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Accelerating

minimizations in ensemble variational assimilation

G. Desroziers and L. Berre

Météo-France and CNRS

2/18

Outline

1.

Ensemble Variational assimilation

2.

Accelerating minimizations

3.

Conclusion and future work

3/18

Outline

1.

Ensemble Variational assimilation

2.

Accelerating minimizations

3.

Conclusion and future work

4/18 The operational

Météo

-France ensemble Var assimilation

1st operational implementation

of En Var assim. (2008; ECMWF, 2010- 11). Six perturbed members, T399 L70 (50 km), with global

4D-Var

Arpege.

Spatial filtering

of error variances.

Inflation

of ensemble B/ model error contributions.

Flow-dependent

background error variances in 4D-Var (and EnDA), for minimizations (all variables) and observation QC. Initialization of Météo-France ensemble prediction by EnDA.

Flow-dependent

background error correlations experimented. 5/18

Background

error correlations using EnDAand wavelets

Wavelet-implied horizontal length-scales (in km),

for wind near 500 hPa, averaged over a 4-day period. (Varella et al 2011b, and also Fisher 2003,

Deckmyn and Berre 2005, Pannekoucke et al 2007)

6/18

Outline

1.

Ensemble Variational assimilation

2.

Accelerating minimizations

3.

Conclusion and future work

7/18

Ensemble variational assimilation

Minimize N cost-functions J

n, n=1, N, with perturbed innovations d n: J n(ddddx n) = 1/2ddddx nT B-1 ddddx n+ 1/2 (d n-H nddddx n)TR-1(d n-H nddddx n), withB and R background and observation error matrices, andd n=y o+ R 1/2 hhhhon- H(M(x bn)), withhhhh ona vector of random numbers. x an = x bn + ddddx n

Perturbed backgrounds for the next analyses:

x bn+ = M( x an) + Q 1/2 hhhhmn , with hhhh mn a vector of random numbers and Q model error covariance matrix. 8/18

Hessian matrixof the assimilation problem

Hessian of the cost-function:

J"" = B

-1+ H

TR-1H.

Bad conditioning of J"": very slow (or no) convergence.

Cost-function withB

1/2 preconditioning ( ddddx = B 1/2 cccc):

J(cccc) = 1/2cccc

Tcccc+ 1/2 (d-H B

1/2 cccc)

TR-1(d-H B

1/2 cccc).

Hessian of the cost-function:

J"" = I + B

T/2 H

TR-1H B

1/2 Far better conditioning and convergence!(Lorenc 1988, Haben et al 2011) 9/18

Lanczos algorithm

Generate iteratively a set of K orthonormal vectorsqsuch as Q KT J"" Q K= T k, whereQ K= (q

1 q2 ... q

K), and T

Kis a tri-diagonal matrix.

The extremal eigenvalues of T

Kquickly converge towards the extremal

eigenvalues of J"". If T K= Y KLLLL KYKT is the eigendecomposition of T

K, the Ritz vectors are

obtained with Z K= Q KYk and the Ritz pairs (z k,l k) approximate the eigenpairs of J"". 10/18

Lanczos algorithm/ Conjugate gradient

Use of the Lanczos vectors to get the solution of the variational problem: c ccc K = cccc 0+ Q KWWWW K.

Optimal coefficients WWWW

kshould make the gradient of J vanish atcccc K:

J"(cccc

K) = J"(cccc

0)+ J"" (cccc

K-cccc

0) = J"(cccc

0)+J"" Q

KWWWW K = 0, which gives W WWW K = - ( Q KT J"" Q K)-1Q KT

J"(cccc

0) = -T K-1 Q KT

J"(cccc

0), and then c ccc K = cccc 0-Q KT K-1 Q KT

J"(cccc

0).

Same solution as after K iterations of a Conjugate Gradient algorithm.(Paige and Saunders 1975, Fisher 1998)

11/18▪

Minimizations with

- unperturbed innovations dand - perturbed innovations d nhave basically the same Hessians:

J""(d) = I + B

T/2 H

TR-1H B

1/2 , J""(d n) = I + B T/2 H nT R-1H nB1/2 , The solution obtained for the " unperturbed » problem c ccc K = cccc 0-Q K( Q KT J"" Q K)-1Q KT

J"(cccc

0, d) can be transposed to the " perturbed » minimization c ccc K,n = cccc 0-Q K( Q KT J"" Q K)-1Q KT

J"(cccc

0, d n) to improve its starting point.

Accelerating

a " perturbed

» minimization

using " unperturbed» Lanczos vectors 12/18

Accelerating

minimizations using " perturbed» Lanczos vectors If N perturbed minimizations, with K iterations, already performed, then the starting pt of a perturbed (or unpert.) minim. can be written c ccc K = cccc 0+ Q K,N WWWW K,N whereWWWW K,N is a vector of N x K coefficients and Q K,N = (q

1,1 ... q

K,1 ... q 1,N ... q K,N is a matrix containing the N x K Lanczos vectors. Following the same approach as above, the solution canbe expressed: c ccc K,N = cccc 0-Q K,N ( Q K,NT J"" Q K,N )-1Q K,NT

J"(cccc

0).

Matrix Q

K,NT J"" Q K,N is no longer tri-diagonal, but can be easily inverted. 13/18 Accelerating minimizations using N setsof " perturbed» Lanczos vectors (K = 10)

Thin solid line: exact perturbed analysis

Thick dashed line : starting point with NxK = 10x10 vectors n=401 (ds = 100km) p=200 (ds o=50/350km) s b=1, L b=300km s b=0,33/1, L o=0kmquotesdbs_dbs31.pdfusesText_37

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