Explicit Treatment of Model Error Simultaneous 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: Mean horizontal wind Vertical diffusion coefficients Static stability Heating amplitude Heating depth The Sea Breeze Model: Equations * (Aksoy et al. 2005, JGR) * 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 Land Sea Forecast Domain Sponge Layer Sponge Layer Sponge Layer 500 km300 km Grid resolution: Horizontal:4 km Vertical:50 m 2 km 3 km

The Sea Breeze Model: Perfect-model behavior LandSeaLandSea Winds and Streamfunction VorticityTemperature LandSea 48H Forecast Noon Maximum Heating 3 km Surface 250 km 500 km 250 km 500 km 250 km 500 km

LandSeaLandSeaLandSea 51H Forecast 3:00PM Onset of Sea Breeze TemperatureVorticity Winds and Streamfunction 3 km Surface 250 km 500 km 250 km 500 km 250 km 500 km The Sea Breeze Model: Perfect-model behavior

LandSeaLandSea Winds and Streamfunction Vorticity LandSea 54H Forecast 6:00PM Warmest Temperature 3 km Surface 250 km 500 km 250 km 500 km 250 km 500 km The Sea Breeze Model: Perfect-model behavior

LandSeaLandSea Winds and Streamfunction Vorticity LandSea 57H Forecast 9:00PM Strongest Sea Breeze Temperature 3 km Surface 250 km 500 km 250 km 500 km 250 km 500 km The Sea Breeze Model: Perfect-model behavior

LandSeaLandSea Winds and Streamfunction Vorticity LandSea 60H Forecast Midnight Maximum Cooling Temperature 3 km Surface 250 km 500 km 250 km 500 km 250 km 500 km The Sea Breeze Model: Perfect-model behavior

LandSeaLandSea Winds and Streamfunction Vorticity LandSea 63H Forecast 3:00AM Onset of Land Breeze Temperature 3 km Surface 250 km 500 km 250 km 500 km 250 km 500 km The Sea Breeze Model: Perfect-model behavior

LandSeaLandSea Winds and Streamfunction Vorticity LandSea 66H Forecast 6:00AM Coldest Temperature 3 km Surface 250 km 500 km 250 km 500 km 250 km 500 km The Sea Breeze Model: Perfect-model behavior

LandSeaLandSea Winds and Streamfunction Vorticity LandSea 69H Forecast 9:00AM Strongest Land Breeze Temperature 3 km Surface 250 km 500 km 250 km 500 km 250 km 500 km The Sea Breeze Model: Perfect-model behavior

LandSeaLandSea Winds and Streamfunction Vorticity LandSea 72H Forecast Noon Maximum Heating Temperature 3 km Surface 250 km 500 km 250 km 500 km 250 km 500 km The Sea Breeze Model: Perfect-model behavior

Model Error - Enkf Properties (Aksoy et al. 2006, MWR) Observations:Surface buoyancy observations on land Observational error:Standard deviation of 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

Buoyancy Vorticity 3H Prior 3H Posterior Perfect-Model EnKF Results

Buoyancy Vorticity 36H Prior 36H Posterior Perfect-Model EnKF Results

Estimation Performance Buoyancy Vorticity MRE = 60%MRE = 54%

Estimation Performance Mean horizontal windStatic stabilityVorticity diff. coef.Buoyancy diff. coef.Heating amplitudeHeating depth

Mean horizontal wind Parameter Identifiability Static Stability Vorticity Diffusion Coef.Heating Depth :RMS correlation M :Any spatial domain  :Any parameter b :Buoyancy 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 with 55×55 grid-point domain 43 vertical sigma layers with 50 hPa model top Initialized: 00Z 28 Aug 2000 Parameterizations: MRF PBL scheme Grell cumulus scheme with shallow cumulus option Simple-ice microphysical parameterization 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 Cases Initial Mean Error = +0.2Initial Mean Error = +0.65Initial 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