Post on 04-Jan-2016
Uncertainty QuantificationUsing Ensemble Methods:
Predictability of Extremes and Coherent Vortices
Joe TribbiaNCAR
IPAM lecture 15 February 2007
Outline
• General problem of uncertainty prediction
• Reliability prediction as practiced operationally
• Specific problem of extreme events
• Stochastic physics
• Path prediction and shadowing
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
87 1010
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.
fc
0
fcj
reality
pdf(0)
pdf(t)
Temperature Temperature
Forecast time
Sampling strategies for small samples in high dimensional
systems
Bred vectors and Singular vectors
Basic state jet
Singular vector (upper)Bred vector (lower)
Singular vectors are the fastest growing structures into the futureBred 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
The degree of mixing of Z500 isolines is an index of low/high perturbation growth.
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).
Forward looking SVs (possibly)better for extrema
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 ECMWFensemble. A perfect system would have aflat histogram. U shape indicates the system is underpredicting uncertainty.
Rationale for stochastic terms
MOTIVATION:
• 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)
T&B examined the growth of errors due to the impact of unresolvedscales by comparing integrations with identical ICs and differing horizontal resolutions from T170 to T42.
Stochasticity: sub-grid distributionconvection parameterization
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.
‘Stochastic physics’ and the ECMWF EPS
T
t
jjjjj dttePtePteATe0
)],(),(),([)(
)0()0()0( 0 jj eee
),,(),(),,( pPrpP jjj
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.
Buizza et al. (2004)
Spread and forecast skill
Not enough spread
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
Gratuitous Hurricane picture:(easier problem?)
ECMWF uses targeted SVs with stochastic physics for TCs
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.
Reliability diagram for strike probabilities
Old CY28R2 EPSNew CY28R3 EPS
Ensemble prediction of tracks
Simplistic TC track model
• Barotropic model with point vortex
• Metaphor/model of tropical cyclone track
• Ref:Kasahara1963,
Morikawa1960,
Zabusky and McWilliams1982
contcont
cont
s
scont
q
tsK
tsq
qqq
qJtq
)(
))((
))((
0),(
22
0
rr
r
Point vortex stream function
Model simulationPoint vortex in hyperbolic flow
Weak point vortex advected inflow; would be sensitive tovariation in x(0).
Interaction makes the track lessSensitive.
Chris Velden (U.Wisc/CIMSS)
Reality: multi-scale interaction and weather
Water Vapor
Channel
Note the smaller scale structure in tropics
Ensemble of tracksTrack distributionvarying x(0),y(0)and s(0)
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)
))()(())()(())0(( iobsit
iobsi
i ttttJ xxWxxx
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)
J as function of ensemble index and 2-d x-y surface
J(x(0),y(0))
x
y
J_min=0.4436
Bayesian Data Assimilation
Posteriordistributionproportionalto product
EDA: towards a probabilistic analysis & forecast system?
EDA ensemble-mean
EDA perturbed members
High-resolution forecast
Low resolution forecast
• Ensemble assimilation predicts covariance
• Variational smoother gets optimal trajectory
Conclusions
• 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 ?