Ensemble-Kalman Filter (Evensen 1994)

Mesoscale Ensemble-Based Data Assimilation

Fuqing Zhang Texas A&M University, College Station, Texas

Collaborators: Ellie Meng, Altug Aksoy, Chris Snyder, David Dowell, Jenny Sun and John Nielsen-Gammon

• Regional scale [O(1 day) & O(1000km)]: assimilating sounding and surface

observations using a mesoscale model (MM5)

• Storm-scale [O(1 hour) & O(100km)]: assimilating radar observations using

a cloud-resolving model

• Thermally-forced circulation: assimilating only surface observations for

simultaneous state and parameter estimation using a 2D sea-breeze model

Ensemble-Kalman Filter (Evensen 1994)

Use ensemble forecast to estimate flow-dependent background error covariance

t=t0-t t=t0

ensemble forecast

x1a

xNa

obs y x1a

EnKF

xNa

x1f

xNf

xa = xf + BHT(HBHT+R) -1(y-Hxf)

Kalman Filter (Kalman 1960)

Uses all available information in order to produce the most accurate possible description of the state of the flow. Also

provides the uncertainty in the state of the flow resulting from the uncertainties in the various sources of information.

ensemble forecast

x1f

xNf

t=t0+t

Vertical velocity at 5km(colored) and surface cold pool (black lines, every 2K)

Storm-scale EnKF with Simulated Radar OBS (Snyder and Zhang 2003; Zhang, Snyder and Sun 2004)

Assimilating Vr if dBZ>12 every 5 minutes; no storm in initial ensemble

Truth

EnKF

Black curves: EnKF analyses at the tower location Gray curves: Independent observations from the instrumented tower Open circles: Samples from the dual-Doppler analysis.

Storm-Scale EnKF with Real Radar OBS (Dowell, Zhang, Wicker, Snyder and Crook 2004)

Assimilating Vr from one radar verify against dual-doppler analysis and tower data

Experimental Design: Regional-scale EnKF(Zhang, Meng and Aksoy 2004; Meng and Zhang 2004)

• Forecast model: MM5, 30-km grid spacing over CONUS domain (190x120x27)

• Case in study: the “surprise” snowstorm of 24-26 January 2000

• A 20-member ensemble: initiated at 00Z 24 Jan with random but balanced

perturbations using MM5 3Dvar background error statistics (Barker et al. 2003)

• Perfect-model OSSE: truth as one of the ensemble members; no model error

• OBS type: sounding obs of u, v, T from truth run at (300 km)2 spacing, every 12h

surface obs of u, v, T from truth run at (60 km)2 spacing, every 3h

• OBS error: 1 K for T and 2 m/s for u&v; uncorrelated

• Square-root sequential EnKF: OBS assimilated one by one; OBS not perturbed

• Radius of influence: 1800 km with Gaspari and Cohn (1999) cutoff

• Variance relaxation: mixing prior and posterior variances (Zhang, Snyder and Sun 2004)

Forecast Experiment: Truth (above) vs. Forecast (below) Model-derived reflectivity (colored) and MSLP (blue lines, every 2 hPa)

EnKF Performance: Forecast Error (above) vs. Analysis Error (below)

Errors in MSLP (every 0.5hPa) and surface winds (full barb, 5m/s)

EnKF Performance: Forecast Error (above) vs. Analysis Error (below)

RMS error of difference total energy (every 2m/s); 2 2 21 1

N 2RM_DTE = (u' + v' + kT' )

EnKF Performance: Forecast Error (dotted) vs. Analysis Error (solid)

vertical error distributions at 0 (green), 12 (red), 24 (blue) and 36 h (black)

EnKF Performance: Time Evolution of EnKF Analysis Error (solid) vs. Forecast Error (dotted) and ensemble spread (gray)

EnKF Performance: Spectral Analysis of Forecast Error (dotted) vs. Analysis Error (solid) at 0 (green), 12 (red), 24 (blue) and 36 (black) h

Sensitivity to OBS Accuracy, Coverage and Availability Analysis Error (solid) vs. CNTL (gray) and Forecast Error (dotted)

Sensitivity to Model Errors (Meng and Zhang 2004) Analysis Error (blue) vs. CNTL (green) and Forecast Errors (dotted)

Resolution Error: truth is produced with 10-km grid spacing; 30km used for the ensembles

Parameterization Error: truth is produced with Grell CPS; KF CPS used for the ensembles

Summary of Regional-scale EnKF

• EnKF assimilation of sounding and surface observations proved

effective for meso- and regional scales

• EnKF can reduce the analysis errors by as much as 80% for u, v, T

and p; it is relatively less efficient for w and q whose errors have

stronger smaller-scale components

• EnKF is most effective in correcting errors in larger-scale growing

structures; less effective in correcting errors in smaller, marginally

resolvable scales which also have faster error growth and shorter

predictability

• EnKF performance can be significantly degraded if an imperfect model

is used, suggesting a need for the explicit treatment of model errors

2

2

22

2

( ) ,

( ) .b

bu u w

t x z x z

b b b bu u w N w Q

t x z z

• Model: Two-dimensional, irrotational, incompressible flow with prognostic variables (perturbation) temperature (b′) and vorticity (η′)

• Explicit heating function:

0/10

0

tan cos ( ) ,2

z zxQ A e t t

x

• Parameters estimated: Mean horizontal wind; vertical mixing coefficients;

Static stability; Heating amplitude; Heating depth

Simultaneous State and Parameter EstimationExplicit treatment of model error in a 2D sea-breeze model

(Aksoy, Zhang, Nielsen-Gammon and Epifanio 2004; Aksoy, Zhang and Nielsen-Gammon 2004)

EnKF Performance: Thermally-forced (Sea-Breeze) Circulation Assimilating only perturbation temperature over land surface

Bu

oya

ncy

Vo

rtic

ity

Prior

Prior

Posterior

Posterior

u b

2N 0A 0z

EnKF Performance: Explicit treatment of Model ErrorsEstimating simultaneously six imperfect model parameters and the state

Concluding Remarks

• EnKF demonstrated to be promising for convective-, meso-/regional scale

data assimilation

– Why: ensemble forecasting provides the best estimate of the state, the

associated uncertainty, and the flow-dependent background error

covariance

• EnKF demonstrated to be promising for explicit treatment of model error

through simultaneous state and parameter estimation

• Remaining issues: ensemble generation, complex model errors, boundary

conditions, large data volume, correlated OBS error, …, etc.