Physical processes in adjoint models: potential ... - ECMWF · ECMWF, Reading Linearized models in NWP • different applications: – variational data assimilation incremental 4D-Var

ECMWF, Reading

Physical processes in adjoint models:

potential pitfalls and benefits

Marta JANISKOVÁ ECMWF

[email protected]

Thanks to: P.Lopez, F. Chevallier A. Benedetti, J.-N. Thépaut, M. Leutbecher

mailto:[email protected]

ECMWF, Reading

Linearized models in NWP

• different applications:– variational data assimilation incremental 4D-Var at ECMWF

– singular vector computations initial perturbations for EPS

– sensitivity analysis forecast errors

• first applications with adiabatic linearized model

• nowadays, including the physical processes in the linearized model

4D-Var – Four-dimensional Variational Data Assimilation EPS – Ensemble Prediction System

ECMWF, Reading

Linearized model with physical processes

Including physical processes can:

• in variational data assimilation:– reduce spin-up– provide a better agreement between the model and data– produce an initial atmospheric state more consistent with physical processes– allow the use of new (satellite) observations (rain, clouds, soil moisture, …)

• in singular vector computations:– help to represent some atmospheric features

(processes in PBL, tropical instabilities, development of baroclinic instabilities, …)

• in sensitivity analysis:– allow a reduction of forecast error

• adjoint of physical processes can also be used for:– model parameter estimation– sensitivity of the parametrization scheme to input parameters

ECMWF, Reading

Development of a physical package

• for important applications:– incremental 4D-Var (ECMWF, Météo-France),– simplified gradients in 4D-Var (Zupanski 1993),– the initial perturbations computed for EPS (ECMWF),

linearized versions of forecast models are run at lower resolution

the linear model can be “not tangent” to the full model

(different resolution and geometry, different physics)

simplified approaches as a way to include progressively physicalprocesses in TL and AD models

• simplifications done with the aim to have a physical package:– simple – for the linearization of the model equations

– regular – to avoid strong non-linearities and thresholds

– realistic enough– computationally affordable

TL – tangent linear AD – adjoint

ECMWF, Reading

Full nonlinear vs. simplified physical parametrizations

In NWP – a tendency to develop more and more sophisticated physical parametrizations they may contain more discontinuities

For the “perturbation” model – more important to describe basic physical tendencies while avoiding the problem of discontinuities

Level of simplifications and/or required complexity depends on:

• which level of improvement is expected (for different variables, vertical and horizontalresolution, …)

• which type of observations should be assimilated• necessity to remove threshold processes

Different ways of simplifications:

• development of simplified physics (for instance, gaining from experience with simpler parametrization schemes used in history)

• applying only part of linearization

ECMWF, Reading

Problems with including physics in adjoint models

• Development – requires substantial resources

• Validation – must be very thorough(for non-linear, tangent-linear and adjoint versions)

• Computational cost – may be very high

• Non-linear and threshold nature of physical processes(affecting the range of validity of the tangent-linear approximation)

ECMWF, Reading

Validation of the physical parametrizations

Non-linear model:

• Forecast runs with particular modified/simplified physical parametrization schemes

Comparison:finite differences (FD) ↔ tangent-linear (TL) integration

( ) ( ) ( )( )guess firstfganalysisan

fganfgan MMM

==

−′↔−

,

xxxx

• examination of the accuracy of the linearization

• classical validation (TL - Taylor formula, AD - test of adjoint identity)

Tangent-linear (TL) and adjoint (AD) model:

Singular vectors:

• Computation of singular vectors to find out whether the new schemes do notproduce spurious unstable modes.

ECMWF, Reading

Importance of the regularization of TL model

• physical processes are characterized by:

* threshold processes:

• discontinuities of some functions describing the physical processes

(some on/off processes)• discontinuities of the derivative of a continuous function

* strong nonlinearities

ECMWF, Reading

WHY REGULARIZATION IS IMPORTANT

Without any treatment of most serious threshold processes, the TL approximation can turn to be useless.

89.651

86.428

6.593

3.103

3.0792.761

2.525

2.296

2.240

2.179

2.166

1.998

1.819

1.340

1.238

1.232

1.204

*******

-11.257

-8.562-5.838

-4.039

-3.314

-2.279

-2.163

-2.061

-2.041

-1.891

-1.782

-1.572

-1.318

-0.988

89.651

86.428

6.593

3.103

3.0792.761

2.525

2.296

2.240

2.179

2.166

1.998

1.819

1.340

1.238

1.232

1.204

*******

-11.257

-8.562-5.838

-4.039

-3.314

-2.279

-2.163

-2.061

-2.041

-1.891

-1.782

-1.572

-1.318

-0.988

lv31 T* 1999-03-15 12h fc t+6 - TL with vdif (no regularization applied) [cont.int: 0.5e+07]

ECMWF, Reading

SPURIOUS UNSTABLE MODES PRODUCED BY THE LINEARIZED PHYSICS The first singular vectors located around the cyclone (58oE, 18oS) computed

at the resolution T95 (Barkmeijer, 2002)

20OS20

OS

0O 0 O

40 OE

40 OE 60OE

60OE

SL diabatic T95

20OS20

OS

0O 0O

40OE

40OE 60 OE

60 OE

lv19 VO 2001-01-04 12h

all linearized physical parametrization schemes

-0.2 10 - 6

20°S

20°S

0° 0°

40°E

40°E 60°E

60°E

0.1 10-6

20°S

20°S

0° 0°

40°E

40°E 60°E

60°E

0.1 10-6

withoutthe linearized large-scaleprecipitationscheme

ECMWF, Reading

Potential source of problem

dyNL

dx

dyTL

ECMWF, Reading

Possible solution, but …

dx

dy N

L

dy T

L1

dy T

L2

ECMWF, Reading

… may just postpone the problem and influence the performance of NL scheme

dx1

dy N

L

dy T

L2

dy T

L3

dx2

ECMWF, Reading

dx

dy N

L

dy T

L1

dy T

L2

However, the better the model → the smaller the increments

ECMWF, Reading

Examples of regularizations and simplifications (1)

Regularization of vertical diffusion scheme:

• perturbation of the exchange coefficients (which are function of the Richardson

number Ri) is neglected, K’ = 0 (Mahfouf, 1999)

• reduced perturbation of the exchange coefficients (Janisková et al., 1999):

– original computation of Ri modified in order to modify/reduce f’(Ri), or

– reducing a derivative, f’(Ri), by factor 10 in the central part (around the point of singularity )

-20. -10. 0. 10. 20. Ri number

0.010

0.000

f(Ri)

-0.010

-0.020

Function of the Richardson number

ECMWF, Reading

Examples of regularizations and simplifications (2)

• reduction of the time step to 10 seconds to guarantee stable time integrations of the associated TL model (Zhu and Kamachi, 2000)

• selective regularization of the exchange coefficients K based on the linearization

error and a criterion for the numerical stability (Laroche et al., 2002)

ECMWF, Reading

Comparison: FULL TL – K’=0 – reduced K’ – selective K’ (Laroche et al.2002)

RMS linearization errors for the potential temperature perturbations at the 1st level above the surface

∆t = 1800 s

δ = 1 FULL TLM selective K’ reduced K’

K’=0

∆t = 60 sδ = 0.001

∆t = 60 sδ = 1

ECMWF, Reading

Importance of the regularization of TL model

• physical processes are characterized by:

* threshold processes:

• discontinuities of some functions describing the physical processes

(some on/off processes)• discontinuities of the derivative of a continuous function

* strong nonlinearities

• regularizations help to remove the most important threshold processes inphysical parametrizations which can effect the range of validity of the tangent linear approximation

• after solving the threshold problems

clear advantage of the diabatic TL evolution of errors compared to the adiabatic evolution

ECMWF, Reading

Zonal wind increments at model level ~ 1000 hPa [ 24-hour integration]

TLADIAB

TLWSPHYS

FD

TLADIABSVD

TLADIAB – adiabatic TL model

TLADIABSVD – TL model with very simple vertical diffusion (Buizza 1994)

TLWSPHYS – TL model with the whole set of simplified physics (Mahfouf 1999)

ECMWF, Reading

Simplified physical parametrizations

ECMWF LINEARIZED PHYSICS

vertical diffusion

gravity wave drag

radiation

simplified LW

SW and LW schemes

(cloud accounted)

dry processes

deep convection

large-scale condensation

mass fluxconvection

statisticalcloud scheme

to be

replaced by

to be

replaced by

moist processes

ECMWF, Reading

Tangent-linear diagnostics

Comparison:finite differences (FD) ↔ tangent-linear (TL) integration

( ) ( ) ( )( )guess firstfganalysisan

fganfgan MMM

==

−′↔−

,

xxxx

Diagnostics:

• mean absolute errors:

• relative error

( ) ( )[ ] ( ) fganfgan MMM xxxx −′−−=ε

%100.

REF

REFEXP

ε

εε −

ECMWF, Reading

Impact of physical processes

80ON 60ON 40ON 20ON 0O 20OS 40OS 60OS 80OS60

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6080N 60N 40N 20N 0 20S 40S 60S 80S

εEXP - εREF

relative improvement

EXP

REF = ADIAB

[%]

pTemperature

adiabsvd

ECMWF, Reading



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εEXP - εREF

EXP


REF = ADIAB

[%]

pTemperature

adiabsvd || vdif

ECMWF, Reading



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80N 60N 40N 20N 0 20S 40S 60S 80S

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εEXP - εREF

EXP


REF = ADIAB

[%]

pTemperature

adiabsvd || vdif + gwd

ECMWF, Reading



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80N 60N 40N 20N 0 20S 40S 60S 80S

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εEXP - εREF

EXP


REF = ADIAB

[%]

pTemperature

adiabsvd || vdif + gwd + radold

ECMWF, Reading



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10

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80N 60N 40N 20N 0 20S 40S 60S 80S

10

20

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εEXP - εREF

EXP


REF = ADIAB

[%]

pTemperature

adiabsvd || vdif + gwd + radold + lsp

ECMWF, Reading



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10

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80N 60N 40N 20N 0 20S 40S 60S 80S

10

20

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εEXP - εREF

EXP


REF = ADIAB

[%]

pTemperature

adiabsvd || vdif + gwd + radold + lsp + conv

ECMWF, Reading



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10

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80N 60N 40N 20N 0 20S 40S 60S 80S

10

20

30

40

50

60

εEXP - εREF

EXP


REF = ADIAB

[%]

pTemperature

adiabsvd || vdif + gwd + lsp + conv + radnew

ECMWF, Reading



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10

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80N 60N 40N 20N 0 20S 40S 60S 80S

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εEXP - εREF

|FD||FD - TL|

REF = ADIAB

EXP

relative improvement [%]

pTemperature

adiabsvd || vdif + gwd + conv + radnew + cl_new

ECMWF, Reading

Zonal wind

Specific humidityrelative improvement


X

X

|FD||FD - TL|

|FD||FD - TL|

[%]

[%]

X

X

εEXP - εREF

εEXP - εREF


ECMWF, Reading

Temperature: 3.8 %


50

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p

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Zonal wind: 6.1 %

Specifichumidity: 7.7 %

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6080N 60N 40N 20N 0 20S 40S 60S 80S

80N 60N 40N 20N 0 20S 40S 60S 80S 80N 60N 40N 20N 0 20S 40S 60S 80S

εEXP - εREF

REF = ADIAB

p yImpact of moist processes (lsp + conv)

ECMWF, Reading

Physics in 4D-Var

( ) 00 , xx δδ ttM ii'=

• In incremental 4D-Var, the objective function is minimized in terms of increments:

with the model state defined at any time ti as: ( ) bbb00, , xxxxx ttM iiiii =+= δ

← tangent linear model

( ) ( ) ( )iiii

n

iiii HH d)(d)(

21

21 1

00

100 −−+= −

=

− ∑ xRxxBxx TT δδδδδ ''J

( )boiiii Hy x−=d

• 4D-Var can be then approximated to the first order as minimizing:

where is the innovation vector

( ) ( )ii'ii

n

iii Htt d)(,

21 1

000

10

−+=∇ −

=

− ∑ xRHMxB TTx δδδ J

Gradient of the objective function to be minimized:

J0xδ∇

id

ixδ

J0xδ∇

← computed with the non-linear model at high resolution using full physics ← M

← computed with the tangent-linear model at low resolution using simplified physics ← M’

← computed with a low resolution adjoint model using simplified physics ← MT

ECMWF, Reading

Impact of the linearized physics in 4D-Var (1)

• comparisons of the operational version of 4D-Var against the version without linearized physics included shows:

– positive impact on analysis and forecast

FORECAST VERIFICATION – 500 hPa GEOPOTENTIAL period: 15/11/2000 – 13/12/2000

root mean square error – 29 cases

N.HEM S.HEM

oper RD 4v no phys

ECMWF, Reading

Impact of the linearized physics in 4D-Var (2)

– reducing spin-up problem when using physical processes

Time evolution of global precipitation in the tropical belt [30S, 30N] averaged over 14 forecasts issued from 4D-Var assimilation

ECMWF, Reading

Impact of the linearized physics in 4D-Var (3) 1-DAY FORECAST ERROR OF 500 hPa GEOPOTENTIAL HEIGHT

OPER vs. NEWRAD (27/08/2001 t+24)

A1:

FC_OPER –

ANAL_OPER

A2:

FC_NEWRAD –

ANAL_OPER

A2 – A1

75.4

-30.3

63.8

-23.3

0.2

-17.9

-10.0

ECMWF, Reading

1D-Var assimilation of observations related to the physical processes

• For a given observation yo, 1D-Var searches for the model state x=(T,qv) that minimizesthe objective function:

B = background error covariance matrixR = observation and representativeness error covariance matrixH = nonlinear observation operator (model space observation space)

(physical parametrization schemes, microwave radiative transfer model, reflectivity model, …)

)(()((21)()(

21)( 11 oobb yxRyxxxBxxx TT −−+−−= −− )H)HJ

Background term Observation term

• The minimization requires an estimation of the gradient of the objective function:

)(()()( 11 ob yxRHxxBx T −+−=∇ −− )HJ

• The operator HT can be obtained:– explicitly (Jacobian matrix)

– using the adjoint technique

ECMWF, Reading

Precipitation assimilation at ECMWF

A bit of history:• Work on precipitation assimilation at ECMWF initiated by Mahfouf and Marécal.

• 1D-Var on TMI and SSM/I rainfall rates (RR) (M&M 2000).

• Indirect ‘1D-Var + 4D-Var’ assimilation of RR more robust than direct 4D-Var.

• ‘1D-Var + 4D-Var’ assimilation of RR is able to improve humidity but also thedynamics in the forecasts (M&M 2002).

More recent developments:• New simplified convection scheme (Lopez 2003)• New simplified cloud scheme (Tompkins & Janisková 2003) used in 1D-Var• Microwave Radiative Transfer Model (Bauer, Moreau 2002)

• Assimilation experiments of direct measurements from TRMM and SSM/I (TB or Z) instead of indirect retrievals of rainfall rates, in a ‘1D-Var + 4D-Var’ framework.

Goal: To assimilate observations related to precipitation and clouds in ECMWF’s 4D-Var system including parameterizations of atmospheric moist processes.

TMI – TRMM Microwave Imager, TRMM – Tropical Rainfall Measuring Mission SSM/I – Special Sensor Microwave/Imager

ECMWF, Reading

“1D-Var+4D-Var” assimilation of observations related to precipitation

4D-Var

1D-Varmoist physics

moist physics + radiative transfer

background T,qv background T,qv

“Observed” rainfall rates

Retrieval algorithm (2A12,2A25)

1D-Var on TBs or reflectivities 1D-Var on TMI or PR rain rates

Observations interpolated on model’s T511 Gaussian grid

TMI TBs or

TRMM-PR reflectivities

“TCWVobs”=TCWVbg+mzδqv

ECMWF, Reading

1D-Var on TMI data (Lopez and Moreau, 2003)

Background

PATER obs 1D-Var/RR PATER

1D-Var/TB

Tropical Cyclone Zoe (26 December 2002 @1200 UTC)

1D-Var on TMI Rain Rates / Brightness Temperatures

Surface rainfall rates (mm h-1)

ECMWF, Reading

1D-Var on TRMM/ Precipitation Radar data (Benedetti, 2003)

2A25 Rain Background Rain 1D-Var Analysed Rain

2A25 Reflect. Background Reflect. 1D-Var Analysed Reflect.

Tropical Cyclone Zoe (26 December 2002 @1200 UTC)

Vertical cross-section of rain rates (top, mm h-1) and reflectivities (bottom, dBZ): observed (left), background (middle), and analysed (right).

Black isolines on right panels = 1D-Var specific humidity increments.

ECMWF, Reading

“1D-Var+4D-Var” assimilation of TRMM-PR rain rates/reflectivities:Impact on analysed and forecast TCWV and MSLP (Experiment – Control)

(Tropical Cyclone Zoe, 26-28 December 2002)

Analysis26/12/2002 03UTC

with

PR

rain

rate

s

Forecast26/12/2002 03UTC t+9

Forecast26/12/2002 03UTC t+57

with

PR

refle

ctiv

ities

ECMWF, Reading

1D-Var assimilation of ARM observations (1)

ARM SGP, May 1999 - observations: - surface downward longwave radiation (LWD),- total column water vapour (TCWV) - cloud liquid water path (LWP)

Observation operator includes: - shortwave and longwave radiation schemes - diagnostic cloud scheme

positive values = improvement

TCWV LWD

LWP SWD

| FG - OBS | - | ANAL - OBS |

ECMWF, Reading

1D-Var assimilation of ARM observations (2)

Cloud %

| 20. | 21. | 22. | 23. | 24. | 25. | 26. | 960.

880.

800.

720.

640.

560.

480.

Pressu

re (h

Pa)

1 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 - 100

Cloud %

| 20. | 21. | 22. | 23. | 24. | 25. | 26. | 960.

880.

800.

720.

640.

560.

480.

Pressu

re (h

Pa)

Cloud %

| 20. | 21. | 22. | 23. | 24. | 25. | 26. |Days since 19990501 00UTC

960.

880.

800.

720.

640.

560.

480.

Pressu

re (h

Pa)

OBS

FG

AN

Sensitivity of LWD to T, q and cloud fraction

ARM – Atmospheric Radiation Measurement

Time series of the cloud fraction (%) for the period 20-26 May 1999.

ECMWF, Reading

Cloudy AIRS Tbs and 4D-Var (Chevallier, 2003)

• 4D-Var assumes that the forward operator is linear in the vicinity of the background

Fairly true for cloudy upper tropospheric channels

6.3 µm4.5 µm14.3 µm

PDF of correlations betweenH.dx and H[x+dx] – H[x]Hemispheric data

• x = T, q profile• H = cloud scheme

+ RTTOVCLD• dx = perturbation

AIRS – Advanced Infrared Sounder

ECMWF, Reading

Cloudy AIRS Tbs and 1D-Var (Chevallier, 2003)

• Linear 1D-Var retrievalsobservations = 35 upper tropospheric AIRS channelsperformed only if clouds are detected in more than 13 channels

Validation:

1D-Var vs European radiosondesNov 2002 and Feb 2003

• If T<243K use Vaisala RS90 only• ~ 250 matches in upper troposphere • ~ 2300 matches in lower troposphere

ECMWF, Reading

20°S

10°S

5

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90

100-48-36-24-12012

243648

7260

8496108120

-48-36-24-1201224

3648

7260

8496108120

10oS

20oS

10oS

20oS

60oE 80oE

Probability that the cyclone KALUNDE will pass within 120 km radius during the next 120 hours

(06/03/2003 12 UTC)

numbers – real position of the cyclone at the certain hour green line – control T255 forecast (unperturbed member of ensemble)

Tropical SV with adiabsvd

Tropical SV with wsphys

Tropical singular vectors (Leutbecher and Van Der Grijn, 2003)

ECMWF, Reading

Sensitivity of the parametrization scheme to input variables

• using the adjoint technique

• adjoint FT of the linear operator F provides the gradient of an objective function Jwith respect to x (input variables) given the gradient of J with respect to y(output variables):

xF

x x ∂∂

=∂∂ JJ T or JJ yxx F ∇=∇ T

xx ∂∂

=∇F

J

∂F / ∂T sensitivity to: temperature ∂F / ∂q spec.humidity ∂F / ∂a cloud cover ∂F / ∂qlw cloud lwc ∂F / ∂qiw cloud iwc

EXAMPLE: sensitivity of radiation scheme - the gradient with respect to y of unity size (i.e., perturbation of some of the radiation fluxes with ± 1 W.m-2)

• experiments done in the global model:– potential for a thorough evaluation of the relative importance of different variables

for parametrization scheme– investigation of spatial and temporal patterns of sensitivity variations

ECMWF, Reading

Sensitivity of the shortwave upward radiation flux at the TOA with respect to specific humidity [W.m-2/g.kg-1] CLEAR SKY


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10dF/dq_swc 20001215 1200 t+24 Expver h1sc (180.0W-180.0E)

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21

12

. 1 . 1

./. 1

−−

−−

→=

=

mWkggdq

kggmWdqdF

winter spring

summer - 0.0125

- 0.025

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el le

vels

mod

el le

vels

ECMWF, Reading

Level 44 ~ 700 hPa

WINTER

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dF/dq_swc 15 December 2000 12UTC ECMWF t+24 Level 44

Sensitivity of the shortwave upward radiation flux at the TOA with respect to specific humidity [W.m-2/g.kg-1] CLEAR SKY

ECMWF, Reading

Sensitivity of the shortwave/longwave upward radiation flux at the TOA with respect to cloud fraction [W.m-2/cloudfr]


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15dF/dc_sw 20001215 1200 t+24 Expver h1sw (180.0W-180.0E)

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.12.0. 5 1

/. 5

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mWdcmWdc

cloudfrmWdcdF


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15dF/dc_lw 20001215 1200 t+24 Expver h1lw (180.0W-180.0E)

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WINTER

shortwave radiation

longwave radiation

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80N 60N 40N 20N 0 20S 40S 60S 80S

80N 60N 40N 20N 0 20S 40S 60S 80S

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mod

el le

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ECMWF, Reading

Summary

• Positive impact from including linearized physical parametrization schemes(into the assimilating model, singular vector computations used in EPS)has been demonstrated in experimental and operational runs.

• Adjoint of physical processes can also be used for sensitivity studies and modelparameter estimation.

• Physical parametrizations become important components in recent variationaldata assimilation systems.

• Some care must be taken when deriving the linearized parametrization schemes

(regularizations/simplifications).

• This is particularly true for the assimilation of observations related to precipitation,clouds and soil moisture, to which a lot of effort is currently devoted.

• One cannot also forget technical difficulties and time-consuming adjointdevelopment → reliable and efficient automatic tool for adjoint coding would be useful.

ECMWF, Reading ZERO BENEFIT




WE AREHERE


WE ARE HERE

MAXIMUM POTENTIAL BENEFIT

ECMWF, Reading

FORECAST VERIFICATION – 500 hPa GEOPOTENTIAL period: 11/05/2001 – 26/05/2001

(4D- Var experiments with the new linearized radiation: lin_rad) root mean square error – 16 cases

Northern Hemisphere North America

Documents

Physical processes in adjoint models: potential ... - ECMWF · ECMWF, Reading Linearized models in NWP • different applications: – variational data assimilation incremental 4D-Var