implementation of spatial correlations in the background …...errera and m enard bascoe becm sparc...

Implementation of spatial correlations in the backgrounderror covariance matrix in the BASCOE system

Quentin ErreraBelgium Institute for Space Aeronomy (BIRA-IASB)

[email protected]

Richard MenardEnvironment Canada

SPARC DA WorkshopJune 20-22, 2011

Errera and Menard BASCOE BECM SPARC DAW 2011 1 / 21

Introduction

The Belgium Assimilation System for Chemical ObsErvations (BASCOE)

4D-Var system dedicated to assimilation of stratospheric chemicalobservations

57 chemical species interact through 200 chemical reactions

All species advected, usually by ECMWF dynamical fields

The system includes a simple PSC parameterization

Typical resolution: 3.75◦ lon × 2.5◦ lat × 37 levels (0.1 hPa to thesurface)

Up to now, B has always been considered diagonal, with a fixed errorusually between 20% to 50% of the background field

This system has been used successfully to assimilate UARS MLS,MIPAS, GOMOS and EOS MLS


Motivations

Why to implement spatial correlations in B?

NRT assimilation of EOS MLS is done by BASCOE within MACC.Correlations in B might allow to improve the analyses AND also allowto increase the resolution ⇒ Double improvement

We would also like to compare 4D-Var and EnKF. For this purpose,an optimal 4D-Var system is required.


Outline

1 New formulation of BASCOE

2 Numerical test: assimilation of one pseudo observation

3 Real test: assimilation of EOS MLS ozone


Control Variable Transform (1/4)

The classical formulation of 4D-Var (i.e. not incremental), as previouslyimplemented in BASCOE, aims at minimizing the objective function J:

J(x(t0)) =1

2[x(t0)− xb(t0)]T B−1[x(t0)− xb(t0)] (1)

+1

2

N∆t∑i=0

[yi − H(x(ti ))]T R−1i [yi − H(x(ti ))]

where:

xb(t0) and B are, respectively, the background model state (at time step 0) and itsassociated error covariance matrix

yi and Ri are, respectively, the observation vector and its associated errorcovariance matrix at time step i , N∆t being the number of model time step

x(ti ) is the model state vector at time step i and is calculated from the previoustime step by the model operator M: x(ti ) = Mi−1[x(ti−1)]

H is the observation operator that maps the model state into the observation space

In the following, Jb and Jo will refer to the background and observation terms of J



1 As the typical dimension of x is around 106, a full B matrix is of size1012. This is far too large for current computers.

2 The specification of the elements of such a matrix requires a hugeamount of a priori information, more than available.

For those reasons, it is necessary to reduce the problem. Following themethod used by meteorological centers [Bannister, 2008a,b, QJ, andreferences therein], we apply the following control variable transform(CVT):

x− xb︸︷︷︸δx

= Lχ (2)

whereB = LLT (3)



The objective function:

Jb(x(t0)) =1

2[x(t0)− xb(t0)]T B−1[x(t0)− xb(t0)] (4)

Jo(x(t0)) =1

2

N∆t∑i=0

[yi − H(x(ti ))]T R−1i [yi − H(x(ti ))] (5)

becomes:

Jb(χ(t0)) =1

2χ(t0)Tχ(t0) (6)

Jo(χ(t0)) =1

2

N∆t∑i=0

[di − H(Lχ(ti ))]T R−1i [di − H(Lχ(ti ))] (7)

where di = yi − H[xb(t0)]



Efficient minimization of J requires ∇J. In the classical formulation, wehad:

∇Jb = B−1[x(t0)− xb(t0)] (8)

∇Jo =

N∆t∑i=0

MT1→0MT

2→1 . . .MTi→i−1

(∂H(x(ti ))

∂x(ti )

)T

R−1i [yi − H(x(ti ))]

(9)

In contrast, with the new control variable χ, we have:

∇Jb = χ0 (10)

∇Jo = LTN∆t∑i=0

MT1→0MT

2→1 . . .MTi→i−1

(∂H(x(ti ))

∂x(ti )

)T

R−1i [yi − H(x(ti ))]. (11)


Formulation of L (1/3)

Again, as done by meteorological centers [Bannister, 2008a,b, QJ], weconstruct the BECM from a set of very sparce matrices. In our case, L isgiven by

L = GΣSΛ1/2 (12)

where

Λ is the correlation matrix defined in the spectral space; χ′ = Λ1/2χ

S operates the transformation from the spectral space to the gaussiangrid. The control variable χ is thus a set of spherical harmoniccoefficients; δx′′ = Sχ′

Σ is the background error variance matrix; δx′ = Σδx′′

G operates the transformation from the gaussian grid to the lat/longrid of BASCOE; δx ≡ x− xb = Gδx′



By considering homogeneous and isotropic correlations, Λ is a blockdiagonal matrix in the spectral space. For example, if Nlat = 2, Nlon = 4and Nlev = 3 and if we organize χ as follow, Λ takes the form:

χ =

χ100

χ200

χ300

χ11−1

χ21−1

χ31−1

χ110

χ210

χ310

χ111

χ211

χ311

, Λ =

q10 c12

0 c130

c120 q2

0 c230

c130 c23

0 q30

q11 c12

1 c131

c121 q2

1 c231

c131 c23

1 q31

q11 c12

1 c131

c121 q2

1 c231

c131 c23

1 q31

q11 c12

1 c131

c121 q2

1 c231

c131 c23

1 q31

χlnm are the expansion of x in spherical harmonics; l is the level index,

n is the total wave number and m is the zonal wave number

qln are the horizontal correlation coefficients, depending on the level l

and on the total wavenumber n (but not on the zonal wavenumber m)

cli ,ljn are the vertical correlation coefficients, depending on the pair of

levels (li , lj) and the total wavenumber n



Current formulation of L in BASCOE

We assume that cli ,ljnp = c

li ,ljnq , i.e. vertical correlations are indendent of

the total wave number n.⇒ vertical and horizontal correlations are separableHorizontal correlations: gaussian and second order autoregressive(SOAR) correlations can be modelled, given a correlation length scaleLh (in km)Vertical correlations: gaussian correlations can be modelled, given Lv

(in level units)

−150 −100 −50 0 50 100 1500

0.2

0.4

0.6

0.8

1

Longitude [deg]

Correlation models with Lh = 2000 km

GaussSOAR


Assimilation of a single pseudo observation (1/2)

The new formulation of BASCOE is tested with the assimilation of a singlepseudo observation:

Model grid size: 41 latitudes × 80 longitudes × 9 levels

Uniform background field and variances: xb = 1; Σ = 0.1I

One observation is located on a grid point (20,40,5);yo = 1.2;σo = 0.001

Gaussian vertical and horizontal correlations. Lv = 1; Lh = 1200km


Assimilation of a single pseudo observation (2/2)

−1 0 10246

1

1.2

Level 3

−1 0 10246

1

1.2

Level 4

−1 0 10246

1

1.2

Level 5

−1 0 10246

1

1.2

Level 6

−1 0 10246

1

1.2

Latitude [rad]

Level 7

Longitude [rad]

0 60 120 180 240 300 3601

1.05

1.1

1.15

1.2

Longitude [deg]

Ana

lyse

s

Cross section of the Analyses along the longitude

L

a priori = 1200 km

Lfitted

= 1205.2465 km

AnalysesFitted analyses

−60 −30 0 30 601

1.05

1.1

1.15

1.2L

a priori = 1200 km

Lfitted

= 1204.9665 km

Cross section of the Analyses along the latitude

Ana

lyse

sLatitude [deg]

1 1.05 1.1 1.15 1.2

0

2

4

6

8

10

Cross section of the Analyses along along the altitude

Analyses

Ver

tical

Lev

els

La priori

= 1L

fitted = 1


Assimilation of EOS MLS Ozone observations (3/9)

BASCOE setup:

Resolution: 2◦ lat × 2◦ lon × 37 levels

Only O3 is considered and the chemistry is turned off in order toreduce the CPU time. Ozone data are rejected above 0.5 hPa.

wind and T◦ are taken from ERA-Interim

EOS MLS O3 data are assimilated between December 2004 and March2005



To make the system optimal, the error parameters must be tuned

This is done using the innovations (Hollingsworth and Lonnberg,1986, Tellus):

xb = xtruth + εb

yo = Hxtruth + εo

then:yo −Hxb = εo −Hεb

Assuming that observationserrors are horizontallyuncorrelated and thatbackground errors arespattially correlated,auto-covariance of I can beused to estimates the errors.

Thus: 〈I (i , i), I (i , i)〉 = (σb)2 + (σo)2

〈I (i , j), I (i , j)〉 = (σb)2ρ(rij)



Application of Hollingsworth and Lonnberg on BASCOE

Step 1: Assimilation of EOS MLS (Dec 2004) without spatialcorrelations, σb = 30%; σo = 15%⇒ EXP 1




Step 2: Estimating the errors parameters from EXP 1





0 500 1000 1500 2000 2500−1

0

1

2

3

4

5x 10

−14 Estimated covariance at 68.1292 hPa on 15dec2004

Gauss: Lgauss

=608km; σb=1.30e−07 ; σo=1.54e−07 ; resnorm=8.62e−03

SOAR: Lsoar

=356km; σb=1.38e−07 ; σo=1.47e−07 ; resnorm=4.16e−03

Distance [km]

Cov

aria

nce

[vm

r2 ]

Estimated covFitted GaussFitted SOAR





10−16

10−14

10−12

10−1

100

101

102

103

Error Variance [vmr]

Pre

ssur

e [h

Pa]

Obs Gauss

Obs SOAR

Obs MLS

Background Gauss

Background SOAR

0 500 1000 1500 2000 2500

10−1

100

101

102

103

Horizontal Correlation Length Scale [km]

Date = 15dec2004

GaussSOAR

0 0.05 0.1 0.15

10−1

100

101

102

103

Residual Norm of the fit [vmr]

Errors form EXP 1

GaussSOAR




Step 3: Re-assimilation of EOS MLS (Dec 2004).

σb and σo provided by tuning of EXP 1Horizontal correlations: SOAR with Lh = 600 kmVertical correlations: gaussian with Lv = 1.5 levelThis is EXP 2





10−16

10−14

10−12

10−1

100

101

102

103


Pre

ssur

e [h

Pa]

Obs Gauss

Obs SOAR

Obs MLS

Background Gauss

Background SOAR

0 500 1000 1500 2000 2500

10−1

100

101

102

103


Date = 15dec2004

GaussSOAR

0 0.05 0.1 0.15

10−1

100

101

102

103


Errors form EXP 2

GaussSOAR





σb and σo provided by tuning of EXP 2Horizontal correlations: SOAR with Lh tuned from EXP 2Vertical correlations: gaussian with Lv = 1.5 levelThis is EXP 3




Step 6: Estimating the errors parameters from EXP 3 and checkingfor convergence

⇒ Convergence not reached





⇒ Convergence not reached

10−16

10−14

10−12

10−1

100

101

102

103


Pre

ssur

e [h

Pa]

Obs Gauss

Obs SOAR

Obs MLS

Background Gauss

Background SOAR

0 500 1000 1500 2000 2500

10−1

100

101

102

103


Date = 15dec2004

GaussSOAR

0 0.05 0.1 0.15

10−1

100

101

102

103


Errors form EXP 2

GaussSOAR




Step 6: Estimating the errors parameters from EXP 3 and checkingfor convergence ⇒ Convergence not reached

10−16

10−14

10−12

10−1

100

101

102

103


Pre

ssur

e [h

Pa]

Obs Gauss

Obs SOAR

Obs MLS

Background Gauss

Background SOAR

0 500 1000 1500 2000 2500

10−1

100

101

102

103


Date = 15dec2004

GaussSOAR

0 0.05 0.1 0.15

10−1

100

101

102

103


Errors form EXP 3

GaussSOAR





σb and σo provided by tunning of EXP 3Horizontal correlations: SOAR with Lh tuned from EXP 3Vertical correlations: gaussian with Lv = 1.5 levelThis is EXP 4





⇒ Convergence Ok





⇒ Convergence Ok

10−16

10−14

10−12

10−1

100

101

102

103


Pre

ssur

e [h

Pa]

Obs Gauss

Obs SOAR

Obs MLS

Background Gauss

Background SOAR

0 500 1000 1500 2000 2500

10−1

100

101

102

103


Date = 15dec2004

GaussSOAR

0 0.05 0.1 0.15

10−1

100

101

102

103


Errors form EXP 2

GaussSOAR





⇒ Convergence Ok

10−16

10−14

10−12

10−1

100

101

102

103


Pre

ssur

e [h

Pa]

Obs Gauss

Obs SOAR

Obs MLS

Background Gauss

Background SOAR

0 500 1000 1500 2000 2500

10−1

100

101

102

103


Date = 15dec2004

GaussSOAR

0 0.05 0.1 0.15

10−1

100

101

102

103


Errors form EXP 3

GaussSOAR




Step 8: Estimating the errors parameters from EXP 4 and checkingfor convergence ⇒ Convergence Ok

10−16

10−14

10−12

10−1

100

101

102

103


Pre

ssur

e [h

Pa]

Obs Gauss

Obs SOAR

Obs MLS

Background Gauss

Background SOAR

0 500 1000 1500 2000 2500

10−1

100

101

102

103


Date = 15dec2004

GaussSOAR

0 0.05 0.1 0.15

10−1

100

101

102

103


Errors form EXP 4

GaussSOAR




New Exp 5

σb = 30%; σo = 15%Horizontal correlations: SOAR with Lh = 600 kmVertical correlations: gaussian with Lv = 1.5 levelThis is EXP 5




Step 9: Checking whether J/p has reached its theoretical value of 0.5

J(x(t0)) =1

2[x(t0)− xb(t0)]T B−1[x(t0)− xb(t0)]

+1

2

N∆t∑i=0




Application of Hollingsworth and Lonnberg on BASCOEStep 9: Checking whether J/p has reached its theoretical value of 0.5

J(x(t0)) =1

2[x(t0)− xb(t0)]T B−1[x(t0)− xb(t0)]

+1

2

N∆t∑i=0


Dec Jan Feb Mar10

−1

100

101

102

J/p

Dec Jan Feb Mar

102

103

104

105

106

Jb

EXP 1

EXP 2

EXP 3

EXP 4

EXP 5



OmF: Deviation of the BASCOE background fields to EOS MLSobservations Period considered: 01Jan2005 - 31Mar2005

−50−30−10 10 30 50

0.1

1

10

100

Bias [−90°,−60°]

[%]

Pre

ssur

e [h

Pa]

0 10 20 30 40 50

0.1

1

10

100

Std. Dev. [−90°,−60°]

[%]

Pre

ssur

e [h

Pa]

−50−30−10 10 30 50

0.1

1

10

100

Bias [−60°,−30°]

[%]

0 10 20 30 40 50

0.1

1

10

100

Std. Dev. [−60°,−30°]

[%]

−50−30−10 10 30 50

0.1

1

10

100

Bias [−30°,30°]

[%]

0 10 20 30 40 50

0.1

1

10

100

Std. Dev. [−30°,30°]

[%]

−50−30−10 10 30 50

0.1

1

10

100

Bias [30°,60°]

[%]

0 10 20 30 40 50

0.1

1

10

100

Std. Dev. [30°,60°]

[%]

−50−30−10 10 30 50

0.1

1

10

100

Bias [60°,90°]

[%]

0 10 20 30 40 50

0.1

1

10

100

Std. Dev. [60°,90°]

[%]

EXP 1

EXP 2

EXP 3

EXP 4

EXP 5

OmF (MLS − BG BASCOE)



OmA: Deviation of the BASCOE analyses fields from independentACE-FTS observations Period considered: 01Jan2005 - 31Mar2005

−50−30−10 10 30 50

0.1

1

10

100

Bias [−90°,−60°]

[%]

Pre

ssur

e [h

Pa]

0 10 20 30 40 50

0.1

1

10

100

Std. Dev. [−90°,−60°]

[%]

Pre

ssur

e [h

Pa]

−50−30−10 10 30 50

0.1

1

10

100

Bias [−60°,−30°]

[%]

0 10 20 30 40 50

0.1

1

10

100

Std. Dev. [−60°,−30°]

[%]

−50−30−10 10 30 50

0.1

1

10

100

Bias [−30°,30°]

[%]

0 10 20 30 40 50

0.1

1

10

100

Std. Dev. [−30°,30°]

[%]

−50−30−10 10 30 50

0.1

1

10

100

Bias [30°,60°]

[%]

0 10 20 30 40 50

0.1

1

10

100

Std. Dev. [30°,60°]

[%]

−50−30−10 10 30 50

0.1

1

10

100

Bias [60°,90°]

[%]

0 10 20 30 40 50

0.1

1

10

100

Std. Dev. [60°,90°]

[%]

EXP 1

EXP 2

EXP 3

EXP 4

EXP 5

OmA (ACEFTS − ANA BASCOE)



Analyses on 22 Feb 2005

Longitude

Latit

ude

Analyses [ppmv] for EXP−1 at 9.8925 hPa

3

4

5

6

7

8

9

10

11

Longitude

Latit

ude


3

4

5

6

7

8

9

10

11

Longitude

Latit

ude


3

4

5

6

7

8

9

10

11

Longitude

Latit

ude


3

4

5

6

7

8

9

10

11



Increments on 22 Feb 2005

Longitude

Latit

ude

Analysis Increments [%] for EXP−1 at 9.8925 hPa

−6

−4

−2

0

2

4

6

Longitude

Latit

ude


−6

−4

−2

0

2

4

6

Longitude

Latit

ude


−5

0

5

Longitude

Latit

ude


−5

0

5


Conclusions

1 Spatial correlation implemented in BASCOE with a new controlvariable defined in the spectral space

2 Homogeneous and isotropic horizontal AND vertical correlations aremodelled

3 Case studies with EOS MLS O3 data show:

Errors parameters are successfully tuned using Hollingsworth andLonnber’s methodTuned error parameters allow to improve the lower stratosphere


implementation of spatial correlations in the background …...errera and m enard bascoe becm sparc...

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