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Observation representativeness error

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Observation representativeness error
ECMWF model spectra
Application to ADM sampling mode
and Joint-OSSE
LWG, Destin (Fl) 27/1/2009
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
LWG, Destin (Fl) 27/1/2009
Observation weight in data assimilation
observation
Data assimilation
Atmospheric analysis
NWP model
• 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
• 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
• For simplicity, assume the observation directly
related to a model parameter and located on a
model grid point: H=I
пѓ� K=I
• y = Hxt + = xt (no observation error is assumed)
• xa=xb+I(xt-xb) = xt
пѓ� The analysed state equals the true atmospheric
state at the measurement location
� Sounds good ……….. 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
• Model perfectly fits observation, but no constraint elsewhere (overfitting)
LWG, Destin (Fl) 27/1/2009
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.
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
• 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.
R пЂЅ R instrument пЂ« R representa
tiveness
• 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
пѓ� Wind spectra follow a
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
• 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
• 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
LWG, Destin (Fl) 27/1/2009
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:
п‚Ґ
пѓІ
E0k
пЂ­5 / 3
dk пЂЅ 3 . 21 m s
2
-2
4eпЂ­6
– 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
ADM representativeness error
burst mode
along track
• Assumption: along and across track
variabilities are independent and of equal size
• 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
пѓ© пѓ¦ sample length пѓ¶
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
ADM information content
• 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
• Square model area of 2,500 km2, 25 km model grid, 10000 model grid points
• single layer at 500 hPa
B
• 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) – burst mode
sampling
Observation impact = 0.52
LWG, Destin (Fl) 27/1/2009
R
A
ADM continuous mode
• Pulse repetition frequency: 50 Hz (100 Hz for burst mode)
• Same energy per shot
пѓ� 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
LWG, Destin (Fl) 27/1/2009
Numerical example (3) – 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
• Assume that models ARE capable to resolve 50 km scales; LB=50 km
LWG, Destin (Fl) 27/1/2009
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
LWG, Destin (Fl) 27/1/2009
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
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)
• 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
LWG, Destin (Fl) 27/1/2009
Backup slides
LWG, Destin (Fl) 27/1/2009
Effective model resolution
• 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
• 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
• 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
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.
• Observations should “feed” 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)
• Wind component variability
– 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
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 hPa
50 km (Granada)
representativeness
error (ms-1)
пЃі0 of ADM
(ms-1)
sampled variance
(m2s-2 )
1.7 (Granada)
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
ADM impact
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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