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Background Error
Daryl T. Kleist*[email protected]
National Monsoon Mission Scoping WorkshopIITM, Pune, India 11-15 April 2011
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Background Error
• B specification vital for controlling amplitude and structure for correction to model first guess (background)
• Covariance matrix– Controls influence distance– Contains multivariate information
• Typically estimated a-prior offline
co1T
ob1
VarT
bVar 2
1
2
1JJ yHxRyHxxxBxxx
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Variables for GSI-GFSAnalysis
• Background errors defined in terms of analysis variable– Streamfunction (Ψ)– Unbalanced Velocity Potential (χunbalanced)– Unbalanced Virtual Temperature (Tunbalanced)– Unbalanced Surface Pressure (Psunbalanced)– Relative Humidity
• Two options
– Ozone mixing ratio– Cloud water mixing ratio– Skin temperature
• Analyzed, but not passed onto GFS model3
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Balanced analysis variables
• χ = χunbalanced + A Ψ
• T = Tunbalanced + B Ψ
• Ps = Psunbalanced + C Ψ
• Streamfunction is a key variable defining a large percentage temperature and surface pressure
• A, B, C are empirical matrices (estimated with linear regression) to project stream function increment onto balanced component of other variables
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Multivariate Variable Definition
Tb = B ; b = A ; Psb = C
Projection of at vertical level 25 onto vertical profile of balanced temperature (G25)
Percentage of full temperature variance explained by the balance projection
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Multivariate B
Cross Section at 180o
u increment (black, interval 0.1 ms-1 ) and T increment (color, interval 0.02K) from GSI
Single zonal wind observation (1.0 ms-1 O-F and error)
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Elements needed for Bin GSI
• For each analysis variable (latitude/level)– Amplitude (variance)– Recursive filter parameters
• Horizontal length scale (km, for Gaussian)
• Vertical length scale (grid units, for Gaussian)– 3D variables only
• Additionally, balance coefficients– A, B, and C from previous slides
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Estimating Background Error
• NMC Method*– Lagged forecast pairs (i.e. 24/24 hr forecasts valid at same time)– Assume: Linear error growth– Easy to generate statistics from operational (old) forecast pairs
• Ensemble Method– Ensemble differences of forecasts– Assume: Ensemble represents actual error
• Observation Method– Difference between forecast and observations– Difficulties: observation coverage and multivariate components
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Stream FunctionStandard Deviation
• Function of latitude and height
• Larger in midlatitudes than in the tropics
• Larger in Southern Hemisphere than Northern Hemisphere
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Standard Deviation
• Divergent wind variance maximum in upper tropospheric tropics
• Large temperature variances near surface in extratropics
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StreamfunctionLength Scales
• Generally smaller scales in the tropics• Horizontal scales more uniform (latitude) than vertical
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Fat-Tailed Spectrum
• Sum of three Gaussians used in horizontal
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Moisture Variable
• Option 1– Pseudo-RH
• Option 2*– Normalized relative humidity– Multivariate with temperature and pressure– Standard Deviation a function of background relative humidity
bbbb
b
T
q
q
p
pRH
RH
RH
TRH
1b
• Holm (2002) ECMWF Tech. Memo
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Normalized PseudoRH
• Figure 23 in Holm (2002)
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Flow Dependent B (variances only)
• One motivation for GSI was to permit flow dependent variability in background error
• Take advantage of FGAT (guess at multiple times) to modify variances based on 9h-3h differences– Variance increased in regions of large tendency– Variance decreased in regions of small tendency– Global mean variance ~ preserved
• Perform reweighting on streamfunction, velocity potential, virtual temperature, and surface pressure only
Saha, S., et al., 2010: The NCEP Climate Forecast System Reanalysis. Bull. Amer. Meteor. Soc., 91, 1015-1057.
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Example of Variance Reweighting
Surface pressure backgrounderror standard deviation fields
a) with flow dependent re-scaling
b) without re-scaling
Valid: 00 UTC November 2007
a)
b)
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Flow-Dependence
• Although flow-dependent variances are used, confined to be a rescaling of fixed estimate based on time tendencies
– No cross-variable or length scale information used
– Does not necessarily capture ‘errors of the day’
• Plots valid 00 UTC 12 September 2008
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Summary
• Background error key component to data assimilation system
• A-prior, off-line estimates are typically used– NMC method for NCEP/GFS
• Can be cumbersome and require substantial testing/tuning
• Ensemble and Hybrid methods are the future (for 3D and 4D applications)– See Hybrid DA Talk
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