Setup of 4D VAR inverse modelling systemfor atmospheric CH4

using the TM5 adjoint model

Peter Bergamaschi

Climate Change Unit

Institute for Environment

and Sustainability(IES)

Joint Research Center

Ispra, Italy

Maarten Krol

Institute for Marine and

Atmospheric Research

Utrecht, Netherlands

adjoint model

forward model:

tangent linear model (TLM):

adjoint model:

))...)x(Μ(Μ(....Μ)xΜ(x tttttt nn 00110

00110ΜΜΜΜ tttttt x~~~

x~

xnn

Μ

Μ~

j,t

i

ox

M~

ijΜ

T~~ΜΜ

Μ~

yΜxyxΜ ~~

Linearisation of non-linear model

transpose of TLM

TM5 adjoint model

forward model:

tangent linear model (TLM):

adjoint model:

))...)x(Μ(Μ(....Μ)xΜ(x tttttt nn 00110

00110ΜΜΜΜ tttttt x~~~

x~

xnn

Μ

Μ~

j,t

i

ox

M~

ijΜ

T~~ΜΜ

Μ~

yΜxyxΜ ~~

TM5 is linear

- slopes - limits: off

- offline chemistry

TM5 represents already tangent linear model

TM5 adjoint: manual coding

- transpose of each model operator

- revert order of operators

- provide passive, but required variables correctly (air mass)

Tt

Tt

Tt

Tttt nn

~~~~~~110011

ΜΜΜ ΜΜΜ

Linearisation of non-linear model

transpose of TLM

adjoint model - applications

efficient way to calculate full Jacobian matrix, if:

number (obs) << number (parameters)

3-D back trajectories

Systems with many observations AND many parameters:

Minimization of cost function:

4D-VAR

adjoint model

Analytical solution of inverse problem

Iterative approximation:

for VERY complex systems (e.g. numerical weather prediction)

cost function

nt

tttobs

Ttobsapriori,tt

Tapriori,ttt )t(xH)t(yE

~)t(xH)t(yxxB

~xx)x(J

1

00000

1211

21

cost function

parameters observations

)lm,im(e

.

.

),(e

lm,jm,imrm

.

.

,,rm

x t

t

t 11

111

0

0

0

Model prediction: Observation operator H

observations

)t(xH)t(y tM

Initial mixing ratio

emission

gradient of cost function

n

t

t

tttobstttt-tapriori,ttx )t(xH)t(yE

~....xxB

~J

0

0000

1TTTT1 HMMM

minimization

[Bouttier and Courtier, 1999]

dimension parameter space:

TM5: ~1 E4 - 1 E5

(ECMWF(T511) : ~1 E 7 )

Minimisation algorithms:

- M1QN3 [Gilbert, Lemarechal, 1995]

- ECMWF conjugate gradient

4D VAR data assimilation system

gradient test

alpha DJ1 DJ2 t 1-t

0.01000000000000 -704.73790893311582-2078.99670025996011 2.95002819332835 -1.95002819332835

0.00100000000000 -70.47379089331153 -84.21637937355342 1.19500282737800 -0.19500282737800

0.00010000000000 -7.04737908933116 -7.18480503077902 1.01950029077560 -0.01950029077560

0.00001000000000 -0.70473790893312 -0.70611217403768 1.00195003715161 -0.00195003715161

0.00000100000000 -0.07047379089331 -0.07048753401415 1.00019501038134 -0.00019501038134

0.00000010000000 -0.00704737908933 -0.00704751656733 1.00001950767696 -0.00001950767696

0.00000001000000 -0.00070473790893 -0.00070473926218 1.00000192020693 -0.00000192020693

0.00000000100000 -0.00007047379089 -0.00007047377832 0.99999982162240 0.00000017837760

0.00000000010000 -0.00000704737909 -0.00000704734254 0.99999481310498 0.00000518689502

example for 3 days integration

JxwithxJ

)x(J)xx(Jlimt

1

0

- (1-t) is reaching ~ 1E-7 -> proof for correct coding of adjoint

- slight deterioration for longer integration times (1E-4 for 45 days integration), due to numerical effects

test of 4DVAR system using synthetic observations

Create synthetic observations:

- station data

- total columns

25

1

25

1

l

l

)l,j,i(m

)l,j,i(rm)j,i(col

"brute force inversion" (-> test of 4DVAR system performance rather than realistic simulation with expected real data)

- Assume global availabiltity of column data, uncertainty 0.1 ppb

- Assume very high uncertainty for emissions (emission x 1 E4)

4DVAR inversion using synthetic observations (global grid 6x4)

Artificial increase of CH4 emissions over Germany by 30 %

-> a priori emissions

4D VAR inversion

emission inventory used to create synthetic observations

-> "true emissions"

4DVAR inversion using synthetic observations (global grid 6x4)

Artificial increase of CH4 emissions over Germany by 30 %

-> a priori emissions

4D VAR inversion

emission inventory used to create synthetic observations

-> "true emissions"

4D VAR inversion returns inventory very close to "true emission inventory'

4DVAR inversion - analysis increments (global grid 6x4)

4D VAR inversion

artificial increase

a priori - SYNOBS

a priori - a posteriori

4DVAR inversion using synthetic observations (European zoom 3x2)

emission inventory used to create synthetic observations

-> "true emissions"

Artificial increase of CH4 emissions over Germany by 30 %

-> a priori emissions

4D VAR inversion

4D VAR inversion returns inventory very close to "true emission inventory'

4DVAR inversion - analysis increments (European zoom 3x2)

4D VAR inversion

artificial increase

a priori - a posteriori

a priori - SYNOBS

4DVAR inversion using synthetic observations (European zoom 1x1)

emission inventory used to create synthetic observations

-> "true emissions"

Artificial increase of CH4 emissions over Germany by 30 %

-> a priori emissions

4D VAR inversion

4D VAR inversion returns inventory very close to "true emission inventory'

4DVAR inversion - analysis increments (European zoom 1x1)

4D VAR inversion

artificial increase

a priori - SYNOBS

a priori - a posteriori

minimisation algorithm (M1QN3)

minimisation cost function (m1qn3)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100 120

iteration

co

st

fun

cti

on

J

J para

J_obs

J_total

minimisation cost function (m1qn3)

0

0.00002

0.00004

0.00006

0.00008

0.0001

0 20 40 60 80 100 120

iteration

co

st

fun

cti

on

J

J para

J_obs

J_total

Good convergence within ~20 - ~100 iterations

Convergence dependent on:

- preconditioning

- parameter set

- observations

- length of assimilation period

Conclusions

• demonstration for correct coding of TM5 adjoint (gradient test)

• setup of 4DVAR system for flexible incorporation various types of observational data (monitoring stations, satellite measurements)

• test of 4DVAR system using synthetic observations -> recovery of "true emissions"

conclusions

Next steps

• finalisation of 4DVAR framework (many details still to be improved)

• further tests with synthetic observations (test of system performance, test of sampling strategies)

• minimisation algorithms / preconditioning

• parameter covariances (correlations)

• a posteriori uncertainty reduction estimates

• …….

• real observations

• longer assimilation periods (-> meteo preprocessing)

next steps