Toward a Real Time Mesoscale Ensemble Kalman Filter

Toward a Real Time Mesoscale Ensemble Kalman Filter Gregory J. Hakim Dept. of Atmospheric Sciences, University of Washington http: //www. atmos. washington. edu/~hakim “Analysis PDF of Record” Collaborators: Ryan Torn (UW) Sebastien Dirren (UW) Chris Snyder (NCAR)

Two Distinct AOR Priorities 1) NDFD forecast verification. • • Nationwide analyses; no critical delivery time? An a posteriori approach could use max data. Better centralized for uniformity? No distribution costs; no hard deadlines. 2) Real-time mesoscale analyses & forecasts. • • • Regional analyses & short (<12 h) forecasts. Delivery time critical; use available data. A distributed/regional approach is helpful? • • DA community resource (cf. MM 5, WRF, etc. ) En. KF appears well suited.

Summary of Ensemble Kalman Filter (En. KF) Algorithm (1) Ensemble forecast provides background estimate & statistics (Pb) for new analyses. (2) Ensemble analysis with new observations. (3) Ensemble forecast to arbitrary future time.

Strengths & Weakness of En. KF • Probabilistic analyses & probabilistic forecasts. – prob. forecasts widely embraced. – prob. analyses don’t yet exist. – account for ensemble variance in NDFD verification? • Straightforward implementation; ~parallelization. • Do not need – background error covariance models. – adjoint models (cf. 4 DVAR). • Weakness: Rank deficient covariance matrices. – ensemble may need to be very large. – Many ways to boost rank for small ensembles O(100).

Synoptic Scale Example • Weather Research and Forecasting Model (WRF). – 100 km grid spacing; 28 vertical levels. • Assimilate 250 surface pressure obs ONLY. • Perfect model assumption. – Observations = truth run + noise.

Surface Pressure Errors

~500 mb Height Errors

Boundary Conditions (Ryan Torn)

Ensemble Surface Pressure & q

Ensemble 500 h. Pa Height & PV

Surface Cov(P, Plow) & Cov (V, Plow)

Cov(Z 500, Plow) & Cov (V 500, Plow)

Covariance Convergence

500 mb j Covariance, Ne = 20

500 mb j Covariance, Ne = 40

500 mb j Covariance, Ne = 60

500 mb j Covariance, Ne = 80

500 mb j Covariance, Ne = 100

Mesoscale Examples • • • 12 km grid spacing, 38 vertical levels. 3 -class microphysics. TKE boundary layer scheme. 60 ensemble members. Assimilate surface pressure observations. – Hourly observations. – Drawn from truth run plus noise. – Realistic surface station distribution.

Observation Network

Surface Pressure Error Snapshot

Mesoscale Covariances 12 Z January 24, 2004 Camano Island Radar |V 950|-qr covariance

Surface Pressure Covariance Land Ocean

Toward a Real-Time Mesoscale En. KF Prototype • Surface observations (U, V, T, RH, green) • Radiosondes (U, V, T, RH) • Scatterometer winds (U, V over ocean, red) • ACARS (U, V, T)

Summary Ensemble Kalman filter AOR opportunities: – – Ensemble mesoscale analyses & short-term forecasts. Lowers barriers-to-entry for DA. Regional DA (prototype in progress at UW). Community DA resource (cf. MM 5, WRF). Background Error Covariances: – Automatic & flow dependent with En. KF. • Cloud field analyses no more difficult than, e. g. , 500 h. Pa height. • Optimal ensemble size? – Vary strongly in space & time. • Difficult to assume mesoscale covariances, unlike synoptic scale.

THE END

En. KF Sampling Issues Problem #1: “under-dispersive” ensembles. • overweight background relative to observations. Solution: Inflate K by a scalar constant. Problem #2: spurious far-field covariances. • affect analysis far from observation. Solution: Localize K with a window function.

Computational & Plotting Domains

Analysis-update Equation analysis = prior + weighted observations

Traditional Kalman Filter Problem A forecast of Pb is needed for next analysis. Problem: Pb is huge (N x N) and cannot be evolved directly. Solution: estimate Pb from an ensemble forecast. • “Ensemble” KF (En. KF).

Current DA and Ensemble Forecasting 3 D/4 Dvar: Pb is ~ flow independent. – – Assumed spatial influence of observations. Assumed field relationships (e. g. wind—pressure balance). Pb assumptions for mesoscale are less clear. Deterministic: a single analysis is produced. Ensemble forecasts – perturbed deterministic analyses (SVs, bred modes). En. KF: unifies DA & ensemble forecasting.

Kalman Gain

Application of Localization

Ensemble Tropopause q

Cov(qtrop, Plow) & Cov (Vtrop, Plow)

Synoptic Observation Network

Application to an Extratropical Cyclone 23 March 2003.

Motivation • Probabilistic forecasts well accepted. – e. g. forecast ensembles. • Genuine probabilistic analyses are lacking. – singular vectors & bred modes are proxies. • Solving this problem creates opportunities. – probabilities: structural & dynamical information. – old: dynamics data assimilation. – new: data assimilation dynamics.

Ensemble Statistics Ensemble-estimated covariance between x and y: cov(x, y) = (x – x) (y – y)T. Here, we normalize y by s(y). • cov(x, y) has units of x. • linear response in x given one-s change in y. • take y = surface pressure in the low center.

Mesoscale Challenges • Cloud fields & precipitation. – No time for “spin up. ” • Complex topography. • Background covariances vary strongly in space & time. – can’t rely on geostrophic or hydrostatic balance. • Boundary conditions on limited-area domains.