setup of 4d var inverse modelling system for atmospheric ch 4 using the tm5 adjoint model peter...
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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
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