Application and interpretation of adjoint-derived

sensitivities in synoptic-case studies

Michael C. Morgan

University of Wisconsin-Madison

Acknowledgements

Linda Keller

Kate La Casse

Dr. Hyun Mee Kim (KMA)

Daryl T. Kleist (NCEP/NOAA)

Goals

• Describe what an adjoint model is• Demonstrate adjoint applications to

– Synoptic case studies– Diagnosis of ‘key’ analysis errors– Data assimilation

• Discuss interesting research problems for which adjoint-based tools might have some utility

Goals

• Provide synoptic interpretations for selected forecast sensitivity gradients

• Describe the “evolution” of sensitivities with respect to the forecast trajectory

• Present a useful technique to display sensitivities with respect to vector quantities

• Discuss interesting research problems for which adjoint-based tools might have some utility

Relationship between the nonlinear model and its adjoint

'inx '

outx

inxR

outx R

LinearModel

AdjointModel

inx outxNonlinearModel

)( outxR

How might adjoints be used?

TOHR

RRR ff

fff ..

xx

xxx

0xPx

xx

xx

,,f

ff

ff

RRRRR

0T x

xP

,f

RR

An adjoint model is useful in the estimation of a change in response function associated with arbitrary, but small changes in the input to the linearized model.

adjoint model input perturbation

Application #1: Synoptic case studies

Impact studies

vs.

Sensitivity studies

Impact studies or “what if?” experiments

• Impact studies involve studying the effects a specific initial and/or boundary perturbation (x0) to an NWP model has on some aspect of a forecast.

• While these perturbations are often chosen based on “synoptic intuition”, typically the precise choice of the location and structure of the imposed initial perturbations is not known.

• The chosen perturbations may have very little impact on the weather system of interest.

• As these studies are performed to assess the importance of a particular synoptic feature, many integrations are needed to yield useful results.

Modeling System Used

• MM5 Adjoint Modeling System (Zou et al. 1997) with non-linear model state vector:

)( vqpTwvux ,',,,,

)('

R,

R,

R,

R,

RRpTwvux

• All sensitivities were calculated by integrating the adjoint model “backwards” using dry dynamics, about a moist basic state.

• The corresponding adjoint model state vector is:

Description of Case 1 and response functions

• Cold frontal passage over the upper midwest during the 36h period beginning 1200 UTC 10 April 2003

• Sensitivity gradients were calculated for the 36 hour MM5 forecast from Eta model initial conditions at 1200 UTC 10 April 2003 for three response functions: – 1) average temperature over WI– 2) average north-south temperature difference

over northern WI– 3) average zonal wind over WI

Mean sea level pressure and temperature (=0.85)

Sensitivity with respect to initial conditions at 1200 UTC 10 April 2003

u

R

v

R

TR

wR

36h temperature sensitivity evolution

700 hPa sensitivities with respect to u and v valid at

1200 UTC 11 April 2003 (f24)

u

R

v

R

700 hPa sensitivities with respect to u and v valid at

1200 UTC 11 April 2003 (f24)

u

R

v

R

Sensitivity with respect to derived variables

inx R

inxfR

Adjoint of f -1

inxf inxf -1

v,u

R

vR

,uR

Inversion

Adjoint of Inversion

)( vqpTwvux ,',,,,

)('

R,

R,

R,

R,

RRpTwvux

700 hPa sensitivity gradients valid at 1200 UTC 11 April 2003 (f24)

u

R

v

R

R

R

Description of Case 2 and response function

Impact study of McTaggart-Cowan (2002)

Initial state (MSLP and 925hPa )

Initial state (250:300 hPa PV)

Forecast evolution

Final state

Sensitivity of 48h KE to vorticity

Application #2: Identification of ‘key’ analysis errors

0

100 x

Cxx

Rnew

If the response function chosen is a (quadratic) measure of forecast error, the output of the adjoint model provides a means of changing the initial conditions to determine an initial condition which will minimize the forecast error

VERIFYING ANALYSIS

Rabier et al. (1996)

DAY-5 FORECAST

11 April 1994 ECMWF forecast bust

Control and perturbed analyses

Evolution of ‘key’ analysis

errors

Rabier et al. (1996)

VERIFYING ANALYSIS

Rabier et al. (1996)

DAY-5 FORECAST

“OPTIMAL” FORECAST

Application #3: 4DVAR data assimilation

Application #3: 4DVAR data assimilation

Application #3: 4DVAR data assimilation

La CASsE STUDY1200 UTC 13 February 2001

NCEP final analysis (mslp) and ship and buoy observations of wind (ms-1)

and mean sea level pressure

NCEP final analysis (blue) and 36 hour MM5 forecast (red) mslp

Water vapor image andsatellite-derived wind vectors (ms-1)

0600 UTC 12 February 2001 300 hPa (yellow) and 400 hPa (blue)

Assimilation in sensitive regions1200 UTC 13 February 2001

NCEP final analysis (blue) and 36 hour MM5 forecast (red) mslp

Observations in sensitive regions assimilated at 0600 UTC

All observations assimilated at 0600 UTC

Assimilation in insensitive regions

36 hour forecast mslp (cont. – assim.)

Observations in insensitive regions assimilated at 0600 UTC

1200 UTC 13 February 2001

25,000

20,000

15,000

10,000

5,000

0

Num

ber

of o

bser

vatio

ns

Questions?

Real-time forecast sensitivities may be found at

http://helios.aos.wisc.edu