Page 1 The Construction and Use of Linear Models in Large-scale Data Assimilation Tim Payne...

Page 1

The Construction and Use of Linear Models in Large-scale Data Assimilation

Tim Payne

Large-Scale Inverse Problems and Applications in the Earth Sciences

October 24th 2011

© Crown copyright 2011 Page 2Page 2

Part I.

The Construction of Linear Models in Data Assimilation


Notation


Update-Prediction Cycle


First Strategy – exact evolution of covariances


Second strategy – EKF using tangent-linear


Third strategy – EKF using best linear approximation


Explicit formula for best linear approximation


Basic properties of best linear approximation


Cloud function

T

Seek a rational basis on which to 'regularise' in 4D-Var the Smith cloud scheme of 1990,

where cloud fraction is expressed as a function of

( , )=

(1 ) ( , )

Cloud fraction and its de

sat LN

c sat L

C

q q T pQ

RH q T p

N

212

212

rivative w.r.t. Q are

0 1 0 1

(1 ) 1 0 (1 ) 1 0'

1 (1 ) 0 1 (1 ) 0 1

1 1 0 1

How should one regularise this function?

Met Office ha

N N

N N N N

N N N N

N N

for Q for Q

Q for Q Q for QC C

Q for Q Q for Q

for Q for Q

21N2d used regularisation [1 tanh(2 )] with derivativeC'=sech (2Q )NC Q


Smith Cloud Scheme with ‘ad hoc’ Regularisation

212

212

0 1

(1 ) 1 0

1 (1 ) 0 1

1 1

N

N N

N N

N

for Q

Q for QC

Q for Q

for Q

0 1

(1 ) 1 0'

(1 ) 0 1

0 1

N

N N

N N

N

for Q

Q for QC

Q for Q

for Q

12Standard regularisation [1 tanh(2 )]NC Q 2' sech (2 )NC Q


Smith Cloud Scheme with ‘Optimal’ Regularisation

1

. 1 Gaussian

~ (0, )

1 | |pdf ( ) exp

2

Eg p

Q N

xf x

| |

Opt reg LS:

E{C(Q+ )}

1( )exp( )

2x

Q

C Q x dx

2

| |3

E{C(Q+ ) Q}Opt reg PF:

1( ) exp( )

2x

Q

C Q x x dx

N N N N N

N N NN N N N N 2

N

Optimal regularisation: C(Q + Q ) F(Q )+T(Q ) Q

E[C(Q + Q ) Q ]If E( Q ) 0 then F(Q )=E[C(Q + Q )] and T(Q )=

E[ Q ]


Incremental 4D-Var


Options for linearisation step required for incremental 4D-Var


Gain matrix implied by each option


Advantage of BLA over TL in incremental 4D-Var


Pseudo Chain-rule for best linear approximation


Use of best linear approximation in EKF


The prior covariance implied by different approximations


Prior covariance using best linear estimate


Prior covariance using the best linear estimate always underestimates the true prior


The Duffing Map


100,000 iterates of Duffing Map


Reminder of EKF algorithm


Prior covariance for Duffing Map


Mean square analysis error in Duffing map: TL and best linear estimate compared


Part II.

The Use of Linear Models in Data Assimilation


Linearisation error in 4D-Var as used in real numerical weather prediction models



Linear model for evolution of increments


Linearisation error as a stochastic error


Issues in forming EKF


Signal model for system with time correlated linearisation error


EKF with time correlated linearisation error


Parameters for filter including linearistion error


Example: L95, nearly perfect full model, persistence for linear model


Example: L95, nearly perfect full model, persistence for linear model, results


Variational version: weak constraint 4D-Var allowing for time correlated linearisation error


Remarks on variational form


Long window weak constraint 4D-Var allowing for linearisation error, same example


Summary to Part I


Summary to Part II


The End

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