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Predicting
uncertainty in forecasts of weather and climate
(Also published as ECMWF Technical Memorandum No. 294)
By T.N. Palmer
Research Department, ECMWF
November 1999
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Abstract
The predictability of weather and climate forecasts is determined by the projection of uncertainties in both initial conditions and model formulation onto flow-dependent instabilities of the chaotic climate attractor. Since it is essential to be able to estimate the impact of such uncertainties on forecast accuracy, no weather or climate prediction can be considered complete without a forecast of the associated flow-dependent predictability. The problem of predicting uncertainty can be posed in terms of the Liouville equation for the growth of initial uncertainty, or a form of Fokker-Planck equation if model uncertainties are also taken into account. However, in practice, the problem is approached using ensembles of integrations of comprehensive weather and climate prediction models, with explicit perturbations to both initial conditions and model formulation; the resulting ensemble of forecasts can be interpreted as a probabilistic prediction.
Many of the difficulties in forecasting predictability arise from the large dimensionality of the climate system, and special techniques to generate ensemble perturbations have been developed. Special emphasis is placed on the use of singular-vector methods to determine the linearly unstable component of the initial probability density function. Methods to sample uncertainties in model formulation are also described. Practical ensemble prediction systems for prediction on timescales of days (weather forecasts), seasons (including predictions of El Niño) and decades (including climate change projections) are described, and examples of resulting probabilistic forecast products shown. Methods to evaluate the skill of these probabilistic forecasts are outlined. By using ensemble forecasts as input to a simple decision-model analysis, it is shown that probability forecasts of weather and climate have greater potential economic value than corresponding single deterministic forecasts with uncertain accuracy.
Table of contents
1 . Introduction
| 1.1 Overview |
| 1.2 Scope |
2 . The Liouville equation
3 . The probability density function of initial error
4 . Representing uncertainty in model formulation
5 . Error growth in the linear and nonlinear phase
| 5.1 Singular vectors, eigenvectors and Lyapunov vectors |
| 5.2 Error dynamics and scale cascades |
6 . Applications of singular vectors
| 6.1 Data assimilation |
| 6.2 Chaotic control of the observing system |
| 6.3 The response to external forcing: paleoclimate and anthropogenic climate change |
| 6.4 Initialising ensemble forecasts |
7 . Forecasting uncertainty by ensemble prediction
| 7.1 Global weather prediction: from 1-10 days |
| 7.2 Seasonal to interannual prediction |
| 7.3 Decadal prediction and anthropogenic climate change |
8 . Verifying forecasts of uncertainty
| 8.1 The Brier score and its decomposition |
| 8.2 Relative operating characteristic |
9 . The economic value of predicting uncertainty
10 . Concluding remarks
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