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Uncertainty Quantification Using Ensemble Methods: Predictability of Extremes and Coherent Vortices. Joe Tribbia NCAR IPAM lecture 15 February 2007. Outline. General problem of uncertainty prediction Reliability prediction as practiced operationally Specific problem of extreme events

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uncertainty quantification using ensemble methods predictability of extremes and coherent vortices

Uncertainty QuantificationUsing Ensemble Methods:Predictability of Extremes and Coherent Vortices

Joe Tribbia


IPAM lecture

15 February 2007

  • General problem of uncertainty prediction
  • Reliability prediction as practiced operationally
  • Specific problem of extreme events
  • Stochastic physics
  • Path prediction and shadowing
uncertainty prediction
Uncertainty prediction

Prior to 1990 all numerical weather forecasts deterministic (n.b. Pitcher and Epstein,1974)

  • Post 1990 Modus Operandi: Numerically forecast weather and its uncertainty (0-10 day) time range
  • Gigantic numerical model, dynamical system: degrees of freedom
  • Uncertainty prediction obtained from ensemble of <100 forecasts with representative initial condition uncertainty
the probabilistic approach to nwp ensemble prediction








Forecast time

The probabilistic approach to NWP: ensemble prediction

A complete description of the weather prediction problem can be stated in terms of the time evolution of an appropriate probability density function (pdf).

Ensemble prediction based on a finite number of deterministic integration appears to be the only feasible method to predict the PDF beyond the range of linear growth.

bred vectors and singular vectors
Bred vectors and Singular vectors

Singular vector (upper)

Bred vector (lower)

Basic state jet

Singular vectors are the fastest growing structures into the future

Bred vectors are the fastest growing structures in the past.

Operational centers battled over which was superior.

NB: Inconsistencies in initial error will disappear with Ens KF

predictability is flow dependent spaghetti plots
Predictability is flow dependent: spaghetti plots

The degree of mixing of Z500 isolines is an index of low/high perturbation growth.

the atmosphere exhibits a chaotic behavior an example
The atmosphere exhibits a chaotic behavior: an example

A dynamical system shows a chaotic behavior if most orbits that pass close to each other at some point do not remain close to it as time progresses.

This is illustrated by the forecasts of the storm that hit northern Europe on 4 December 1999.

4 Dec 1999, 00UTC : verifying analysis (top-left) and t+132h ensemble forecasts of mean-sea-level pressure started from slightly different initial conditions (i.e. from initially very close points).

quantifying known unknowns model error
Quantifying known unknowns:model error

Ensemble prediction demonstrated that IC error was important but the imperfection of models needed to be accounted for in any UQ for weather prediction

Rank histogram shows the verification of

72hr temperature predictions with ECMWF

ensemble. A perfect system would have a

flat histogram. U shape indicates the

system is underpredicting uncertainty.

rationale for stochastic terms
Rationale for stochastic terms


  • Traditional dimensional reduction/closure-account for unresolved scales
  • Weather uncertainty prediction-should take into account all sources of uncertainty in particular model error
  • May induce extremes
growth of model error t b
Growth of model error (T&B)

T&B examined the growth of errors due to the impact of unresolved

scales by comparing integrations with identical ICs and differing

horizontal resolutions from T170 to T42.

stochastic physics and the ecmwf eps
‘Stochastic physics’ and the ECMWF EPS

Each ensemble member evolution is given by the time integration

of the perturbed model equations starting from the perturbed initial conditions

The model tendency perturbation is defined at each grid point by

where rj(x) is a set of random numbers.


Spread and forecast skill

Not enough spread

Buizza et al. (2004)

Figure 6. May-June-July 2002 average RMS error of the ensemble-mean (solid lines) and ensemble standard deviation (dotted lines) of the EC-EPS (green lines), the MSC-EPS (red lines) and the NCEP-EPS (black lines). Values refer to the 500 hPa geopotential height over the northern hemisphere latitudinal band 20º-80ºN.

bad news for extremes
  • Even with stochastic forcing, predicted (conditional) distribution deficient in wings
  • SVs need unrepresentative amplitude to represent total initial uncertainty
  • Stochastic forcing can alleviate under-dispersion but masks model rectifiable(?) model variability deficiencies
tr svs target areas impact of the sep 04 change

Reliability diagram for strike probabilities

Old CY28R2 EPS

New CY28R3 EPS

TR-SVs’ target areas: impact of the Sep ’04 change

Results based on 44 cases (from 3 Aug to 15 Sep 2004) indicate that the implemented changes in the computation of the tropical areas has a positive impact on the reliability diagram of strike probability.

simplistic tc track model
Simplistic TC track model
  • Barotropic model with point vortex
  • Metaphor/model of tropical cyclone track
  • Ref:Kasahara1963,


Zabusky and McWilliams1982

model simulation point vortex in hyperbolic flow
Model simulationPoint vortex in hyperbolic flow

Weak point

vortex advected in

flow; would

be sensitive to

variation in x(0).

Interaction makes

the track less



Reality: multi-scale interaction and weather

Water Vapor Channel

ChrisVelden (U.Wisc/CIMSS)

Note the smaller scale structure in tropics

ensemble of tracks
Ensemble of tracks

Track distribution

varying x(0),y(0)

and s(0)

variational shadowing
Variational shadowing
  • Shadowing trajectories needed to separate model errors from observational errors
  • Objective measure of trajectory accuracy
  • Four dimensional variational minimization of cost J(x)
use ensemble to minimize cost function j 1 d slices
Use ensemble to minimize cost function J :1-d slices

J is strongly dependent on x(0); weakly dependent on y(0) and s(0)

bayesian data assimilation
Bayesian Data Assimilation




to product

eda towards a probabilistic analysis forecast system

EDA perturbed members

EDA ensemble-mean

High-resolution forecast

Low resolution forecast

EDA: towards a probabilistic analysis & forecast system?
  • Ensemble assimilation predicts covariance
  • Variational smoother gets optimal trajectory
  • Ensemble techniques offer method of uncertainty/predictability prediction
  • Can be tailored for extrema, but extremes must exist in the ensemble (i.e. seeds in the conditional distribution)
  • Stochastic terms needed to inflate ensemble variance
  • Shadowing can be used to ensure that verifying analysis is part of model repertoire and calibrate model errors to rationally gauge stochastic terms.
  • Ensemble can be used to solve variational problem . Can this be generalized for small ensemble-large dimensions ?