Transcript hu2007 6435
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 • 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: 10 7 10 8 degrees of freedom • Uncertainty prediction obtained from ensemble of <100 forecasts with representative initial condition uncertainty 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). Temperature Temperature fcj fc 0 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. reality pdf(0) Forecast time pdf(t) 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 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 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 ECMWF ensemble. A perfect system would have a flat histogram. U shape indicates the system is underpredicting uncertainty. Rationale for stochastic terms MOTIVATION: • Traditional dimensional reduction/closureaccount 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 unresolved scales by comparing integrations with identical ICs and differing horizontal resolutions from T170 to T42. Stochasticity: sub-grid distribution convection parameterization ‘Stochastic physics’ and the ECMWF EPS Each ensemble member evolution is given by the time integration T e j (T ) [ A(e j , t ) P(e j , t ) Pj (e j , t )]dt t 0 of the perturbed model equations starting from the perturbed initial conditions e j (0) e0 (0) e j (0) The model tendency perturbation is defined at each grid point by Pj ( , , p) r j ( , ) Pj ( , , p) 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 ECEPS (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 underdispersion 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 EPS New 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 q t J ( , q ) 0 q q cont q s q s s (r (t )) cont sK 0 ( r r (t ) ) q cont ( 2 2 ) cont Point vortex stream function Model simulation Point vortex in hyperbolic flow Weak point vortex advected in flow; would be sensitive to variation in x(0). Interaction makes the track less Sensitive. Reality: multi-scale interaction and weather Water Vapor Channel Chris Velden (U.Wisc/CIMSS) Note the smaller scale structure in tropics Ensemble of tracks Track distribution varying 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) J (x(0)) (x(t i ) x obs (t i )) t W(x(t i ) x obs (t i )) i 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)) y J_min=0.4436 x Bayesian Data Assimilation Posterior distribution proportional to product EDA: towards a probabilistic analysis & forecast system? • Ensemble assimilation predicts covariance • Variational smoother gets optimal trajectory EDA perturbed members EDA ensemblemean High-resolution forecast Low resolution forecast 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 ?