Introduction to Data Assimilation: Lecture 2

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Introduction to Data Assimilation Lecture 2

• Saroja Polavarapu
• Meteorological Research Division Environment
  Canada

PIMS Summer School, Victoria. July 14-18, 2008
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Outline of lecture 2

1. Covariance modelling Part 1
2. Initialization (Filtering of analyses)
3. Basic estimation theory
4. 3D-Variational Assimilation (3Dvar)

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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!
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A simple demonstration of filtering properties of
B matrix
K
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Choose a correlation function and obs increment
shape then compute analysis increments
cos(x)
cos(2x)
sobs/sb 0.5
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cos(5x)
cos(6x)
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cos(7x)
cos(9x)
cos(8x)
cos(10x)
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1. Covariance Modelling

1. Innovations method
2. NMC-method
3. Ensemble method

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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

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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

l
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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)
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Obs and Forecast error variances
Mitchell et al. (1990)
Mitchell et al. (1990)
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Vertical correlations of forecast error
Height
Lonnberg and Hollingsworth (1986)
Non-divergent wind
Hollingsworth and Lonnberg (1986)
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Multivariate correlations
Bouttier and Courtier www.ecmwf.int 2002
Mitchell et al. (1990)
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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
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Correlations are not isotropic
Gustafsson (1981)
Daley (1991)
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Are correlations separable?
If so, correlation length should be Independent
of height.
Mitchell et al. (1990)
Lonnberg and Hollingsworth (1986)
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• 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.
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2. Initialization

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

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Balance in data assimilation
Daley 1991
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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 1980s a sophisticated initialization
  scheme based on Normal modes of the model
  equations was developed and used operationally
  with OI.

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Nonlinear Normal Mode Initialization (NNMI)
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The slow manifold
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Daley 1991
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Digital Filter Initialization (DFI)
Lynch and Huang (1992)
N12, Dt30 min
Tc6 h
Tc8 h
Fillion et al. (1995)
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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)
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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)
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Some signals in the forecast e.g. tides should
NOT be destroyed by NNMI! So filter analysis
increments only
Semi-diurnal mode has amplitude seen in free
model run, if anl increments are filtered
Seaman et al. (1995)
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3. A bit of Estimation theory(will lead us to
3D-Var)
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a posteriori p.d.f.
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Data Selection
From ECMWF training course available at
www.ecmwf.int
Bouttier and Courtier (2002)
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The effect of data selection
PSAS
OI
Cohn et al. (1998)
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The effect of data selection
Cohn et al. (1998)
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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.

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3D-Var Preconditioning
(1)

• Hessian of cost function is B-1 HTR-1H
• To avoid computing B-1 in (1), change control
  variable to dxLc so first term in (1) becomes ½
  ccT and we minimize w.r.t. c. Here dxx-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

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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)
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To assimilate radiances directly, H includes an
instrument-specific radiative transfer model
Normalized AMSU weighting functions
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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)
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Operational weather centers used 3D-Var
from1990s 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, 505-516  
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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.