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Background Error

Background Error. Daryl T. Kleist* daryl.kleist@noaa.gov. National Monsoon Mission Scoping Workshop IITM, Pune, India 11-15 April 2011. 1. Background Error. B specification vital for controlling amplitude and structure for correction to model first guess (background) Covariance matrix

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Background Error

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  1. Background Error Daryl T. Kleist*daryl.kleist@noaa.gov National Monsoon Mission Scoping Workshop IITM, Pune, India 11-15 April 2011 1

  2. 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

  3. 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 model 3

  4. Balanced analysis variables χ = χunbalanced + AΨ T = Tunbalanced + BΨ Ps = Psunbalanced + CΨ 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 4

  5. Multivariate Variable Definition Tb = B ; b = A ; Psb = C Percentage of full temperature variance explained by the balance projection Projection of  at vertical level 25 onto vertical profile of balanced temperature (G25) 5

  6. Multivariate B Single zonal wind observation (1.0 ms-1 O-F and error) Cross Section at 180o u increment (black, interval 0.1 ms-1 ) and T increment (color, interval 0.02K) from GSI

  7. 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

  8. 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

  9. Stream FunctionStandard Deviation • Function of latitude and height • Larger in midlatitudes than in the tropics • Larger in Southern Hemisphere than Northern Hemisphere

  10. Standard Deviation • Divergent wind variance maximum in upper tropospheric tropics • Large temperature variances near surface in extratropics

  11. StreamfunctionLength Scales • Generally smaller scales in the tropics • Horizontal scales more uniform (latitude) than vertical

  12. Fat-Tailed Spectrum • Sum of three Gaussians used in horizontal

  13. Moisture Variable • Option 1 • Pseudo-RH • Option 2* • Normalized relative humidity • Multivariate with temperature and pressure • Standard Deviation a function of background relative humidity • Holm (2002) ECMWF Tech. Memo

  14. Normalized PseudoRH • Figure 23 in Holm (2002)

  15. 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.

  16. Example of Variance Reweighting a) b) • Surface pressure background • error standard deviation • fields • with flow dependent re-scaling • without re-scaling • Valid: 00 UTC November 2007

  17. 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

  18. 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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