Use of observations in data assimilation Grald Desroziers - - PDF document

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Use of observations in data assimilation Grald Desroziers Mto-France, Toulouse, France Outline Introduction Optimizing observation error statistics Ensembles based on a perturbation of observations Impact of observations

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Use of observations in data assimilation

Gérald Desroziers Météo-France, Toulouse, France

Outline

  • Introduction
  • Optimizing observation error statistics
  • Ensembles based on a perturbation of observations
  • Impact of observations on analyses and forecasts
  • Conclusion and perspectives
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Data coverage 05/09/03 09–15 UTC (courtesy J-.N. Thépaut)

Radiosondes Pilots and profilers Aircraft Synops and ships Buoys ATOVS Satobs Geo radiances Scatterometer SSM/I Ozone

Satellites

(EUMETSAT)

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5 10 15 20 25 30 35 40 45 50 55

  • No. of sources

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 Year

Number of satellite sources used at ECMWF

AEOLUS SMOS TRMM CHAMP/GRACE COSMIC METOP MTSAT rad MTSAT winds JASON GOES rad METEOSAT rad GMS winds GOES winds METEOSAT winds AQUA TERRA QSCAT ENVISAT ERS DMSP NOAA

In 2007, ECMWF uses ~ 40 different satellite data sources

(courtesy J-.N.Thépaut, ECMWF)

General formalism

  • Statistical linear estimation :

xa = xb + δx = xb + K d = xb + BHT (HBHT+R)-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(εo) = 0.
  • If not, they have to be preliminarily de-biased,
  • r de-biasing can be made along the minimization

(Derber and Wu, 1998; Dee, 2005; Auligné, 2007).

  • Observation error variances are supposed to be known

( diagonal elements of R = E(εoεoT) ).

  • Observation errors are supposed to be un-correlated :

( non-diagonal elements of E(εoεoT) = 0 ),

  • but, the representation of observation error correlations is also

investigated (Fisher, 2006) .

Outline

  • Introduction
  • Optimizing observation error statistics
  • Ensembles based on a perturbation of observations
  • Impact of observations on analyses and forecasts
  • Conclusion and perspectives
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A posteriori diagnostics

  • Is the system consistent?
  • We should have

E[J(xa) ] = p, p = total number of observations,

  • but also

E[Jo

i(xa) ] = pi – Tr(Ri

  • 1/2 H i A H i

T Ri

  • 1/2 ),

pi : number of observations associated with Jo

i

(Talagrand, 1999) .

  • Computation of optimal E[Jo

i(xa) ] by a Monte-Carlo procedure is

possible. (Desroziers and Ivanov, 2001) .

Application : optimization of R

(Chapnik, et al, 2004; Buehner, 2005) Optimization of HIRS σo One tries to obtain E[Jo

i (xa)] = (E[Jo i (xa)])opt.

by adjusting the σo

i

∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙

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Diagnostics /observations

yo

εa εo

xt

εb

xb xa

d doa d = yo – H (xb) doa = yo – H (xa) = (I-HK) d dab = H (xa) – H (xb) = HK d E[doa dT] = (I-HK) E[d dT] = R E[dab dT] = HK E[d dT] = HBHT <ε,ε’> = E[ε ε’] dab (Desroziers et al, 2005)

Implementation in 4D-Var

and For any subset i

  • bservations, simply compute

pi

with

∑

=

− − =

pi j i b j

  • j

b j a j

p y y y y

1

/ ) )( (

This is nearly cost-free and can be computed, a posteriori, over one or several analyses.

pi

i T i ab b i

/ ) ( ) (

2

d d σ =

∑

=

− − =

pi j i b j

  • j

a j

  • j

p y y y y

1

/ ) )( ( pi

i T i

  • a
  • i

/ ) ( ) (

2

d d σ =

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Implementation in 4D-Var

Q(TEMP) U(TEMP)

Computation over four 6h 4D-Var analyses (one day)

Outline

  • Introduction
  • Optimizing observation error statistics
  • Ensembles based on a perturbation of observations
  • 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 date. (Evensen, 1997; Fisher, 2004; Berre et al, 2007)

(Ehrendorfer, 2006)

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 εa: xa = (I – KH) xb + K xo εa = (I – KH) εb + K εo The same equation also holds for the analysis perturbation: ea = (I – KH) eb + K eo , with eo = R1/2 η and η a vector of random numbers Covariance matrix of analysis error: Pa = E(ea eaT) = (I – KH) E(eb ebT) (I – KH)T + K E(eo eoT) KT

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Background error standard-deviations

Over a month Vorticity at 500 hPa For a particular date 08/12/2006 00H 6 member ensemble Vorticity at 500 hPa

Validation/tuning of ensemble variances

Background errors eb can be projected to observation space by applying the observation operator: H εb.

Using several eb, ensemble variances in observation

space can be computed and compared to

  • variances of innovations (d) minus variances of observation

errors

  • or background error variances given by statistics of dab x d

cross products

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Ensemble / diagnosed bg error std-dev HIRS 5 (28/8/6 00h)

Ensemble (6 members) Diagnosed from dab x d cross-products

Outline

  • Introduction
  • Optimizing observation error statistics
  • Ensembles based on a perturbation of observations
  • 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 :

ri

DFS = Tr(Ki Hi) = E[Jo i(xa)]

  • Reduction of error projected onto a variable/area:

ri

P = Tr(P Ki Hi B PT)

  • Reduction of error evolved by a forecast model:

ri

PM = Tr(P M Ki Hi B MT PT) = Tr(L Ki Hi B LT)

(Cardinali, 2003; Fisher, 2003; Chapnik et al, 2006)

Randomized estimates of error reduction on analyses and forecasts

) ( L B H K L

T i i

tr r =

It can be shown that

). ( K L L B H

T i

tr =

This can be estimated by a randomization procedure:

j

  • T

j i T j

  • i

i i

r ) ( ) (

1

y K L L B H R y δ δ

∑

−

≈

where

j

  • )

( y δ

is a vector of observation perturbations and

j a)

( x δ

the corresponding perturbation on the analysis.

j a T i i j T j

  • L

L B B H R ) ( ) (

' * 2 / 2 / 1 1

x y δ δ

−

∑

≈

(Fisher, 2003; Desroziers et al, 2005)

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Degree of Freedom for Signal (DFS)

AIREP_T AIREP_U AMSU SATOB PILOT TEMP_Q TEMP_T TEMP_U 1000 2000 3000 lat>20°

  • 20°<lat<20°

lat<-20°

(Chapnik et al, 2006)

Error variance reduction

% of error variance reduction for T 850 hPa by area and observation type

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Conclusion and perspectives

  • Observation operators allow the use of a wide range of observations
  • Statistics on observation errors:

– variances not perfectly known, but may be tuned by using optimality criteria

  • Simulation of analysis and background errors:
  • can be obtained by ensembles based on a perturbation of observations
  • variance of ensembles might be validated in the space of observations
  • Measure of the impact of observations:

– how observations contribute to the reduction of the uncertainty in the analyzed and forecast state? – which are the most useful observations? –

  • bservation impacts are by-products of ensembles based on a perturbation
  • f observations