Uncertainty dynamics and predictability in chaotic systems

Smith, L; Ziehmann, Christine; Fraedrich, Klaus

doi:10.1256/smsqj.56004

Cited by 56 publications

(51 citation statements)

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“…(c) The curve R(t) is slightly oscillatory. Feature (c) has been observed by many other researchers (see, e.g., [44]). It is caused by the circular motions along either of the scrolls of the Lorenz attractor, and is related to the so-called hidden frequency phenomenon (see [13,18,35]).…”

Section: Case Studiessupporting

confidence: 60%

Quantifying dynamical predictability: the pseudo-ensemble approach

Gao¹,

Tung

Hu³

2009

Chin. Ann. Math. Ser. B

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The ensemble technique has been widely used in numerical weather prediction and extended-range forecasting. Current approaches to evaluate the predictability using the ensemble technique can be divided into two major groups. One is dynamical, including generating Lyapunov vectors, bred vectors, and singular vectors, sampling the fastest error-growing directions of the phase space, and examining the dependence of prediction efficiency on ensemble size. The other is statistical, including distributional analysis and quantifying prediction utility by the Shannon entropy and the relative entropy. Currently, with simple models, one could choose as many ensembles as possible, with each ensemble containing a large number of members. When the forecast models become increasingly complicated, however, one would only be able to afford a small number of ensembles, each with limited number of members, thus sacrificing estimation accuracy of the forecast errors.To uncover connections between different information theoretic approaches and between dynamical and statistical approaches, we propose an ( , τ )-entropy and scale-dependent Lyapunov exponent -based general theoretical framework to quantify information loss in ensemble forecasting. More importantly, to tremendously expedite computations, reduce data storage, and improve forecasting accuracy, we propose a technique for constructing a large number of "pseudo" ensembles from one single solution or scalar dataset. This pseudo-ensemble technique appears to be applicable under rather general conditions, one important situation being that observational data are available but the exact dynamical model is unknown.

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Section: Case Studiessupporting

confidence: 60%

Quantifying dynamical predictability: the pseudo-ensemble approach

Gao¹,

Tung

Hu³

2009

Chin. Ann. Math. Ser. B

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“…25 We will show later that the behavior of the spread of the ensemble with the different parameters of the model is associated, not only with the intrinsic nonlinear behavior of the Lorenz model, but also with a possible grouping of the Lorenz oscillators into clusters of synchronized cells. Here, we define a cluster as a ͑i͒ connected interval of cells ͑ii͒ wherein each pair of cells satisfies ͉ i Ϫ j ͉ϽT h , where T h is some threshold, below it, cells are considered to be synchronized.…”

Section: ͑7͒mentioning

confidence: 93%

Spatiotemporal stochastic forcing effects in an ensemble consisting of arrays of diffusively coupled Lorenz cells

Lorenzo

Santos

Pérez‐Muñuzuri

2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Spatiotemporal stochastic forcing of an ensemble system consisting of chaotic Lorenz cells diffusively coupled is analyzed. The nontrivial effects of time and length correlations on the ensemble mean error and spread are studied and the implications to new trends in weather forecast methodologies are discussed. A maximum for the forecast scores is observed to occur for specific values of time and length correlations. This maximum is studied in terms of an interplay between the natural scales occurring in the system and the noise parameters.

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“…This implies that, even with a perfect model, any uncertainty in the initial state leads to growing forecast errors and eventual failure of the forecast. It is often stated that forecast errors will grow exponentially; generally this is not what happens (Smith et al 1999)-it is only what happens on average. Even on the attractor, states can move closer together before moving apart.…”

Section: A Lessons From Lorenzmentioning

confidence: 99%

The Geometry of Model Error

Judd¹,

Reynolds

Rosmond

et al. 2008

Journal of the Atmospheric Sciences

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This paper investigates the nature of model error in complex deterministic nonlinear systems such as weather forecasting models. Forecasting systems incorporate two components, a forecast model and a data assimilation method. The latter projects a collection of observations of reality into a model state. Key features of model error can be understood in terms of geometric properties of the data projection and a model attracting manifold. Model error can be resolved into two components: a projection error, which can be understood as the model's attractor being in the wrong location given the data projection, and direction error, which can be understood as the trajectories of the model moving in the wrong direction compared to the projection of reality into model space. This investigation introduces some new tools and concepts, including the shadowing filter, causal and noncausal shadow analyses, and various geometric diagnostics. Various properties of forecast errors and model errors are described with reference to low-dimensional systems, like Lorenz's equations; then, an operational weather forecasting system is shown to have the same predicted behavior. The concepts and tools introduced show promise for the diagnosis of model error and the improvement of ensemble forecasting systems.

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Uncertainty dynamics and predictability in chaotic systems

Cited by 56 publications

References 0 publications

Quantifying dynamical predictability: the pseudo-ensemble approach

Quantifying dynamical predictability: the pseudo-ensemble approach

Spatiotemporal stochastic forcing effects in an ensemble consisting of arrays of diffusively coupled Lorenz cells

The Geometry of Model Error

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