Toward a Real Time Mesoscale Ensemble Kalman Filter
Gregory J. HakimDept. of Atmospheric Sciences, University of Washington
Collaborators: Ryan Torn (UW) Sebastien Dirren (UW) Chris Snyder (NCAR)
“Analysis PDF of Record”
http://www.atmos.washington.edu/~hakim
Two Distinct AOR Priorities1) 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. MM5, WRF, etc.)
• EnKF appears well suited.
Summary of Ensemble Kalman Filter (EnKF) 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 EnKF• 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. 4DVAR).
• 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 &
Ensemble 500 hPa Height & PV
Surface Cov(P, Plow) & Cov (V, Plow)
Cov(Z500, Plow) & Cov (V500, Plow)
Covariance Convergence
500 mb Covariance, Ne = 20
500 mb Covariance, Ne = 40
500 mb Covariance, Ne = 60
500 mb Covariance, Ne = 80
500 mb 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
Camano Island Radar |V950|-qr covariance
12 Z January 24, 2004
Surface Pressure Covariance
OceanLand
Toward a Real-Time Mesoscale EnKF 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. MM5, WRF).
Background Error Covariances:– Automatic & flow dependent with EnKF.
• Cloud field analyses no more difficult than, e.g., 500 hPa height.• Optimal ensemble size?
– Vary strongly in space & time.• Difficult to assume mesoscale covariances, unlike synoptic scale.
THE END
EnKF 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 (EnKF).
Current DA and Ensemble Forecasting
3D/4Dvar: 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).
EnKF: unifies DA & ensemble forecasting.
Kalman Gain
Application of Localization
Ensemble Tropopause
Cov(trop, 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 (y).• cov(x, y) has units of x.• linear response in x given one- 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.