lwg, destin (fl) 27/1/2009 observation representativeness error ecmwf model spectra application to...

LWG, Destin (Fl) 27/1/2009

Observation representativeness error

ECMWF model spectra

Application to ADM sampling modeand Joint-OSSE


Motivation

• ESA is re-considering burst vs. continuous mode for ADM-Aeolus

• Information content of various sampling modes for NWP

• Effective model resolution– Number of degrees of freedom of a model

• ADM observation representation– Observations should represent this model resolution

– ADM representativeness error


Observation weight in data assimilation

• Observation impact in atmospheric analysis is determined by the relative weight of the observation and the model in the analysis

observation

NWP model

Data assimilation Atmospheric analysis

1TT

)(

RHBHBHK

HK bba xyxx


Perfect observation

• Perfect observation has no observation error: R=0

• For simplicity, assume the observation directly related to a model parameter and located on a model grid point: H=I

K=I

• y = Hxt + = xt (no observation error is assumed)

• xa=xb+I(xt-xb) = xt

The analysed state equals the true atmospheric state at the measurement location

Sounds good ……….. or?

1TT

)(

RHBHBHK

HK bba xyxx


Perfect observation

• Model perfectly fits observation, but no constraint elsewhere (overfitting)

The model state is a smooth representation of the real atmospheric state

assimilation of perfect observation


Perfect observation

• What goes wrong?

• Model information (including information from observations in previous cycles) is ignored

• Model is forced to fit the small-scale structures present in the (point) observation

• But– model is a smooth representation of the real atmosphere, not

representing small-scale features

– Small-scale structures are not well treated by the model (noise) and should be avoided in the NWP analysis step.

Weight given to the observation is too large How to determine a more appropriate weight?


Observation representativeness error

• Representativeness error = the small scale atmospheric variability which is sampled by individual observations, but which the model is incapable of representing

• To avoid ingesting small-scale structures in the model state, the impact (weight) of the observation in the analysis is reduced by increasing the observation error with the representativeness error, i.e.,

• observation error variance = measurement error variance + representativeness error variance.

• How to determine the observation representativeness error?

tivenessrepresentainstrument RRR


Wave number spectra near tropopause

k-5/3

k-3

500 km

5000 kmcyclones

2 kmshifted

Nastrom and Gage (1985)

GASP aircraft data near tropopause

Wind spectra follow a k-5/3 spectrum for horizontal spatial scales below 500 km

atmospheric variability (m2s-2)is found by the surface below the spectrum


ECMWF model spectra

ECMWF model does not well resolvethe atmospheric variability on scales smaller than ~300 km

Lorenc curve (1992): k-5/3 atmosphere wind variability spectrum (ESA study by Lorenc on ADM) based on Nastrom and Gage

3/50)( kEkE

1000 hPa

500 hPa

ECMWF (2008, T799)

Power law and amplitude determine unresolved model variance


ECMWF comment (1)


ECMWF comment (2)


ECMWF comment (3)


Illustration representativeness error model

• Resolved wind variability: ECMWF and scatterometer

kR

CC

C

kG

KT

k -2

-3

W 25% wind variance

difference

4 times less windvariance

Half of wind variance

10.000 1000 100 10 Wave Number [km]

L o g

W i n d

S p

D e n s i t y

300

kRC

Jur Vogelzang (2006)


Tropical cyclone Ike

ECMWF T799 ~ 25 km

HARMONIE ~ 2.5 km

HARMONIE

More small-scale structures in high-resolution (LAM) models


Implication for Joint OSSE

• Nature run (NR): ECMWF T511/T799– Lacking atmospheric variability on scales smaller than ~250km

• Simulate atmospheric variability for missing NR scales– representativeness error

• Observation simulation:

o = intpol(NR) + instrument error + representativeness error


Model resolution cell

• Introduce Model Resolution Cell (MRC):– spatial scales below the MRC are not well resolved by the model

– ECMWF model: MRC ~250km

– unresolved wind variability:

– UKMO 1992: unresolved wind variability: 3.95 m2s-2

64

2-23/50 sm 21.3

e

dkkE

computational grids of global NWP models have increased substantially over the last 15 years,

but the horizontal scales that are resolved by these models have increased to a much lesser extent


ADM representativeness error

• Assumption: along and across track variabilities are independent and of equal size

• Total error error variance

o 2 = r2across + r2

along + m2/N

MRC

across track

alon

g tr

ack

burst mode

MRC

continuous moderepresentativeness error

instrument error ~ photon counts

23/2

23/2

2along

1

MRC

length sample5.0

MRC

length sample15.0 r

Nrr

22across 5.0 rr with r2 = atmospheric variability in MRC

Increasing the sample length reduces the along track representativeness error !


ADM information content

• Analysis equations

)( bba xyxx HK

HBRHBHBHBA1TT

)cov(

)cov(

)cov(

yy

xx

xx

tb

ta

R

B

A

)trace(

)trace()trace(impactn observatio

B

AB

Observation impact [0,1]; 0: no impact, 1: maximum impact (analysis equals true atmosphere)


Numerical example

• Square model area of 2,500 km2, 25 km model grid, 10000 model grid points• single layer at 500 hPa• No clouds

2

2

2

)(

2 e),( B

ji

L

xx

bji

B

b = 2.5 ms-1

LB = 250 km

T2r

2m

repm

, HHI

RRR

ji

,e),(with

),(),(

2

2

2

)(

2

O

ji

L

yy

r

ji

jiji

R

B


Numerical example (2) – burst mode

sampling R A

Observation impact = 0.52


ADM continuous mode

• Pulse repetition frequency: 50 Hz (100 Hz for burst mode)

• Same energy per shot Double the energy along a 200 km track in continuous mode

• Continuous mode offers more flexibility− 50/100/200/ …. km accumulation

− 50/100/200/ …. km observation distance

• Increasing the accumulation length reduces the representativeness error

• BUT, observation correlation increases with decreasing observation distance


Numerical example (3) – continuous mode

sampling R A

100 km accumulation,100 km spacing



observation impact = 0.61



Closely separated observations => highly correlated => reduced impact


LAM model resolving small-scales

• Assume that models ARE capable to resolve 50 km scales; LB=50 km


LAM model resolving small-scales – ctd.

0.24

0.50

Models capable of resolving small-scale structures => high effective model resolution => small representativeness errors, closely separated observations are less correlated => continuous mode substantially better than burst mode


Conclusion

• Spatial scales that can be resolved by global NWP models has not decreased a lot over the last 15 years; model resolution cell ~ 250 – 300 km Burst mode is still a useful scenario, despite the increased model grid resolution 100 km accumulations provide independent information on model degrees of

freedom (model resolution cells)

• The quality of ADM-Aeolus HLOS winds is expected to be better, on average, in continuous mode than in burst mode– About double the energy is transmitted into the atmosphere

– Similar instrument noise (for 100 km accumulation)

– Reduced representativeness error

• Continuous mode offers a variety of accumulation scenarios (possibly depending on cloud coverage)– More advanced processing needed to get the maximum out of it


Backup slides


Effective model resolution

• Effective model resolution is not the same as model grid mesh size

• Effective model resolution is related to the spatial scales that can be resolved by the model

Model grid mesh size

ECMWF 1992: 100 km grid box ECMWF 2008: 25 km grid box ECMWF 2010: 15 km grid box


Model resolution cell/representativeness error summary

• Model resolution ~ number of degrees of freedom of the model• Number of degrees of freedom is limited because

– Limited computer capacity– Limited observation coverage to measure atmosphere non-linearity model is a smooth representation of the real atmosphere, not representing small-

scale features area (MRC) mean variables (model of a model) Small-scale structures are not well treated by the model (noise) and should be

avoided in the NWP analysis step.

• Observations should “feed” these degrees of freedom, i.e. the area mean model variables Observed scales smaller than the MRC (model resolution cell) are treated as

noise, i.e. the representativeness error

Representativeness error small scale variability which is sampled by an observation, but which the model is incapable of representing


Model resolution (3)

• Wind component variability– integration of the spectra in the

previous image Lorenc curve

P (hPa) MRC size (km) T unresolved wind variability (m2s-2)

resolved wind variability (m2s-

2)

1000 340 59 3.94 1.3

500 263 76 3.30 1.0

250 312 64 3.72 1.2

Model resolution cell (MRC) spatial scales below the MRC are not well resolved by the model

MRC

computational grids of global NWP models have increased substantially over the last 15 years, but the horizontal scales that are resolved by these models have increased to a much lesser extent


ADM representativeness error (2)

• Numerical example ADM HLOS error:– ADM burst mode: sample length = 50 km

– ADM continuous mode : sample length = 100, 170 km

– m2/14 = 1.64 (ms-1)2 ~ 1 ms-1 LOS observation error standard deviation

– r2 = 3.3 (ms-1)2

– MRC = 250 km

500 hParepresentativeness

error (ms-1)0 of ADM

(ms-1)

sampled variance (m2s-2 )

NWP resolved(% of sampled)

50 km (Granada) 1.7 (Granada) 2.36 0.53 1

50 km burst (2008) 1.66 (0.84 r2)1/2 2.33 0.53 1

100 km continuous 1.57 (0.75 r2)1/2 2.27 0.83 4

170 km continuous 1.44 (0.63 r2)1/2 1.91 1.22 16


ADM impact

doubling of the energy in continuous mode does not double the additional impact as compared to burst mode.

Observation correlation reduces impact of individual observations (redundancy of sampling the degrees of freedom)

Highly correlated observations (last row) should be avoided

Observation length (km) Observation spacing (km) Number of observations Obs. Impact

50 200 12 0.3948

100 200 12 0.4472

200 200 12 0.5136

50 100 25 0.4444

100 100 25 0.4822

50 50 50 0.4685

lwg, destin (fl) 27/1/2009 observation representativeness error ecmwf model spectra application to...

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