Using Model Reduction in Data Assimilation Met Office Website - - PDF document

Using Model Reduction in Data Assimilation

Nancy Nichols, Amos Lawless

The University of Reading

Caroline Boess, Angelika Bunse-Gerstner

Cerfacs & The University of Bremen

Met Office Website

Outline

• Incremental 4D variational assimilation
• Model reduction in incremental 4DVar
• Oblique projection using balanced truncation
• Numerical experiments
• Conclusions

4D-Var Nonlinear Problem

) ] [ ( ) ] [ ( ) ( ) ( 2 1 ) ( min

1

• i

i i n i i T

• i

i i b T b

H H J y x R y x x x B x x x

1

• )

, , (

0 x

x t t S

i i

subject to

• i

i i b

H R B y x

• Background state (prior)
• Observations
• Observation operator
• Background error covariance matrix
• Observation error covariance matrix
• Observation

Time Temperature Background xb

Incremental 4D-Var

Analysis

Solve by iteration a sequence of linear least squares problems that approximate the nonlinear problem.

Incremental 4D-Var

Set (usually equal to background) For k = 0, …, K find: Solve inner loop linear minimization problem:

) (

x

) , , (

) ( ) ( k i k i

t t S x x

• )

( ) ( ) 1 ( k k k

x x x

• )

( ) ( ] [ min

) ( 1 ) ( ) ( ) ( i k i i n i i T i k i i k k

J d x H R d x H x

• ]

[

) (k i i i i

H x y d

• subject to

Update:

,

• )

( ) ( ) ( ] [

) ( 1 ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( i k i i n i i T i k i i k b k T k b k k k

J d x H R d x H x x B x x x

• min
• On each outer iteration the linear least squares problem

is solved subject to the linearized dynamical system In practice this problem is too computationally expensive to solve. Approximations to the inner minimization problem are therefore used.

i i i i i i

x H d x M x

• 1

N i

• x
• N

N i

• M

N p i

• Incremental 4D-Var without approximations is

equivalent to a Gauss-Newton iteration for nonlinear least squares problems.

• In operational implementation the solution

procedure is approximated: – Truncate inner loop iterations – Use an approximate linear system model

• Theoretical convergence results obtained by

reference to Gauss-Newton method (QJRMS, SIOPT).

Previous Results Low order incremental 4D-Var

Aim: approximate the linearized system by a low order inner problem of size r << N. Define: Linear restriction operators Low order variables Prolongation operators where and is a projection operator.

N r T i

• U

i T i i

x U x

• ˆ

r N i

• V

r i T i

I V U

• T

i iU

V

A restricted version of the dynamical linear system is then given by where approximates Mi and approximates Hi Then a low order inner minimization is solved subject to the low order linear system.

1 1 1

ˆ ˆ ˆ ˆ ˆ

• i

i i i i i

x H d x M x

• r
• x

ˆ

• r

r

• M

ˆ

r p

• ˆ

T i i i

U M V ˆ

T i iU

H ˆ

Low Order Assimilation Problem

Set (usually equal to background) For k = 0, …, K find: Solve low order inner loop minimization problem:

) (

x ) , , (

) ( ) ( k i k i

t t S x x

• )

( ) ( ) (

) , , (

k k i k i

t t x x L x

• )

( ) ( ) 1 ( k k k

x x x

• )

( ) ( ] [ min

) ( 1 ) ( ) ( ) ( i k i i n i i T i k i i k k

J d x H R d x H x

• ]

[

) (k i i i i

H x y d

• with

Update:

,

• )

( ) ( ) ( ] [

) ( 1 ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( i k i i n i i T i k i i k b k T k b k k k

J d x H R d x H x x B x x x

• min
• )

ˆ ˆ ( ) ˆ ˆ ( ) ˆ ( ˆ ˆ ] ˆ [

) ( 1 ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( i k i i n i i T i k i i k b T k T k b T k k k

J d x H R d x H x U x B x U x x

• min
• )

( ) ( ) 1 (

ˆ k

k k

x V x x

How are the operators UiT and Vi chosen so that the solution of the reduced problem is accurate?

Two approaches:

• 1. Standard operational technique: The restriction UiT

is a low resolution spatial operator and the prolongation operator Vi represents spatial interpolation.

• 2. New method: The projections are based on optimal

model reduction techniques.

Optimal Reduced Order Models

• Find approximate linear system models

using optimal reduced order modeling techniques from control theory to improve the efficiency of the incremental 4DVar method.

• Test feasibility of approach in comparison

with low resolution models using a simple shallow water flow model.

Aim:

Model Reduction via Oblique Projections

Given: Find: projections U, V with UT V = Ir , r << N, such that the output of the reduced order system minimizes: (over all inputs with expected norm equal to a constant)

Balanced truncation

Balanced truncation removes states that are least affected by inputs and that have least effect on outputs (in a statistical sense). There are 2 steps:

• 1. Balancing – Transform system to one in which these

states are the same.

• 2. Truncation – Truncate states related to the smallest

singular values of the transformed covariance matrices (Hankel singular values). Projected system exactly matches the largest Hankel singular values of the full system.

Balanced Truncation

Find: such that where is diagonal and Then: near optimal projections are given by

Reduced Order Assimilation Problem

subject to The reduced order inner loop problem is to minimize

,

and set

Why might we expect a benefit?

• The model reduction approach tries to match the

input-output response of the whole system, allowing for the system dynamics, the

• bservations and the error covariances.
• The use of a low resolution model ignores some
• f this information.

Does this help in the data assimilation problem? D ) D(ln D D

• x

u t x h g x t u

• We discretize using a semi-implicit semi-Lagrangian

scheme and linearize to get linear model (TLM). Nonlinear continuous equations with

x u t t

• D

D

1D Shallow Water Model

Methodology

• Define an initial random perturbation x0 from a

distribution B0.

• Calculate ‘true’ solution by solving full linear least

squares problem.

• Calculate ‘observations’ di=H xi for 5 steps (t=0 to t=5)
• Compare solutions solving with

– Low resolution linear model. – Reduced order model.

• Size of full dimension is 400.

Numerical Experiments - Error Norms

, .

from TLM model

• bservations at every other point

quite realistic covariance matrix

Error between exact and approximate analysis for 1-D SWE model

Low Res Model of order = 200 vs Reduced Model of order = 80 Low Res Model of order = 200 vs Reduced Model of order = 200

Component of state

Red (dotted) = Low Res Model Green (dashed) = Reduced Rank Model

Log Error Component of state Log Error

22

Comparison of Error Norms Low resolution vs Reduced order models

23

Comparison of Error Norms Low resolution vs Reduced order models

24

Comparison of Error Norms Low resolution vs Reduced order models

Comparison of Model Eigenvalues

(a) (b) (c) Eigenvalues plotted on the complex plane for (a) full resolution model; (b) low resolution model of order 200; (c) reduced rank model of order 200.

Importance of B Matrix

Red (dotted) = Low Res Model Green (dashed) = Reduced Rank Model Low Res Model of order = 200 vs Reduced Model of order = 200

Log Error Component of state

Errors where covariance B0 is not used in model reduction

Conclusions Conclusions

• Reduced rank linear models obtained by
• ptimal reduction techniques give more

accurate analyses than low resolution linear models that are currently used in practice.

Conclusions

• Reduced rank linear models obtained by
• ptimal reduction techniques give more

accurate analyses than low resolution linear models that are currently used in practice.

• Incorporating the background and
• bservation error covariance information is

necessary to achieve good results

Conclusions

• Reduced rank linear models obtained by
• ptimal reduction techniques give more

accurate analyses than low resolution linear models that are currently used in practice.

• Incorporating the background and
• bservation error covariance information is

necessary to achieve good results

• Reduced order systems capture the
• ptimal growth behaviour of the model more

accurately than low resolution models

• to obtain efficient model reduction techniques

for use in data assimilation

• to demonstrate convergence of the

Incremental 4DVar method using low order models. Work in progress:

http://www.maths.rdg.ac.uk/

Future Work:

• Ensemble Square Root Filters
• Conservation of Dynamical Properties
• High Resolution Local Area Models
• Multiple Timescales / Coupled Systems
• Correlated Observations
• Multi-scale 4DVar Optimization

http://www.maths.rdg.ac.uk/