theme 'gwr': data assimilation - university of...

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Theme ”GWR”: Data Assimilation

Melina Freitag

Department of Mathematical Sciences

University of Bath

BICS Away Day16th September 2008

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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

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Introduction

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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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

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Introduction

Data Assimilation in NWP

Estimate the state of the atmosphere xi.

Observations y

Satellites

Ships and buoys

Surface stations

Aeroplanes

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Introduction



A priori information xB

background state (usualprevious forecast)

Observations y

Satellites

Ships and buoys

Surface stations

Aeroplanes

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Introduction





Models

a model how the atmosphereevolves in time (imperfect)

xi+1 = M(xi)

Observations y

Satellites

Ships and buoys

Surface stations

Aeroplanes

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Introduction





Models


xi+1 = M(xi)

a function linking model spaceand observation space(imperfect)

yi = H(xi)

Observations y

Satellites

Ships and buoys

Surface stations

Aeroplanes

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Introduction





Models


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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Introduction


Underdeterminacy

Size of the state vector x: 432 × 320 × 50 × 7 = O(107)

Number of observations (size of y): O(105 − 106)

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Introduction

Schematics of DA

Figure: Background state xB

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Introduction

Schematics of DA

Figure: Observations y

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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)

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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

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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



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.

unconstrained quadratic optimisation problem (easier).

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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

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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

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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

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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

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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

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Examples


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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Examples


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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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

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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)

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Examples


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

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Examples


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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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

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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

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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

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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

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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

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Examples

Other implemented models

Three-Body problem

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Examples


Three-Body problem

Lorenz model (3D)

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Examples


Three-Body problem

Lorenz model (3D)

Lorenz95 model (n-dimensional)

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Examples


Three-Body problem

Lorenz model (3D)

Lorenz95 model (n-dimensional)

Lorenz95 model (n-dimensional) with 2 timescales

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Problems and Issues

Problems with Data Assimilation

DA is computational very expensive, one cycle is much more expensivethan the actual forecast

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Problems and Issues



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”)

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Problems and Issues




× 107


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

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Problems and Issues




× 107


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

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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



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4D-Var minimises

J(x0) = (x0 − xB0 )T

B−1(x0 − x

B0 ) +

nX

i=0

(yi − Hi(xi))TR



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.

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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)

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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

.

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J(x0) = (x0 − xB0 )T

B−1(x0 − x

B0 ) + (y − H(x0))

TR

−1(y − H(x0))

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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

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J(x0) = (x0 − xB0 )T

B−1(x0 − x

B0 ) + (y − H(x0))

TR

−1(y − H(x0))



OFR and define new variable z := F−1/2

B (x0 − xB0 )

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J(x0) = (x0 − xB0 )T

B−1(x0 − x

B0 ) + (y − H(x0))

TR

−1(y − H(x0))




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.

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J(x0) = (x0 − xB0 )T

B−1(x0 − x

B0 ) + (y − H(x0))

TR

−1(y − H(x0))




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!

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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)

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use regularisation methods from image processing, for exampleL1 regularisation to improve forecasts

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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

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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)

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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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