1. CV3 background error statistics and tuning experiments
with WRFDA
Yong-Run Guo
NCAR/MMM
Presented in Central Weather Bureau, Taiwan
8 April 2011

2. 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 Z to Z 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

3. References
Wu, Wan-Shu, R. James Purser, and David F Parrish, 2002: Three-Demesional Analysis with Spatially Inhomogeneous Covariances. Mon. Wea. Rev., 130,
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,
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,
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,
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,

4. Introduction of CV3 BE

5. 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.

6. Incremental WRF-Var and Preconditioning
Define analysis increments: xa = xb + I x’
Solve incremental cost function:
where y’ = Hx’,
Define preconditioned control variable v space transform x’=Uv (CVT)
where U transform CAREFULLY chosen to satisfy B = UUT .
Bannister (2008): Part II, Table 1, p.1975.
Choose (at least assume) control variable components with uncorrelated errors:

7. Background Error Estimation for WRF-Var
Assume background error covariance estimated by model perturbations x’ :
Two ways of defining x’ in utility gen_be:
The NMC-method (Parrish and Derber 1992):
where e.g. t2=24hr, t1=12hr forecasts…
…or ensemble perturbations (Fisher 2003):
Tuning via innovation vector statistics and/or variational methods (??)

8. 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,
U is a orthogonal transform, i.e. UTU=I, to transform a vector between space and an alternative space. Given that the background error covariance matrix B in , we have
If the eigenvectors are the columns of U, and eigenvalues are the elements of , the -matrix in its eigenrepresentation is diagonal.
B-matrix is square and symmetric, The eigenvalues are real-valued and eigenvectors are mutually orthogonal.
Covariance matrices are positive semi-definite. It means that the background term in the cost function, is convex or flat in all directions in state space, so the minimum of the whole cost function exists.

9. WRF-Var Control Variable Transform
Define control variables:
V(i,j,k)

10. Balance Via Statistical Regression
Regression Coefficients after Wu et al (2002):
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
Wu et al. (2002): Appendix
Daley (1993), Eq.(4.3.10) P110.

11. Regression coefficients

12. Regression coefficients

13. Standard deviation

14. Horizontal length-scale

15. 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.

16. 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 : (4,967,859)2
CV3 BE (NCEP Global):
192 latitudes, 42 sigma levels:
Regression coefficients: x42 = 8,064, x42x42 = 338,688, x42 = 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

17. Approaches to tune CV3 BE

18. Approach I for CV3 BE tuning
The formulation for normalization of standard deviation is
 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 scale-length, 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.

19. CV5 Tuning factors in TWRF
CV3 Tuning factors in 2 loops Exp
Factor
Loop 1
Loop 2
Loop3
Var_scaling1
1.50
1.00
0.50
Var_scaling2
Var_scaling3
Var_scaling4
Var_scaling5
Len_scaling1
0.25
Len_scaling2
Len_scaling3
Len_scaling4
Len_scaling5
0.20
Factor
Loop 1
Loop 2
Var2 HL VL
as1(3)
0.063
0.75
1.5
0.016
0.50
as2(3)
as3(3)
0.220
1.00
as4(3)
0.230
2.00
1.50
as5(3)
0.270
0.068
0.40
Based on the formulation of variance:
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

20. 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

21. Approach II for CV3 BE tuning
Use TWRF CV5 BE as the target: standard deviation and length-scale
Two outer-loops for CV3 BE with the 1st tuning factors same as before (OP211)
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
Analysis time is Z
Background error variance for the standard (one outer-loop) single ob test:
However, if BE is kept same, the above equation is meaningless after the 1st outer loop because the (a-b) equals to zero starting the 2nd outer loop. But if BE is changed in the different outer loops, we can still use the above equation to diagnose a presumable by completing all the outer loops with the final analysis as a.

22. Scale-length determination in the single ob test:
B(r)
B(0)
0.882xσ2 S

23. ob
Params.
TWRF(CV5)
CV3 BE I
CV3 BE II u
a-b (m/s)
0.818
0.729
0.839
S.L (km)
71.07
124.78
77.23 t
a-b (o)
0.445
0.129
0.441
S.L. (km)
21.06
98.98
21.61 q
a-b (g/kg)
0.623
0.371
0.678
13.33
76.76
17.45 p
a-b (Pa)
41.9
20.7
40.4
98.48
109.11
97.64
Increments (a-b) and scale-lengths for single ob tests wit different BEs
Tuning factors for CV3 BE with approach I and II
CV3 BE I
CV3 BE II
Factor
Loop 1
Loop 2
Loop1
Loop2
Var2 HL VL
as1
0.063
0.75
1.5
0.016
0.50
0.003
0.20
as2
as3
0.220
1.00
0.950
0.30
as4
0.230
2.00
1.50
as5
0.270
0.068
0.40
1.300
0.575

24. Reults
Single ob tests
Morakot forecasts

25. 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) = m/s
S.L. = km
CV3 BE I
2-outer-loops
U(a – b) = m/s
S.L. = km
CV3 BE II
2-outer-loops
U(a – b) = m/s
S.L. = km

26. 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. = km
CV3 BE I
2-outer-loops
t (a – b) = 0.129o
S.L. = km
CV3 BE II
2-outer-loops
t (a – b) = 0.441o
S.L. = km

27. 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) = g/kg
S.L. = km
CV3 BE I
2-outer-loops
q (a – b) = g/kg
S.L. = km
CV3 BE II
2-outer-loops
q (a – b) = g/kg
S.L. = km

28. South-North Cross-section of increments of u, p, θ, and q
from single p(148,55,1) tests:
o - b = 100 Pa,
TWRF:
CV5 3-outer-loops
p (a – b) = 41.9 Pa
S.L. = km
CV3 BE I
2-outer-loops
p (a – b) = 20.7 Pa
S.L. = km
CV3 BE II
2-outer-loops
p (a – b) = 40.4 Pa
S.L. = km

29. 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 Z to Z 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

30. 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

31. 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
CV3LP2ii
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

32. 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.

33. on Morakot track forecast
GSPRO Impact
on Morakot track forecast

34. 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 km to km.

35. END
THANK YOU