Nicolas Daget1, Anthony T' Weaver1

1
Background-error covariance estimation using an
ensemble 3D-Var

• Nicolas Daget1, Anthony T. Weaver1
• and Magdalena A. Balmaseda2
• 1CERFACS, Toulouse
• 2ECMWF, Reading

2
Motivation

• An ensemble 3D-Var system has been developed for
  the ENSEMBLES project to provide ocean analyses
  for initializing climate forecasts (seasonal to
  decadal).
• The ensemble is produced using perturbed surface
  forcing fluxes (windstress, SST, precipitation).
• The ensemble is assumed to sample uncertainty in
  ocean initial conditions.
• An ensemble of ocean simulations can provide
  valuable flow-dependent information about
  background error.
• We can use this information in the assimilation
  system to update the background-error covariance
  matrix.
• To avoid underestimating the spread (standard
  deviation of background error), the assimilated
  ocean observations should also be perturbed (as
  in a stochastic EnKF).

3
The assimilation method 3D-Var FGAT

• On a given assimilation cycle 3D-Var FGAT solves
• Notation

is an estimate of the obs.-error cov. matrix
4
T-observation error standard deviation at
50m(Computing using Fu et al. (1993), Fukumori
et al. (1999) method)
5
Schematic illustration of the 3D-Var cycling
procedure (for a given ensemble member)
tN t0 10 days
IAU weights are specified to produce a smooth
transition from one cycle to the next.
6
Background-error covariance matrix

• The background-error covariance matrix is
  formulated as

• is a linear balance operator that
  transforms (approximately) uncorrelated
  variables into balanced variables (Weaver et al.
  2005, QJRMS).
• 
  for the uncorrelated variables.
• is a diffusion operator (square-root of a
  correlation operator) acting on the uncorrelated
  variables (Weaver and Courtier 2001, QJRMS).
• ? We are initially using the ensemble method to
  obtain flow-dependent estimates of ,
  although the method can be generalized to
  estimate other covariance parameters as well.

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Background-error covariance matrix

• In practice, is defined implicitly by
  making a change of control variable
• so that
• Our assumption is that

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Theoretical basis for ensemble estimation

• It can be shown that, to first order, (e.g., see
  ager et al. 2005, QJRMS Berre et al. 2006,
  Tellus)
• The evolution of the ensemble difference fields
  in a cycled ensemble analysis/forecast system is
  the same as that of the true error fields.
• If the covariance matrix of the input
  perturbations equals that of the true input
  errors then the covariance matrix of the
  difference between ensemble members
  (analysis/forecast) is equal to twice the
  covariance matrix of the true errors
  (analysis/forecast).

9
Schematic illustration of the 3D-Var ensemble
system(with updating of the background-error
standard deviations)
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Strategy for constructing the ensemble

• 4 random forcing perturbations
• available per day

• Observation perturbations
• N( 0, R )

• 9 ensemble members per
• assimilation cycle

• To reduce sampling noise, variances are
  estimated using a sliding window
• of 9 cycles (90 days) gt 81 members.

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1993-2000 time-series of the globally-averaged
standard deviation of the model-minus-observation
misfit for 1) control (no d.a.) 2) background
and 3) analysis
Temperature
Salinity
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Impact of flow-dependent ensemble-estimated
variances on the T backgroundminus-observation
(BmO) statistics
(1)
(2)
(1) (2)
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Impact on SSH variability
NW Extratropical Atlantic
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Impact on heat content variability
Standard deviation of heat content in top 300m
with
with
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Impact on equatorial currents
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Impact on error growth ( BmO AmO ) / BmO
temperature
salinity
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Impact on globally-averaged standard-deviation of
the temperature BmO
too small
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Impact of increasing ensemble variance in upper
100m on globally-averaged temperature BmO
inflated
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Summary

• An ensemble method can be combined with a
  variational data assimilation system to produce
  flow-dependent estimates of the background-error
  covariances.
• Here, the ensembles have been used to update the
  standard deviations of background error.
• Extensions to update other parameters of the
  covariance model (e.g., directional length scales
  of the diffusion operator) can also be envisaged.
• The method has been applied in a cycled 3D-Var
  reanalysis (1993-2000)
• The ensemble size was increased using a sliding
  window trade-off between flow dependence
  versus stable statistics
• Locally, the ensemble variances can have a
  significant positive impact compared to a
  simplified parameterization of the variances.
• Local inflation (near the surface) of the
  variances was required to get globally
  satisfactory results.
• Results can be improved with better input
  perturbations (e.g., to account for model error).
  

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Summary

• Extension to 4D-Var is possible, but expensive.
• The extra cost of running ensembles may be
  justified if the ensembles can serve different
  purposes simultaneously ensemble forecasting,
  (re)analysis uncertainty estimation, covariance
  estimation.