Introduction to data assimilation: Lecture 3

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Introduction to data assimilation Lecture 3
Saroja Polavarapu Meteorological Research
Division Environment Canada
PIMS Institute, Victoria, 14-18 July 2008
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OUTLINE

1. Covariance modelling 2,3
2. 4D-Variational assimilation
3. Nonlinear dynamics
4. Constrained variational data assimilation

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Covariance Modelling

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

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2. NMC-method

• Need global statistics
• N. American radiosonde network is only 4000 km in
  extent defining only up to wavenumber 10.
  Vertical and horizontal resolution is too coarse.
• A posteriori justification compare resulting
  statistics with those obtained using other methods

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The NMC-method
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-48
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• Compares 24-h and 48-h forecasts valid at same
  time
• Provides global, multivariate corr. with full
  vertical and spectral resolution
• Not used for variances
• Assumes forecast differences approximate forecast
  error

Why 24 48 ?

• 24-h start forecast avoids spin-up problems
• 24-h period is short enough to claim similarity
  with 0-6 h forecast error. Final difference is
  scaled by an empirical factor
• 24-h period long enough that the forecasts are
  dissimilar despite lack of data to update the
  starting analysis
• 0-6 h forecast differences reflect assumptions
  made in OI background error covariances

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A posteriori justification compare NMC results
to innovation-method results
Horizontal correlation length scale
Innovations
NMC
Rabier et al. (1998)
Hollingsworth and Lonnberg (1986)
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Different vertical correlation lengths for
different wavenumbers
Different horizontal correlation lengths for
different vertical levels
Rabier et al. (1998)
Rabier et al. (1998)
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Properties of the NMC-methodBouttier (1994)

• For linear H, no model error, 6-h forecast
  difference, can compare NMC P calc. to what
  Kalman Filter suggests.
• NMC-method breaks down if there is no data
  between launch of 2 forecasts. With no data P is
  under-estimated
• For dense, good quality hor. uncorr. obs, P is
  over-estimated
• For obs at every gridpoint, where obs and bkgd
  error variances are equal, the NMC-method P
  estimate is equivalent to that from the KF.

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NMC-method usage
Center Region Reference
NCEP U.S.A. Parrish Derber 1991
ECMWF Europe Rabier et al.1998
CMC Canada Gauthier et al. 1999
Met Office U.K. Ingleby et al. 1996
BMRC Australia Steinle et al. 1995
Meteo-Fr. France Desroziers et al. 1995
Later replaced by ensemble-based methods
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3. Ensemble-based methods of covariance
estimation
These methods attempt to simulate error of actual
assimilation systems by perturbing obs and
background states with specified errors and
computing ensemble spread
Belo Pereira and Berre (2006)
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Comparing NMC and ensemble-based method results
Horizontal correlation length scales are longer
with NMC method
temperature
vorticity
Belo Pereira and Berre (2006)
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Vertical correlations are too deep with NMC method
Vertical correlations of temperature background
error (at level 21, 500 hPa)
Belo Pereira and Berre (2006)
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Y at 250 hPa
T at 500 hPa
Background error standard deviations
Specified NMC STD are independent of longitude
Ensemble-based STD show reduced error in data
dense regions
Time averaged background errors from actual EnsKF
is used as reference
Buehner (2005)
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Ensemble-method usage
Center Region Reference
ECMWF Europe Fisher 2003
CMC Canada Buehner 2005
Meteo-Fr. France Berre et al. 2006
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2. Four-Dimensional variational data assimilation
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Extension to the time dimension

• 3D DA schemes make sense when all obs are taken
  at the same time (e.g. radiosondes).
• But they dont take full advantage of
  measurements which have high temporal resolution
  (satellite obs, profilers, aircraft, etc.).

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4D-Variational assimilation
Analysis trajectory
Background trajectory
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The benefit of temporal information
4D-Var experiment with obs every time step at
only 1 of 128 grid points
Initial guess field misplaces front
Dotted red line is 3D-Var solution
With time series of obs from 1 station only, the
frontal position is corrected
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Analysis trajectory
Background trajectory
4D-Var algorithm
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Minimization algorithm
http//www-rocq.inria.fr/estime/modulopt/optimizat
ion-routines/m1qn3/m1qn3.html

• M1QN3
• Gilbert Lemaréchal 1989
• limited memory quasi-Newton technique (the L-BFGS
  method of J. Nocedal)
• designed for very large scale problems

Minimization of a quadratic cost function J(x).
The gradient of the cost function and the cost
function itself are supplied to a minimization
algorithm which determines how to change x to get
a lower cost.
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4D-Var as described

• Assumes NWP model is perfect
• Complex nonlinear relationships between analysis
  variables are permitted
• Aids in reducing underdeterminacy problem
• Needs TLM and ADJ models for NWP model
• DA scheme now intimately tied to NWP model
• Is expensive
• Adjoint model about 1-2 times CPU of NWP model.
  One iterationNWP run adj run. Typically 50
  iterations.

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Circled term is 1 column of B matrix, i.e. a
vector
LHS is a vector
Term in ( ) is a scalar
Predictability error
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Geopotential height analysis increments at the
end of a 24-h assimilation period due to 1 obs
500 hPa

• 4D-Var single obs experiments show
• The shape of analysis increments depends on
  location of obs
• The spreading of information is flow dependent

1000 hPa
4D-Var 1 height obs at (42N,170.6E,850
hPa) Changes shape with height
3D-Var 1 height obs at (42N,180E,500 hPa) No
change of shape with height
Thépaut et al. (1996)
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Why does 4D-Var beat 3D-Var?
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3. Complications due to nonlinear dynamics
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Highly nonlinear dynamics
Lorenz (1963) equations
for
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• If assimilation window is too long, 4D-Var fails

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Miller et al. (1994)
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Length of 4D-Var assimilation window
The longer the assimilation window, the greater
the number of local minina in the cost function
Miller et al. (1994)
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Optimal assimilation period

• examine ability to fill in small scales
  through downscale energy cascade
• barotropic vorticity equation
• Perfect model, observations
• Initial guess for trajectory is completely
  decorrelated from truth

3 days
12 days
Nonlinear time scale is TNL9
Tanguay et al. (1995)
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3 days
1.5 days
Obs at large scales only
Downscale transfer of information to unobserved
scales
6 days
9 days
Upscale propagation of error to observed scales
Tanguay et al. (1995)
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Incremental Approach Courtier et al. (1994)

• TLM will be valid for large scales but not for
  some smaller scales
• So, solve for analysis increments at lower
  resolution. Write 4D-Var cost in terms of
  increments (departures from background).
• Use of lower resolution filters scales and
  processes not well forecast by TLM
• Forecast model in cost function is then TLM model
• Cost function is purely quadratic
• Use of lower resolution reduces cost of 4D-Var
• Compute the innovation (z-H(x)) at full
  resolution
• Solve a series (2-3) 4D-Var problems, updating
  the background between each one

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4. Constrained variational data assimilation
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Does 4D-Var inherently produce balanced analyses?

• 4D-Var tries to find the model state which best
  fits the observations in a time window
• The model contains many modes at its disposal,
  for use in fitting observations Rossby waves,
  gravity waves,
• If the obs contain high frequency signals (which
  they will), the model will use as many gravity
  waves as needed to fit the obs

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Strong Constraints

• Minimize J(x0) subject to the constraints

Necessary and sufficient conditions for x0 to be
a minimum are
Projection onto constraint tangent
Gill, Murray, Wright (1981)
Hessian of constraints
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Weak Constraints
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4DVAR with NNMI strong constraint

• owing to the iterative and approximate character
  of the initialization algorithm, the condition
  dG/dt 0 cannot in practice be enforced as
  an exact constraint.
• Courtier and Talagrand (1990)

Thépaut and Courtier (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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4DVAR with DFI Strong Constraint

• Because filter is not perfect, some inversion of
  intermediate scale noise occurs, but DFI as a
  strong constraint suppresses small scale noise.

Polavarapu et al. (2000)
4DVAR with DFI Weak Constraint

• Introduced by Gustafsson (1993)
• Weak constraint can control small scale noise
  (Polavarapu et al. 2000)
• Implemented operationally at Météo-France
  (Gauthier and Thépaut 2001)

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Disadvantages of 4D-Var

• Model specific (Needs TLM and ADJ)
• The U.K. Met Office uses Perturbation Forecast
  Model and its Adjoint
• Assumes NWP model is perfect.
• Weak constraint formulations relax this
  assumption. Already under investigation at
  ECMWF (see Tremolet QJ papers)
• Expensive. 2-3 x CPU of NWP model per iteration,
  with 50 iterations per outer loop
• Computing power keeps increasing
• European Centre for Medium Range Weather
  Forecasting

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4D-Var Challenges

• Obtaining fast, efficient large-scale
  optimization routines
• Extracting analysis error covariance A-1
  B-1 HTR-1H
• Want to know MAMT to learn about forecast error
  levels
• Cycling 4D-Var (Using evolved covariance at end
  of one assimilation window to start next
  assimilation cycle.)
• Estimating and incorporating model error

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Weather centers using 4D-var operationally
Center Region Opera-tional Ref.
ECMWF Europe Jan. 1999 Rabier et al. (1999)
Météo- France France 2000 Gauthier and Thepaut (2001)
JMA Japan Mar. 2002 http//www.jma.go.jp/jma/jma-eng/jma-center/nwp/NAPS-8_DDB_spec.txt
Met Office U.K. Mar. 2003 Rawlins et al. (2007)
CMC Canada Mar. 2005 Gauthier et al. (2007)
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Uppala et al. (2005, QJ)
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Exciting but missed topics

• Ensemble Kalman Filter
• Operational at CMC for Ensemble prediction system
• Combining variational and Ensemble techniques
• WWRP/THOPEX workshop on 4D-Var and Ensemble
  Kalman Filter Inter-comparisons, Buenos Aires,
  Argentina, 10-13 Nov. 2008 http//4dvarenkf.cima.f
  cen.uba.ar/
• Operational ensemble/variational assimiliation
  system at Météo-France on July 1, 2008. Ref
  Berre et al. (2007)

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Final Summary

• The atmospheric data assimilation problem is
  characterized by huge, nonlinear systems and
  insufficient observations.
• Because the math of the linear estimation problem
  is well known, the key to progress is using
  atmospheric physics to make the right
  approximations
• There has been considerable improvement in
  forecast skill in the past 2.5 decades, partly
  due to improvements in data assimilation systems.

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The End