The surface drifting buoys, or drifters, of the Global Drifter Program (GDP) are predominantly tracked by the Argos positioning system, providing drifter locations with O(100 m) errors at nonuniform temporal intervals, with an average interval of 1.2 h since January 2005. This data set is thus a rich and global source of information on high‐frequency and small‐scale oceanic processes, yet is still relatively understudied because of the challenges associated with its large size and sampling characteristics. A methodology is described to produce a new high‐resolution global data set since 2005, consisting of drifter locations and velocities estimated at hourly intervals, along with their respective errors. Locations and velocities are obtained by modeling locally in time trajectories as a first‐order polynomial with coefficients obtained by maximizing a likelihood function. This function is derived by modeling the Argos location errors with t location‐scale probability distribution functions. The methodology is motivated by analyzing 82 drifters tracked contemporaneously by Argos and by the Global Positioning System, where the latter is assumed to provide true locations. A global spectral analysis of the velocity variance from the new data set reveals a sharply defined ridge of energy closely following the inertial frequency as a function of latitude, distinct energy peaks near diurnal and semidiurnal frequencies, as well as higher‐frequency peaks located near tidal harmonics as well as near replicates of the inertial frequency. Compared to the spectra that can be obtained using the standard 6‐hourly GDP product, the new data set contains up to 100% more spectral energy at some latitudes.

Abstract. Stochastic processes exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm), with the spectral slope at high frequencies being associated with the degree of small-scale roughness or fractal dimension. However, a broad class of real-world signals have a high-frequency slope, like fBm, but a plateau in the vicinity of zero frequency. This lowfrequency plateau, it is shown, implies that the temporal integral of the process exhibits diffusive behavior, dispersing from its initial location at a constant rate. Such processes are not well modeled by fBm, which has a singularity at zero frequency corresponding to an unbounded rate of dispersion. A more appropriate stochastic model is a much lesser-known random process called the Matérn process, which is shown herein to be a damped version of fractional Brownian motion. This article first provides a thorough introduction to fractional Brownian motion, then examines the details of the Matérn process and its relationship to fBm. An algorithm for the simulation of the Matérn process in O(N log N ) operations is given. Unlike fBm, the Matérn process is found to provide an excellent match to modeling velocities from particle trajectories in an application to two-dimensional fluid turbulence.

Summary The paper proposes stochastic models for the analysis of ocean surface trajectories obtained from freely drifting satellite‐tracked instruments. The time series models proposed are used to summarize large multivariate data sets and to infer important physical parameters of inertial oscillations and other ocean processes. Non‐stationary time series methods are employed to account for the spatiotemporal variability of each trajectory. Because the data sets 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 by using semiparametric 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.

The Whittle likelihood is a widely used and computationally efficient pseudo-likelihood. However, it is known to produce biased parameter estimates for large classes of models. We propose a method for de-biasing Whittle estimates for second-order stationary stochastic processes. The de-biased Whittle likelihood can be computed in the same O(n log n) operations as the standard approach. We demonstrate the superior performance of the method in simulation studies and in application to a large-scale oceanographic dataset, where in both cases the de-biased approach reduces bias by up to two orders of magnitude, achieving estimates that are close to exact maximum likelihood, at a fraction of the computational cost. We prove that the method yields estimates that are consistent at an optimal convergence rate of n −1/2 , under weaker assumptions than standard theory, where we do not require that the power spectral density is continuous in frequency. We describe how the method can be easily combined with standard methods of bias reduction, such as tapering and differencing, to further reduce bias in parameter estimates.

We propose a simple stochastic process for modeling improper or noncircular complex-valued signals. The process is a natural extension of a complex-valued autoregressive process, extended to include a widely linear autoregressive term. This process can then capture elliptical, as opposed to circular, stochastic oscillations in a bivariate signal. The process is order one and is more parsimonious than alternative stochastic modeling approaches in the literature. We provide conditions for stationarity, and derive the form of the covariance and relation sequence of this model. We describe how parameter estimation can be efficiently performed both in the time and frequency domain. We demonstrate the practical utility of the process in capturing elliptical oscillations that are naturally present in seismic signals.Comment: Link to published version: http://ieeexplore.ieee.org/abstract/document/7539658