Tbilisi, 10.07.2014 GGSWBS'14 Optimization for inverse modelling Ketevan Kasradze 1 Hendrik Elbern...

Tbilisi, 10.07.2014 GGSWBS'14

Optimization for inverse modelling

Ketevan Kasradze1 Hendrik Elbern1,2 [email protected] [email protected]

and the Chemical Data Assimilation group of RIU

1 Rhenish Institute for Environmental Research at the University of Cologne, Germany2 Institute for Energy and Climate Research -Troposphere, Germany


Atmospheric layers

3/18


Atmospheric layers

3/18

SACA

DA


SACADA assimilation-system

BackgroundMeteorological

ECMWF analysesTrace gas

observations

Analysis

SACADA

PREPDWD GME

CTM

CTMadDiffusion

L-BFGS


Horizontal GME Grid

9/18

• ~147km between the grid points• 23 042 grid points per Model layer


Add

itio

nal r

efin

emen

t tro

posp

here

/low

er

stra

tosp

here

SACADA Vertical Grid54 layer

CRISTA-NF

MLS


HNO3

4.11.2005

~137hPa

12 Uhr UTC

MLS

15


SCOUT-O3 campaign Stratospheric-Climate Links with Emphasis on the UTLS - O3

November-December 2005

AMMA-campaign African Monsoon Multidisciplinary Analyses

29.07.2006 -17.08.2006

12/18


N

iii

Tiib

Tb yxMHRyxMHxxBxxxJ

00

100

100 2

1

2

1)(

Cost function

Vector of observations

Observation error covariance matrix

Projection operator

Background

Model operator

SACADA assimilation-system4D-Var

Background error covariance matrixBECM ~ 1012 ~ 80 Terrabyte


N

iii

Tiib

Tb yxMHRyxMHxxBxxxJ

00

100

100 2

1

2

1)(

GradientAdjoint Model

N

iii

Tibx yHxRHMxxBxJ

0

1*0

10 )(

0



N

iii

Tiib

Tb yxMHRyxMHxxBxxxJ

00

100

100 2

1

2

1)(

)(),(, 000 0xJxJx x 0x

Quasi-Newton method L-BFGS

N

iii

Tibx yHxRHMxxBxJ

0

1*0

10 )(

0



N

iii

Tiib

Tb yxMHRyxMHxxBxxxJ

00

100

100 2

1

2

1)(

)(),(, 000 0xJxJx x 0x


N

iii

Tibx yHxRHMxxBxJ

0

1*0

10 )(

0

Background error covariance matrixBECM ~ 1012 ~ 80 Terrabyte



Radius of Influence ((de-)correlation length):Extending the information from an observation location

Textbook: horizontal influence radius Laround a measurement site,to be based on a priori statistical assessments

Lverticalcut

L

Horizontal structure function,to be stored as a column of the forecast error covariance matrix

diffusion operatorconstruction

For atmospheric chemistry covariance modelling the diffusion approach is advocated:•localisation intrinsically performed•sharp gradients easily feasible•matrix square roots for preconditioning straightforward to calculate; no inversion needed

Background error covariance matrix formulation


Isopleths of the cost function and transformed cost function and minimisation steps

Minimisation by mere gradients, quasi-Newon method L-BFGS(Large dimensional Broyden Fletcher Goldfarb Shanno), and preconditioned (transformed) L-BFGS application

concentration species 1 transformed species 1

conc

entr

atio

n sp

ecie

s 2

tran

sfor

med

spe

cies

2


Solution: Diffusion Approach

Transformation of cost-function:

=> Inverse of B and B-1/2 are not needed, if xb= 1. guess.

2 outstanding problems:1. With linear estimation: How to treat the background error covariance

matrix B (O(1012))?2. How can this be treated for preconditioning? (need B-1, B1/2, B-1/2) With

variational methods:

minimisation procedure




Background



Background Observation: 3 ppm Ozone


Analysis (B diagonal)





Observation: 3 ppm Ozone

Analysis increment isotropic correlation

The increment in initial values is spread out to neighbouring grid-points depending on the correlations that are known / assumed.

Background


Assumption: Strong correlation along isolines of Potential Vorticity Enhancement of diffusion flow-dependent BECM

Diffusion can be generalised to account for inhomogeneous and anisotropic correlations: Stratospheric case


use PV field for anisotropiccorrelation modelling



Analysis increment



N

iii

Tiib

Tb yxMHRyxMHxxBxxxJ

00

100

100 2

1

2

1)(

)(),(, 000 0xJxJx x 0x


N

iii

Tibx yHxRHMxxBxJ

0

1*0

10 )(

0

Adjoint Model



Construction of the adjoint code(3 different possible pathways)

forward model(forward differential equation)

algorithm(solver)

code

backward model(backward differential equation)

adjoint algorithm(adjoint solver)

adjoint code


Adjoint model

).()( 00,11,21,1 xMMMxMx iiiii

A numerical model integration over a time interval [t0; ti]

0,11,21, MMMM iii

Accordingly, the tangent linear of this sequence of model operators is given by

****

1,2,10,1

iiiiMMMM i

Thus, the adjoint model operator Mi propagates the gradient of the cost function with respect to xi backwards in time, to deliver the gradient of the cost function with respect to x0.


Adjoint model example

yaxz *2**

ayx

y

x

z

y

x

RRF2

33 ,:

...

...

...

JAC


Adjoint model example

yaxz *2**

ayx

y

x

z

y

x

RRF2

33 ,:

0

2

:

000

10

201**

**

*

*

*

*

*

*

*

** azy

xzx

z

y

x

M

z

y

x

Fa

x

M


N

iii

Tiib

Tb yxMHRyxMHxxBxxxJ

00

100

100 2

1

2

1)(

)(),(, 000 0xJxJx x 0x


N

iii

Tibx yHxRHMxxBxJ

0

1*0

10 )(

0

Limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm



Gradient of the cost function h

n

Tx x

h

x

hxDhxghgradhgrad ,

1

Hessian of the cost function

2

2

1

2

1

2

21

2

nn

nT

x

h

xx

h

xx

h

x

h

xDgxDDhxH


BFGS algorithm (2)

From an initial guess x0 and an approximate Hessian matrix H0

the following steps are repeated as xk converges to the solution.

1.Obtain a direction sk by solving:

2.Perform a line search to find an acceptable step size in the

direction found in the first step, then update

3.Set

4.

5.

Convergence can be checked by observing the norm of

the gradient, .

kkk xhsH k

kkkk sxx 1

kkk sp kkk xhxhq 1

kkTk

kTkkk

kTk

Tkk

kk pHp

HppH

pq

qqHH 11

kxh


BFGS example with MATLAB

222 1100, xxyyxf


BFGS example with MATLAB

it= 40 f=1.497581e-13 ||g||=1.726061e-05 sig=1.200 step=BFGSit= 41 f=5.990317e-15 ||g||=3.452127e-06

Successful termination with ||g||<1.000000e-08*max(1,||g0||):

222 1100, xxyyxf


Thank you for your attention!

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Vielen Dank für Ihre Aufmerksamkeit!

Tbilisi, 10.07.2014 GGSWBS'14 Optimization for inverse modelling Ketevan Kasradze 1 Hendrik Elbern...

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