1. Introduction to Data Assimilation: Lecture 2
Saroja Polavarapu
Meteorological Research Division Environment Canada
PIMS Summer School, Victoria. July 14-18, 2008

2. Outline of lecture 2 Covariance modelling – Part 1
Initialization (Filtering of analyses)
Basic estimation theory
3D-Variational Assimilation (3Dvar)

3. Background error covariance matrix filters analysis increments
Analysis increments (xa – xb) are a linear combination of columns of B
Properties of B determine filtering properties of assimilation scheme!

4. A simple demonstration of filtering properties of B matrixK

6. Choose a correlation function and obs increment shape then compute analysis increments
cos(x)
cos(2x)
sobs/sb = 0.5

7. cos(3x)
cos(4x)
cos(5x)
cos(6x)

8. cos(7x)
cos(9x)
cos(8x)
cos(10x)

9. 1. Covariance Modelling
Innovations method
NMC-method
Ensemble method

10. Background error covariance matrix
If x is 108, Pb is 108 x 108.
With 106 obs, cannot estimate Pb.
Need to model Pb.
The fewer the parameters in the model,
the easier to estimate them, but
less likely the model is to be valid

11. 1. Innovations method Historically used for Optimal Interpolation
(e.g. Hollingsworth and Lonnberg 1986,
Lonnberg and Hollingsworth 1986, Mitchell et al. 1990)
Typical assumptions:
separability of horizontal and vertical correlations
Homogeneity
Isotropy r i j q l m r i j m l

12. Assume homogeneous, isotropic correlation model. Choose a
Background error
Instrument+
representativeness
Choose obs s.t. these terms =0
Dec. 15/87-Mar. 15/88
radiosonde data.
Model: CMC T59L20
Assume homogeneous, isotropic
correlation model. Choose a
continuous function r(r) which has
only a few parameters such as
L, correlation length scale. Plot all
innovations as a function of distance
only and fit the function to the data.
Mitchell et al. (1990)

13. Obs and Forecast error variances
Mitchell et al. (1990)
Mitchell et al. (1990)

14. Vertical correlations
of forecast error
Height
Lonnberg and Hollingsworth (1986)
Non-divergent wind
Hollingsworth and Lonnberg (1986)

17. Multivariate correlations
Bouttier and Courtier
2002
Mitchell et al. (1990)

18. Covariances are not homogeneous
If covariances are homogeneous,
variances are independent of space
Covariances are not homogeneous
If correlations are homogeneous,
correlation lengths are independent
of location
Correlations are not homogeneous

19. Correlations are not isotropic
Gustafsson (1981)
Daley (1991)

20. Are correlations separable?
If so, correlation length should be
Independent of height.
Lonnberg and Hollingsworth (1986)
Mitchell et al. (1990)

21. Covariance modelling assumptions:
No correlations between background and obs errors
No horizontal correlation of obs errors
Homogeneous, isotropic horizontal background error correlations
Separability of vertical and horizontal background error correlations
None of our assumptions are really correct. Therefore
Optimal Interpolation is not optimal so it is often called
Statistical Interpolation.

22. 2. Initialization Nonlinear Normal Mode (NNMI)
Digital Filter Initialization (DFI)
Filtering of analysis increments

23. Balance in data assimilation
Starting from analysis with insufficient consideration of balance, get noise
This is they type of noise that made Richardson’s forecast appear unrealistic
NNMI greatly reduces the noise but captures the signal well.
Daley 1991

24. The “initialization” step
Integrating a model from an analysis leads to motion on fast scales
Mostly evident in surface pressure tendency, divergence and can affect precipitation forecasts
6-h forecasts are used to quality check obs, so if noisy could lead to rejection of good obs or acceptance of bad obs
Historically, after the analysis step, a separate “initialization” step was done to remove fast motions
In the 1980’s a sophisticated “initialization” scheme based on Normal modes of the model equations was developed and used operationally with OI.

25. Nonlinear Normal Mode Initialization (NNMI)
Consider model
Determine modes
Separate R and G
Project onto G
Define balance
Solution 1.
E_G includes deep, high-frequency gravitational modes.
E_R includes rotational modes but also shallow or low frequency gravitational modes.
In early NNMI, c_R was held fixed, although that was not optimal as Saroja will desribe

26. The slow manifold G R S A N L

27. Starting from analysis with insufficient consideration of balance, get noise
This is they type of noise that made Richardson’s forecast appear unrealistic
NNMI greatly reduces the noise but captures the signal well.
Daley 1991

28. Digital Filter Initialization (DFI)
Lynch and Huang (1992)
N=12, Dt=30 min
Tc=6 h
Tc=8 h
Fillion et al. (1995)

29. Combining Analysis and Initialization steps
Doing an analysis brings you closer to the data.
Doing an initialization moves you farther from the data.
Gravity modes N
Rossby modes
Daley (1986)

30. Variational Normal model initialization
Daley (1978), Tribbia (1982), Fillion and Temperton (1989), etc.
Minimize I such that uI, vI, fI stays on M.
Daley (1986)

31. So filter analysis increments only
Some signals in the forecast e.g. tides should NOT be destroyed by NNMI!
So filter analysis increments only
Distance from origin gives amplitude of mode, angle from pos real axis its phase. Circular trajectory about origin: propagting linear wave. Circular trajectory with offset: a stationary component equal to the magnitude of the offset is also present.
Semi-diurnal mode has amplitude seen in free model run, if anl increments are filtered
Seaman et al. (1995)

32. 3. A bit of Estimation theory (will lead us to 3D-Var)

33. a posteriori p.d.f.

39. Data Selection From: ECMWF training course available at www.ecmwf.int
Bouttier and Courtier (2002)

40. The effect of data selection
PSAS OI
Cohn et al. (1998)

41. The effect of
data selection
Cohn et al. (1998)

42. Advantages of 3D-var
Obs and model variables can be nonlinearly related.
H(X), H, HT need to be calculated for each obs type
No separate inversion of data needed – can directly assimilate radiances
Flexible choice of model variables, e.g. spectral coefficents
No data selection is needed.

43. 3D-Var Preconditioning
(1)
Hessian of cost function is B-1 + HTR-1H
To avoid computing B-1 in (1), change control variable to dx=Lc so first term in (1) becomes ½ ccT and we minimize w.r.t. c. Here dx=x-xb
After change of variable, Hessian is I + term
If no obs, preconditioner is great, but with more obs, or more accurate obs, it loses its advantage

44. With covariances in spectral space,
longer correlation lengths scales are
permitted in the stratosphere
With flexibility of choice of obs,
can assimilate many new types
of obs such as scatterometer
Andersson et al. (1998)
Andersson et al. (1998)

45. To assimilate radiances directly, H includes an instrument-specific radiative transfer model14 13 12 11 10 9 8 7 6 5
Normalized AMSU
weighting functions

46. Impact of Direct Assimilation of Radiances
Anomaly = difference between forecast and climatolgy
Anomaly correlation = pattern correlation between forecast anomalies and
verifying analyses
1974 – improved
NESDIS VTPR
Retrievals
1978 – TOVS
retrievals
Kalnay et al. (1998)

47. Operational weather centers used 3D-Var from1990’s Center Region
*Later replaced by 4D-Var
Center
Region
Operational
Ref.
NCEP
U.S.A.
June 1991
Parrish& Derber (1991)
ECMWF*
Europe
Jan. 1996
Courtier et al. (1997)
CMC*
Canada
June 1997
Gauthier et al. (1998)
Met Office*
U.K.
Mar. 1999
Lorenc et al. (2000)
DAO
NASA
1997
Cohn et al. (1997)
NRL
US Navy
2000?
Daley& Barker (2001)
JMA*
Japan
Sept. 2001
Takeuchi et al. (2004) SPIE proceedings,5234,   

48. Summary (Lecture 2)
Estimation theory provides mathematical basis for DA. Optimality principles presume knowledge of error statistics.
For Gaussian errors, 3D-var and OI are equivalent in theory, but different in practice
3D-var allows easy extension for nonlinearly related obs and model variables. Also allows more flexibility in choice of analysis variables.
3D-var does not require data selection so analyses are in better balance.
Improvement of 3D-var over OI is not statistically significant for same obs. Systematic improvement of 3DVAR over OI in stratosphere and S. Hemisphere. Scores continue to improve as more obs types are added.