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Daniel Hodyss

Abstract

A 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 estimate varies with the innovation and the moments of the distribution. These and other implications of this new view of ensemble state estimation in nonlinear systems are illustrated in simple scalar systems as well as on the Lorenz attractor.

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Daniel Hodyss

Abstract

A 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, the new algorithm reduces to the performance of the ensemble Kalman filter.

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Daniel Hodyss
and
Eric Hendricks

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 work for the predictability of downstream impacts from recurving TCs are briefly described.

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Daniel Hodyss
and
Matthias Morzfeld

Abstract

Localization is the key component to the successful application of ensemble data assimilation (DA) to high-dimensional problems in the geosciences. We study the impact of sampling error and its amelioration through localization using both analytical development and numerical experiments. Specifically, we show how sampling error in covariance estimates accumulates and spreads throughout the entire domain during the computation of the Kalman gain. This results in a bias, which is the dominant issue in unlocalized ensemble DA, and, surprisingly, we find that it depends directly on the number of independent observations but only indirectly on the state dimension. Our derivations and experiments further make it clear that an important aspect of localization is a significant reduction of bias in the Kalman gain, which in turn leads to an increased accuracy of ensemble DA. We illustrate our findings on a variety of simplified linear and nonlinear test problems, including a cycling ensemble Kalman filter applied to the Lorenz-96 model.

Significance Statement

The dampening of long-range correlations has been the key to the success of ensemble data assimilation in global numerical weather prediction. In this paper, we show how noise in covariance estimates propagates through the state estimation process and corrupts state estimates. We show that this noise results in a bias and that this bias depends on the number of observations and not, as might be expected, on the state dimension. We go on to show how dampening long-range covariances through a process referred to as “localization” helps to mitigate the detrimental effects of this sampling noise.

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Matthias Morzfeld
and
Daniel Hodyss

Abstract

Covariance localization has been the key to the success of ensemble data assimilation in high dimensional problems, especially in global numerical weather prediction. We review and synthesize optimal and adaptive localization methods that are rooted in sampling error theory and that are defined by optimality criteria, e.g., minimizing errors in forecast covariances or in the Kalman gain. As an immediate result, we note that all optimal localization methods follow a universal law and are indeed quite similar. We confirm the similarity of the various schemes in idealized numerical experiments, where we observe that all localization schemes we test—optimal and nonadaptive schemes—perform quite similarly in a wide array of problems. We explain this perhaps surprising finding with mathematical rigor on an idealized class of problems, first put forward by Bickel and others to study the collapse of particle filters. In combination, the numerical experiments and the theory show that the most important attribute of a localization scheme is the well-known property that one should dampen spurious long-range correlations. The details of the correlation structure, and whether or not these details are used to construct the localization, have a much smaller effect on posterior state errors.

Significance Statement

Covariance localization has been the key to the success of ensemble data assimilation (DA) in global numerical weather prediction. In this paper, we synthesize a large body of literature on optimal localization and then report and explain a number of surprising observations.

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Craig H. Bishop
and
Daniel Hodyss

Abstract

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.

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Daniel Hodyss
and
William F. Campbell

Abstract

The 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.

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Richard Grotjahn
,
Daniel Hodyss
, and
Cris Castello

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.

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Daniel Hodyss
and
David S. Nolan

Abstract

A linear anelastic-vortex model is derived using assumptions appropriate to waves on vortices with scales similar to tropical cyclones. The equation set is derived through application of a multiple-scaling technique, such that the radial variations of the thermodynamic fields are incorporated into the reference state. The primary assumption required for the model is that the horizontal variations in the thermodynamic variables describing the reference state are appreciably longer than the waves on the vortex. This new version of the anelastic system makes no approximation to the requirements for hydrostatic and gradient wind balance, or the buoyancy frequency, in the core of the vortex. A small but measurable improvement in the performance of the new equation set is demonstrated through simulations of gravity waves and vortex–Rossby waves in a baroclinic vortex.

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Daniel Hodyss
and
Terrence R. Nathan

Abstract

A theory is presented that addresses the connection between low-frequency wave packets (LFWPs) and the formation and decay of coherent structures (CSs) in large-scale atmospheric flow. Using a weakly nonlinear evolution equation as well as the nonlinear barotropic vorticity equation, the coalescence of LFWPs into CSs is shown to require packet configurations for which there is a convergent group velocity field. These LFWP configurations, which are consistent with observations, have shorter wave groups with faster group velocities upstream of longer wave groups with slower group velocities. These wave group configurations are explained by carrying out a kinematic analysis of wave focusing, whereby a collection of wave groups focus at some point in space and time to form a large amplitude wave packet having a single wave front. The wave focusing and the subsequent formation of CSs are enhanced by zonal variations in the background flow, while nonlinearity extends the lifetimes of the CSs. These results are discussed in light of observed blocking formation in the Atlantic–European and South Pacific regions.

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