Embed Size (px)
Citation preview
Sensitivity Studies (1):
Motivation
Theoretical background.
Sensitivity of the Lorenz model.
Thomas JungECMWF, Reading, UK
Time Evolution of Forecast Errors
Initial Conditions: Error Growth
Initial Time
Final Time
FC
Erro
r
Re
du
ce
d F
C E
rror
Model Error Growth
Initial Time
FC
Erro
r
Re
du
ce
d F
C E
rror
Final Time
Time
Linear Growth of Errors
)(xGdt
xd
Nonlinear model equation
fxLdt
xd
Linearized model equation
(3) ),0(
(2) ),0(
(1) ),0(),0(
0
0
0
0
dssR
fxtRx
dsfsRxtRx
t
t
t
t
Solution to the linearized model equation
Linear Initial Perturbation Growth
For a perfect model with f=0 we have:
0
0
012
01
1
),0(
)1,0()1,2(),1(
)1,0()2,1()2,1(
)1,0(
(1) )1,(
xtR
xRttRttRx
xRRxRx
xRx
xttRx
t
tt
Linear Model Perturbation Growth
For perfect initial conditions we have:
12
1
0
210223
10112
01
),1(
)3,2()2,1(),1(
)2,1()2,1(),1(
)3,2()2,1()3,2()3,2(
)2,1()2,1(
tt
t
ffttR
fRttRttR
fRttRttRx
ffRfRRfxRx
ffRfxRx
fx
Gradient of the Forecast Error w.r.t. Initial Conditions (Sensitivity)
. ofadjoint theis e wher
,J
obtain we; of Because (5)
.;
;;
Now, (4)
)()(
:assumption modelperfect ion approximatlinear Use(3)
;2/1
:for function diagnostic a Define (2)
hrs) 48(terror forecast thebe Let )1(
*
*0
00
0*
0
0000
RR
ePR
xJJ
xePRJ
xPRePxPePJ
xRxGxxGx
ePePJ
e
xxe
t
t
ttt
t
tt
t
refttt
Sensitivity Gradient and Norms
dpdTTcpTRvuxCx
ePCPMCJ
ePCPRCJ
xCxxCx
rrpsra
t
t
)/()/()(ln21 ;
:used is normenergy total thensapplicatiocurrent most In
:by gradientsy sensitivit forcing theand
:bygiven are gradientsy sensitivit Then the
time.finalat ; and timeinitialat norm thedefine ;Let
2222
1**1
00
1**1
00
10
Interpretation of the Gradient
)ln
,,,,(,,,,,,,,,
0
spppp p
J
T
J
v
J
u
J
T
JJ
Key-Analysis Errors
forth so and
Jon perturbati find (4)
! of decrease
.at J and of valuenew (3)
J (2)
J (1)
:follows as worksprocedureon minimizati The
;2
1
:minimize that onsperturbati ofset a are Errors analysis-Key
100
10
0000
000
00
*00
000
x
J
xxJ
x
eR
xRexRexJ
t
tt
Key-Analysis Errors: Schematic
1x
)(Function Cost xJ
2xxx 0
0x
1x 3x
2x
Lorenz System
bZXYZ
YrXXZY
YXX
Discrete Lorenz System: Tangent Linear Approximation
111
1111
11
)1(
)1(
)1(
tttt
ttttt
ttt
ZtbYXtZ
ZXtYtXtrY
YtXtX
Discrete Lorenz System:
11111
11111
11
)1()()(
)()1()(
)()1(
tttttt
tttttt
ttt
ZtbYXtXYtZ
ZXtYtXZttrY
YtXtX
Tangent Linear Lorenz System:
0000 )()( xRxGxxGxt
Discrete Lorenz System: Tangent Linear Approximation
t
t
t
t
t
t
t
t
t
t
t
t
Z
Y
X
Z
Y
X
R
Z
Y
X
Z
Y
X
1
1
1
1
1
1
000
000
000
1
1
01000
000
000
100
010
001
11
11
tbXtYt
XttZttr
ttR
tt
tt
Discrete Lorenz System: The Adjoint
adt
adt
adt
adt
adt
adt
T
adt
adt
adt
adt
adt
adt
Z
Y
X
Z
Y
X
R
Z
Y
X
Z
Y
X
1
1
1
1
1
1
)0,0,0(,,
)1()(0
)()1()(
)()()1(
111
111
1111
adt
adt
adt
adt
adtt
adt
adt
adt
adtt
adt
adt
adt
adt
adtt
adtt
adt
adt
adt
ZYX
ZtbYXtXZZ
ZXtYtXtYY
ZYtYZttrXtXX
Discrete adjoint Lorenz system:
Testing the Tangent Linear Hypothesis:Lorenz Model
?)()( 0000 xRxGxxG
Hypothesis:
G: Nonlinear Model
R: Tangent Linear Propagator
Experimental Design
First the Lorenz model is run for a long period.
This forecast serves a truth (“nature”).
Then, the Lorenz model is run from the “truth” using slightly perturbed initial conditions. This is meant to mimic analysis error.
Random number were used to generate the initial conditions.
To mimic model error the model parameters of the Lorenz model were perturbed and “weather forecasts” with the erroneous model were carried out using perfect initial conditions.
Sensitivity Gradients
Analysis vs. Key-Analysis Errors
Key-Analysis Errors Analysis Errors
Analysis vs. Key-Analysis Errors
Analysis vs. Key-Analysis Errors
Analysis vs. Key-Analysis Errors
Sensitivity: Initial vs Model Error
Initial Error Only Model Error Only
Sensitivity: Initial vs Model Error
Cost Function Reduction
FC Error SFC Error
Forecast Errors
Skill: Regular vs. Sensitivity Forecast
Conclusions I
Sensitivity gradients are being determined by integrating the short-term forecast error backwards in time using the adjoint model.
It is possible to determine the sensitivity of the short-term forecast error w.r.t. the initial conditions and model tendencies.
Key-analysis errors are obtained through minimization of the cost function. They are supposed to represent the part of the analysis and model errors, respectively, which are largely responsible for the forecast error.
Conclusions II
The sensitivity technique has been further illustrated using the Lorenz system.
For this simple system it has been shown that key-analysis errors reflect growing parts of analysis errors.
A cautionary example has been presented showing that key-analysis errors might be misleading if model errors significantly contribute to forecast errors.
Bibliography
Barkmeijer et al., 1999: 3D-Var Hessian SVs… QJRMS, 125, 2333ff.
Barkmeijer et al. 2003: Forcing singular vectors and other sensitive model structures. QJRMS, 129, 2401-2423.
Corti and Palmer, 1997: Sensitivity analysis of atmospheric low-frequency variability. QJRMS, 123, 2425-2447.
Errico, 1997: What is an adjoint model? BAMS, 78, 2577-2591.
Gelaro et al, 1998: Sensitivity analysis of forecast errors and the contruction of optimal perturbations using singular vectors, JAS, 55, 1012-1037.
Giering and Kaminski, 1996: Recipes for adjoint code construction. Max-Planck-Institut fuer Meteorologie, Report No. 212, Hamburg, Germany.
Gilmour et al., 2001: Linear regime duration: Is 24 hours a long time in synoptic weather forecasting? JAS, 58, 3525-3539.
Klinker et al, 1998: Estimation of key analysis errors using the adjoint technique. QJRMS, 124, 1909-1923.
Mahfouf, 1999: Influence of physical processes on the tangent-linear approximation. Tellus, 51A, 147-166.
Orrell et al., 2001: Model error in weather forecasting. Nonlin. Proc. Geophys., 8, 357ff.
Palmer et al., 1998: Singular vectors, metrics, and adaptive observations. JAS, 55, 633-653.
Palmer, 1993: Extended-range prediction and the Lorenz model. BAMS, 74, 49ff.
Palmer, 2000: Predicting uncertainty in forecasts of weather and climate. Rep. Prog. Phys., 63, 71ff.
Rabier et al., 1996: Sensitivity of forecast errors to initial conditions. QJRMS, 122, 121-150.
D’Andrea and Vautard, 2000: Reducing systematic model error by empirically correcting model errors. Tellus, 52A, 21-41.
Interpretation of Sensitivity Gradients
ectors.singular v theontoerror analysis theof sprojection theare
theand alues;singular v ingcorrespond theare the;propagator
linearngent adjoint ta theof ectorssingular v theare thewhere
,
:shown that becan It
2
1
20
i
i
i
N
iiii
c
v
vcJ
Therefore, if the analysis error has a white spectrum in the expansion, then the sensitivity pattern is dominated by the singular vectors with largest amplification factors.
