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
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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.
Talk overview:
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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?
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It is the covariance of (forecast-truth) for analysis control
variables
BE = <(x-xt), (x-xt)>
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Since “truth” is not known, BE needs to be estimated
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Common methods to estimate BE
a) Innovation Based approach
b) NMC Method: (x-xt) ≈ (xt1 - xt2)
(Forecast differences valid for same time)
c) Ensemble Method: (x-xt) ≈ (xens - <xens>)
(Ensemble - Ensemble mean)
d) Flow dependent (adaptive approach)
Role of BE in WRF-Var cost Function:
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Basic WRF-Var cost function (J):
J = 1/2[ (x-xb)TB-1(x-xb) +(yo - H(x))TR-1(yo-H(x))) ]
x - Analysis control variable
xb - Background (FG)
B - Background Error covariance
H - Full (Non-linear) Forward Observation Operator
yo - Observations
R - Observation error covariance
Role of BE:
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BE is used for preconditioning the analysis equation by
representing it as follows
B = U UT
with a suitable choice of U
U = UpUvUh
for which the variable   Ux’ has
Uh Horizontal Transform
Uv Vertical Transform
Up Physical Transform
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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.
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
The background error covariance matrix is defined implicitly by U
How BE is represented?
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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 UhTUvTUpT
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  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:
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Computes various components of BE statistics needed for the
WRF-Var
WRFDA/var/gen_be/*.f90
WRFDA/var/da/da_gen_be
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Resides in WRF-Var top directory
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Designed both for NMC and Ensemble methods
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Consists of five stages (stage 0 - stage 4)
WRFDA/var/gen_be>
Makefile
gen_be_cov2d.f90
gen_be_cov3d.f90
gen_be_diags.f90
gen_be_diags_read.f90
gen_be_ensmean.f90
gen_be_ensrf.f90
gen_be_ep1.f90
gen_be_ep2.f90
gen_be_etkf.f90
gen_be_read_regcoeffs.f90
be_method=‘NMC’
be_method=‘ENS’
ls
gen_be_stage0_wrf.f90
gen_be_stage1.f90
gen_be_stage1_1dvar.f90
gen_be_stage2.f90
gen_be_stage2_1dvar.f90
gen_be_stage2a.f90
gen_be_stage3.f90
gen_be_stage4_global.f90
gen_be_stage4_regional.f90
gen_spectra.f90
“gen_be” - Stage0:
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Computes ( , ) from (u,v)
Forms desired differences for the following fields


T
q
ps
- Stream Function
- Velocity potential
- Temperature
- Relative Humidity
- Surface Pressure
“gen_be” - Stage1:
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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
Relative Humidity
Surface Pressure
(´)
(´)
(T´)
(q´)
(ps´)
“gen_be” bins structure
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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.
“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
Wu et al., 2002
b
= C´
The summation over the vertical
Tb(k)
= ∑lG(k,l) ´(l)
index relates to the integral
(hydrostatic) relationship between
ps_b
= ∑k W(k) ´(k)
mass fields and the wind fields.
• Computes unbalanced part (stage 2a)
 u´
= ´ - b
Tu´
= T´ - Tb
ps_u´
= ps´ - ps_b
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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.
WRF-Var Balance constraints
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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
b   /   
Tb  T / T  T
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psb  ps / ps  ps

“gen_be” - Stage3:
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Reads “gen_be_stage3” namelist
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Removes mean for u´, Tu´ & ps_u´
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Computes eigenvector and eigen values for vertical error
covariance matrix of ´ , u´, Tu´ & q
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Computes variance of ps_u´
Processing: For each variable: compute vertical component of B,
perform eigenvector decomposition B  EET, 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:
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Reads “gen_be_stage4” namelist
For each variable & each eigen mode for regional option,
computes “lengthscale (s)”
B(r)  B(0)exp{r 2 /8s2}
1
y(r)  2 2[ln(B(0)/B(r)] 2  r /s
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For global option, computes “power spectrum (Dn)”
Dn 
   
n
m 2
n
F
m n
0 2
n
F
n

 

 2 Re(Fnm )  Im(Fnm )
m 1
2
2
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
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 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
Reads in ASCII fort.1* output from gen_be/gen_be_diags_read.f90
Namelist - WRFVAR11
Lat_stats_option: .False. (default). Only set it true when be.dat is
computed with i-dependence (approximately latitude-dependence) .
lat_stats_option  .true.
Use latitude-dependent regression coefficients
lat_stats_option  .false.
Use domain-averaged regression coefficients
lat_stats_option=.false.
lat_stats_option=.true.
700 hPa temperature increment
lat_stats_option=.false.
lat_stats_option=.true.
20 hPa temperature increment
Single observation test
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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
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“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
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Horizontal component of BE can be tuned with following
namelist parameters
LEN_SCALING1 - 5 (Length scaling parameters)
VAR_SCALING1 - 5 (Variance scaling parameters)
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Vertical component of BE can be tuned with following namelist
parameter
MAX_VERT_VAR1 - 5
(Vertical variance parameters)
Tuning BE
No tuning
Len_scaling1=0.25
Tuning BE
No tuning
Len_scaling1 & 2 =0.25