Sensitivity Studies (1):

Motivation

Theoretical background.

Sensitivity of the Lorenz model.

Thomas JungECMWF, Reading, UK

(thomas.jung@ecmwf.int)

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.