Application **and** interpretation **of**adjoint-derivedsensitivities in synoptic-casestudiesMichael C. Morgan**University** **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 model**and** 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 **of**ten 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 **of**Inversion←∂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 **of**forecast 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** ship**and** buoy observations **of** wind (ms -1 )**and** mean sea level pressureNCEP final analysis (blue) **and**36 hour **MM5** forecast (red) mslp

Water vapor image **and**satellite-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