1. BE, gen_be and Single ob experiments Syed RH Rizvi National Center For Atmospheric Research NCAR/MMM, Boulder, CO-80307, USA Email: rizvi@ucar.edu Syed RH Rizvi National Center For Atmospheric Research NCAR/MMM, Boulder, CO-80307, USA Email: rizvi@ucar.edu  A Description of the Advanced Research WRF Version 3 (chapter 9) NCAR/TN-475+STR NCAR TECHNICAL NOTE  gen_be technote by Dale Barker  Wu, W. -S., R. J. Purser, and D. F. Parrish, 2002: Three-Dimensional Variational Analysis with Spatially Inhomogeneous Covariances. Mon. Wea. Rev., 130, 2905-2916.

2. Talk overview: What is Background Error (BE) ? Role of BE in WRF-Var Importance of BE How is it computed (“gen_be” utility)? Impact of Background Error on minimization Single Observation Tests Tuning of background error

3. Where BE fits in WRF-modelling System

4. What is BE? It is the covariance of (forecast-truth) for analysis control variables BE = Since “truth” is not known, BE needs to be estimated Common methods to estimate BE a) Innovation Based approach b) NMC Method: (x-x t ) ≈ (x t1 - x t2 ) (Forecast differences valid for same time) c) Ensemble Method: (x-x t ) ≈ (x ens - ) (Ensemble - Ensemble mean) d) Flow dependent (adaptive approach)

5. Role of BE in WRF-Var cost Function: Basic WRF-Var cost function (J): J = 1/2[ (x-x b ) T B -1 (x-x b ) +(y o - H(x)) T R -1 (y o -H(x))) ] x - Analysis control variable x b - Background (FG) B- Background Error covariance H- Full (Non-linear) Forward Observation Operator y o - Observations R- Observation error covariance

6. Role of BE: BE is used for preconditioning the analysis equation by representing it as follows B = U U T with a suitable choice of U U = U p U v U h U h Horizontal Transform U v Vertical Transform U p Physical Transform Horizontal transformation (U h ) is via Regional ----- Recursive filters Global ----- Power spectrum Vertical transformation (U v ) is via EOF’s Physical transformation (U p ) depends upon the choice of the analysis control variable for which the variable  Ux’ has covariance matrix equal to the identity matrix. A natural way to build U is as a sequence of steps, each of which removes some correlation from the background error. The background error covariance matrix is defined implicitly by U

7. How BE is represented? In true sense the size of B is typically of the order of 10 7 x10 7 Thus it is not possible to handle such huge matrix Size of B is reduced by designing the analysis control variables in such a way that cross covariance between these variables are minimum Currently the analysis control variables for WRF-Var are the amplitudes of EOF’s of stream function (  ) Unbalanced part of velocity potential (  u ) Unbalanced part of temperature (T u ) Relative Humidity (q) Unbalanced part of surface pressure (p sfc_u ) With this choice of analysis control variables off-diagonal elements of BE is very small and thus its size typically reduces to the order of 10 7

8. How BE is represented Contd. UpUp B UvUv UhUh.... B = U p U v U h U h T U v T U p T

9. Basic statistical parameters of BE Corresponding to each control variables, following parameters constitutes the basic statistics of BE a) Regression Coefficient for estimating balanced (statistical) part of Velocity potential, Temperature and Surface pressure b) Eigen vectors and Eigen values c) Scalelength for regional and power spectrum for global option In WRF-Var, background error covariances are specified in a control variable space related to the model-space x’ via a control variable transform U defined by x’  Uv  U p U v U h v In gen_be, the inverse transform v  U -1 x’  U h -1 U v -1 U p -1 x’ is performed in order to accumulate statistics for each component of the control vector v.

10. WRF-Var “gen_be” utility: Computes various components of BE statistics needed for the WRF-Var Resides in WRF-Var top directory Designed both for NMC and Ensemble methods Consists of five stages (stage 0 - stage 4) WRFDA/var/gen_be > ls Makefile gen_be_stage0_wrf.f90 gen_be_cov2d.f90 gen_be_stage1.f90 gen_be_cov3d.f90 gen_be_stage1_1dvar.f90 gen_be_diags.f90 gen_be_stage2.f90 gen_be_diags_read.f90 gen_be_stage2_1dvar.f90 gen_be_ensmean.f90 gen_be_stage2a.f90 gen_be_ensrf.f90 gen_be_stage3.f90 gen_be_ep1.f90 gen_be_stage4_global.f90 gen_be_ep2.f90 gen_be_stage4_regional.f90 gen_be_etkf.f90 gen_spectra.f90 gen_be_read_regcoeffs.f90 be_method=‘NMC’ be_method=‘ENS’ WRFDA/var/gen_be/*.f90 WRFDA/var/da/da_gen_be

11. “gen_be” - Stage0: Computes ( ,  ) from (u,v) Forms desired differences for the following fields  - Stream Function  - Velocity potential T- Temperature q- Relative Humidity p s - Surface Pressure

12. “gen_be” - Stage1: Reads “gen_be_stage1” namelist Fixes “bins” for computing BE statistics Computes “mean” of the differences formed in stage0 Removes respective “mean” and forms perturbations for Stream Function(  ´) Velocity potential(  ´) Temperature (T´) Relative Humidity (q´) Surface Pressure(p s ´)

13. “gen_be” bins structure Currently “gen_be” utility has provisions of following seven (0-6) “bin_types” 0: No binning (each grid point is a bin) 1: mean in X-direction (Each latitude is a bin) 2: bins with binwidth_lat/binwidth_hgt 3: bins with binwidth_lat/nk 4: bins with binwidth_lat/nk (binwidth_lat (integer) is defined in terms of latitudinal grid points) 5: bins with all horizontal points (nk bins) 6: Average over all points (only 1 bin) nk - Number of vertical levels Default option is “bin_type=5” Only bin_type=1 and bin_type=5 have been tested. bin_type=1 results seem questionable.

14. “gen_be” - Stage2 & 2a: Reads “gen_be_stage2” namelist Reads field written in stage1 and computes covariance of the respective fields Computes regression coefficient & balanced part of , T & p s  b = C  ´ T b (k)= ∑ l G(k,l)  ´(l) p s_b = ∑ k W(k)  ´(k) Computes unbalanced part (stage 2a)  u ´ =  ´ -  b T u ´ = T´ - T b p s_u ´ = p s ´ - p s_b Wu et al., 2002 Regression coefficients G(k,l) and W(k) are computed individually for each bin in order to allow representation of differences between e.g. polar, mid-latitude, and tropical dynamical and physical processes. The scalar coefficient C used to estimate velocity potential errors from those of streamfunction is calculated as a function of height to represent e.g. the positive correlation between divergence and vorticity in the PBL. The summation over the vertical index relates to the integral (hydrostatic) relationship between mass fields and the wind fields.

15. WRF-Var Balance constraints WRF-Var imposes statistical balance constraints between Stream Function & Velocity potential Stream Function & Temperature Stream Function & Surface Pressure How good are these balanced constraints? Based on KMA global model

16. “gen_be” - Stage3: Reads “gen_be_stage3” namelist Removes mean for  u ´, T u ´ & p s_u ´ Computes eigenvector and eigen values for vertical error covariance matrix of  ´,  u ´, T u ´ & q Computes variance of p s_u ´ Processing: For each variable: compute vertical component of B, perform eigenvector decomposition B  E  E T, and compute projections of fields onto eigenvectors. Output: eigenvectors E, eigenvalues , and projected 3D (i,j,m) control variable fields for calculation of horizontal correlations.

17. “gen_be” - Stage4: Reads “gen_be_stage4” namelist For each variable & each eigen mode for regional option, computes “lengthscale (s)” For global option, computes “power spectrum (D n )” In WRF-Var (regional application), a recursive filter is used to provide the horizontal correlations. The error covariance input to the recursive filter is a correlation lengthscale and the number of passes through the recursive filter. Processing: perform linear regression of horizontal correlations to calculate recursive filter lengthscales

18. var/graphics/ncl/gen_be gen_be_corr_ps.ncl gen_be_corr_yz.ncl gen_be_corr_z.ncl gen_be_global_evals.ncl gen_be_global_evecs.ncl gen_be_lengthscales.ncl Reads in ASCII fort.1* output from gen_be/gen_be_diags_read.f90 Reads in ASCII ps_u.ps.dat, chi_u.chi.dat, t_u.t.dat, ASCII output from gen_be/gen_be_cov2d.f90 and gen_be/gen_be_cov3d.f90

19. Lat_stats_option:.False. (default). Only set it true when be.dat is computed with i-dependence (approximately latitude-dependence). Namelist - WRFVAR11 lat_stats_option .true. Use latitude-dependent regression coefficients lat_stats_option .false. Use domain-averaged regression coefficients

21. lat_stats_option=.false.lat_stats_option=.true. 700 hPa temperature increment

22. lat_stats_option=.false.lat_stats_option=.true. 20 hPa temperature increment

23. Single observation test Through single observation, one can understand a) structure of BE b) It identifies the “shortfalls” of BE c) It gives a broad guidelines for tuning BE “single obs utility” or “psot” may be activated by setting the following namelist parameters num_pseudo = 1 pseudo_var=“ Variable name” like ”U”, “T”, “P”, etc. pseudo_x = “X-coordinate of the observation” pseudo_y = “Y-coordinate of the observation” pseudo_z = “Z-coordinate of the observation” pseudo_val= “Observation value”, departure from FG” pseudo_err= “Observation error”

24. Single Observation (U) test

25. Single Observation (U) test - different BE

26. BE tuning Horizontal component of BE can be tuned with following namelist parameters LEN_SCALING1 - 5 (Length scaling parameters) VAR_SCALING1 - 5 (Variance scaling parameters) Vertical component of BE can be tuned with following namelist parameter MAX_VERT_VAR1 - 5 (Vertical variance parameters)

27. Tuning BE Len_scaling1=0.25 No tuning

28. Tuning BE Len_scaling1 & 2 =0.25No tuning