11

Background Error

Daryl T. Kleist*daryl.kleist@noaa.gov

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

4

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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