Abstract:
A modulated oscillation in two or three dimensions can be represented as the trajectory traced out in space by a particle orbiting an ellipse, the properties of which vary as a function of time. Generalizing ideas from signal analysis, the signal variability can be described in terms of kinematic quantities, the instantaneous moments, that formalize our intuitive notions of time-varying frequency and amplitude. On the other hand, if we observed an ellipse evolving in space we would seek to describe it in terms of its physical moments, such as angular momentum, moment of inertia, etc. The main result of this paper is to show that the two sets of moments are identical. Most significantly, an essential physical quantity---the circulation---is the same as the product of the two most important kinematic quantities, the instantaneous frequency and the squared instantaneous amplitude. In addition to providing a rich set of geometric tools for the analysis of nonstationary oscillations in two or three dimensions, this result also has implications for the practical problem of inferring physical ellipse parameters from the trajectory of a single particle on the ellipse periphery, as is frequently encountered in the study of vortex motions.

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.

Abstract:
The generalizations of instantaneous frequency and instantaneous bandwidth to a bivariate signal are derived. These are uniquely defined whether the signal is represented as a pair of real-valued signals, or as one analytic and one anti-analytic signal. A nonstationary but oscillatory bivariate signal has a natural representation as an ellipse whose properties evolve in time, and this representation provides a simple geometric interpretation for the bivariate instantaneous moments. The bivariate bandwidth is shown to consist of three terms measuring the degree of instability of the time-varying ellipse: amplitude modulation with fixed eccentricity, eccentricity modulation, and orientation modulation or precession. An application to the analysis of data from a free-drifting oceanographic float is presented and discussed.

Abstract:
The concept of a common modulated oscillation spanning multiple time series is formalized, a method for the recovery of such a signal from potentially noisy observations is proposed, and the time-varying bias properties of the recovery method are derived. The method, an extension of wavelet ridge analysis to the multivariate case, identifies the common oscillation by seeking, at each point in time, a frequency for which a bandpassed version of the signal obtains a local maximum in power. The lowest-order bias is shown to involve a quantity, termed the instantaneous curvature, which measures the strength of local quadratic modulation of the signal after demodulation by the common oscillation frequency. The bias can be made to be small if the analysis filter, or wavelet, can be chosen such that the signal's instantaneous curvature changes little over the filter time scale. An application is presented to the detection of vortex motions in a set of freely-drifting oceanographic instruments tracking the ocean currents.

Abstract:
An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of non-negligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias.

Abstract:
The generalized Morse wavelets are shown to constitute a superfamily that essentially encompasses all other commonly used analytic wavelets, subsuming eight apparently distinct types of analysis filters into a single common form. This superfamily of analytic wavelets provides a framework for systematically investigating wavelet suitability for various applications. In addition to a parameter controlling the time-domain duration or Fourier-domain bandwidth, the wavelet {\em shape} with fixed bandwidth may be modified by varying a second parameter, called $\gamma$. For integer values of $\gamma$, the most symmetric, most nearly Gaussian, and generally most time-frequency concentrated member of the superfamily is found to occur for $\gamma=3$. These wavelets, known as "Airy wavelets," capture the essential idea of popular Morlet wavelet, while avoiding its deficiencies. They may be recommended as an ideal starting point for general purpose use.

Abstract:
In many application areas a goal of statistical analysis is to obtain parametric models for observed time series. Bivariate time series arising on an equal footing are often represented as complex-valued processes. A special subclass exhibiting rotational symmetry, known as proper processes, has been particularly well studied. We introduce a simple model for an improper complex-valued autoregressive process. The process is a natural extension of the proper complex-valued autoregressive process, which is extended here to permit elliptical oscillations in a bivariate signal. The process we propose is Markovian and order one, such that the process is only dependent on its value at the previous time step. We detail how the process relates to bivariate autoregressive processes, and provide conditions for stationarity. We derive the form of the covariance and relation sequence, and describe how inference can be efficiently performed in the frequency domain. We demonstrate the practical utility of the process in capturing elliptical motions that are naturally present in seismic time series signals.

Abstract:
A method for extracting time-varying oscillatory motions from time series records is applied to Lagrangian trajectories from a numerical model of eddies generated by an unstable equivalent barotropic jet on a beta plane. An oscillation in a Lagrangian trajectory is represented mathematically as the signal traced out as a particle orbits a time-varying ellipse, a model which captures wavelike motions as well as the displacement signal of a particle trapped in an evolving vortex. Such oscillatory features can be separated from the turbulent background flow through an analysis founded upon a complex-valued wavelet transform of the trajectory. Application of the method to a set of one hundred modeled trajectories shows that the oscillatory motions of Lagrangian particles orbiting vortex cores appear to be extracted very well by the method, which depends upon only a handful of free parameters and which requires no operator intervention. Furthermore, vortex motions are clearly distinguished from wavelike meandering of the jet---the former are high frequency, nearly circular signals, while the latter are linear in polarization and at much lower frequencies. This suggests that the proposed method can be useful for identifying and studying vortex and wave properties in large Lagrangian datasets. In particular, the eccentricity of the oscillatory displacement signals, a quantity which is not normally considered in Lagrangian studies, emerges as an informative diagnostic for characterizing qualitatively different types of motion.

Abstract:
Subsurface float and moored observations are presented to show for the first time the formation and propagation of anticyclonic submesoscale coherent vortices that transport relatively cold, fresh subpolar water to the interior subtropical North Atlantic. Acoustically tracked RAFOS floats released in the southward-flowing Western Boundary Current at the exit of the Labrador Sea reveal the formation of three of these eddies at the southern tip of the Grand Banks (42 N, 50 W). Using a recently developed method to detect eddies in float trajectories and estimate their kinematic properties, it was found that the eddies had average rotation periods of 5--7 days at radii of 1025 km, with mean rotation speeds of up to 0.3 m/s. One especially long-lived (5.1 months) eddy crossed under the Gulf Stream path and translated southwestward in the subtropical recirculation to at least 35 N, where it hit one of the Corner Rise Seamounts. Velocity, temperature and salinity measurements from a nine-month deployment of two moorings south of the Gulf Stream at 38 N, 50 W reveal the passage of at least two eddies with similar hydrographic and kinematic properties. The core temperature and salinity properties of the eddies imply their formation at intermediate levels of the Labrador Current south of the Tail of the Grand Banks. These observations confirm earlier speculation that eddies form in this region and transport anomalously cold, low-salinity water directly into the subtropical interior. Possible formation mechanisms and potential importance of these eddies to interior ventilation and the equatorward spreading of Labrador Sea Water are discussed.

Abstract:
This paper proposes stochastic models for the analysis of ocean surface trajectories obtained from freely-drifting satellite-tracked instruments. The proposed time series models are used to summarise large multivariate datasets and infer important physical parameters of inertial oscillations and other ocean processes. Nonstationary time series methods are employed to account for the spatiotemporal variability of each trajectory. Because the datasets are large, we construct computationally efficient methods through the use of frequency-domain modelling and estimation, with the data expressed as complex-valued time series. We detail how practical issues related to sampling and model misspecification may be addressed using semi-parametric techniques for time series, and we demonstrate the effectiveness of our stochastic models through application to both real-world data and to numerical model output.