Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting

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Data assimilation schemes in numerical weather
forecasting and their link with ensemble
forecasting
Gérald Desroziers Météo-France, Toulouse, France
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Outline

• Numerical weather prediction
• Data assimilation
• A posteriori diagnostics optimizing error
  statistics
• Ensemble assimilation
• Impact of observations on analyses and forecasts
• Conclusion and perspectives

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Outline

• Numerical weather prediction
• Data assimilation
• A posteriori diagnostics optimizing error
  statistics
• Ensemble assimilation
• Impact of observations on analyses and forecasts
• Conclusion and perspectives

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Numerical Weather Prediction at Météo-France
DX 10 km
Global Arpège model DX 15 km
Arome DX 2,5 km
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Initial condition problem
Prévision état à t0 24h
État atmosphérique à t0
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Outline

• Numerical weather prediction
• Data assimilation
• A posteriori diagnostics optimizing error
  statistics
• Ensemble assimilation
• Impact of observations on analyses and forecasts
• Conclusion and perspectives

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Synops and ships
Buoys
Data coverage 05/09/03 0915 UTC (courtesy J-.N.
Thépaut)
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Satellites
(EUMETSAT)
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Satellite data sources
(courtesy J-.N.Thépaut, ECMWF)
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General formalism

• Statistical linear estimation
• xa xb dx xb K d xb BHT
  (HBHTR)-1 d,
• with d yo H (xb ), innovation, K, gain
  matrix,
• B et R, covariances of background and
  observation errors,
• H is called  observation operator  (Lorenc,
  1986),
• It is most often explicit,
• It can be non-linear (satellite observations)
• It can include an error,
• Variational schemes require linearized and
  adjoint observation operators,
• 4D-Var generalizes the notion of  observation
  operator  .

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Statistical hypotheses

• Observations are supposed un-biased E(eo) 0.
• If not, they have to be preliminarly de-biased,
• or de-biasing can be made along the minimization
• (Derber and Wu, 1998 Dee, 2005 Auligné,
  2007).
• Oservation error variances are supposed to be
  known
• ( diagonal elements of R E(eoeoT) ).
• Observation errors are supposed to be
  un-correlated
• ( non-diagonal elements of E(eoeoT) 0 ),
• but, the representation of observation error
  correlations is also investigated (Fisher, 2006)
  .

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Implementation

• Variational formulation
• minimization of J(dx) dxT B-1 dx (d-H dx)T
  R-1 (d-H dx)
• Computation of J development and use of adjoint
  operators
• 4D-Var
• generalized observation operator H addition of
  forecast model M.
• Cost reduction  low resolution increment dx
• (Courtier, Thépaut et Hollingsworth, 1994)

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4D-Var principle
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Outline

• Numerical weather prediction
• Data assimilation
• A posteriori diagnostics optimizing error
  statistics
• Ensemble assimilation
• Impact of observations on analyses and forecasts
• Conclusion and perspectives

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A posteriori diagnostics

• Is the system consistent?
• We should have
• EJ(xa) p,
• p total number of observations,
• but also
• EJoi(xa) pi Tr(Ri-1/2 H i A H iT Ri-1/2 ),
• pi number of observations associated with
  Joi
• (Talagrand, 1999) .
• Computation of optimal EJoi(xa) by a
  Monte-Carlo procedure is
• possible.
• (Desroziers et Ivanov, 2001) .

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Application optimisation of R


One tries to obtain EJoi (xa) (EJoi
(xa))opt. by adjusting the soi








Optimisation of HIRS so








(Chapnik, et al, 2004 Buehner, 2005)
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Outline

• Numerical weather prediction
• Data assimilation
• A posteriori diagnostics optimizing error
  statistics
• Ensemble assimilation
• Impact of observations on analyses and forecasts
• Conclusion and perspectives

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Ensemble of perturbed analyses

• Simulation of the estimation errors
• along analyses and forecasts.
• Documentation of error covariances
• over a long period (a month/ a season),
• for a particular day.

(Evensen, 1997 Fisher, 2004 Berre et al, 2007)
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Ensembles Based on a perturbation of observations

• The same analysis equation and (sub-optimal)
  operators K and H
• are involved in the equations of xa and ea
• xa (I KH) xb K xo
• ea (I KH) eb K eo
• The same equation also holds for the analysis
  perturbation
• pa (I KH) pb K po

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Background error standard-deviations
Over a month Vorticity at 500 hPa
For a particular date 08/12/2006 00H Vorticity
at 500 hPa
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Ensemble assimilationerrors 08/12/2006 06UTC
500 hPa vorticity error
surface pressure
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Ensemble assimilationerrors 15/02/2008 12UTC
850 hPa vorticity error (shaded) sea surface
level pressure (isoligns)
(Montroty, 2008)
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Outline

• Numerical weather prediction
• Data assimilation
• A posteriori diagnostics optimizing error
  statistics
• Ensemble assimilation
• Impact of observations on analyses and forecasts
• Conclusion and perspectives

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Measure of the impact of observations

• Total reduction of estimation error variance
• r Tr(K H B)
• Reduction due to observation set i
• ri Tr(Ki Hi B)
• Variance reduction normalized by B
• riDFS Tr(Ki Hi)
• Reduction of error projected onto a
  variable/area
• riP Tr(P Ki Hi B PT)
• Reduction of error evolved by a forecast model
• riPM Tr(P M Ki Hi B MT PT) Tr(L Ki Hi B LT)

(Cardinali, 2003 Fisher, 2003 Chapnik et al,
2006)
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Randomized estimates of error reduction on
analyses and forecasts
It can be shown that
This can be estimated by a randomization
procedure
is a vector of observation perturbations and
where
the corresponding perturbation on the analysis.
(Fisher, 2003 Desroziers et al, 2005)
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Degree of Freedom for Signal (DFS)
01/06/2008 00H
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Error variance reduction
of error variance reduction for T 850 hPa by
area and observation type
(Desroziers et al, 2005)
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Outline

• Numerical weather prediction
• Data assimilation
• A posteriori diagnostics optimizing error
  statistics
• Ensemble assimilation
• Impact of observations on analyses and forecasts
• Conclusion and perspectives

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Conclusion and perspectives

• Importance of the notion of  observation
  operator 
• most often explicit,
• rarely statistical
• Large size problems
• state vector 107
• observations 106
• Ensemble assimilation
• estimation error covariances
• measure of the impact of observations
• link with Ensemble forecasting
• ( 40 members of 96h forecasts)