Background Error Daryl T Kleist daryl kleistnoaa gov

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 – Controls influence distance – Contains multivariate information • Typically estimated a-prior offline 2

Variables for GSI-GFS Analysis • 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

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

Multivariate Variable Definition Tb = B ; b = A ; Psb = C Projection of at vertical level 25 onto vertical profile of balanced temperature (G 25) Percentage of full temperature variance explained by the balance projection 55

Multivariate B Single zonal wind observation (1. 0 ms-1 O-F and error) Cross Section at 180 o u increment (black, interval 0. 1 ms-1 ) and T increment (color, interval 0. 02 K) from GSI 6

Elements needed for B in 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) – 3 D variables only • Additionally, balance coefficients – A, B, and C from previous slides 7

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 8

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

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

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

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

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 13

Normalized Pseudo. RH • Figure 23 in Holm (2002) 14

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 9 h-3 h 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. 15

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

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 17

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 3 D and 4 D applications) – See Hybrid DA Talk 18