Introduction to Data Assimilation with 4D-Var and its...
Transcript of Introduction to Data Assimilation with 4D-Var and its...
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Introduction to Data Assimilation with 4D-Var and itsrelation to Tikhonov regularisation
Melina Freitag
Department of Mathematical Sciences
University of Bath
INVERT Centre Bath22nd August 2008
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1 Introduction
2 Variational Data AssimilationLeast square estimationExamplesKalman FilterProblems and Issues
3 Tikhonov regularisation
4 Plan and work in progress
![Page 3: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/3.jpg)
Outline
1 Introduction
2 Variational Data AssimilationLeast square estimationExamplesKalman FilterProblems and Issues
3 Tikhonov regularisation
4 Plan and work in progress
![Page 4: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/4.jpg)
What is Data Assimilation?
Loose definition
Estimation and prediction (analysis) of an unknown, true state bycombining observations and system dynamics (model output).
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What is Data Assimilation?
Loose definition
Estimation and prediction (analysis) of an unknown, true state bycombining observations and system dynamics (model output).
Some examples
Navigation
Medical imaging
Numerical weather prediction
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Data Assimilation in NWP
Estimate the state of the atmosphere xi.
Observations y
Satellites
Ships and buoys
Surface stations
Aeroplanes
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Data Assimilation in NWP
Estimate the state of the atmosphere xi.
A priori information xB
background state (usualprevious forecast)
Observations y
Satellites
Ships and buoys
Surface stations
Aeroplanes
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Data Assimilation in NWP
Estimate the state of the atmosphere xi.
A priori information xB
background state (usualprevious forecast)
Models
a model how the atmosphereevolves in time (imperfect)
xi+1 = M(xi)
Observations y
Satellites
Ships and buoys
Surface stations
Aeroplanes
![Page 9: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/9.jpg)
Data Assimilation in NWP
Estimate the state of the atmosphere xi.
A priori information xB
background state (usualprevious forecast)
Models
a model how the atmosphereevolves in time (imperfect)
xi+1 = M(xi)
a function linking model spaceand observation space(imperfect)
yi = H(xi)
Observations y
Satellites
Ships and buoys
Surface stations
Aeroplanes
![Page 10: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/10.jpg)
Data Assimilation in NWP
Estimate the state of the atmosphere xi.
A priori information xB
background state (usualprevious forecast)
Models
a model how the atmosphereevolves in time (imperfect)
xi+1 = M(xi)
a function linking model spaceand observation space(imperfect)
yi = H(xi)
Observations y
Satellites
Ships and buoys
Surface stations
Aeroplanes
Assimilation algorithms
used to find an (approximate)state of the atmosphere xi attimes i (usually i = 0)
using this state a forecast forfuture states of the atmospherecan be obtained
xA: Analysis (estimation of thetrue state after the DA)
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Schematics of DA
Figure: Background state xB
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Schematics of DA
Figure: Observations y
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Schematics of DA
Figure: Analysis xA (consistent with observations and model dynamics)
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Data Assimilation in NWP
Underdeterminacy
Size of the state vector x: 432 × 320 × 50 × 7 = O(107)
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Data Assimilation in NWP
Underdeterminacy
Size of the state vector x: 432 × 320 × 50 × 7 = O(107)
Number of observations (size of y): O(105 − 106)
![Page 16: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/16.jpg)
Data Assimilation in NWP
Underdeterminacy
Size of the state vector x: 432 × 320 × 50 × 7 = O(107)
Number of observations (size of y): O(105 − 106)
Operator H (nonlinear!) maps from state space into observationsspace: y = H(x)
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Any easy scheme
Cressman analysis
Figure: Copyright:ECMWF
At each time step i
xA(k) = x
B(k)+
Pnl=1
w(lk)(y(l) − xB(l))Pn
l=1w(lk)
w(lk) = max
„
0,R2 − d2
lk
R2 + d2lk
«
dlk measures the distance between pointsl and k.
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Outline
1 Introduction
2 Variational Data AssimilationLeast square estimationExamplesKalman FilterProblems and Issues
3 Tikhonov regularisation
4 Plan and work in progress
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Data Assimilation in NWP
Estimate the state of the atmosphere xi.
Apriori information xB
background state (usualprevious forecast) has errors!
Models
a model how the atmosphereevolves in time (imperfect)
xi+1 = M(xi) + error
a function linking model spaceand observation space(imperfect)
yi = H(xi) + error
Observations y has errors!
Satellites
Ships and buoys
Surface stations
Airplanes
Assimilation algorithms
used to find an (approximate)state of the atmosphere xi attimes i (usually i = 0)
using this state a forecast forfuture states of the atmospherecan be obtained
xA: Analysis (estimation of thetrue state after the DA)
![Page 20: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/20.jpg)
Error variables
Modelling the errors
background error εB = xB − xTruth of average εB and covariance
B = (εB − εB)(εB − εB)T
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Error variables
Modelling the errors
background error εB = xB − xTruth of average εB and covariance
B = (εB − εB)(εB − εB)T
observation error εO = y − H(xTruth) of average εO and covariance
R = (εO − εO)(εO − εO)T
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Error variables
Modelling the errors
background error εB = xB − xTruth of average εB and covariance
B = (εB − εB)(εB − εB)T
observation error εO = y − H(xTruth) of average εO and covariance
R = (εO − εO)(εO − εO)T
analysis error εA = xA − xTruth of average εA and covariance
A = (εA − εA)(εA − εA)T
measure of the analysis error that we want to minimise
tr(A) = ‖εA − εA‖2
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Assumptions
Linearised observation operator: H(x) − H(xB) = H(x− xB)
Nontrivial errors: B, R are positive definite
Unbiased errors: xB − xTruth = y − H(xTruth) = 0
Uncorrelated errors: (xB − xTruth)(y − H(xTruth))T = 0
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Optimal least-squares estimater
Cost function
Solution of the variational optimisation problem xA = arg minJ(x) where
J(x) = (x − xB)T
B−1(x− x
B) + (y − H(x))TR
−1(y − H(x))
= JB(x) + JO(x)
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Optimal least-squares estimater
Cost function
Solution of the variational optimisation problem xA = arg minJ(x) where
J(x) = (x − xB)T
B−1(x− x
B) + (y − H(x))TR
−1(y − H(x))
= JB(x) + JO(x)
Interpolation equations
xA = x
B + K(y − H(xB)), where
K = BHT (HBH
T + R)−1K . . . gain matrix
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Conditional probabilities
Non-Gaussian PDF’s (probability density function)
P (x) is a priori PDF (background)
P (y|x) is the observation PDF (likelihood of the observations givenbackground x)
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Conditional probabilities
Non-Gaussian PDF’s (probability density function)
P (x) is a priori PDF (background)
P (y|x) is the observation PDF (likelihood of the observations givenbackground x)
P (x|y) conditional probability of the model state given theobservations, Bayes theorem:
argx max P (x|y) = argx maxP (y|x)P (x)
P (y)
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Conditional probabilities
Non-Gaussian PDF’s (probability density function)
P (x) is a priori PDF (background)
P (y|x) is the observation PDF (likelihood of the observations givenbackground x)
P (x|y) conditional probability of the model state given theobservations, Bayes theorem:
argx max P (x|y) = argx maxP (y|x)P (x)
P (y)
Gaussian PDF’s
P (x|y) = c1 exp“
−(x− xB)T
B−1(x − x
B)”
·
c2 exp“
−(y − H(x))TR
−1(y − H(x))”
xA is the maximum a posteriori estimator of xTruth. Maximising P (x|y)equivalent to minimising J(x)
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A simple scalar illustration
Room temperature
T O observation with standard deviation σO
T B background with standard deviation σB
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A simple scalar illustration
Room temperature
T O observation with standard deviation σO
T B background with standard deviation σB
T A = T B + k(T O − T B) with error variance σ2A = (1 − k)2σ2
B + k2σ2O
optimal k which minimises error variance
k =σ2
B
σ2B + σ2
O
equivalent to minimising
J(T ) =(T − T B)2
σ2B
+(T − T O)2
σ2O
and then1
σ2A
=1
σ2B
+1
σ2O
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Optimal interpolation
Optimal interpolation
Computation ofx
A = xB + K(y − H(xB))
K = BHT (HBH
T + R)−1K . . . gain matrix
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Optimal interpolation
Optimal interpolation
Computation ofx
A = xB + K(y − H(xB))
K = BHT (HBH
T + R)−1K . . . gain matrix
expensive!
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Three-dimensional variational assimilation (3D-Var)
3D-Var
Minimisation of
J(x) = (x − xB)T
B−1(x− x
B) + (y − H(x))TR
−1(y − H(x))
= JB(x) + JO(x)
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Three-dimensional variational assimilation (3D-Var)
3D-Var
Minimisation of
J(x) = (x − xB)T
B−1(x− x
B) + (y − H(x))TR
−1(y − H(x))
= JB(x) + JO(x)
avoids computation of K by using a descent algorithm s
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Four-dimensional variational assimilation (4D-Var)
Minimise the cost function
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) +
nX
i=0
(yi − Hi(xi))TR
−1i (yi − Hi(xi))
subject to model dynamics xi = M0→ix0
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Four-dimensional variational assimilation (4D-Var)
Minimise the cost function
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) +
nX
i=0
(yi − Hi(xi))TR
−1i (yi − Hi(xi))
subject to model dynamics xi = M0→ix0
Figure: Copyright:ECMWF
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4D-Var analysis
Model dynamics
Strong constraint: model states xi are subject to
xi = M0→ix0
nonlinear constraint optimisation problem (hard!)
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4D-Var analysis
Model dynamics
Strong constraint: model states xi are subject to
xi = M0→ix0
nonlinear constraint optimisation problem (hard!)
Simplifications
Causality (forecast expressed as product of intermediate forecast steps)
xi = Mi,i−1Mi−1,i−2 . . . M1,0x0
Tangent linear hypothesis (H and M can be linearised)
yi−Hi(xi) = yi−Hi(M0→ix0) = yi−Hi(M0→ixB0 )−HiM0→i(x0−x
B0 )
M is the tangent linear model.
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4D-Var analysis
Model dynamics
Strong constraint: model states xi are subject to
xi = M0→ix0
nonlinear constraint optimisation problem (hard!)
Simplifications
Causality (forecast expressed as product of intermediate forecast steps)
xi = Mi,i−1Mi−1,i−2 . . . M1,0x0
Tangent linear hypothesis (H and M can be linearised)
yi−Hi(xi) = yi−Hi(M0→ix0) = yi−Hi(M0→ixB0 )−HiM0→i(x0−x
B0 )
M is the tangent linear model.
unconstrained quadratic optimisation problem (easier).
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Minimisation of the 4D-Var cost function
Efficient implementation of J and ∇J :
forecast state xi = Mi,i−1Mi−1,i−2 . . . M1,0x0
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Minimisation of the 4D-Var cost function
Efficient implementation of J and ∇J :
forecast state xi = Mi,i−1Mi−1,i−2 . . . M1,0x0
normalised departures di = R−1i (yi − Hi(xi))
cost function JOi = (yi − Hi(xi))T di
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Minimisation of the 4D-Var cost function
Efficient implementation of J and ∇J :
forecast state xi = Mi,i−1Mi−1,i−2 . . . M1,0x0
normalised departures di = R−1i (yi − Hi(xi))
cost function JOi = (yi − Hi(xi))T di
∇J is calculated by
−1
2∇JO = −
1
2
nX
i=0
∇JOi
=n
X
i=0
MT1,0 . . .M
Ti,i−1H
Ti di
= HT0 d0 + M
T1,0[H
T1 d1 + M2,1[H
T2 d2 + . . . + M
Tn,n−1H
Tndn] . . .]
![Page 43: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/43.jpg)
Minimisation of the 4D-Var cost function
Efficient implementation of J and ∇J :
forecast state xi = Mi,i−1Mi−1,i−2 . . . M1,0x0
normalised departures di = R−1i (yi − Hi(xi))
cost function JOi = (yi − Hi(xi))T di
∇J is calculated by
−1
2∇JO = −
1
2
nX
i=0
∇JOi
=n
X
i=0
MT1,0 . . .M
Ti,i−1H
Ti di
= HT0 d0 + M
T1,0[H
T1 d1 + M2,1[H
T2 d2 + . . . + M
Tn,n−1H
Tndn] . . .]
initialise adjoint variable xn = 0 and then xi−1 = MTi,i−1(xi + HT
i di)etc., . . . x0 = − 1
2∇JO
![Page 44: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/44.jpg)
Minimisation of the 4D-Var cost function
Efficient implementation of J and ∇J :
forecast state xi = Mi,i−1Mi−1,i−2 . . . M1,0x0
normalised departures di = R−1i (yi − Hi(xi))
cost function JOi = (yi − Hi(xi))T di
∇J is calculated by
−1
2∇JO = −
1
2
nX
i=0
∇JOi
=n
X
i=0
MT1,0 . . .M
Ti,i−1H
Ti di
= HT0 d0 + M
T1,0[H
T1 d1 + M2,1[H
T2 d2 + . . . + M
Tn,n−1H
Tndn] . . .]
initialise adjoint variable xn = 0 and then xi−1 = MTi,i−1(xi + HT
i di)etc., . . . x0 = − 1
2∇JO
Further simplifications
preconditioning with B = LLT (transform into control variable space)so that x = L−1x
Incremental 4D-Var
![Page 45: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/45.jpg)
Example - Three-Body Problem
Motion of three bodies in a plane, two position (q) and two momentum (p)coordinates for each body α = 1, 2, 3
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Example - Three-Body Problem
Motion of three bodies in a plane, two position (q) and two momentum (p)coordinates for each body α = 1, 2, 3
Equations of motion
H(q,p) =1
2
X
α
|pα|2
mα−
X X
α<β
mαmβ
|qα − qβ|
dqα
dt=
∂H
∂pα
dpα
dt= −
∂H
∂qα
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Example - Three-Body problem
solver: partitioned Runge-Kutta scheme with time step h = 0.001
observations are taken as noise from the truth trajectory
background is given from a previous forecast
![Page 48: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/48.jpg)
Example - Three-Body problem
solver: partitioned Runge-Kutta scheme with time step h = 0.001
observations are taken as noise from the truth trajectory
background is given from a previous forecast
assimilation window is taken 300 time steps
minimisation of cost function J using a Gauss-Newton method(neglecting all second derivatives)
∇J(x0) = 0
∇∇J(xj0)∆x
j0 = −∇J(xj
0), xj+10 = x
j0 + ∆x
j0
subsequent forecast is take 3000 time steps
R is diagonal with variances between 10−3 and 10−5
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Changing the masses of the bodies
DA needs Model error!
ms = 1.0 → ms = 1.1
mp = 0.1 → mp = 0.11
mm = 0.01 → mm = 0.011
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Changing the masses of the bodies
DA needs Model error!
ms = 1.0 → ms = 1.1
mp = 0.1 → mp = 0.11
mm = 0.01 → mm = 0.011
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Changing the masses of the bodies
0 500 1000 1500 2000 2500 3000 35000
0.5
1
1.5
2
2.5
Time step
RM
S e
rror
before assimilation
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Changing the masses of the bodies
0 500 1000 1500 2000 2500 3000 35000
0.5
1
1.5
2
2.5
Time step
RM
S e
rror
before assimilationafter assimilation
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Changing the masses of the bodies
0 500 1000 1500 2000 2500 3000 35000
0.5
1
1.5
2
2.5
Time step
RM
S e
rror
before assimilationafter assimilation
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Changing the masses of the bodies
0 500 1000 1500 2000 2500 3000 35000
0.5
1
1.5
2
2.5
Time step
RM
S e
rror
before assimilationafter assimilation
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Changing the masses of the bodies
0 500 1000 1500 2000 2500 3000 35000
0.5
1
1.5
2
2.5
Time step
RM
S e
rror
before assimilationafter assimilation
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Root mean square error over whole assimilation window
1 2 3 4 50
0.05
0.1sun position
1 2 3 4 50
0.5
1planet position
1 2 3 4 50
1
2moon position
1 2 3 4 50
0.1
0.2sun momentum
1 2 3 4 50
0.2
0.4planet momentum
1 2 3 4 50
0.1
0.2moon momentum
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Changing numerical method
Truth trajectory: 4th order Runge-Kutta method with local truncationerror O(∆t5)
Model trajectory: Explicit Euler method with local truncation errorO(∆t2)
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Changing numerical method
0 500 1000 1500 2000 2500 3000 35000
0.05
0.1
0.15
0.2
0.25
Time step
RM
S e
rror
before assimilation
![Page 59: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/59.jpg)
Changing numerical method
0 500 1000 1500 2000 2500 3000 35000
0.05
0.1
0.15
0.2
0.25
Time step
RM
S e
rror
before assimilationafter assimilation
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Changing numerical method
0 500 1000 1500 2000 2500 3000 35000
0.05
0.1
0.15
0.2
0.25
Time step
RM
S e
rror
before assimilationafter assimilation
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Changing numerical method
0 500 1000 1500 2000 2500 3000 35000
0.05
0.1
0.15
0.2
0.25
Time step
RM
S e
rror
before assimilationafter assimilation
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Root mean square error over whole assimilation window
1 2 3 40
0.01
0.02sun position
1 2 3 40
0.1
0.2planet position
1 2 3 40
0.5
1moon position
1 2 3 40
0.01
0.02sun momentum
1 2 3 40
0.05
0.1planet momentum
1 2 3 40
0.05
0.1moon momentum
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Less observations - observations in sun only
1 2 3 4 50
0.05
0.1sun position
1 2 3 4 50
10
20planet position
1 2 3 4 50
500
1000moon position
1 2 3 4 50
0.5
1sun momentum
1 2 3 4 50
5planet momentum
1 2 3 4 50
5
10moon momentum
![Page 64: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/64.jpg)
Less observations - observations in planet only
1 2 3 4 50
100
200sun position
1 2 3 4 50
2
4planet position
1 2 3 4 50
5000moon position
1 2 3 4 50
20
40sun momentum
1 2 3 4 50
0.5
1planet momentum
1 2 3 4 50
20
40moon momentum
![Page 65: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/65.jpg)
Less observations - observations in moon only
1 2 3 4 50
200
400sun position
1 2 3 4 50
500
1000planet position
1 2 3 4 50
200
400moon position
1 2 3 4 50
100
200sun momentum
1 2 3 4 50
100
200planet momentum
1 2 3 4 50
5
10moon momentum
![Page 66: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/66.jpg)
Less observations - observations in sun and planet only
1 2 3 4 50
0.05
0.1sun position
1 2 3 4 50
1
2planet position
1 2 3 4 50
200
400moon position
1 2 3 4 50
0.1
0.2sun momentum
1 2 3 4 50
0.5
1planet momentum
1 2 3 4 50
5moon momentum
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Less observations - observations in sun and moon only
1 2 3 4 50
0.5
1sun position
1 2 3 4 50
500
1000planet position
1 2 3 4 50
5moon position
1 2 3 4 50
1
2sun momentum
1 2 3 4 50
50
100planet momentum
1 2 3 4 50
0.2
0.4moon momentum
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Less observations - observations in planet and moon only
1 2 3 4 50
5
10sun position
1 2 3 4 50
0.5
1planet position
1 2 3 4 50
1
2moon position
1 2 3 4 50
20
40sun momentum
1 2 3 4 50
0.2
0.4planet momentum
1 2 3 4 50
0.1
0.2moon momentum
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The Kalman Filter Algorithm
Sequential data assimilation, background is provided by the forecastthat starts from the previous analysis
covariance matrices BF , BA
forecast/model error xTruthi+1 = Mi+1,ix
Truthi + ηi where ηi ∼ N (0,Qi),
assumed to be uncorrelated to analysis error of previous forecast
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The Kalman Filter Algorithm
Sequential data assimilation, background is provided by the forecastthat starts from the previous analysis
covariance matrices BF , BA
forecast/model error xTruthi+1 = Mi+1,ix
Truthi + ηi where ηi ∼ N (0,Qi),
assumed to be uncorrelated to analysis error of previous forecast
State and error covariance forecast
State forecast xFi+1 = Mi+1,ix
Ai
Error covariance forecast BFi+1 = Mi+1,iB
Ai M
Ti+1,i + Qi
![Page 71: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/71.jpg)
The Kalman Filter Algorithm
Sequential data assimilation, background is provided by the forecastthat starts from the previous analysis
covariance matrices BF , BA
forecast/model error xTruthi+1 = Mi+1,ix
Truthi + ηi where ηi ∼ N (0,Qi),
assumed to be uncorrelated to analysis error of previous forecast
State and error covariance forecast
State forecast xFi+1 = Mi+1,ix
Ai
Error covariance forecast BFi+1 = Mi+1,iB
Ai M
Ti+1,i + Qi
State and error covariance analysis
Kalman gain Ki = BFi H
Ti (HiB
Fi H
Ti + Ri)
−1
State analysis xAi = x
Fi + Ki(yi − Hix
Fi )
Error covariance of analysis BAi = (I −KiHi)B
Fi
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The Kalman Filter Algorithm
Extended Kalman Filter
Extension of the Kalman Filter Algorithm to nonlinear observationoperators H and nonlinear model dynamics M , where both H and M arelinearised.
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The Kalman Filter Algorithm
Extended Kalman Filter
Extension of the Kalman Filter Algorithm to nonlinear observationoperators H and nonlinear model dynamics M , where both H and M arelinearised.
Equivalence 4D-Var Kalman Filter
Assume
Qi = 0, ∀i (no model error)
both 4D-Var and the Kalman filter use the same initial input data
H and M are linear,
then 4D-Var and the Kalman Filter produce the same state estimate xA atthe end of the assimilation window.
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RMS error over whole assimilation window - using 4D-Var
1 2 3 4 50
0.05
0.1sun position
1 2 3 4 50
0.5
1planet position
1 2 3 4 50
1
2moon position
1 2 3 4 50
0.1
0.2sun momentum
1 2 3 4 50
0.2
0.4planet momentum
1 2 3 4 50
0.1
0.2moon momentum
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RMS error over whole assimilation window - using Kalman Filter
1 2 3 4 50
0.05
0.1sun position
1 2 3 4 50
0.5
1planet position
1 2 3 4 50
1
2moon position
1 2 3 4 50
0.1
0.2sun momentum
1 2 3 4 50
0.2
0.4planet momentum
1 2 3 4 50
0.1
0.2moon momentum
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Example - Three-Body Problem
solver: partitioned Runge-Kutta scheme with time step h = 0.001
observations are taken as noise from the truth trajectory
background is given from a perturbed initial condition
assimilation window is taken 300 time steps
minimisation of cost function J using a Gauss-Newton method(neglecting all second derivatives)
application of 4D-Var
![Page 77: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/77.jpg)
Example - Three-Body Problem
solver: partitioned Runge-Kutta scheme with time step h = 0.001
observations are taken as noise from the truth trajectory
background is given from a perturbed initial condition
assimilation window is taken 300 time steps
minimisation of cost function J using a Gauss-Newton method(neglecting all second derivatives)
application of 4D-Var
Compare using B = I with using a flow-dependent matrix B which wasgenerated by a Kalman Filter before the assimilation starts (see G.Inverarity (2007))
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Example - Three-Body Problem
2.4 2.5 2.6 2.7 2.8 2.9 3
x 104
0
0.02
0.04
0.06
0.08
0.1
0.12
Time step
RM
S e
rror
before assimilation
after assimilation
Figure: 4D-Var with B = I
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Example - Three-Body Problem
2.4 2.5 2.6 2.7 2.8 2.9 3
x 104
0
0.02
0.04
0.06
0.08
0.1
0.12
Time step
RM
S e
rror
before assimilation
after assimilation
Figure: 4D-Var with B = I
2.4 2.5 2.6 2.7 2.8 2.9 3
x 104
0
0.005
0.01
0.015
0.02
0.025
0.03
Time step
RM
S e
rror
before assimilation
after assimilation
Figure: 4D-Var with B = PA
![Page 80: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/80.jpg)
Problems with Data Assimilation
DA is computational very expensive, one cycle is much more expensivethan the actual forecast
![Page 81: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/81.jpg)
Problems with Data Assimilation
DA is computational very expensive, one cycle is much more expensivethan the actual forecast
estimation and storage of the B-matrix is hardin operational DA B is about 107
× 107
B should be flow-dependent but in practice often staticB needs to be modeled and diagonalised since B−1 too expensive tocompute (”control variable transform”)
![Page 82: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/82.jpg)
Problems with Data Assimilation
DA is computational very expensive, one cycle is much more expensivethan the actual forecast
estimation and storage of the B-matrix is hardin operational DA B is about 107
× 107
B should be flow-dependent but in practice often staticB needs to be modeled and diagonalised since B−1 too expensive tocompute (”control variable transform”)
many assumptions are not validerrors non-Gaussian, data have biasesforward model operator M is not exact and also non-linear and systemdynamics are chaoticminimisation of the cost function needs close initial guess, smallassimilation window
![Page 83: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/83.jpg)
Problems with Data Assimilation
DA is computational very expensive, one cycle is much more expensivethan the actual forecast
estimation and storage of the B-matrix is hardin operational DA B is about 107
× 107
B should be flow-dependent but in practice often staticB needs to be modeled and diagonalised since B−1 too expensive tocompute (”control variable transform”)
many assumptions are not validerrors non-Gaussian, data have biasesforward model operator M is not exact and also non-linear and systemdynamics are chaoticminimisation of the cost function needs close initial guess, smallassimilation window
model error not included
![Page 84: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/84.jpg)
Outline
1 Introduction
2 Variational Data AssimilationLeast square estimationExamplesKalman FilterProblems and Issues
3 Tikhonov regularisation
4 Plan and work in progress
![Page 85: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/85.jpg)
Relation between 4D-Var and Tikhonov regularisation
4D-Var minimises
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) +
nX
i=0
(yi − Hi(xi))TR
−1i (yi − Hi(xi))
subject to model dynamics xi = M0→ix0
![Page 86: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/86.jpg)
Relation between 4D-Var and Tikhonov regularisation
4D-Var minimises
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) +
nX
i=0
(yi − Hi(xi))TR
−1i (yi − Hi(xi))
subject to model dynamics xi = M0→ix0
or
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) + (y − H(x0))
TR
−1(y − H(x0))
whereH = [HT
0 , (H1M(t1, t0))T, . . . (HnM(tn, t0))
T ]T
y = [yT0 , . . . ,y
Tn ]
and R is block diagonal with Ri on diagonal.
![Page 87: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/87.jpg)
Relation between 4D-Var and Tikhonov regularisation
Solution to the optimisation problem
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) + (y − H(x0))
TR
−1(y − H(x0))
is given by
x0 = xB0 + (B−1 + H
TR
−1H)−1
HTR
−1d, d = H(xB
0 − y)
![Page 88: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/88.jpg)
Relation between 4D-Var and Tikhonov regularisation
Solution to the optimisation problem
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) + (y − H(x0))
TR
−1(y − H(x0))
is given by
x0 = xB0 + (B−1 + H
TR
−1H)−1
HTR
−1d, d = H(xB
0 − y)
Singular value decomposition
Assume B = σ2BI and R = σ2
OI and define the SVD of the observabilitymatrix H
H = UΛVT
Then the optimal analysis can be written as
x0 = xB0 +
X
j
λ2j
µ2 + λ2j
uTj d
λjvj
where µ2 =σ2
O
σ2B
.
![Page 89: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/89.jpg)
Relation between 4D-Var and Tikhonov regularisation
Solution to the optimisation problem
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) + (y − H(x0))
TR
−1(y − H(x0))
![Page 90: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/90.jpg)
Relation between 4D-Var and Tikhonov regularisation
Solution to the optimisation problem
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) + (y − H(x0))
TR
−1(y − H(x0))
Variable transformations
B = σ2BFB and R = σ2
OFR and
![Page 91: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/91.jpg)
Relation between 4D-Var and Tikhonov regularisation
Solution to the optimisation problem
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) + (y − H(x0))
TR
−1(y − H(x0))
Variable transformations
B = σ2BFB and R = σ2
OFR and define new variable z := F−1/2
B (x0 − xB0 )
![Page 92: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/92.jpg)
Relation between 4D-Var and Tikhonov regularisation
Solution to the optimisation problem
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) + (y − H(x0))
TR
−1(y − H(x0))
Variable transformations
B = σ2BFB and R = σ2
OFR and define new variable z := F−1/2
B (x0 − xB0 )
J(z) = µ2‖z‖2
2 + ‖F−1/2
R d − F−1/2
R HF−1/2
B z‖22
µ2 can be interpreted as a regularisation parameter.
![Page 93: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/93.jpg)
Relation between 4D-Var and Tikhonov regularisation
Solution to the optimisation problem
J(x0) = (x0 − xB0 )T
B−1(x0 − x
B0 ) + (y − H(x0))
TR
−1(y − H(x0))
Variable transformations
B = σ2BFB and R = σ2
OFR and define new variable z := F−1/2
B (x0 − xB0 )
J(z) = µ2‖z‖2
2 + ‖F−1/2
R d − F−1/2
R HF−1/2
B z‖22
µ2 can be interpreted as a regularisation parameter.This is the well-known Tikhonov regularisation!
![Page 94: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/94.jpg)
Outline
1 Introduction
2 Variational Data AssimilationLeast square estimationExamplesKalman FilterProblems and Issues
3 Tikhonov regularisation
4 Plan and work in progress
![Page 95: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/95.jpg)
Met Office research and plans
include several time scales (to model the atmosphere)
![Page 96: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/96.jpg)
Met Office research and plans
include several time scales (to model the atmosphere)
improve the representation of multiscale behaviour in theatmosphere in existing DA methods
improve the forecast of small scale features (like convective storms)
![Page 97: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/97.jpg)
Met Office research and plans
include several time scales (to model the atmosphere)
improve the representation of multiscale behaviour in theatmosphere in existing DA methods
improve the forecast of small scale features (like convective storms)
compare assimilation algorithms and optimisation strategies to reduceexisting errors
use regularisation methods from image processing, for exampleL1 regularisation to improve forecasts
![Page 98: Introduction to Data Assimilation with 4D-Var and its ...people.bath.ac.uk/mamamf/talks/invert08.pdf · Introduction to Data Assimilation with 4D-Var and its relation to Tikhonov](https://reader033.fdocuments.net/reader033/viewer/2022052801/5f11b0359cf28628006ac878/html5/thumbnails/98.jpg)
Met Office research and plans
include several time scales (to model the atmosphere)
improve the representation of multiscale behaviour in theatmosphere in existing DA methods
improve the forecast of small scale features (like convective storms)
compare assimilation algorithms and optimisation strategies to reduceexisting errors
use regularisation methods from image processing, for exampleL1 regularisation to improve forecasts
identify and analyse model error and analyse influence of this modelerror onto the DA scheme