1. Assimilation of Cabauw observations in a single-column model using an ensemble-kalman filter Peter Baas

2. Summary status • 1D RACMO physics with dynamics from 3D RACMO available for Cabauw point; • Prototype for ensemble Kalman filtering • System for Cabauw column available. RK observation group talk, 4th December 2007, by Reinder Ronda

3. Content • Some theory • enKF implementation • Results • Motivation: “To arrive at the best estimate of the state of the atmospheric boundary layer at Cabauw by combining local observations with a state of the art atmospheric model.”

4. Best linear unbiased estimator Forecast Observation Analysis Minimum variance:

5. For multiple dimensions… • Instead of scalar Tf now vector xf • Instead of scalar To now vector y • Instead of scalar sa2, sf2, so2, now matrix Pa, Pf, R • Observed variables may not correspond directly to model variables • Given model-space vector x, the vector H(x) can always be compared directly to the observation vector y Recall Scalar case:

6. (Extended) Kalman filter • Kalman filter: Hk and Mtk-1tkarelinear • Extended Kalman filter: Hk and Mtk-1tkare nonlinearbut linearized! • Unlike other assimilation techniques no need to specify Pb – it is generated and propagated by the filter using model dynamics.

7. Ensemble Kalman Filter • But… Kalman filter impractical for large dimension systems:  Computational reasons  Linearization leads to errors Solution: ensemble Kalman filter: • Error covariance matrix and Kalman gain calculated from ensemble members, e.g. • No tangent linear and adjoint operators needed! • Initial conditions ensemble members taken from normal distribution

8. enKF in practice…

9. enKF problem1: filter divergence • I.e., the spread between the ensemble members becomes too small • Consequently, the observations are neglected • Sampling errors • Model attractor ≠ nature attractor • Solution: inflation!

10. Covariance inflation • Expected separation prior – observations = • Inflating increases expected separation  Increases ‘apparent’ consistency prior and observation • Inflation factor function of D, sf and so

11. enKF problem 2: localization • Spurious correlations between greatly distant grid points • Result from noise in estimate of Pf through an ensemble • Solved by multiplying with localization function

12. enKF SCM system • C31r1 • Initialized at 12 UTC • Simulation time 48 h • Time-step 10 min • Each hour analysis is performed • Driven by 3D RACMO runs • 50 ensemble members • Always parallel run without assimilation is done for comparison •  Initial conditions •  Forcings

13. Initial conditions • Create perturbed profiles of u, v, T, q, Tskin, qskin,Ts, qs for each ensemble member with realistic correlations • Monthly correlations are derived from 3-year driverfile archive T q T T

14. Initial conditions • Generate random perturbation matrix with correct correlations (for each variable mean=0 and std=1). • Specify stdevs: su= sv=1m/s, sT=sTskin=sTs=1K, sq=0.5g/kg, sqskin=sqs=0.02m3/m3. • Calculate profiles • E.g. for every ensemble member: T(z,ens#) = 3D(z) + RM(T(z),ens#) * std(T) * max(0 ; 1-z/4000)

15. 1st 2nd 3rd 4th 5th SCM 48 h Forcings • Assumption: the uncertainty in the forcings is well represented by the variability in different forecasts valid at the same time • 48 h forecast with SCM, using 72 h RACMO files as forcing  Always 3 3D forecasts valid each moment of SCM run: • New forcing fields are calculated from average and stdev • For each ensnr multiplied by random number from normal distribution N~(0,1)

16. Results single cases (1) Thick: enKF Thin: Empty 20080508 20080511 • Does enKF give better correspondence with data? • 20080508: model spread has disappeared at 4 h, clouds increase ensemble spead at 30 h • 20050511: enKF adapts to ‘irregular’ observations BAD! GOOD!

17. Results single cases (2) 20080508 20080511 • 20080508: indeed, it’s the clouds that go wrong • 20080511: now the clouds go better than in Empty

18. Results single cases (3) 20080511 • Updates not only close to surface • Clouds increase model spread / promote larger updates  Gaussianity questionable!

19. Results composite (1) • Averaged time series over 31 48h-simulations for May 2008 • enKF (black) vs Empty (red), 10 (full) and 200 m (dashed)

20. Results composite (2)

21. Results composite (3) Summary • Averaged time series show slight improvement. • Bias and rms scores enKF consequently better.

22. To conclude • A prototype enKF system was extended and tested • enKF system reduces bias and rms • enKF can adapt better to sudden changes • Monitoring • Model evaluation • Improving forecasts

23. Baysian view f(y|x) f(x|y) f(x) • 3D var cost function that is minimized iteratively • (3D var avoids computation of K) • (Pf must be specified)