Mathematics of Data Assimilation- Complex Systems in ...mamamf/talks/melinabics.pdf · Mathematics of Data Assimilation-Complex Systems in Numerical Weather Prediction Melina Freitag

Mathematics of Data Assimilation-Complex Systems in Numerical Weather Prediction

Melina Freitag

Department of Mathematical Sciences

University of Bath

BICS meeting ’The Maths of Complex Systems’6th February 2008

1 Introduction

2 Basic concepts

3 Variational Data AssimilationLeast square estimationKalman Filter

4 Problems

5 Plan

Outline

1 Introduction

2 Basic concepts


4 Problems

5 Plan

What is Data Assimilation?

Loose definition

Estimation and prediction (analysis) of an unknown, true state bycombining observations and system dynamics (model output).


Loose definition


Some examples

Navigation (collect observations and produce velocity corrections)

Geophysics (values of some model parameter must be obtained fromthe observed data)

Medical imaging


Loose definition


Some examples

Navigation (collect observations and produce velocity corrections)

Geophysics (values of some model parameter must be obtained fromthe observed data)

Medical imaging

Numerical weather prediction

The atmosphere

The atmosphere is a complex system!Mathematical modelling, observations and mathematical DataAssimilation help to understand this complex multi-scale system

Figure: http://www.usoe.k12.ut.us

The atmosphere

The atmosphere is a complex system!Mathematical modelling, observations and mathematical DataAssimilation help to understand this complex multi-scale system

Figure: http://www.usoe.k12.ut.us

⇒”Aspects of Ionosphere modelling”Nathan (see talk later this afternoon)

The weather (NWP)

Data Assimilation in NWP

Estimate the state of the atmosphere xi at a certain time/certain times i.

A priori information xB

background state (usualprevious forecast)





Observations y

Satellites

Ships and buoys

Surface stations

Airplanes





Models

a model how the atmosphereevolves in time (imperfect)

xi+1 = M(xi)

Observations y

Satellites

Ships and buoys

Surface stations

Airplanes





Models


xi+1 = M(xi)

a function linking model spaceand observation space(imperfect)

yi = H(xi)

Observations y

Satellites

Ships and buoys

Surface stations

Airplanes





Models


xi+1 = M(xi)


yi = H(xi)

Observations y

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/these states aforecast for future states of theatmosphere can be obtained

Outline

1 Introduction

2 Basic concepts


4 Problems

5 Plan

Schematics of DA

Figure: Background state xB

Schematics of DA

Figure: Observations y

Schematics of DA

Figure: Analysis xA (consistent with observations and model dynamics)


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)

Notation

xTruth: True state

xB: Background state (taken from previous forecast)

xA: Analysis (estimation of the true state after the DA)

Outline

1 Introduction

2 Basic concepts


4 Problems

5 Plan


We are looking for the state of the atmosphere xi at a certain time/certaintimes i.

Apriori information xB

background state (usualprevious forecast) has errors!

Models


xi+1 = M(xi) + error


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/these states aforecast for future states of theatmosphere can be obtained

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

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

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

Optimal least-squares estimater

Cost function minimisation (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)

B−1 expensive!

Optimal least-squares estimater

Cost function minimisation (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)

B−1 expensive!

Interpolation equations

xA = x

B + K(y − H(xB)), where

K = BHT (HBH

T + R)−1K . . . gain matrix

expensive!

Four-dimensional variational assimilation (4D-Var)

Minimise the cost function

J(x0) = (x0 − xB)T

B−1(x0 − x

B) +nX

i=0

(yi − Hi(xi))TR

−1i (yi − Hi(xi))

subject to model dynamics xi = M0→ix0

Four-dimensional variational assimilation (4D-Var)

Minimise the cost function

J(x0) = (x0 − xB)T

B−1(x0 − x

B) +nX

i=0

(yi − Hi(xi))TR

−1i (yi − Hi(xi))

subject to model dynamics xi = M0→ix0

Figure: Copyright:ECMWF

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

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

∇J(x0) = 0

∇∇J(xj0)∆x

j0 = −∇J(xj

0), xj+10 = x

j0 + ∆x

j0

subsequent forecast is take 5000 time steps

Example- Three-Body problem

Figure: Truth trajectory


500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500−0.05

0

0.05

0.1

0.15

0.2

Time step

x po

sitio

n of

the

sun

Solution for position of the sun

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500−1

−0.5

0

0.5

1

Time step

x po

sitio

n of

the

moo

n

Solution for position of the moon

Truth x

Observation x

Background x

Figure: Truth trajectory with observations and background


500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500−0.05

0

0.05

0.1

0.15

0.2

Time step

x po

sitio

n of

the

sun

Solution for position of the sun

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500−1

−0.5

0

0.5

1

Time step

x po

sitio

n of

the

moo

n

Solution for position of the moon

Truth x

Observation x

Background x

Analysis x (after EKF)

Figure: Analysis


500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000

0.05

0.1

0.15

0.2

Time step

RM

S e

rror

before assimilation

after assimilation

Figure: RMS error


500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000

0.05

0.1

0.15

0.2

Time step

RM

S e

rror

before assimilation

after assimilation

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

sun before

planet before

moon before

sun after

planet after

moon after

Figure: RMS error

The Kalman Filter Algorithm

Sequential data assimilation

covariance matrices are updated at each step PF , PA

State and error covariance forecast

State forecast xFi+1 = Mi+1,ix

Ai

Error covariance forecast PFi+1 = Mi+1,iP

Ai M

Ti+1,i + Qi

The Kalman Filter Algorithm

Sequential data assimilation

covariance matrices are updated at each step PF , PA

State and error covariance forecast

State forecast xFi+1 = Mi+1,ix

Ai

Error covariance forecast PFi+1 = Mi+1,iP

Ai M

Ti+1,i + Qi

State and error covariance analysis

Kalman gain Ki = PFi H

Ti (HiP

Fi H

Ti + Ri)

−1

State analysis xAi = x

Fi + Ki(yi − Hix

Fi )

Error covariance of analysis PAi = (I− KiHi)P

Fi


same setup as before

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))


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

Outline

1 Introduction

2 Basic concepts


4 Problems

5 Plan

Problems with Data Assimilation

DA is computational very expensive, one cycle is much more expensivethan the actual forecast (many applications of the tangent linear modeland adjoint)



estimation 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”)




× 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




× 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

Outline

1 Introduction

2 Basic concepts


4 Problems

5 Plan

Model error and perturbation error

Figure: One assimilation window (6 hours)

xF1 − x

Truth1 = Mappr(x

A0 ) − M(xTruth

0 )

= Mappr(xA0 ) − Mappr(x

Truth0 )

| {z }

Perturbation error

+Mappr(xTruth0 ) − M(xTruth

0 )| {z }

Model error

Model error and perturbation error

Figure: Model error and perturbation error

Model error

Figure: Perturbation error after 7 hours (Copyright: MetOffice)

Model error

Figure: Perturbation error after 12 hours (Copyright: MetOffice)

Model error

Figure: Model error after 12 hours (Copyright: MetOffice)

Plan

design a simple chaotic model of reduced order (Lorenz model)

include several time scales (to model the atmosphere)

Plan



include model error and analyse influence of this model error onto theDA scheme

analyse the influence of the error made by the numericalapproximation (part of the model error) on the error in the DA scheme

investigate several assimilation algorithms and optimisation strategiesto reduce existing errors

Plan






improve the representation of multi-scale behaviour in the atmospherein existing DA methods

improve the forecast of small scale features (like convective storms)

Plan






improve the representation of multi-scale behaviour in the atmospherein existing DA methods

improve the forecast of small scale features (like convective storms)

Theme D: Numerical methods for multi-scale modelling

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Mathematics of Data Assimilation- Complex Systems in ...mamamf/talks/melinabics.pdf · Mathematics of Data Assimilation-Complex Systems in Numerical Weather Prediction Melina Freitag