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03/17/09 1
DATA ASSIMILATION ALGORITHMS
Arnold Heemink
Delft University of Technology
Joint work with Martin Verlaan, Remus Hanea and Alina Barbu
03/17/09 2
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
03/17/09 23
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
03/17/09 24
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−=
03/17/09 25
Nonlinearity measure V versus RMS error for
different algorithms (Lorenz3 model)
03/17/09 26
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
03/17/09 27
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 =
03/17/09 28
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
03/17/09 29
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
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