Explicit Treatment of Model Error Simultaneous State and Parameter Estimation with an Ensemble Kalman Filter Altuğ Aksoy*, Fuqing Zhang, and John W. Nielsen-Gammon

Explicit Treatment of Model ErrorSimultaneous State and Parameter

Estimation with an Ensemble Kalman Filter

Altuğ Aksoy*, Fuqing Zhang, and John W. Nielsen-Gammon

Texas A&M University

* Current affiliation: National Center for Atmospheric Research

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

• Explicit heating function:

• Estimated model parameters:

0/10

0

1 1tan cos ( )

2z zx

Q A e t tx

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

Mean horizontal wind

Vertical diffusion coefficients

Static stability

Heating amplitude

Heating depth

The Sea Breeze Model: Equations *(Aksoy et al. 2005, JGR)

0

b g

u

z

* Similar to Rotunno’s (1983) linear approach

The Sea Breeze Model: Numerics

• Model domain:

• Numerical features:• Leapfrog time integration

• Cranck-Nicholson implicit trapezoidal vertical diffusion

• Rayleigh-damping sponge layers for vorticity

• Second-order lagged horizontal diffusion for both model variables

• Asselin-type filtering to control computational mode of the leapfrog scheme

LandSea

ForecastDomain

SpongeLayer

SpongeLayer

SpongeLayer

500 km300 km 300 km

Grid resolution:

Horizontal: 4 kmVertical: 50 m

2 km

3 km

The Sea Breeze Model: Perfect-model behavior

LandSea

LandSea

Winds and Streamfunction

VorticityTemperature

LandSea

48H Forecast

NoonMaximumHeating

3 km 3 km

3 km

Surface Surface

Surface

250 km 500 km 250 km 500 km

250 km 500 km

LandSea

LandSeaLandSea

51H Forecast

3:00PMOnset of

Sea Breeze

Temperature Vorticity


3 km 3 km

3 km

Surface Surface

Surface

250 km 500 km 250 km 500 km

250 km 500 km


LandSea

LandSea


Vorticity

LandSea

54H Forecast

6:00PMWarmest

Temperature

Temperature3 km 3 km

3 km

Surface Surface

Surface

250 km 500 km 250 km 500 km

250 km 500 km


LandSea

LandSea


Vorticity

LandSea

57H Forecast

9:00PMStrongest

Sea Breeze


3 km

Surface Surface

Surface

250 km 500 km 250 km 500 km

250 km 500 km


LandSea

LandSea


Vorticity

LandSea

60H Forecast

MidnightMaximumCooling


3 km

Surface Surface

Surface

250 km 500 km 250 km 500 km

250 km 500 km


LandSea

LandSea


Vorticity

LandSea

63H Forecast

3:00AMOnset of

Land Breeze


3 km

Surface Surface

Surface

250 km 500 km 250 km 500 km

250 km 500 km


LandSea

LandSea


Vorticity

LandSea

66H Forecast

6:00AMColdest

Temperature


3 km

Surface Surface

Surface

250 km 500 km 250 km 500 km

250 km 500 km


LandSea

LandSea


Vorticity

LandSea

69H Forecast

9:00AMStrongest

Land Breeze


3 km

Surface Surface

Surface

250 km 500 km 250 km 500 km

250 km 500 km


LandSea

LandSea


Vorticity

LandSea

72H Forecast

NoonMaximumHeating


3 km

Surface Surface

Surface

250 km 500 km 250 km 500 km

250 km 500 km


Model Error - Enkf Properties (Aksoy et al. 2006, MWR)

• Observations: Surface buoyancy observations on land

• Observational error: Standard deviation of 10-3 ms-2

• Observation spacing: 40 km (10 grid points)

• Ensemble size: 50 members

• Ensemble initialization: Perturbations from model climatology

• Covariance localization: Gaspari and Cohn’s (1999) fifth-order correlation function with 100

grid-point radius of influence

• Observation processing: Sequential with no correlation between observation errors (Snyder and Zhang 2003)

• Filter: Square-root after Whitaker and Hamill (2002) with no perturbed observations

Bu

oya

ncy

Vo

rtic

ity

3H Prior

3H Prior

3H Posterior

3H Posterior

Perfect-Model EnKF Results

Bu

oya

ncy

Vo

rtic

ity

36H Prior

36H Prior

36H Posterior

36H Posterior

Perfect-Model EnKF Results

Estimation Performance

Buoyancy Vorticity

Experiment Best Case

Worst Case Best Case

RMSE - RMSEMRE 100

RMSE - RMSE

MRE = 60% MRE = 54%

Estimation Performance

Mean horizontal wind Static stability Vorticity diff. coef.

Buoyancy diff. coef. Heating amplitude Heating depth

Mean horizontal wind

Parameter Identifiability

1/ 2

2

1

1cor( , ) .

M

ii

r bM

Static Stability

Vorticity Diffusion Coef. Heating Depth

: RMS correlationM : Any spatial domain : Any parameterb : Buoyancy

r

• Distinct differences among parameters

• Static stability and heating depth sensitive to observation location

• Vort. diff. coef. with smallest correlation, appears to exhibit smallest identifiability

MM5 Experiments: Experimental Setup (Aksoy et al. 2006, GRL, submitted)

• 36-km resolution with55×55 grid-point domain

• 43 vertical sigma layers with50 hPa model top

• Initialized: 00Z 28 Aug 2000

• Parameterizations:• MRF PBL scheme

• Grell cumulus scheme withshallow cumulus option

• Simple-ice microphysicalparameterization

• Prognostic variables• Winds (u, v, w )

• Temperature (T )

• Water vapor mixing ratio (q )

• Pressure perturbation (p ‘ )

Control Forecast

•Evidence of the clockwise turning of winds

•Penetration of the temperature and moisture gradients inland during the sea breeze phase

•Well-established return flow during the land breeze phase

Sea breeze phase (7pm local)

Land breeze phase

Ensemble and Filter Properties

• Ensemble size: 40 members

• Observations: Surface and sounding observations of u, v, and T

• Observational error: Std. deviations of 2 ms-1 for u and v ; 1 K for T

• Observation spacing: 72 km for surface, 324 km for sounding

• Covariance localization: Gaspari and Cohn’s (1999) fifth-order correlation function with 30 grid-

point radius of influence

• Observation processing: Sequential (Snyder and Zhang 2003)

• Filter: Square-root (Whitaker and Hamill 2002)

Parameter Estimation Details

• Not attempting to identify individual error sources within the PBL scheme associated with different empirical parameters:

– Multiplier (m) of the vertical eddy mixing coefficient implanted into the MRF PBL code → for a value of 1.0, the original MRF PBL computation is simply repeated

• Variance limit applied at 1/4 of initial parameter error

• Updating is carried out spatially:

– Prior parameter value converted to 2-d matrix assumed at surface

– Spatial updating is performed with same covariance localization properties as the updating of the state

– Updated 2-d distribution is averaged to obtain posterior global parameter value

Correlation Signal – (T,m) and (U,m)

•Relatively strong overall correlation signal with both temperature and winds (signal strength “comparable” to idealized sea breeze model experiments)

•Spatially and temporally varying correlation structure

•Stronger signal near the surface

•Smaller-scale variability with horizontal winds

Estimation Performance – 3 CasesInitial Mean Error = +0.2 Initial Mean Error = +0.65 Initial Mean Error = -0.3

Concluding Remarks

• EnKF demonstrated to be promising for explicit treatment of model error through simultaneous state and parameter estimation

• Lessons from the idealized sea breeze model experiments:

– Sensitivity to observation location, radius of influence, and variance limit is parameter-specific

– Counter-acting correlations do lead to identifiability issues with some parameter pairs (do we really need to estimate every single parameter?)

• A more global approach to the MRF PBL scheme in MM5 appears to be responding well

• Updating of a global parameter through observations that contain spatial information is an issue and does lead to divergence as number of observations increases:

– We have approached this problem through our “spatial updating” technique – ad hoc but effective

Documents

Explicit Treatment of Model Error Simultaneous State and Parameter Estimation with an Ensemble Kalman Filter Altuğ Aksoy*, Fuqing Zhang, and John W. Nielsen-Gammon