lwg, destin (fl) 27/1/2009 observation representativeness error ecmwf model spectra application to...
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LWG, Destin (Fl) 27/1/2009
Observation representativeness error
ECMWF model spectra
Application to ADM sampling modeand 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 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
LWG, Destin (Fl) 27/1/2009
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
LWG, Destin (Fl) 27/1/2009
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
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.
Weight given to the observation is too large 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.
• How to determine the observation representativeness error?
tivenessrepresentainstrument RRR
LWG, Destin (Fl) 27/1/2009
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
LWG, Destin (Fl) 27/1/2009
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
LWG, Destin (Fl) 27/1/2009
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
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)
LWG, Destin (Fl) 27/1/2009
Tropical cyclone Ike
ECMWF T799 ~ 25 km
HARMONIE ~ 2.5 km
HARMONIE
More small-scale structures in high-resolution (LAM) models
LWG, Destin (Fl) 27/1/2009
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:
– 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
LWG, Destin (Fl) 27/1/2009
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 !
LWG, Destin (Fl) 27/1/2009
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)
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• 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
LWG, Destin (Fl) 27/1/2009
Numerical example (2) – burst mode
sampling R A
Observation impact = 0.52
LWG, Destin (Fl) 27/1/2009
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
LWG, Destin (Fl) 27/1/2009
Numerical example (3) – continuous mode
sampling R A
100 km accumulation,100 km spacing
200 km accumulation,200 km spacing
50 km accumulation,50 km spacing
observation impact = 0.61
observation impact = 0.63
observation impact = 0.60
Closely separated observations => highly correlated => reduced impact
LWG, Destin (Fl) 27/1/2009
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 are less correlated => continuous mode substantially better than burst mode
LWG, Destin (Fl) 27/1/2009
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
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
• 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
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 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
LWG, Destin (Fl) 27/1/2009
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
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 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
LWG, Destin (Fl) 27/1/2009
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