BE, gen_be and Single ob experiments Syed RH Rizvi National Center For Atmospheric Research NCAR/MMM, Boulder, CO-80307, USA Email: [email protected]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.
BE, gen_be and Single ob experiments. Syed RH Rizvi National Center For Atmospheric Research NCAR/MMM, Boulder, CO-80307, USA Email : [email protected]. A Description of the Advanced Research WRF Version 3 (chapter 9) NCAR/TN - 475+STR NCAR TECHNICAL NOTE gen_be technote by Dale Barker - PowerPoint PPT Presentation
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BE, gen_be andSingle ob experiments
Syed RH RizviNational Center For Atmospheric Research
• 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
Where BE fits in WRF-modelling System
What is BE?• It is the covariance of (forecast-truth) for analysis control
variablesBE = <(x-xt), (x-xt)>
• Since “truth” is not known, BE needs to be estimated
• Common methods to estimate BE a) Innovation Based approach
b) NMC Method: (x-xt) ≈ (xt1 - xt2) (Forecast differences valid for same time)
• Horizontal transformation (Uh) is via Regional ----- Recursive filters Global ----- Power spectrum• Vertical transformation (Uv) is via EOF’s • Physical transformation (Up) 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
How BE is represented?• In true sense the size of B is typically of the order of 107x107
• 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 (Tu) Relative Humidity (q) Unbalanced part of surface pressure (psfc_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 107
How BE is represented Contd.
Up
B
Uv
Uh. . . .
B = UpUvUh UhTUv
TUpT
Basic statistical parameters of BECorresponding 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 valuesc) 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 UpUvUhv
In gen_be, the inverse transform v U-1 x’ Uh
-1 Uv-1 Up
-1 x’ is performed in order to accumulate statistics for each component of the control vector v.
WRF-Var “gen_be” utility:• Computes various components of BE statistics needed for the
“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
“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_hgt3: bins with binwidth_lat/nk4: 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.
“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 & ps
b = C´ Tb(k)= ∑lG(k,l) ´(l) ps_b = ∑k W(k) ´(k)
• Computes unbalanced part (stage 2a) u´ = ´ - b
Tu´ = T´ - Tb
ps_u´ = ps´ - ps_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.
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
• Computes eigenvector and eigen values for vertical error covariance matrix of ´ , u´, Tu´ & q
• Computes variance of ps_u´
Processing: For each variable: compute vertical component of B, perform eigenvector decomposition B EET, 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.
“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 (Dn)” €
B(r) = B(0)exp{−r2 /8s2}
€
y(r) = 2 2[ln(B(0) /B(r)]12 = r /s
€
Dn = Fnm( )2=
m=−n
n
∑ Fn0( )2+ 2 Re(Fn
m )( )2+ Im(Fn
m )( )2
[ ]m=1
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
Single observation test• Through single observation, one can understand
a) structure of BE b) It identifies the “shortfalls” of BEc) 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”
Single Observation (U) test
Single Observation (U) test - different BE
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