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.