tbilisi, 10.07.2014 ggswbs'14 optimization for inverse modelling ketevan kasradze 1 hendrik...
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Tbilisi, 10.07.2014 GGSWBS'14
Optimization for inverse modelling
Ketevan Kasradze1 Hendrik Elbern1,2 kk@riu.uni-koeln.de he@riu.uni-koeln.de
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
Tbilisi, 10.07.2014 GGSWBS'14
Atmospheric layers
3/18
Tbilisi, 10.07.2014 GGSWBS'14
Atmospheric layers
3/18
SACA
DA
Tbilisi, 10.07.2014 GGSWBS'14
SACADA assimilation-system
BackgroundMeteorological
ECMWF analysesTrace gas
observations
Analysis
SACADA
PREPDWD GME
CTM
CTMadDiffusion
L-BFGS
Tbilisi, 10.07.2014 GGSWBS'14
Horizontal GME Grid
9/18
• ~147km between the grid points• 23 042 grid points per Model layer
Tbilisi, 10.07.2014 GGSWBS'14
Add
itio
nal r
efin
emen
t tro
posp
here
/low
er
stra
tosp
here
SACADA Vertical Grid54 layer
CRISTA-NF
MLS
Tbilisi, 10.07.2014 GGSWBS'14
HNO3
4.11.2005
~137hPa
12 Uhr UTC
MLS
15
Tbilisi, 10.07.2014 GGSWBS'14
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
Tbilisi, 10.07.2014 GGSWBS'14
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
Tbilisi, 10.07.2014 GGSWBS'14
N
iii
Tiib
Tb yxMHRyxMHxxBxxxJ
00
100
100 2
1
2
1)(
GradientAdjoint Model
N
iii
Tibx yHxRHMxxBxJ
0
1*0
10 )(
0
SACADA assimilation-system4D-Var
Tbilisi, 10.07.2014 GGSWBS'14
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
SACADA assimilation-system4D-Var
Tbilisi, 10.07.2014 GGSWBS'14
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
Background error covariance matrixBECM ~ 1012 ~ 80 Terrabyte
SACADA assimilation-system4D-Var
Tbilisi, 10.07.2014 GGSWBS'14
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
Tbilisi, 10.07.2014 GGSWBS'14
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
Tbilisi, 10.07.2014 GGSWBS'14
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 error covariance matrix formulation
Tbilisi, 10.07.2014 GGSWBS'14
Background error covariance matrix formulation
Background
Tbilisi, 10.07.2014 GGSWBS'14
Background error covariance matrix formulation
Background Observation: 3 ppm Ozone
Tbilisi, 10.07.2014 GGSWBS'14
Analysis (B diagonal)
Background error covariance matrix formulation
Background Observation: 3 ppm Ozone
Tbilisi, 10.07.2014 GGSWBS'14
Background error covariance matrix formulation
Background Observation: 3 ppm Ozone
Tbilisi, 10.07.2014 GGSWBS'14
Background error covariance matrix formulation
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
Tbilisi, 10.07.2014 GGSWBS'14
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
Background error covariance matrix formulation
use PV field for anisotropiccorrelation modelling
Tbilisi, 10.07.2014 GGSWBS'14
Background Observation: 3 ppm Ozone
Background error covariance matrix formulation
Tbilisi, 10.07.2014 GGSWBS'14
Background Observation: 3 ppm Ozone
Analysis increment
Background error covariance matrix formulation
Tbilisi, 10.07.2014 GGSWBS'14
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
Adjoint Model
SACADA assimilation-system4D-Var
Tbilisi, 10.07.2014 GGSWBS'14
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
Tbilisi, 10.07.2014 GGSWBS'14
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.
Tbilisi, 10.07.2014 GGSWBS'14
Adjoint model example
yaxz *2**
ayx
y
x
z
y
x
RRF2
33 ,:
...
...
...
JAC
Tbilisi, 10.07.2014 GGSWBS'14
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
Tbilisi, 10.07.2014 GGSWBS'14
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
Limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm
SACADA assimilation-system4D-Var
Tbilisi, 10.07.2014 GGSWBS'14
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
Tbilisi, 10.07.2014 GGSWBS'14
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
Tbilisi, 10.07.2014 GGSWBS'14
BFGS example with MATLAB
222 1100, xxyyxf
Tbilisi, 10.07.2014 GGSWBS'14
BFGS example with MATLAB
222 1100, xxyyxf
Tbilisi, 10.07.2014 GGSWBS'14
BFGS example with MATLAB
222 1100, xxyyxf
Tbilisi, 10.07.2014 GGSWBS'14
BFGS example with MATLAB
222 1100, xxyyxf
Tbilisi, 10.07.2014 GGSWBS'14
BFGS example with MATLAB
222 1100, xxyyxf
Tbilisi, 10.07.2014 GGSWBS'14
BFGS example with MATLAB
222 1100, xxyyxf
Tbilisi, 10.07.2014 GGSWBS'14
BFGS example with MATLAB
222 1100, xxyyxf
Tbilisi, 10.07.2014 GGSWBS'14
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
Tbilisi, 10.07.2014 GGSWBS'14
Thank you for your attention!
!გმადლობთ ყურადღებისათვის
Vielen Dank für Ihre Aufmerksamkeit!
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