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DATA ASSIMILATION ALGORITHMS

Arnold Heemink

Delft University of Technology

Joint work with Martin Verlaan, Remus Hanea and Alina Barbu

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Overview Introduction Kalman filtering Ensemble Kalman filter algorithms for

large scale systems: - stochastic scheme: EnKF - deterministic schemes: Reduced Rank

filters - semi-deterministic schemes: ESRF Nonlinearity of the data assimilation

problem: which algorithm is the most suitable for a given application?

An ensemble approach to variational data assimilation

Concluding remarks

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State space model

The (non linear) physics:

where X is the state, p is vector of uncertain parameters, f represents the (numerical) model, G is a noise input matrix and W is zero mean system noise with covariance Q

The measurements:

where M is the measurement matrix and V is zero mean measurement noise with covariance R

kkk WkGkpXfX )(),,(1 +=+

kkk VXkMZ += )(

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Linear dynamics F(k) and constant parameters: State estimation using

Kalman filtering

A recursive algorithm for k=1,2,… to determine :

ak

ak

fk

fk

P

X

P

X Optimal estimate of the state at time k using measurements up to and including k-1

Covariance matrix of the estimation error

Optimal estimate of the state at time k using measurements up to and including k

Covariance matrix of the estimation error

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Kalman filter algorithm

1

1

1

)]()()([)()(

)]()([

])()[(

)1()1()1()1()1(

,)1(

−

−

−

+=

−=

−+=

−−−+−−=

−=

kRkMPkMkMPkK

PkMkKIP

XkMZkKXX

kGkQkGkFPkFP

XkFX

Tfk

Tfk

fk

ak

fkk

fk

ak

TTak

fk

ak

fk

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Second-order truncated filter algorithm for nonlinear systems with dimension n

1

1

1 1,1

2

21

1

)]()()([)()(

)]()([

])()[(

)1()1()1()1()1(

)())(()(

1

−

−

= =−−

+=

−=

−+=

−−−+−−=

∂∂

∂+= ∑ ∑−

kRkMPkMkMPkK

PkMkKIP

XkMZkKXX

kGkQkGkFPkFP

Pxx

fXfX

Tfk

Tfk

fk

ak

fkk

fk

ak

TTak

fk

n

i

n

jji

ak

Xji

ll

akl

fk

ak

F(k) is now the tangent linear model

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SQRT formulation of the covariance P

Define S according to P=SS’And rewrite the algorithm in terms of S

Advantages:

-SS’ always positive definite-S can be approximated by a matrix with reduced

number of columns

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Stochastic scheme:Ensemble Kalman filter (EnKF)

To represent the probability density of the state

estimate N ensemble members are chosen

randomly:

...]1

ˆ[...

ˆ 1

−−=

= ∑

N

xS

x

i

iN

ξξ

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Ensemble Kalman filter

Each ensemble member is propagated using the original (non linear) model, no tangent linear model is required

Errors are of statistical nature Errors decrease very slowly with large sample

size Computational effort required is approximately N

model simulations

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Deterministic schemesReduced Rank square root filtering (RRSQRT)

The square root matrix S is defined according to

P=SS’ where S are the q leading EOF’s of P:

S is generally of very low rank: q<<n

ii Sx

x

εξξ

+==

ˆ

ˆ0

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Reduced-rank Kalman filter

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Reduced Rank filter

Each ensemble member is propagated using the original (non linear) model, no tangent linear model is required

Errors are caused by truncation of the eigenvectors

The algorithm is sensitive to filter divergence problems and, therefore q has to be chosen sufficiently large

Computational effort required is approximately q+1 model simulations + eigenvalue decomposition (~q³)

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Grid of Ozone prediction model

Application to atmospheric chemistry model

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Comparison between the different approaches

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Semi-deterministic schemes: Ensemble Square Root Filters (ESRF)

An alternative way to solve the measurement update step is:

The general solution is given by:

where T is an ensemble transform matrix.

RHSHSR

SMSRMSISSSPTff

TffTffTaaa

+=

−== −

)(

)]()([)( 1

)]()([ 1 fTfT

fa

MSRMSITT

TSS−−=

=

The symmetric ESRF

Unbiased updated ensemble mean

Minimum analysis increment:

fa XX −

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Application to a two dimensional transporttransport model based on the advection diffusion equation

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Performance ESRF

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Performance RR SQRT filter

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Application to the Lorenz 40 model

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Lorenz40 model: ESRF

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Lorenz40 model: RRSQRT

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Nonlinearity in data assimilation problems

Nonlinearity in data assimilation problems may

introduce filter divergence. It depends on:- Nonlinearity of the underlying model- Covariance of the system noise process- Amount of measurement information

Nonlinearity in the data assimilation problem can be

a guideline to choose to most suitable algorithm

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A measure for nonlinearity

The only difference between a first order extended

Kalman filter and a second order truncated filter is

a bias correction term b

In order to evaluate the relative importance of the

bias b compared to the uncertainty of the state

estimate P we propose the non linearity measure V

bPbV T 1−=

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Nonlinearity measure V versus RMS error for

different algorithms (Lorenz3 model)

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Strong constraint variational data assimilation

If we solve the uncoupled system:

where F(k) is the tangent linear model,the gradient of the criterion can be computed by:

Very efficient in combination with a gradient-based optimization scheme. BUT: we need the adjoint implementation!

0,

))(()()(

),,(

00

11

1

==−+=

=−

+

+

K

kkT

kT

k

kk

vxX

XkMZRkMvkFv

kpXfX

p

fv

p

J Kk

k

Tk ∂

∂−=∂∂ ∑

=

=0

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A model reduction approach to data assimilation

Consider the q dimensional sub space:

And project the tangent linear approximation of the original model onto this sub space:

We now have an explicit (approximate) system description of the model variations including its adjoint!The sub space can be determined by computing the EOF (Empirical Orthogonal Functions) of an ensemble of model simulations

kkk

kT

k

vrPkMZ

rPkFPr

+==+

])([

])([1

...][... jpP =

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A model reduction approach to data assimilation

Run the original model and determine the residuals in the data points

Generate an ensemble of N model and choose a set of “snapshots” from this ensemble

Determine the q dominant EOF's: sub space P

Project original model onto P. This requires another q model simulations. The adjoint is now available too.

Perform the optimization in reduced space and obtain the new parameter estimates

Repeat the process from the start if necessary

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Some remarks

Very efficient in case the simulation period of the ensemble of model simulation is very small compared to the calibration period

The amount of measurements should not be very large

Not very sensitive to local minimaWill not find the exact minimum of the original

problem

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Concluding remarks

Semi-deterministic schemes: The symmetric ESRF is very attractive

Deterministic schemes: The symmetric RRSQRT is very attractive

For some type of applications the adjoint implementation in 4Dvar can be avoided using model reduction.

More error analysis of the algorithms is needed