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# Observation representativeness error

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```Observation representativeness error
ECMWF model spectra
and Joint-OSSE
LWG, Destin (Fl) 27/1/2009
Motivation
вЂў ESA is re-considering burst vs. continuous mode for
вЂў Information content of various sampling modes for NWP
вЂў Effective model resolution
вЂ“ Number of degrees of freedom of a model
вЂ“ Observations should represent this model resolution
LWG, Destin (Fl) 27/1/2009
Observation weight in data assimilation
observation
Data assimilation
Atmospheric analysis
NWP model
вЂў Observation impact in atmospheric analysis is determined by the relative
weight of the observation and the model in the analysis
x a пЂЅ xb пЂ« K ( y пЂ­ H xb )
K пЂЅ BH
LWG, Destin (Fl) 27/1/2009
T
пЃ›HBH
T
пЂ«R
пЃќ
пЂ­1
Perfect observation
вЂў Perfect observation has no observation error:
x a пЂЅ xb пЂ« K ( y пЂ­ H xb )
R=0
пЂ­1
T
T
K пЂЅ BH пЃ›HBH пЂ« R пЃќ
вЂў For simplicity, assume the observation directly
related to a model parameter and located on a
model grid point: H=I
пѓ� K=I
вЂў y = Hxt +пЃҐ = xt (no observation error is assumed)
вЂў xa=xb+I(xt-xb) = xt
пѓ� The analysed state equals the true atmospheric
state at the measurement location
пѓ� Sounds good вЂ¦вЂ¦вЂ¦.. or?
LWG, Destin (Fl) 27/1/2009
Perfect observation
The model state is a smooth representation of the real atmospheric state
assimilation of perfect observation
вЂў Model perfectly fits observation, but no constraint elsewhere (overfitting)
LWG, Destin (Fl) 27/1/2009
Perfect observation
вЂў What goes wrong?
вЂў Model information (including information from
observations in previous cycles) is ignored
вЂў Model is forced to fit the small-scale structures present in
the (point) observation
вЂў But
вЂ“ model is a smooth representation of the real atmosphere, not
representing small-scale features
вЂ“ Small-scale structures are not well treated by the model (noise) and
should be avoided in the NWP analysis step.
пѓ� Weight given to the observation is too large
пѓ� How to determine a more appropriate weight?
LWG, Destin (Fl) 27/1/2009
Observation representativeness error
вЂў Representativeness error = the small scale atmospheric variability
which is sampled by individual observations, but which the model
is incapable of representing
вЂў 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.,
вЂў observation error variance = measurement error variance +
representativeness error variance.
R пЂЅ R instrument пЂ« R representa
tiveness
вЂў How to determine the observation representativeness error?
LWG, Destin (Fl) 27/1/2009
Wave number spectra near tropopause
5000 km
cyclones
k-3
Nastrom and Gage (1985)
GASP aircraft data
near tropopause
500 km
2 km
LWG, Destin (Fl) 27/1/2009
k-5/3
shifted
k-5/3 spectrum for horizontal
spatial scales below 500 km
пѓ� atmospheric variability (m2s-2)
is found by the surface below
the spectrum
ECMWF model spectra
пЂ­5 / 3
Lorenc curve (1992):
E (k ) пЂЅ E0k
k-5/3 atmosphere wind
variability spectrum
(ESA study by Lorenc on ADM)
based on Nastrom and Gage
1000 hPa
500 hPa
ECMWF (2008, T799)
ECMWF model does not well resolve
the atmospheric variability
on scales smaller than ~300 km
LWG, Destin (Fl) 27/1/2009
Power law and amplitude determine
unresolved model variance
ECMWF comment (1)
LWG, Destin (Fl) 27/1/2009
ECMWF comment (2)
LWG, Destin (Fl) 27/1/2009
ECMWF comment (3)
LWG, Destin (Fl) 27/1/2009
Illustration representativeness error model
вЂў Resolved wind variability: ECMWF and scatterometer
k -2
L
o
g
Half of wind
variance
W
i
n
d
25% wind variance
difference
S
p
W
D
e
n
s
i
t
y
3
4 times less
windvariance
KT
kk
R RC
CC
C
kG
10.000
1000
300
Wave Number [km]
LWG, Destin (Fl) 27/1/2009
100
10
Jur Vogelzang (2006)
Tropical cyclone Ike
ECMWF T799 ~ 25 km
HARMONIE
More small-scale structures in high-resolution (LAM) models
LWG, Destin (Fl)~27/1/2009
HARMONIE
2.5 km
Implication for Joint OSSE
вЂў Nature run (NR): ECMWF T511/T799
вЂ“ Lacking atmospheric variability on scales smaller than ~250km
вЂў Simulate atmospheric variability for missing NR scales
вЂ“ representativeness error
вЂў Observation simulation:
o = intpol(NR) + instrument error + representativeness error
LWG, Destin (Fl) 27/1/2009
Model resolution cell
вЂў Introduce Model Resolution Cell (MRC):
вЂ“ spatial scales below the MRC are not well resolved by the model
вЂ“ ECMWF model: MRC ~250km
вЂ“ unresolved wind variability:
п‚Ґ
пѓІ
E0k
пЂ­5 / 3
dk пЂЅ 3 . 21 m s
2
-2
4eпЂ­6
вЂ“ UKMO 1992: unresolved wind variability: 3.95 m2s-2
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
LWG, Destin (Fl) 27/1/2009
burst mode
along track
вЂў Assumption: along and across track
variabilities are independent and of equal size
вЂў Total error error variance
o 2 = r2across + r2along + m2/N
across track
representativeness error
continuous mode
instrument error ~ photon counts
racross пЂЅ 0 . 5 r
2
2
along
r
2
with r2 = atmospheric variability in MRC
пЂЅ 0 . 5 пѓЄ1 пЂ­ пѓ§
пѓ·
MRC
пѓё
пѓЄпѓ« пѓЁ
2/3
пѓ№ 2
пѓ¦ sample length пѓ¶
пѓ·
пѓє r пЂ« 0 .5пѓ§
MRC
пѓЁ
пѓё
пѓєпѓ»
MRC
2/3
MRC
1
r
2
N
Increasing the sample length reduces the along track representativeness error !
LWG, Destin (Fl) 27/1/2009
вЂў Analysis equations
A пЂЅ cov( x a пЂ­ x t )
B пЂЅ cov( x b пЂ­ x t )
x a пЂЅ xb пЂ« K ( y пЂ­ H xb )
A пЂЅ B пЂ­ BH
observatio n impact
пЂЅ
T
пЃ›HBH
T
пЂ«R
пЃќ
пЂ­1
R пЂЅ cov( y пЂ­ y )
HB
trace( B ) пЂ­ trace( A )
trace( B )
Observation impact пѓЋ [0,1];
0: no impact,
1: maximum impact (analysis equals true atmosphere)
LWG, Destin (Fl) 27/1/2009
Numerical example
вЂў Square model area of 2,500 km2, 25 km model grid, 10000 model grid points
вЂў single layer at 500 hPa
B
вЂў No clouds
( xi пЂ­ x j )
B (i , j ) пЂЅ пЃі e
2
b
пЃіb = 2.5 ms-1
R (i , j ) пЂЅ пЃі r пЃІ (i , j )
2
LB = 250 km
( yi пЂ­ y j )
with
пЃІ (i , j ) пЂЅ e
2
2
2 LO
,
R пЂЅ R m пЂ« R rep
пЂЅ пЃі m I пЂ« пЃі r H пЃІ пЂЁ i , j пЂ©H
2
2
2 LB
2
LWG, Destin (Fl) 27/1/2009
T
2
Numerical example (2) вЂ“ burst mode
sampling
Observation impact = 0.52
LWG, Destin (Fl) 27/1/2009
R
A
вЂў Pulse repetition frequency: 50 Hz (100 Hz for burst mode)
вЂў Same energy per shot
пѓ� Double the energy along a 200 km track in continuous mode
вЂў Continuous mode offers more flexibility
в€’ 50/100/200/ вЂ¦. km accumulation
в€’ 50/100/200/ вЂ¦. km observation distance
вЂў Increasing the accumulation length reduces the
representativeness error
вЂў BUT, observation correlation increases with decreasing
observation distance
LWG, Destin (Fl) 27/1/2009
Numerical example (3) вЂ“ continuous mode
sampling
R
A
100 km accumulation,
100 km spacing
observation impact = 0.61
200 km accumulation,
200 km spacing
observation impact = 0.63
50 km accumulation,
50 km spacing
observation impact = 0.60
LWG, Destin
(Fl) 27/1/2009
Closely
separated
observations => highly correlated => reduced impact
LAM model resolving small-scales
вЂў Assume that models ARE capable to resolve 50 km scales; LB=50 km
LWG, Destin (Fl) 27/1/2009
LAM model resolving small-scales вЂ“ ctd.
0.24
0.50
Models capable of resolving small-scale structures => high effective model
resolution
=> small representativeness errors, closely separated observations
LWG, Destin (Fl) 27/1/2009
are less correlated => continuous mode substantially better than burst mode
Conclusion
вЂў Spatial scales that can be resolved by global NWP models has not decreased a
lot over the last 15 years; model resolution cell ~ 250 вЂ“ 300 km
пѓ� Burst mode is still a useful scenario, despite the increased model grid resolution
пѓ� 100 km accumulations provide independent information on model degrees of
freedom (model resolution cells)
вЂў The quality of ADM-Aeolus HLOS winds is expected to be better, on
average, in continuous mode than in burst mode
вЂ“ About double the energy is transmitted into the atmosphere
вЂ“ Similar instrument noise (for 100 km accumulation)
вЂ“ Reduced representativeness error
вЂў Continuous mode offers a variety of accumulation scenarios (possibly
depending on cloud coverage)
вЂ“ More advanced processing needed to get the maximum out of it
LWG, Destin (Fl) 27/1/2009
Backup slides
LWG, Destin (Fl) 27/1/2009
Effective model resolution
вЂў Effective model resolution is not the same as model grid
mesh size
Model grid mesh size
ECMWF 1992: 100 km grid box
ECMWF 2008: 25 km grid box
ECMWF 2010: 15 km grid box
вЂў Effective model resolution is related to the spatial scales
that can be resolved by the model
LWG, Destin (Fl) 27/1/2009
Model resolution cell/representativeness error
summary
вЂў Model resolution ~ number of degrees of freedom of the model
вЂў Number of degrees of freedom is limited because
вЂ“ Limited computer capacity
вЂ“ Limited observation coverage to measure atmosphere non-linearity
пѓ� model is a smooth representation of the real atmosphere, not representing smallscale features пѓћ area (MRC) mean variables (model of a model)
пѓ� Small-scale structures are not well treated by the model (noise) and should be
avoided in the NWP analysis step.
вЂў Observations should вЂњfeedвЂќ these degrees of freedom, i.e. the area mean
model variables
пѓ� 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
LWG, Destin (Fl) 27/1/2009
Model resolution (3)
вЂў Wind component variability
вЂ“ integration of the spectra in the
previous image
Lorenc curve
Model resolution cell (MRC)
spatial scales below the MRC are
not well resolved by the model
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
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
LWG, Destin (Fl) 27/1/2009
вЂў Numerical example ADM HLOS error:
вЂ“
вЂ“
вЂ“
вЂ“
вЂ“
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 hPa
representativeness
error (ms-1)
(ms-1)
sampled variance
(m2s-2 )
2.36
0.53
1
NWP resolved
(% of sampled)
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
LWG, Destin (Fl) 27/1/2009
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
пѓ� doubling of the energy in continuous mode does not double
the additional impact as compared to burst mode.
пѓ� Observation correlation reduces impact of individual observations
(redundancy of sampling the degrees of freedom)
пѓ� Highly correlated observations (last row) should be avoided
LWG, Destin (Fl) 27/1/2009
```
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