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1christoph.schraff@dwd.deKENDASiminaria, ARPA-SIM, Bologna, 22 November 2010

PP KENDA : Contents

the Priority Project KENDA (Km-Scale Ensemble-Based Data Assimilation):

an overview of the new assimilation scheme for the COSMO model

Christoph SchraffDeutscher Wetterdienst, Offenbach, Germany

basic theory

overview on PP KENDA / preliminary tasks + technical implementation

results from preliminary experiments

current work and plans within PP KENDA

2christoph.schraff@dwd.deKENDASiminaria, ARPA-SIM, Bologna, 22 November 2010

after few hours, a forecast of convection is a long-term forecast

deliver probabilistic (pdf) rather than deterministic forecast

need ensemble forecast and data assimilation for convection-permitting NWP

and Ensemble methods allow to account for the errors of the day

Km-scale Ensemble-based Data Assimilation (KENDA):Local Ensemble Transform Kalman Filter (LETKF, a variant of EnKF family),because of its relatively low computational costs

KENDA : Why develop Ensemble-Based Data Assimilation ?

3christoph.schraff@dwd.deKENDASiminaria, ARPA-SIM, Bologna, 22 November 2010

Estimation Theory

observation hO observation error Obackground hB (a-priori info) background error B

Assume: - no systematic errors (no bias) - Gaussian distribution of random errors

( ) ( )22

22)(B

B

O

O hhhhdhdJhJ

+== 0 : of minimum Find

( )

222

22

2

22

2

22

2

111:OBA

A

BOOB

BBB

OB

OO

OB

BA hhhhhh

+=

+

+=+

++

=

error Estimation

estimateBest (in a statistical sense)

( ) ( )2

2

2

2

B

B

O

O hhhhJ+= functionCost measures the distance

to observation and background

Basic theory of data assimilation:estimation theory

( ) ( )

= 22

2

2

expexpB

B

O

O hhhh

density yProbabilitwant solution with min. error variance/ maximum likelihood

4christoph.schraff@dwd.deKENDASiminaria, ARPA-SIM, Bologna, 22 November 2010

Estimation Theory

observation hO observation error Obackground hB background error BAssumptions: - no systematic errors (no bias)

- Gaussian distribution of errors

Generalisationsy vector of N observationsR matrix of observation error covariances x vector of M model grid pts * variables (M>>N)B matrix of background error covariancesH observation operator : projecting

from model space into observation space

( )[ ] ( )[ ] [ ] [ ]BBJ xxBxxxyRxy += 1T1T HHFind x minimising J . Problems:

1. H non-linear only approx. solution iterativ. 2. Matrix to be inverted too large( )

( )

222

22

2

22

2

111:OBA

A

BOBO

OB

BOOB

BBA

hhh

hhhh

+=

+

+=

+

+=

error Estimation

estimateBest (in a statistical sense)

4DVAR: y = ytA yt>tA H(xtA) H(xt) = H(M(xtA)) , M : NWP model measures the distance of the model trajectory to observations within an interval, with t tA

( ) ( )2

2

2

2

B

B

O

O hhhhJ+= functionCost

measures the distance to observation and background

Basic theory of data assimilation:estimation theory

no need to compute A, but then need to specify R, B ! usually by statistics (climatology), can be wrong

( ) ( )[ ]BxyRHHRHBxx H++= 1T11T1 ----BA solution (in model space) : 3DVAR

( ) 11T1 --- HRHBA +=

5christoph.schraff@dwd.deKENDASiminaria, ARPA-SIM, Bologna, 22 November 2010

provides optimal solution to data assimilation problem if no systematic errors Gaussian random errors system linear (M, H ) (practical issue:) problem small enough to allow computing matrix inversions

Basic theory :Kalman filter

[ ] [ ]bbbTaba HxyKxHxyRHPxx OO +=+= 1

( )[ ] 1-1HRH PP -1 += Tba and also compute explicitly analysis error covariances (notation: B Pb )

and then, forecast to the next analysis time (using the forward model M )n

an

bf Mxxx == +1

( )QMPMPP errormodel+== + Tnanbf 1

analysis step: compute analysis of system state as before (K : Kalman Gain)

6christoph.schraff@dwd.deKENDASiminaria, ARPA-SIM, Bologna, 22 November 2010

but also need new analysis ensemble xa(i) in order to cycle the system: apply Kalman gain to each xb(i)

obs must be artificially perturbed to get reasonable spread (stochastic filter) ensemble square root filters, e.g. (L)ETKF :

computes analysis ensemble in a deterministic way (deterministic filter)

idea : choose ensemble (k members) of initial conditions (model states) such thatensemble spread around analysis mean characterises analysis error Pa ; then propagate ensemble to next analysis time using the non-linear model, and use resulting forecast ensemble xb(i) to represent forecast error Pb :

Basic theory :Ensemble Kalman filter (EnKF)

( ) bibbTbbb ik xxXXXP == )(,)1( of column th- wherePb known can compute Pa, K , analysis mean

for : non-linear forecast systems (M NWP model M ) large problems (large state vector, many obs)

remaining assumptions : no systematic errors Gaussian random errors

ax

7christoph.schraff@dwd.deKENDASiminaria, ARPA-SIM, Bologna, 22 November 2010

with model error

model error is not accounted for covariance inflation

Pb = (k 1) Xb (Xb)T has rank (k 1) sampling error for small correlations localise error covariances cannot represent all errors in all degrees of freedom of the global system (model)

locally, model has much less degrees of freedom localisation increase ensemble spread artificially perturbed obs / covariance inflation

initial (analysis) ensemble spread characterises analysis error Pa , but ensemble size is limited, ensemble can only sample but not fully represent errors

Basic theory :Ensemble Kalman filter (EnKF)

ensemble members

area of high probabilitynon-linear

use resulting forecast ensemble xb(i) to represent forecast error Pb :

not represented by ensemble

Local ETKF : compute analysis (transform matrix) for each grid point separatelyusing only nearby obs

8christoph.schraff@dwd.deKENDASiminaria, ARPA-SIM, Bologna, 22 November 2010

ensemble spread is mainly localised over frontal area forecast errors assumed in EnKF are mainly localised over frontal area observation causes analysis increments over frontal area (see right panel) EnKF accounts for errors of the day

Whitaker et al., 2005

Ensemble Kalman filter (EnKF) : a favourable property

use forecast ensemble perturbations Xb(i) to represent forecast error Pb :

9christoph.schraff@dwd.deKENDASiminaria, ARPA-SIM, Bologna, 22 November 2010

for Pb = (k 1) Xb (Xb)T , J(x) is well-defined in sub-space S spanned by Xb

Basic theory :Ensemble Transform KF (Hunt et al.)

( ) ( ) [ ] [ ])()()( 11 xyRxyxxPxxx OO HHJ TbbTb +=

regard Xb as linear transformation from kdim. space S onto S ,then for any vector w in S, Xbw is in S,

if w is Gaussian random vector with mean 0 and covariance (k 1) I ,then is Gaussian with mean and cov. (k 1) Xb (Xb)T bxwXxx bb +=

( ) [ ] [ ])()(1)( 1 wXxyRwXxywww OO bbTbbT HHkJ +++= set up cost function in (low-dimensional !) ensemble space

( ) ( ) ( )space obs. in onsperturbati ensemble : of column th-

, , wherebibb

ibibbibibbbbb

i

HHH

yyYxyyxywYywXx

=

=++)(

)()()()() ,

observation operator H non-linear approximate :

10christoph.schraff@dwd.deKENDASiminaria, ARPA-SIM, Bologna, 22 November 2010

( ) [ ] [ ]wYyyRwYyywww OO bbTbbTkJ += 1)1)(J(w) : has the form of Kalman Filter cost function

solution as above, valid for ensemble mean:

need analysis ensemble members, with ensemble spread (k 1) Xa (Xa)T = Pa ,

choose:

( ))()()( iaabbiaaia WwXxXxx ++=+=[ ]1/2PWWXX awaaba k )1( == where,

and in model space : ( )Tbawbaabba XPXP,wXxx =+=( ) [ ] ( )[ ]-111 )1( bTbawbTbawa k YRY I P,yyRYPw O +==

analysis (mean) and analysis error covariance :

note: Pwa is in low-dimensional ensemble space and can be computed directly makes the scheme very efficient

Basic theory :Ensemble Transform KF (Hunt et al.)

11christoph.schraff@dwd.deKENDASiminaria, ARPA-SIM, Bologna, 22 November 2010

wanted: ensemble of perturbed analyses, to start next cycle of ensemble forecasts analysis mean and each analysis ensemble member is (locally) a linear combination of the forecast ens. members, where those forecast members that match obs well get larger weights; observations reduce spread of ensemble explicit solution for minimisation of cost function in ensemble space provides required transform matrix

perturbed forecasts

ensemble mean forecast

(local, inflated) transform

matrixanalysis mean(computed only in ensemble space)

pert

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