CV3 background error statistics
and tuning experiments
with WRFDA
Yong-Run Guo
NCAR/MMM
Presented in Central Weather Bureau, Taiwan
8 April 2011
CWB/UCAR 2011 project ---- TASK#1
Yong-Run Guo
1 April 2011
1.2 Improve the performance of WRFVar
1.2.1 Conduct additional tests on multiple outer-loop with the variable CV3 BE tuning
factors
•To conduct the TWRF run and CV3 BE outer-loop experiments
Developed the running shell script to include the relocation module, DFI, new KF.
Testing runs starting from 2009080312Z to 2009080718Z with 6h cycling
Experiments conducted:
1, TWRF : CV5 BE with 3 outer-loops (TWRF)  Benchmark
2, CV3LP1: CV3 BE with 1 outer-loop  OP211
3, CV3LP2: CV3 BE with 2 outer-loops, tuning approach-I
4, CV3LP2II: CV3 BE with 2 outer-loops, tuning approach-II
With and without GPSRO data experiments
References
• Wu, Wan-Shu, R. James Purser, and David F Parrish, 2002: Three-Demesional Analysis
with Spatially Inhomogeneous Covariances. Mon. Wea. Rev., 130, 2905-2916.
• Purser, R. James, Wan-Shu Wu, David F. Parrish, and Nigel M. Robert, 2003:Numerical
Aspects of the Application of Recursive Filters to Variational Statistical Analysis,
Part I: Saptial Homogeneous and Isotropic Gaussian Covariances. Mon. Wea.
Rev., 131, 1524-1535.
• Purser, R. James, Wan-Shu Wu, David F. Parrish, and Nigel M. Robert,
2003:Numerical
Aspects of the Application of Recursive Filters to Variational
Statistical Analysis,
Part II: Spatially Inhomogeneous and Anisotropic General
Covariances. Mon.
Wea. Rev., 131, 1536-1548.
• Bannister, R.N., 2008: A review of forecast error covariance statistics in atmospheric
variational data assimilation. I: Characteristics and measurements of forecast
error covariances. Q. J. R. Meteorol. Sco. 134, 1951-1970.
• Bannister, R.N., 2008: A review of forecast error covariance statistics in atmospheric
variational data assimilation. II: Modeling the forecast error covariance
statistics. Q. J. R. Meteorol. Sco., 134, 1971-1996.
Introduction of CV3 BE
Background Error (BE) Estimation in WRF-Var
The number 1 question from WRFDA users is
“What background error covariances are best for my application?”.
Procedure:
 Use default statistics files supplied with code (CV5 BE, CV3 BE, etc.).
 Create you own, once you have run your system for ~a few weeks
(gen_be utility provided with WRFDA).
 Implement, tune, and iterate.
Incremental WRF-Var and Preconditioning

Define analysis increments: xa = xb + I x’

Solve incremental cost function:
1 T 1
1
2
J x  x  B x  d  yn / on2
2
2 n
where y’ = Hx’,

d = y  H(x b )
Define preconditioned control variable v space transform x’=Uv
where U transform CAREFULLY chosen to satisfy B = UUT .
(CVT)

Bannister (2008): Part II, Table 1, p.1975.


Choose (at least assume) control variable components with uncorrelated
errors:
1
1
2
2
Jx  v i  d  y' n /  on2
2 i
2 n
Background Error Estimation for WRF-Var

Assume background error covariance estimated by model perturbations x’ :
B f  (x b  x t )(x b  x t )T  xxT
Two ways of defining x’ in utility gen_be:


The NMC-method
(Parrish and Derber 1992):
B f  xxT  A(x t 2  x t1)(x t 2  x t1)T
where e.g. t2=24hr, t1=12hr forecasts…

…or ensemble
 perturbations (Fisher 2003):
B f  xxT  C(x k  x )(x k  x )T

Tuning via innovation vector statistics and/or variational methods (??)

Mathematical properties of the B-matrix
• Correlation matrix C and variance Σ2, the covariance matrix is formed by
multiplying respective columns and rows of C by the square roots of the
variance, B  C  T ,
  xb  xt .
• U is a orthogonal transform, i.e. UTU=I, to transform a vector between space X
and an alternative Xˆ space. Given that the background error covariance
matrix
B in X , we have Bˆ  U T BU.


• If the eigenvectors are the columns of U, and eigenvalues are the elements
of

, theBˆ -matrix
B inits eigenrepresentation is diagonal.

• B-matrix is square and symmetric, Bij  B ji. The eigenvalues are real-valued
and eigenvectors are mutually orthogonal.


• Covariance matrices are positive semi-definite. It means that the background
convex or flat in all directions in state space, so
term in the cost function, is
the minimum of the whole cost function exists.
WRF-Var Control Variable Transform
x'  Uv  UpUvUhv
Define control variables:
'
r' = q'/ qs Tb, qb, pb
 '  u' +  b' '
V(i,j,k)

T' Tu' +Tb' '

ps' = psu' + psb' '

Balance Via Statistical Regression
Regression Coefficients after Wu et al (2002):
  c
'
b
'
Tb' (k)   G(k,k1) ' (k1)
k1
p'sb  W (k) ' (k)
k
Variances for Ψ, unbalnced χ_u, T_u, Ps_u, and rh

Use
corresponding
regression
coefficients
to compute


unbalanced field. Variances vary with latitude and level.
Vertical length scale and horizontal length scale
1
4
8   2  
Lh   2

2

    

 
Lv   2
2
d

/dx

 
Wu et al. (2002): Appendix
1
2




x 0 
Daley (1993), Eq.(4.3.10) P110.

Regression coefficients
  c
'
b
p'sb  W (k) ' (k)
'
k

Regression coefficients
Tb' (k)   G(k,k1) ' (k1)
k1
Standard deviation
St eamfuction: 
Unbalanced velocity potential
: u
Unbalanced temperature
: tu
Unbalanced surface pressure: PSu
Relative humidit y: rh
Horizontal
length-scale
Vertical
length-scale
The vertical length-scale
in sigma unit
{log(sigma)?}
Only four 3-D control
variables, ψ, χu, tu, and
rh, have the vertical
length-scale.
Size of B-matrix
• CWB doamin: (221x127x44)=1,234,948
4  3-D variables (u, v, t, q) and 1  2-D variable (Ps).
The size of vector η=(xb-xt) is 4x1,234,948+28,067 = 4,967,859.
The size of B-matrix B  T : (4,967,859)2
• CV3 BE (NCEP Global):
42 sigma levels:
192 latitudes,
Regression coefficients: 192x42 = 8,064, 192x42x42 = 338,688, 192x42
= 8,064. Total = 354,816  cross-covariance between variables (balance)
Variances: (4x42 + 1) x 192 = 32,448  standard deviation
Length-scales: (2x4x42 + 1) x 192 = 64,704  auto-covariance of variables
Total = 451,968
Approaches to tune CV3 BE

Approach I
for CV3 BE tuning
The formulation for normalization of standard deviation is
 norm _ k 
 k  as1  samp
hlk  v k,k  M fac
 be%corp, be%corz in WRFDA
where σnorm_k is the normalized standard deviation and will be used in the
transformation of the control variables. σk is the input standard deviation, as1
is the tuning factor for standard deviation, hlk is the tuned horizontal scalelength, vk,k depends on the tuned vertical scale-length, Mfac is the map factor,
and Samp is a factor by which the amplitude at the second sweep is normalized.
Samp is determined by the filter characteristics. The subscript k denotes the
model level k.
So the normalized standard deviation σnorm_k will be obtained by the combined
effect from all tuning factors (as1) for variance, (as2) for horizontal and (as3)
vertical scale-lengths. The caution must be taken to set the proper tuning
factors to get the reasonable increments from each of the outer-loops.
CV5 Tuning factors in TWRF
Factor
CV3 Tuning factors in 2 loops Exp
Loop 1
Loop 2
Loop3
Factor
Var_scaling1
1.50
1.00
0.50
Var_scaling2
1.50
1.00
0.50
as1(3)
Var_scaling3
1.50
1.00
0.50
Var_scaling4
1.00
1.00
Var_scaling5
1.50
Len_scaling1
Loop 1
VL
Var2
0.063
0.75 1.5
0.016
0.50 1.5
as2(3)
0.063
0.75 1.5
0.016
0.50 1.5
0.50
as3(3)
0.220
1.00 1.5
0.063
0.75 1.5
1.00
0.50
as4(3)
0.230
2.00 1.5
0.063
1.50 1.5
1.00
0.50
0.25
as5(3)
0.270
0.50 1.5
0.068
0.40 1.5
Len_scaling2
1.00
0.50
0.25
Len_scaling3
1.00
0.50
0.25
Len_scaling4
1.00
0.50
0.50
Len_scaling5
1.00
0.50
0.20
Var2
Loop 2
HL
HL
VL
Based on the formulation of variance:
as12
as11

hl2
hl1
2

hl  
0.50
2
as12   as11  2    0.063
  0.167  0.028
hl1  
0.75

2
Considering that the (σnorm_k)2 should be smaller than (σnorm_k)1, the as12 should be
further reduced from {0.167}2 to {0.167x0.75}2 = {0.125}2 = 0.016

00
06
12
18
24
30
36
42
48
54
60
66
72
Mean
TWRF
10.2
36.2
52.0
65.3
103.4
127.0
143.9
151.3
158.9
177.2
185.1
211.7
231.2
127.2
CV3LP1
18.9
53.6
88.0
118.1
151.1
168.1
179.1
231.0
281.4
303.7
325.5
357.2
413.2
206.8
CV3LP2
22.1
47.6
71.1
88.4
106.0
130.3
153.0
187.5
238.2
257.5
314.8
315.9
341.0
174.9
Approach II
for CV3 BE tuning
o Use TWRF CV5 BE as the target: standard deviation and length-scale
o Two outer-loops for CV3 BE with the 1st tuning factors same as before (OP211)
o Single ob (near the typhoon center i=148, j=55) tests for u, t, q at level 20 and
p at level 1.
(since CV3 BE is latitude-dependent, we may need to try other the single ob at
points
o Analysis time is 2009080400Z
 Background error variance  2 for the standard (one outer-loop) single ob test:
 b2
ab 2
2 (o  b);
b 0
b
ab 2
ab
 =
o =
 o2
o a
(o  b)  (a  b)
2
b
However, if BE is kept same, the above equation is meaningless after the 1st
 the (a-b) equals to zero starting the 2nd outer loop. But if
outer loop because
BE is changed in the different outer loops, we can still use the above equation
2
to diagnose a presumable  b by completing all the outer loops with the final
analysis as a.
Scale-length determination in the single ob test:
B(r)  B(0)e
 r 2 
  2 
 8s 
 r 2 
y  lnB(r)   2   lnB(0)  m 2 r 2  R  Mx  R
 8s 
2
r
where x  r 2 , M  m 2, R  lnB(0),  2  m 2 r 2,
8s
1
1
Lengthscale: s  


, M0
8m 2
8M
backgrounderror variance: B(0)  e R
B(r)

 s2  B(s)  B(0)  e
 s 2 
  2 
 8s 
  b2  e
 1 
  
 8 
 0.882  b2
B(0)
 b2  a  b
0.882xσ2
S
r
ob
Params.
u
Increments (a-b) and
scale-lengths for single
ob tests wit different BEs
Tuning factors for CV3 BE
with approach I and II
0.818
0.729
0.839
S.L (km)
71.07
124.78
77.23
a-b (o)
0.445
0.129
0.441
S.L. (km)
21.06
98.98
21.61
a-b (g/kg)
0.623
0.371
0.678
S.L. (km)
13.33
76.76
17.45
a-b (Pa)
41.9
20.7
40.4
S.L. (km)
98.48
109.11
97.64
p
CV3 BE I
Factor
Loop 1
Var2
CV3 BE II
Loop 2
VL
Var2
as1
0.063 0.75 1.5
as2
CV3 BE II
a-b (m/s)
t
q
TWRF(CV5) CV3 BE I
Loop1
VL
Var2
0.016
0.50 1.5
0.063 0.75 1.5
0.016
as3
0.220 1.00 1.5
as4
as5
Loop2
VL
Var2
HL
VL
0.063
0.75 1.5
0.003
0.20
1.5
0.50 1.5
0.063
0.75 1.5
0.003
0.20
1.5
0.063
0.75 1.5
0.220
1.00 1.5
0.950
0.30
1.5
0.230 2.00 1.5
0.063
1.50 1.5
0.230
2.00 1.5
0.230
0.20
1.5
0.270 0.50 1.5
0.068
0.40 1.5
0.270
0.50 1.5
1.300
0.575
1.5
HL
HL
HL
Reults
 Single ob tests
 Morakot forecasts
South-North Cross-section of increments of u, p, θ, and q
from single u(148,55,20) tests:
o - b = 1 m/s,
TWRF:
CV5 3-outer-loops
U(a – b) = 0.818 m/s
S.L. = 71.07 km
 o2  1 m /s
CV3 BE I
2-outer-loops
U(a – b) = 0.729 m/s

S.L. = 124.78 km
CV3 BE II
2-outer-loops
U(a – b) = 0.839 m/s
S.L. = 77.23 km
South-North Cross-section of increments of u, p, θ, and q
from single t(148,55,20) tests:
o - b = 1oK,
TWRF:
CV5 3-outer-loops
t (a – b) = 0.445o
S.L. = 21.06 km
 o2  1o K
CV3 BE I
2-outer-loops
t (a – b) = 0.129o

S.L. = 98.98 km
CV3 BE II
2-outer-loops
t (a – b) = 0.441o
S.L. = 21.61 km
South-North Cross-section of increments of u, p, θ, and q
from single q(148,55,20) tests:
o - b = 1 g/kg,
TWRF:
CV5 3-outer-loops
q (a – b) = 0.623 g/kg
S.L. = 13.33 km
 o2  1 g /kg
CV3 BE I
2-outer-loops
q (a – b) = 0.371 g/kg
S.L. = 76.76 km

CV3 BE II
2-outer-loops
q (a – b) = 0.678 g/kg
S.L. = 17.45 km
South-North Cross-section of increments of u, p, θ, and q
from single p(148,55,1) tests:
o - b = 100 Pa,  o2  100 Pa
TWRF:
CV5 3-outer-loops
p (a – b) = 41.9 Pa
S.L. = 98.48 km
CV3 BE I
2-outer-loops
p (a – b) = 20.7 Pa
 = 109.11 km
S.L.
CV3 BE II
2-outer-loops
p (a – b) = 40.4 Pa
S.L. = 97.64 km
CWB/UCAR 2011 project ---- TASK#1
Yong-Run Guo
10 February 2011
1.2 Improve the performance of WRFVar
1.2.1 Conduct additional tests on multiple outer-loop with the variable CV3 BE tuning
factors
•To conduct the TWRF run and CV3 BE outer-loop experiments
Developed the running shell script to include the relocation module and DFI
Testing runs starting from 2009080312Z to 2009080718Z with 6h cycling
Experiments conducted:
1, TWRF : CV5 BE with 3 outer-loops (TWRF)
2, CV3LP1: CV3 BE with 1 outer-loop
3, CV3LP2: CV3 BE with 2 outer-loops, tuning approach-I
4, CV3LP2II: CV3 BE with 2 outer-loops, tuning approach-II
With and without GPSRO data experiments
00
06
12
18
24
30
36
42
48
54
60
66
72
Mean
TWRF
10.2
36.2
52.0
65.3
103.4
127.0
143.9
151.3
158.9
177.2
185.1
211.7
231.2
127.2
CV3LP1
18.9
53.6
88.0
118.1
151.1
168.1
179.1
231.0
281.4
303.7
325.5
357.2
413.2
206.8
CV3LP2
22.1
47.6
71.1
88.4
106.0
130.3
153.0
187.5
238.2
257.5
314.8
315.9
341.0
174.9
00
06
12
18
24
30
36
42
48
54
60
66
72
Mean
TWRF
10.2
36.2
52.0
65.3
103.4
127.0
143.9
151.3
158.9
177.2
185.1
211.7
231.2
127.2
CV3LP1
18.9
53.6
88.0
118.1
151.1
168.1
179.1
231.0
281.4
303.7
325.5
357.2
413.2
206.8
CV3LP2
22.1
47.6
71.1
88.4
106.0
130.3
153.0
187.5
238.2
257.5
314.8
315.9
341.0
174.9
CV3LP2i
i
9.0
38.8
56.5
101.0
132.5
150.4
167.8
183.7
221.2
261.6
282.8
335.4
369.6
177.7
Remarks
There are no significant differences between approach I and II. The reasons
might be:
 The same tuning factors were used in the 1st loop in order to be consistent
with the OP211
 Only variances and scale-lengths could be tuned with the tuning factors. The
statistic balance part cannot be tuned, which are different between the CV5 and
CV3 BE.
 We did not touch the vertical scale tuning. The vertical covariance modeling
between the CV5 and CV3 are totally different.
 We just tuned the CV3 BE based on the single ob tests at ONE points. It is
worth to try single ob tests at other grid-points.
 Here we just proposed two approaches for CV3 BE tuning only for typhoon
Morakot case, which may be worth for more experiments.
 Regional GSI BE (code is ready) is also worth to be tested.
GSPRO Impact
on Morakot track forecast
Impact of GPSRO data on Typhoon MORAKOT track forecast
From 18 UTC 3 to 12 UTC 7 August 2009 with Sixteen 6-h WRFDA/WRF (TWRF)
full cycles
Domain : CWB 45-km operational domain (222x128x45)
TWRF
: Relocation => 3DVAR with 3 CV5-outer-loops => Update_BC =>
DFI => WRF with new-KF
Totally there are 506 GPSRO profiles available during the experimental period.
With GPSRO data assimilated, mean error is reduced from 131.4 km to 114.9 km.
END
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