Abstract

The analysis of the fully three-dimensional and time-varying polarization characteristics of a modulated trivariate, or three-component, oscillation is addressed. The use of the analytic operator enables the instantaneous three-dimensional polarization state of any square-integrable trivariate signal to be uniquely defined. Straightforward expressions are given which permit the ellipse parameters to be recovered from data. The notions of instantaneous frequency and instantaneous bandwidth, generalized to the trivariate case, are related to variations in the ellipse properties. Rates of change of the ellipse parameters are found to be intimately linked to the first few moments of the signal's spectrum, averaged over the three signal components. In particular, the trivariate instantaneous bandwidth---a measure of the instantaneous departure of the signal from a single pure sinusoidal oscillation---is found to contain five contributions: three essentially two-dimensional effects due to the motion of the ellipse within a fixed plane, and two effects due to the motion of the plane containing the ellipse. The resulting analysis method is an informative means of describing nonstationary trivariate signals, as is illustrated with an application to a seismic record.