R - University of Wisconsin MM5 Real Time Forecast and Forecast ...

Application and interpretation ofadjoint-derivedsensitivities in synoptic-casestudiesMichael C. MorganUniversity of Wisconsin-Madison

AcknowledgementsLinda KellerKate La CasseDr. 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 forwhich adjoint-based tools might have someutility

Goals• Provide synoptic interpretations for selectedforecast sensitivity gradients• Describe the “evolution” of sensitivitieswith respect to the forecast trajectory• Present a useful technique to displaysensitivities with respect to vector quantities• Discuss interesting research problems forwhich adjoint-based tools might have someutility

Relationship between the nonlinear modeland its adjointxin→NonlinearModel→ → R x )xout( outx ' in→LinearModel→ x ' out∂ R∂ x in←AdjointModel←∂R∂ x out

∆R=How might adjoints be used?R∂R∂x( x + x′) − R( x′) = ⋅ x′+ H . O Tf f ff .f∂R∂R∂R∆ R ≅ δR = ⋅x′f = , x′f = , Px′0∂x∂x∂xfffadjoint modelδRT ∂R= P , x′∂xf0input perturbationAn adjoint model is useful in the estimation of a changein response function associated with arbitrary, but smallchanges in the input to the linearized model.

Application #1: Synoptic case studiesImpact studiesvs.Sensitivity studies

Impact studies or “what if?” experiments• Impact studies involve studying the effects a specificinitial and/or boundary perturbation (x 0 ) to an NWPmodel has on some aspect of a forecast.• While these perturbations are often chosen based on“synoptic intuition”, typically the precise choice of thelocation and structure of the imposed initialperturbations is not known.• The chosen perturbations may have very little impacton the weather system of interest.• As these studies are performed to assess theimportance of a particular synoptic feature, manyintegrations are needed to yield useful results.

Modeling System Used• MM5 Adjoint Modeling System (Zou et al. 1997)with non-linear model state vector:x = ( u,v,w,T,p', qv)• All sensitivities were calculated by integrating theadjoint model “backwards” using dry dynamics,about a moist basic state.• The corresponding adjoint model state vector is:∂R∂x∂R= (∂u∂R∂R∂R∂R, , , , )∂v∂w∂T∂p'

Description of Case 1 and response functions• Cold frontal passage over the upper midwest duringthe 36h period beginning 1200 UTC 10 April 2003• Sensitivity gradients were calculated for the 36hour MM5 forecast from Eta model initialconditions at 1200 UTC 10 April 2003 for threeresponse functions:– 1) average temperature over WI– 2) average north-south temperature differenceover northern WI– 3) average zonal wind over WI

Mean sea level pressure and temperature (σ=0.85)

Sensitivity with respect to initial conditions at 1200 UTC 10 April 2003∂R∂T∂R∂ w∂R∂u∂R∂v

36h temperature sensitivity evolution

∂R∂u∂R∂v700 hPa sensitivities with respect to u and vvalid at1200 UTC 11 April 2003 (f24)

∂R∂u∂R∂v700 hPa sensitivities with respect to u and vvalid at1200 UTC 11 April 2003 (f24)

Sensitivity with respect to derived variables∂R∂xx = ( u,v,w,T,p', qv)∂R= (∂u∂R∂R∂R∂R, , , , )∂v∂w∂T∂p'∂f∂Rfx in( x in )→( ) ←f -1Adjoint of f -1→ xin←∂ R∂x inζ →Inversion→ uψ , vψ∂R←∂ζAdjoint ofInversion←∂R∂u∂R,∂v

700 hPa sensitivity gradients valid at 1200 UTC 11 April 2003 (f24)∂R∂u∂R∂v∂R∂ζ∂R∂φ

Description of Case 2 and response function

Impact study of McTaggart-Cowan (2002)

Initial state (MSLP and 925hPa θ)

Initial state (250:300 hPa PV)

Forecast evolution

Final state

Sensitivity of 48h KE to vorticity

Application #2: Identification of ‘key’ analysis errorsIf the response function chosen is a (quadratic) measure offorecast error, the output of the adjoint model provides ameans of changing the initial conditions to determine aninitial condition which will minimize the forecast errorxnew0=x0−αC−1∂R∂x0

11 April 1994 ECMWF forecast bustVERIFYING ANALYSISDAY-5 FORECASTRabier et al. (1996)

Control and perturbed analyses

Rabier et al. (1996)Evolution of‘key’ analysiserrors

VERIFYING ANALYSISDAY-5 FORECAST“OPTIMAL” FORECASTRabier et al. (1996)

Application #3: 4DVAR data assimilationinitial timefinal timebackground fieldoriginal forecast

Application #3: 4DVAR data assimilationinitial timefinal timebackground fieldoriginal forecastobservationsobservation error

Application #3: 4DVAR data assimilationinitial timefinal timebackground fieldoriginal forecastobservationsobservation errornew initial conditionsnew forecast

La CASsE STUDY1200 UTC 13 February 2001NCEP final analysis (mslp) and shipand buoy observations of wind (ms -1 )and mean sea level pressureNCEP final analysis (blue) and36 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 2001NCEP final analysis (blue) and 36 hour MM5 forecast (red) mslpAll observations assimilatedat 0600 UTCObservations in sensitiveregions assimilated at 0600 UTC

Assimilation in insensitive regions1200 UTC 13 February 200136 hour forecast mslp (cont. – assim.)25,00020,000Number of observations15,00010,0005,0000Observations in insensitiveregions assimilated at 0600 UTC

Questions?Real-time forecast sensitivities may be found athttp://helios.aos.wisc.edu