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BICS Away Day
Theme ”GWR”: Data Assimilation
Melina Freitag
Department of Mathematical Sciences
University of Bath
BICS Away Day16th September 2008
BICS Away Day
Outline
1 Introduction
2 Variational Data AssimilationExamplesProblems and Issues
3 Relation to Tikhonov regularisation
4 Plan and work in progress
5 External links and talks
BICS Away Day
Introduction
What is Data Assimilation?
Loose definition
Estimation and prediction (analysis) of an unknown, true state bycombining observations and system dynamics (model output).
BICS Away Day
Introduction
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
Geosciences (link to INVERT)
Medical imaging (link to INVERT)
Numerical weather prediction
BICS Away Day
Introduction
Data Assimilation in NWP
Estimate the state of the atmosphere xi.
Observations y
Satellites
Ships and buoys
Surface stations
Aeroplanes
BICS Away Day
Introduction
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
BICS Away Day
Introduction
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
BICS Away Day
Introduction
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
BICS Away Day
Introduction
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)
BICS Away Day
Introduction
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)
BICS Away Day
Introduction
Schematics of DA
Figure: Background state xB
BICS Away Day
Introduction
Schematics of DA
Figure: Observations y
BICS Away Day
Introduction
Schematics of DA
Figure: Analysis xA (consistent with observations and model dynamics)
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Variational Data Assimilation
Optimal least-squares estimater
Cost function - 3D-VAR
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)
BICS Away Day
Variational Data Assimilation
Optimal least-squares estimater
Cost function - 3D-VAR
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 - Optimal Interpolation
xA = x
B + K(y − H(xB)), where
K = BHT (HBH
T + R)−1K . . . gain matrix
BICS Away Day
Variational Data Assimilation
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
BICS Away Day
Variational Data Assimilation
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
BICS Away Day
Variational Data Assimilation
4D-Var analysis
Model dynamics
Strong constraint: model states xi are subject to
xi = M0→ix0
nonlinear constraint optimisation problem (hard!)
BICS Away Day
Variational Data Assimilation
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).
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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)
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
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
BICS Away Day
Variational Data Assimilation
Examples
Other implemented models
Three-Body problem
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Variational Data Assimilation
Examples
Other implemented models
Three-Body problem
Lorenz model (3D)
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Variational Data Assimilation
Examples
Other implemented models
Three-Body problem
Lorenz model (3D)
Lorenz95 model (n-dimensional)
BICS Away Day
Variational Data Assimilation
Examples
Other implemented models
Three-Body problem
Lorenz model (3D)
Lorenz95 model (n-dimensional)
Lorenz95 model (n-dimensional) with 2 timescales
BICS Away Day
Variational Data Assimilation
Problems and Issues
Problems with Data Assimilation
DA is computational very expensive, one cycle is much more expensivethan the actual forecast
BICS Away Day
Variational Data Assimilation
Problems and Issues
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 modelled and diagonalised since B−1 too expensive tocompute (”control variable transform”)
BICS Away Day
Variational Data Assimilation
Problems and Issues
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 modelled 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
BICS Away Day
Variational Data Assimilation
Problems and Issues
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 modelled 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
BICS Away Day
Relation to Tikhonov regularisation
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
BICS Away Day
Relation to Tikhonov regularisation
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.
BICS Away Day
Relation to Tikhonov regularisation
Relation between 4D-Var and Tikhonov regularisation
Solution to the optimisation problem
Linearise about x0 then the 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)
BICS Away Day
Relation to Tikhonov regularisation
Relation between 4D-Var and Tikhonov regularisation
Solution to the optimisation problem
Linearise about x0 then the 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 µ
2 =σ2
O
σ2B
.
BICS Away Day
Relation to Tikhonov regularisation
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))
BICS Away Day
Relation to Tikhonov regularisation
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
BICS Away Day
Relation to Tikhonov regularisation
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 )
BICS Away Day
Relation to Tikhonov regularisation
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.
BICS Away Day
Relation to Tikhonov regularisation
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!
BICS Away Day
Plan and work in progress
Met Office research and plans
improve the representation of multiscale behaviour in theatmosphere in existing DA methods
improve the forecast of small scale features (like convective storms)
BICS Away Day
Plan and work in progress
Met Office research and plans
improve the representation of multiscale behaviour in theatmosphere in existing DA methods
improve the forecast of small scale features (like convective storms)
use regularisation methods from image processing, for exampleL1 regularisation to improve forecasts
BICS Away Day
Plan and work in progress
Met Office research and plans
improve the representation of multiscale behaviour in theatmosphere in existing DA methods
improve the forecast of small scale features (like convective storms)
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
BICS Away Day
External links and talks
Links (external and internal)
MetOffice (4 weeks visit in October 2007, several one-day visits)
INVERT Centre (talk in August 2008)
University of Reading (EPSRC NETWORK: Mathematics of DataAssimilation - A. Lawless, includes Reading, Bath, Oxford, Surrey,Warwick as well as ECMWF, MetOffice, QuinetiQ, Schlumberger, HRWallingford, Environment Agency)
BICS Away Day
External links and talks
Presentations
Bath, BICS Meeting (Maths of Complex Systems) (February 2008)
Bath, NA Seminar (February 2008)
Bath, Postgraduate Seminar (February 2008)
Exeter, Exeter - Bath - Met Office meeting (March 2008)
Copper Mountain, Colorado, USA: Copper Mountain Conference onIterative Methods (April 2008)
Zeuthen, Germany: Householder Symposium XVII (June 2008)
Durham, LMS Durham Symposium: Computational Linear Algebra forPartial Differential Equations (July 2008)
Bath, INVERT Centre (August 2008)
Bath, Bath/RAL Numerical Analysis Day (September 2008)
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