Observation representativeness error ECMWF model spectra Application to ADM sampling mode and Joint-OSSE LWG, Destin (Fl) 27/1/2009 Motivation вЂў ESA is re-considering burst vs. continuous mode for ADM-Aeolus вЂў Information content of various sampling modes for NWP вЂў Effective model resolution вЂ“ Number of degrees of freedom of a model вЂў ADM observation representation вЂ“ Observations should represent this model resolution вЂ“ ADM representativeness error 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 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 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 ADM representativeness error 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 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 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 ADM continuous mode 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 ADM representativeness error (2) 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 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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