III Subjective probabilities. III.4. Adaptive Kalman filtering. Probability Course III:4 Bologna 9-13 February 2015

Transcription

1 III Subjective probabilities III.4. Adaptive Kalman filtering

2 III.4.1 A 2-dimensional Kalman filter system

3 Obs-Tfc= correction Bias? Corr = A(t) Tfc It appears as if the old bias has abruptly changed into a new one

4 Obs-Tfc= correction Bias? Corr = A(t) Tfc It appears as if the old bias has abruptly changed into a new one

5 Obs-Tfc= correction Systematic error Corr = A(t) + B(t) Tfc In reality the systematic error has stayed more or less the same, but defined by two coefficients, A and B Tfc

6 Obs-Tfc= correction The latest verified numerical forecast Corr = A(t) + B(t) Tfc Tfc

7 Obs-Tfc= correction B changes slightly A changes slightly Tfc Corr = A + B Tfc The combination translation (A) and rotation(b) occurs under the variational condition of least effort

8 Obs-Tfc= correction New forecast Systematic error Tfc Suggested correction Corr = A(t) + B(t) Tfc

9 Obs-Tfc= correction Tfc When observations finally start to arrive and make it possible to estimate the coefficients, normally only small adjustments are needed Corr = A(t) + B(t) Tfc

10 The variation of the coefficients indicate significant changes in model and/or environment Land point X 1 coast point sea point X 2 Disappearance of ice and snow cover Sea water cooling, ice will form in due course..

11 III.4.2 Station and grid point can be far away!

12 Observations München 447m Grid point values Feuerkogel 1627nm (1456 m 1362m)

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14 Range before filtering 21 K

15 Range after filtering 27 K

16 Range before filtering 20 K

17 Range after filtering 26 K

18 III.4.3. The 2- or N-dimensional filter does not only correct mean errors ( biases ) but also systematic overand under variability

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21 The Kalman 2 filter improvement of forecast variance after Forecast variance after filtering before Forecast variance before filtering

22 Before Kalman filtering

23 After Kalman filtering

24 III.4.4 Further improvement of the spread

25 Obs-Tfc= correction Corr = A(t) + B(t) Tfc This is actually described by a covariance matrice such as Cov(AA) Cov(AB) Cov(BA) Cov(BB) Tfc

26 The ECMSWF Kalman filters The covariance matrices in Kalman filter used in Ensemble Kalman Filtering only has non-zero values in the diagonals cov(a,a) 0 0 cov(b,b) But that is because its covariance matrices are in dimension not of 2, 3 or 4 but in 10 6

27 Expected error dt = A t + B t Fc The Kalman filter will now provide a 2-dim variance matrix Cov(AB) = ( ) Variance(dT) = E{dT 2 } = cov(a,a) cov(a,b) cov(b,a) cov(b,b) E{(A + B F C ) 2 } = E{A 2 } + E{B 2 } F C2 + 2E{AB} F C yields Var(A) + F C2 Var(B) + 2F C Cov(AB)

28 A practical example: The 2-dim Kalman filter system has found that the error equation Expected error dt = F c provides the best estimation, which for F c =5.0 yields a correction of dt = 1.7. Assume the covariance matrix ( ) cov(a) F c2 cov(b) 2F c cov(ab) Var(dT) = = or a standard deviation dt = 0.45 which, as representing small scale uncertainty, in an ensemble application, should be added to the large scale, synoptic-dynamic uncertainty.

29 III.4.5 The Joseph Form

30 The covariance update equation: My update of coefficient and covariances P = + t/t T ( I k tft ) Pt/t-1 ( I -k tft ) k trtk T t But according to most textbooks: ( I-k f ) T t/t t/t-1 t t P = P

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32 2/16/

33 (1 K t ) > 1

34 END