El Aprendizaje

Predicting uncertainty in forecasts of weather and climate
(Also published as ECMWF Technical Memorandum No. 294)
By T.N. Palmer Research Department November 1999 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. Sinceit 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 modeluncertainties 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 fromthe 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 forprediction 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 thatprobability 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 5.1Singular 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

Meteorological Training Course Lecture Series © ECMWF, 2003 1

Predicting uncertainty in forecasts of weather andclimate

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 uncertainty10 . Concluding remarks

1. INTRODUCTION
1.1 Overview A desirable if not necessary characteristic of any physical model is an ability to make falsifiable predictions. Such predictions are the life blood of meteorology and climate science. Predictions from vast computer models of the atmosphere, integrating the Navier-Stokes equations for a three dimensional multi-constituent multi-phaserotating fluid, and coupled to a representation of the land surface, are continually put to the test through the daily weather forecast (e.g. Bengtsson, 1999; see also http://www.ecmwf.int). On seasonal to interannual timescales, these same models, with 2-way coupling to similar mathematical representations of the global oceans, predict the development of phenomena such as El Niño, with consequences for...