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Zoltan Toth and Tamas Szentimrey

Abstract

A new probability distribution, which has some of the advantages of the normal distribution but avoids the constraint of symmetry undesired in many applications, is presented. The distribution, called binormal, has three parameters, as the standard deviations on the two sides of the most probable value are different, but it also includes the Gaussian distribution as a special case. A test for symmetry and a method for the estimating the parameters are also shown. The distribution will be employed to express probabilistic temperature forecasts, but its climatological application for modeling certain temperature distributions is also useful and recommended.

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Craig H. Bishop and Zoltan Toth

Abstract

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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Zoltan Toth, Steve Albers, and Yuanfu Xie

No abstract available.

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Ming Cai, Eugenia Kalnay, and Zoltan Toth

Abstract

The breeding method is used to obtain the bred vectors (BV) of the Zebiak–Cane (ZC) atmosphere–ocean coupled model. Bred vectors represent a nonlinear, finite-time extension of the leading local Lyapunov vectors of the ZC model. The spatial structure and growth rate of bred vectors are strongly related to the background ENSO evolution of the ZC model. It is equally probable for the BVs to have a positive or negative sign (defined using the Niño-3 index of the BV), though often there is a sign change just before or after an El Niño event. The growth rate (and therefore the spatial coherence) of the BVs peaks several months prior to and after an El Niño event and it is nearly neutral at the mature stage.

Potential applications of bred vectors for ENSO predictions are explored in the context of data assimilation and ensemble forecasting under a perfect model scenario. It is shown that when bred vectors are removed from random initial error fields, forecast errors can be reduced by up to 30%. This suggests that minimizing the projection of the bred vectors on the observation-minus-analysis field may be a beneficial factor to an operational forecast system. The ensemble mean of a pair of forecasts perturbed with positive/negative bred vectors improves the forecast skill, particularly for lead times longer than 6 months, substantially reducing the “spring barrier” for ENSO prediction.

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Malaquias Peña, Zoltan Toth, and Mozheng Wei

Abstract

A variety of ad hoc procedures have been developed to prevent filter divergence in ensemble-based data assimilation schemes. These procedures are necessary to reduce the impacts of sampling errors in the background error covariance matrix derived from a limited-size ensemble. The procedures amount to the introduction of additional noise into the assimilation process, possibly reducing the accuracy of the resulting analyses. The effects of this noise on analysis and forecast performance are investigated in a perfect model scenario. Alternative schemes aimed at controlling the unintended injection of noise are proposed and compared. Improved analysis and forecast accuracy is observed in schemes with minimal alteration to the evolving ensemble-based covariance structure.

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Huug M. Van Den Dool and Zoltan Toth

Abstract

It has been observed by many that skill of categorical forecasts, when decomposed into the contributions from each category separately, tends to be low, if not absent or negative, in the “near normal” (N) category. We have witnessed many discussions as to why it is so difficult to forecast near normal weather, without a satisfactory explanation ever having reached the literature. After presenting some fresh examples, we try to explain this remarkable fact from a number of statistical considerations and from the various definitions of skill. This involves definitions of rms error and skill that are specific for a given anomaly amplitude. There is low skill in the N-class of a 3-category forecast system because a) our forecast methods tend to have an rms error that depends little on forecast amplitude, while the width of the categories for predictands with a near Gaussian distribution is very narrow near the center, and b) it is easier, for the verifying observation, to ‘escape’ from the closed N-class (2-sided escape chance) than from the open ended outer classes. At a different level of explanation, there is lack of skill near the mean because in the definition of skill we compare the method in need of verification to random forecasts as the reference. The latter happens to perform, in the rms sense, best near the mean. Lack of skill near the mean is not restricted to categorical forecasts or to any specific lead time.

Rather than recommending a solution, we caution against the over-interpretation of the notion of skill-by-class. It appears that low skill near the mean is largely a matter of definition and may therefore not require a physical-dynamical explanation. We note that the whole problem is gone when one replaces the random reference forecast by persistence.

We finally note that low skill near the mean has had an element of applying the notion forecasting forecast skill in practice long before it was deduced that we were making a forecast of that skill. We show analytically that as long as the forecast anomaly amplitude is small relative to the forecast rms error, one has to expect the anomaly correlation to increase linearly with forecast magnitude. This has been found empirically by Tracton et al. (1989).

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Zoltan Toth, Yuejian Zhu, and Timothy Marchok

Abstract

In the past decade ensemble forecasting has developed into an integral part of numerical weather prediction. Flow-dependent forecast probability distributions can be readily generated from an ensemble, allowing for the identification of forecast cases with high and low uncertainty. The ability of the NCEP ensemble to distinguish between high and low uncertainty forecast cases is studied here quantitatively. Ensemble mode forecasts, along with traditional higher-resolution control forecasts, are verified in terms of predicting the probability of the true state being in 1 of 10 climatologically equally likely 500-hPa height intervals. A stratification of the forecast cases by the degree of overall agreement among the ensemble members reveals great differences in forecast performance between the cases identified by the ensemble as the least and most uncertain. A new ensemble-based forecast product, the “relative measure of predictability,” is introduced to identify forecasts with below and above average uncertainty. This measure is standardized according to geographical location, the phase of the annual cycle, lead time, and also the position of the forecast value in terms of the climatological frequency distribution. The potential benefits of using this and other ensemble-based measures of predictability is demonstrated through synoptic examples.

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

Abstract

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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Yuejian Zhu, Zoltan Toth, Richard Wobus, David Richardson, and Kenneth Mylne

The potential economic benefit associated with the use of an ensemble of forecasts versus an equivalent or higher-resolution control forecast is discussed. Neither forecast systems are postprocessed, except a simple calibration that is applied to make them reliable. A simple decision-making model is used where all potential users of weather forecasts are characterized by the ratio between the cost of their action to prevent weather-related damages, and the loss that they incur in case they do not protect their operations. It is shown that the ensemble forecast system can be used by a much wider range of users. Furthermore, for many, and for beyond 4-day lead time for all users, the ensemble provides greater potential economic benefit than a control forecast, even if the latter is run at higher horizontal resolution. It is argued that the added benefits derive from 1) the fact that the ensemble provides a more detailed forecast probability distribution, allowing the users to tailor their weather forecast–related actions to their particular cost–loss situation, and 2) the ensemble's ability to differentiate between high-and low-predictability cases. While single forecasts can statistically be supplemented by more detailed probability distributions, it is not clear whether with more sophisticated postprocessing they can identify more and less predictable forecast cases as successfully as ensembles do.

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Joel K. Sivillo, Jon E. Ahlquist, and Zoltan Toth

Abstract

An ensemble forecast is a collection (an ensemble) of forecasts that all verify at the same time. These forecasts are regarded as possible scenarios given the uncertainty associated with forecasting. With such an ensemble, one can address issues that go beyond simply estimating the best forecast. These include estimation of the probability of various events and estimation of the confidence that can be associated with a forecast.

Global ensemble forecasts out to 10 days have been computed at both the U.S. and European central forecasting centers since December 1992. Since 1995, the United States has computed experimental regional ensemble forecasts focusing on smaller-scale forecast uncertainties out to 2 days.

The authors address challenges associated with ensemble forecasting such as 1) formulating an ensemble, 2) choosing the number of forecasts in an ensemble, 3) extracting information from an ensemble of forecasts, 4) displaying information from an ensemble of forecasts, and 5) interpreting ensemble forecasts. Two synoptic- scale examples of ensemble forecasting from the winter of 1995/96 are also shown.

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