Daniel Hodyss

Vertical Covariance Localization for Satellite Radiances in Ensemble Kalman Filters

Abstract A widely used observation space covariance localization method is shown to adversely affect satellite radiance assimilation in ensemble Kalman filters (EnKFs) when compared to model space covariance localization. The two principal problems are that distance and location are not well defined for integrated measurements, and that neighboring satellite channels typically have broad, overlapping weighting functions, which produce true, nonzero correlations that localization in radiance space can incorrectly eliminate. The limitations of the method are illustrated in a 1D conceptual model, consisting of three vertical levels and a two-channel satellite instrument. A more realistic 1D model is subsequently tested, using the 30 vertical levels from the Navy Operational Global Atmospheric Prediction System (NOGAPS), the Advanced Microwave Sounding Unit A (AMSU-A) weighting functions for channels 6–11, and the observation error variance and forecast error covariance from the NRL Atmospheric Variational Da...

Adaptive Ensemble Covariance Localization in Ensemble 4D-VAR State Estimation

An adaptive ensemble covariance localization technique, previously used in ‘‘local’’ forms of the ensemble Kalman filter, is extended to a global ensemble four-dimensional variational data assimilation (4D-VAR) scheme. The purely adaptive part of the localization matrix considered is given by the element-wise square of the correlation matrix of a smoothed ensemble of streamfunction perturbations. It is found that these purely adaptive localization functions have spurious far-field correlations as large as 0.1 with a 128-member ensemble. To attenuate the spurious features of the purely adaptive localization functions, the authors multiply the adaptive localization functions with very broadscale nonadaptive localization functions. Using the Navy’s operational ensemble forecasting system, it is shown that the covariance localization functions obtained by this approach adapt to spatially anisotropic aspects of the flow, move with the flow, and are free of far-field spurious correlations. The scheme is made computationally feasible by (i) a method for inexpensively generating the square root of an adaptively localized global four-dimensional error covariance model in terms of products or modulations of smoothed ensemble perturbations with themselves and with raw ensemble perturbations, and (ii) utilizing algorithms that speed ensemble covariance localization when localization functions are separable, variable-type independent, and/or large scale. In spite of the apparently useful characteristics of adaptive localization, single analysis/forecast experiments assimilating 583 200 observations over both 6- and 12-h data assimilation windows failed to identify any significant difference in the quality of the analyses and forecasts obtained using nonadaptive localization from that obtained using adaptive localization.

Efficient Ensemble Covariance Localization in Variational Data Assimilation

Abstract Previous descriptions of how localized ensemble covariances can be incorporated into variational (VAR) data assimilation (DA) schemes provide few clues as to how this might be done in an efficient way. This article serves to remedy this hiatus in the literature by deriving a computationally efficient algorithm for using nonadaptively localized four-dimensional (4D) or three-dimensional (3D) ensemble covariances in variational DA. The algorithm provides computational advantages whenever (i) the localization function is a separable product of a function of the horizontal coordinate and a function of the vertical coordinate, (ii) and/or the localization length scale is much larger than the model grid spacing, (iii) and/or there are many variable types associated with each grid point, (iv) and/or 4D ensemble covariances are employed.

Impact of Resolution and Design on the U.S. Navy Global Ensemble Performance in the Tropics

The performance of the U.S. Navy globalatmosphericensemble prediction systemis examinedwith a focus on tropical winds and tropical cyclone tracks. Ensembles are run at a triangular truncation of T119, T159, and T239, with 33, 17, and 9 ensemble members, respectively, to evaluate the impact of resolution versus the number of ensemble member tradeoffs on ensemble performance. Results indicate that the T159 and T239 ensemble mean tropical cyclone track errors are significantly smaller than those of the T119 ensemble out to 4days.For ensemble forecastsofupper-andlower-tropospherictropicalwinds,increasingresolutionhasonly a small impact on ensemble mean root-mean-square error for wind speed, but does improve Brier scores for 10-m wind speed at the 5 m s 21 threshold. In addition to the resolution tests, modifications to the ensemble transform initial perturbation methodology and inclusion of stochastic kinetic energy backscatter are also evaluated. Stochastic kinetic energy backscatter substantially increases the ensemble spread and improves Brier scores in the tropics, but for the most part does not significantly reduce ensemble mean tropical cyclone track error.

Ensemble State Estimation for Nonlinear Systems Using Polynomial Expansions in the Innovation

AbstractA new framework is presented for understanding how a nonnormal probability density function (pdf) may affect a state estimate and how one might usefully exploit the nonnormal properties of the pdf when constructing a state estimate. A Bayesian framework is constructed that naturally leads to an expansion of the expected forecast error in a polynomial series consisting of powers of the innovation vector. This polynomial expansion in the innovation reveals a new view of the geometric nature of the state estimation problem. It is shown that this expansion in powers of the innovation provides a direct relationship between a nonnormal pdf describing the likely distribution of states and a normal pdf determined by powers of the forecast error. One implication of this perspective is that when state estimation is performed on a nonnormal pdf it leads to state estimates based on the mean to be nonlinear functions of the innovation. A direct relationship is shown between the degree to which the state estima...

Accounting for Skewness in Ensemble Data Assimilation

AbstractA practical data assimilation algorithm is presented that explicitly accounts for skewness in the prior distribution. The algorithm operates as a global solve (all observations are considered at once) using a minimization-based approach and Schur–Hadamard (elementwise) localization. The central feature of this technique is the squaring of the innovation and the ensemble perturbations so as to create an extended state space that accounts for the second, third, and fourth moments of the prior distribution. This new technique is illustrated in a simple scalar system as well as in a Boussinesq model configured to simulate nonlinearly evolving shear instabilities (Kelvin–Helmholtz waves). It is shown that an ensemble size of at least 100 members is needed to adequately resolve the third and fourth moments required for the algorithm. For ensembles of this size it is shown that this new technique is superior to a state-of-the-art ensemble Kalman filter in situations with significant skewness; otherwise, ...

Do Frontal Cyclones Change Size? Observed Widths of North Pacific Lows

Abstract Wavelet transforms in the longitudinal and latitudinal directions are applied to sea level pressure data for 12 extratropical cyclones. Each low is tracked over time from a stage of small amplitude to a stage of large amplitude. The wavelet transform provides a quantitative, localized estimate of the size of the low pressure. Separate one-dimensional transforms are taken in the longitudinal and latitudinal directions; these are averaged to reduce scale variations created as circular asymmetries rotate around a low center. On average, the size of the lows increases such that the diameter doubles over a 4-day period. These results pass a standard “f test” with greater than 99% confidence. Some implications for theoretical studies are included.

Square Root and Perturbed Observation Ensemble Generation Techniques in Kalman and Quadratic Ensemble Filtering Algorithms

AbstractThe main goal of this work is to present a new square root ensemble generation technique that is consistent with a recently developed extension of Kalman-based linear regression algorithms such that they may perform nonlinear polynomial regression (i.e., includes a quadratically nonlinear term in the mean update equation) and that is applicable to ensemble data assimilation in the geosciences. Along the way the authors present a unification of the theories of square root and perturbed observation (sometimes referred to as stochastic) ensemble generation in data assimilation algorithms configured to perform both linear (Kalman) regression as well as quadratic nonlinear regression. The performance of linear and nonlinear regression algorithms with both ensemble generation techniques is explored in the three-variable Lorenz model as well as in a nonlinear model configured to simulate shear layer instabilities.

Nonmodal and unstable normal mode baroclinic growth as a function of horizontal scale

Abstract Linear, quasi-geostrophic, Cartesian, spectral models based on Grotjahn (1980) are solved as initial-value problems. The basic-state wind flow includes realistic vertical shear in the form of an upper-level jet but no horizontal shear. Two archetype initial vertical structures are selected. One structure, labeled “connected”, develops strong nonmodal growth (NG). The other structure, labeled “separated”, is intended to approximate better conditions prior to observed cyclogenesis. NG is deduced from growth rates of common growth measures: amplitude, total energy, potential enstrophy, and their components. Significant NG may occur, usually early on, before a solution asymptotes to the most unstable normal mode. This study focuses on how the relative amounts of NG and unstable normal mode growth vary for different scales in both horizontal dimensions. The peak NG in most growth measures is greatest for wavelengths much smaller than the most unstable normal mode wavelength. The peak NG occurs earlier as wavelength decreases consistent with relative phase speed and distance arguments applied to constituent eigenmodes coming into favorable superposition. The peak NG is much less at all wavelengths for a separated trough than a connected initial condition (IC), except for the boundary contribution to potential enstrophy. Also, the connected IC has peak NG at shorter wavelengths than the separated IC. The peak NG occurs at a shorter wavelength for amplitude than for total energy. The connected and separated ICs are shown with the horizontal structure of a square wave and for a wave having initially localized structure along the meridional axis but allowed to evolve in that dimension. The main differences are initially localized waves develop larger meridional scale and commensurately larger growth rates. When the meridional structure is allowed to evolve, transient horizontal tilts appear leading most commonly to zonal mean convergence of eddy momentum. Phase speed differences between the main eigenmodes comprising the total solution primarily explain this result.

The Resonant Excitation of Baroclinic Waves by the Divergent Circulation of Recurving Tropical Cyclones

Abstract This paper explores the hypothesis that a tropical cyclone (TC) may produce baroclinic waves through the divergent circulation that arises from its low-level inflow and upper-level outflow. The model setting is a quasigeostrophic (QG) two-layer fluid in which the effect of the tropical cyclone is parameterized through a source term on the QG potential vorticity equation. Equations predicting the spectral subset of baroclinic waves that are excited through linear resonance are derived. The near-TC pattern of the baroclinic waves in the streamfunction field typically takes the form of a ridge–trough couplet whose phase with respect to the TC varies with the speed and direction of the TCs motion vector. The predictions from the linearized theory are verified in two ways: 1) fully nonlinear simulations are shown and 2) comparison is made to the observed upper-level ridge–trough couplets produced by recurving TCs in the Navy’s Operational Global Prediction System (NOGAPS). The implications of this wor...