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Zoltan Toth


This study addresses two questions: 1) whether there are local density maxima and minima in the Northern Hemisphere extratropical wintertime circulation phase space and 2) if so, what the preferred circulation types are. All investigations are based on the null hypothesis that the statistical distribution of circulation patterns in the phase space is a multinormal distribution. If this is a good approximation (as it was shown in a phase-average sense in earlier studies) and the assumed independent variables have equal variance, then the theoretical distribution of circulation patterns can be uniquely described by the (climatological) mean, a single standard deviation, and the number of independent variables (dimension). Having the estimates of all these variables, the local density of the actual circulation data can be easily compared to the theoretical expectation of a multinormal distribution. With randomly generated multinormal samples the significance of such discrepancies can also be tested.

The results at the 1% significance level show that out of the 273 circulation maps investigated in the phase space there are 28(15) that have a higher (lower) local density than that expected from a multinormal distribution. Moreover, the high number of local discrepancies is a statistically clear indication that the circulation data sample cannot come from a symmetric, fully multinormal distribution (global significance). The positive deviation from normal density properties in certain areas of the phase space (preferred maps) is offset by opposite sign deviations in other areas (unpreferred maps), ensuring multinormality only in a phase-average sense. This is clear evidence for the existence of multiple flow regimes in the hemispheric circulation.

As to the second question, the preferred and unpreferred circulation maps were found to cluster around 6 and 5 distinct area of the phase space, respectively. The average of the preferred or unpreferred circulation maps for each cluster was interpreted as an estimate of local density maximum or minimum areas in the phase space. Large changes in the database and the statistical methods made little change in the estimates (especially for local maxima).

The advantages and innovations of the above analysis were the following. 1) The phase space was studied in its full dimensionality. 2) Based on the appropriate null hypothesis (multinormality), the results were presented with a clear determination of statistical significance. 3) Due to the statistically significant results of the local density analysis, the possibility of a physical interpretation of clustering results (preferred circulation types) is guaranteed. A relationship between local maximum points and various boundary conditions is suspected. 4) Lacunar areas or unpreferred types that are theoretically as interesting as the preferred ones have been identified for the first time in the circulation phase space.

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Craig H. Bishop and Zoltan Toth


Suppose that the geographical and temporal resolution of the observational network could be changed on a daily basis. Of all the possible deployments of observational resources, which particular deployment would minimize expected forecast error? The ensemble transform technique answers such questions by using nonlinear ensemble forecasts to rapidly construct ensemble-based approximations to the prediction error covariance matrices associated with a wide range of different possible deployments of observational resources. From these matrices, estimates of the expected forecast error associated with each distinct deployment of observational resources are obtained. The deployment that minimizes the chosen measure of forecast error is deemed optimal.

The technique may also be used to find the perturbation that evolves into the leading eigenvector or singular vector of an ensemble-based prediction error covariance matrix. This time-evolving perturbation “explains” more of the ensemble-based prediction error variance than any other perturbation. It may be interpreted as the fastest growing perturbation on the subspace of ensemble perturbations.

The ensemble-based approximations to the prediction error covariance matrices are constructed from transformation matrices derived from estimates of the analysis error covariance matrices associated with each possible deployment of observational resources. The authors prove that the ensemble transform technique would precisely recover the prediction error covariance matrices associated with each possible deployment of observational resources provided that (i) estimates of the analysis error covariance matrix were precise, (ii) the ensemble perturbations span the vector space of all possible perturbations, and (iii) the evolution of errors were linear and perfectly modeled. In the absence of such precise information, the ensemble transform technique links available information on analysis error covariances associated with different observational networks with error growth estimates contained in the ensemble forecast to estimate the optimal configuration of an adaptive observational network. Tests of the technique will be presented in subsequent publications. Here, the objective is to describe the theoretical basis of the technique and illustrate it with an example from the Fronts and Atlantic Storm Tracks Experiment (FASTEX).

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Jie Feng, Jianping Li, Ruiqiang Ding, and Zoltan Toth


Instabilities play a critical role in understanding atmospheric predictability and improving weather forecasting. The bred vectors (BVs) are dynamically evolved and flow-dependent nonlinear perturbations, indicating the most unstable modes of the underlying flow. Especially over smaller areas, however, BVs with different initial seeds may to some extent be constrained to a small subspace, missing potential forecast error growth along other unstable perturbation directions.

In this paper, the authors study the nonlinear local Lyapunov vectors (NLLVs) that are designed to capture an orthogonal basis spanning the most unstable perturbation subspace, thus potentially ameliorating the limitation of BVs. The NLLVs are theoretically related to the linear Lyapunov vectors (LVs), which also form an orthogonal basis. Like BVs, NLLVs are generated by dynamically evolving perturbations with a full nonlinear model. In simulated forecast experiments, a set of mutually orthogonal NLLVs show an advantage in predicting the structure of forecast error growth when compared to using a set of BVs that are not fully independent. NLLVs are also found to have a higher local dimension, enabling them to better capture localized instabilities, leading to increased ensemble spread.

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