Frequency-Domain Stochastic Modeling of Stationary Bivariate or Complex-Valued Signals

There are three equivalent ways of representing two jointly observed real-valued signals: as a bivariate vector signal, as a single complex-valued signal, or as two analytic signals known as the rotary components. Each representation has unique advantages depending on the system of interest and the application goals. In this paper we provide a joint framework for all three representations in the context of frequency-domain stochastic modeling. This framework allows us to extend many established statistical procedures for bivariate vector time series to complex-valued and rotary representations. These include procedures for parametrically modeling signal coherence, estimating model parameters using the Whittle likelihood, performing semi-parametric modeling, and choosing between classes of nested models using model choice. We also provide a new method of testing for impropriety in complex-valued signals, which tests for noncircular or anisotropic second-order statistical structure when the signal is represented in the complex plane. Finally, we demonstrate the usefulness of our methodology in capturing the anisotropic structure of signals observed from fluid dynamic simulations of turbulence.Comment: To appear in IEEE Transactions on Signal Processin

Leisure Reading Habits: Students Attitudes Toward Their in School Reading Compared to Out of School Reading.

The purpose of this study is to discover how students attitudes toward their in school reading compare to their out of school reading. It is my claim that students will have a much more positive attitude toward the reading that they do outside of school than they do of in school reading. Data was collected using students from three different ELA classrooms grades 10-12, including questionnaire data as well as interviews. The study reveals that a variety of different factors contribute to the difference in reading outside of school compared to the reading that takes place in school. If teachers are able to draw more of a connection between the two, then students will view their in school reading more positively

A Power Variance Test for Nonstationarity in Complex-Valued Signals

We propose a novel algorithm for testing the hypothesis of nonstationarity in complex-valued signals. The implementation uses both the bootstrap and the Fast Fourier Transform such that the algorithm can be efficiently implemented in O(NlogN) time, where N is the length of the observed signal. The test procedure examines the second-order structure and contrasts the observed power variance - i.e. the variability of the instantaneous variance over time - with the expected characteristics of stationary signals generated via the bootstrap method. Our algorithmic procedure is capable of learning different types of nonstationarity, such as jumps or strong sinusoidal components. We illustrate the utility of our test and algorithm through application to turbulent flow data from fluid dynamics

Mathematical approaches to the physics of mesoscale oceanography

The long-term evolution of Gaussian eddies is studied in an equivalent barotropic model using both linear and nonlinear quasi-geostrophic theory in an attempt to understand westward propagating satellite altimetry tracked mesoscale eddies. By examining both individual eddies and a large basin seeded with eddies, it is shown that long term eddy coherence and the zonal wavenumber-frequency power spectral density are best matched by the nonlinear model. Individual characteristics of the eddies including amplitude decay, length decay, zonal and meridional propagation speed of a previously unrecognized quasi-stable state are examined to provide baseline properties for comparison with extended models. An analytical technique is then used for evaluating scales of motion of typical mesoscale eddies in order evaluate the success of existing models and find other more appropriate theories. Starting from the spherical shallow water equations and assuming geostrophic dominance, a potential vorticity conservation law is derived in terms of all four non-dimensional parameters inherent in the equations while retaining the spherical geometry. By retaining freedom in the parameters, the scales can be determined at which various theories remain valid. It is argued that the FP equation equation and a new extension to the FP equation are required to describe the mid-latitude mesoscale eddies. Analytical solutions to the FP equation are sought using the classical and exterior differential systems methods of group foliation. Both methods of group foliation are used to find the cnoidal solution of the Korteweg-de Vries equation, a one-dimensional form of the FP equation. An exact analytical solution is found for the radial FP equation, although it does not appear to be of direct geophysical interest, and a reduced quasi-linear hyperbolic system is derived for the two-dimensional FP equation. The forces driving the slow westward propagation of mesoscale eddies also underly a particle constrained to the surface of the earth, but are quantitatively misunderstood. Starting with a free particle and successively adding constraints, it is shown that the particle's motion is inertial, despite literature to the contrary, and that an accelerometer trapped in inertial motion would not measure an acceleration

A tensor-valued integral theorem for the gradient of a vector field, with a fluid dynamical application

The familiar divergence and Kelvin--Stokes theorem are generalized by a tensor-valued identity that relates the volume integral of the gradient of a vector field to the integral over the bounding surface of the outer product of the vector field with the exterior normal. The importance of this long-established yet little-known result is discussed. In flat two-dimensional space, it reduces to a relationship between an integral over an area and that over its bounding curve, combining the divergence and Kelvin--Stokes theorems together with two related theorems involving the strain, as is shown through a decomposition using a suitable tensor basis. A fluid dynamical application to oceanic observations along the trajectory of a moving platform is given, and potential extensions to geometrically complex surfaces are discussed.Comment: 20 page

Available potential vorticity and the wave-vortex decomposition for arbitrary stratification

We consider a rotating non-hydrostatic flow with arbitrary stratification and argue that 1) the appropriate form of potential vorticity (PV) for this system is in terms of isopycnal deviation and 2) the decomposition into energetically orthogonal solutions is fundamentally a PV-inversion. The new closed-form expression for available potential vorticity (APV) is expressed in terms of isopycnal deviation, following the ideas in Wagner & Young (2015). This form of APV linearizes to quasigeostrophic PV (QGPV) after discarding the nonlinear stretching term and a height nonlinearity, the latter of which is not present in constant stratification. This formulation leads to positive definite definitions of potential enstrophy and total energy expressed in terms of isopycnal deviation, from which the quadratic versions emerge at lowest order. It is exactly these quantities diagonalized by the linear eigenmodes. Internal-gravity waves, geostrophic motions, inertial oscillations, and a mean density anomaly form the energetically and enstrophically orthogonal constituents of flow. The complete state of the fluid can be represented in terms of these physically realizeable modes and determined from the derived projection operators using the horizontal velocity and density anomaly. The projection of the fluid state onto the non-hydrostatic wave modes, reveals that one must first account for the PV portion of the flow before recovering the wave solutions. We apply the physical insights of the decomposition to a mesoscale eddy showing how strict adherence to adiabatic rearrangement places strong constraints on the vertical structure of such eddies, including a skew towards stronger cyclonic eddies in the upper water-column. Finally, the expression for APV is shown to reproduce the height nonlinearity of shallow-water PV, a well know feature that breaks the cyclone-anticyclone symmetry in QGPV

Separating Mesoscale and Submesoscale Flows from Clustered Drifter Trajectories

Drifters deployed in close proximity collectively provide a unique observational data set with which to separate mesoscale and submesoscale flows. In this paper we provide a principled approach for doing so by fitting observed velocities to a local Taylor expansion of the velocity flow field. We demonstrate how to estimate mesoscale and submesoscale quantities that evolve slowly over time, as well as their associated statistical uncertainty. We show that in practice the mesoscale component of our model can explain much first and second-moment variability in drifter velocities, especially at low frequencies. This results in much lower and more meaningful measures of submesoscale diffusivity, which would otherwise be contaminated by unresolved mesoscale flow. We quantify these effects theoretically via computing Lagrangian frequency spectra, and demonstrate the usefulness of our methodology through simulations as well as with real observations from the LatMix deployment of drifters. The outcome of this method is a full Lagrangian decomposition of each drifter trajectory into three components that represent the background, mesoscale, and submesoscale flow

The Evolution and Propagation of Quasigeostrophic Ocean Eddies

The long-term evolution of initially Gaussian eddies is studied in a reduced-gravity shallow-water model using both linear and nonlinear quasigeostrophic theory in an attempt to understand westward-propagating mesoscale eddies observed and tracked by satellite altimetry. By examining both isolated eddies and a large basin seeded with eddies with statistical characteristics consistent with those of observed eddies, it is shown that long-term eddy coherence and the zonal wavenumber–frequency power spectral density are best matched by the nonlinear model. Individual characteristics of the eddies including amplitude decay, horizontal length scale decay, and zonal and meridional propagation speed of a previously unrecognized quasi-stable state are examined. The results show that the meridional deflections from purely westward flow (poleward for cyclones and equatorward for anticyclones) are consistent with satellite observations. Examination of the fluid transport properties of the eddies shows that an inner core of the eddy, defined by the zero relative vorticity contour, contains only fluid from the eddy origin, whereas a surrounding outer ring contains a mixture of ambient fluid from throughout the eddy’s lifetime.Keywords: Quasigeostrophic models, Eddies, Ocean circulation, Satellite observation