CV3 background error statistics and tuning experiments

with WRFDA

Yong-Run GuoNCAR/MMM

Presented in Central Weather Bureau, Taiwan8 April 2011

CWB/UCAR 2011 project ---- TASK#1Yong-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) Benchmark2, CV3LP1: CV3 BE with 1 outer-loop OP2113, CV3LP2: CV3 BE with 2 outer-loops, tuning approach-I4, 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

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.

Background Error (BE) Estimation in WRF-Var

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:

Incremental WRF-Var and Preconditioning

€

J x( ) =1

2′ x TB−1 ′ x +

1

2d − ′ y [ ]n

2/σ on

2

n∑

€

d= y−H(xb )

€

J x( ) =1

2v i

2

i

∑ +1

2d − y'[ ]n

2/σ on

2

n

∑

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 (??)

€

B f = (xb − x t)(xb − x t)T ≈ ′ x ′ x T

Background Error Estimation for WRF-Var

€

B f = ′ x ′ x T ≈ A(x t 2 − x t1)(x t 2 − x t1)T

€

B f = ′ x ′ x T ≈ C(x k − x )(x k − x )T

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.

€

X

€

ˆ X

€

ˆ B = UTBU

€

X

€

ˆ B

€

B

€

Bij = B ji

€

B = ΣCΣ = ηη T , η = x b − x t .

WRF-Var Control Variable Transform

vvx' hvp UUUU ==

Define control variables:

€

ψ'

€

χ'= χ u' + χ b' ψ '( )

€

T'= Tu' +Tb' ψ '( )

€

ps' = psu' + psb' ψ'( )

€

r' = q'/qs Tb,qb, pb( )

V(i,j,k)

Balance Via Statistical Regression

Regression Coefficients after Wu et al (2002):

€

χb' = cψ '

€

Tb' (k) = G(k,k1)ψ ' (k1)

k1

∑

€

psb' = W (k)ψ ' (k)

k

∑

Variances for Ψ, unbalnced χ_u, T_u, Ps_u, and rhUse corresponding regression coefficients to compute unbalanced field. Variances vary with latitude and level.

Vertical length scale and horizontal length scale

€

Lh =8 × σ 2 ψ( )

σ 2∇ 2ψ( )

⎧ ⎨ ⎩

⎫ ⎬ ⎭

1

4

€

Lv =ρφφ

d2ρφφ /dx 2

x =0

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

1

2

Wu et al. (2002): Appendix Daley (1993), Eq.(4.3.10) P110.

Regression coefficients

€

χb' = cψ '

€

psb' = W (k)ψ ' (k)

k

∑

Regression coefficients

€

Tb' (k) = G(k,k1)ψ ' (k1)

k1

∑

Standard deviation

€

Steamfuction : ψ

Unbalanced velocity potential : χ u

Unbalanced temperature : tu

Unbalanced surface pressure : PSu

Relative humidity : 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 : (4,967,859)2

• CV3 BE (NCEP Global):

192 latitudes, 42 sigma levels:

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

€

B = ηη T{ }

Approaches to tune CV3 BE

The formulation for normalization of standard deviation is

€

σ norm _ k =σ k × as1 × samp

hlk × vk,k × M fac

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.

Approach I for CV3 BE tuning

be%corp, be%corz in WRFDA

Factor Loop 1 Loop 2 Loop3

Var_scaling1 1.50 1.00 0.50

Var_scaling2 1.50 1.00 0.50

Var_scaling3 1.50 1.00 0.50

Var_scaling4 1.00 1.00 0.50

Var_scaling5 1.50 1.00 0.50

Len_scaling1 1.00 0.50 0.25

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

Factor Loop 1 Loop 2

Var2 HL VL Var2 HL VL

as1(3) 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

as3(3) 0.220 1.00 1.5 0.063 0.75 1.5

as4(3) 0.230 2.00 1.5 0.063 1.50 1.5

as5(3) 0.270 0.50 1.5 0.068 0.40 1.5

€

as12

as11

=hl2

hl1

as12 = as11 ×hl2

hl1

⎧ ⎨ ⎩

⎫ ⎬ ⎭

2

= 0.063 ×0.50

0.75 ⎧ ⎨ ⎩

⎫ ⎬ ⎭

2

= 0.167{ }2 = 0.028

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

CV5 Tuning factors in TWRF CV3 Tuning factors in 2 loops Exp

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

oUse TWRF CV5 BE as the target: standard deviation and length-scale

oTwo outer-loops for CV3 BE with the 1st tuning factors same as before (OP211)

oSingle 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

oAnalysis time is 2009080400Z

Background error variance for the standard (one outer-loop) single ob test:

€

a − b =σ b

2

σ b2 + σ 0

2 (o − b); σ b2 =

a − b

o − aσ o

2 =a − b

(o − b) − (a − b)σ o

2

€

σ b2

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.

€

σ b2

Scale-length determination in the single ob test:

€

B(r) = B(0)e−

r 2

8s2

⎛

⎝ ⎜

⎞

⎠ ⎟

y = ln B(r)( ) = −r2

8s2

⎛

⎝ ⎜ ⎞

⎠ ⎟ + ln B(0)( ) = m2r2 + R = Mx + R

where x = r2, M = m2, R = ln B(0)( ), −r2

8s2 = m2r2,

Lengthscale : s = −1

8m2 = −1

8M, M < 0

background error var iance : B(0) = eR

r

B(r)

B(0)

€

σ s2 = B(s) = B(0) × e

−s2

8s2

⎛

⎝ ⎜

⎞

⎠ ⎟= σ b

2 × e−

1

8 ⎛ ⎝ ⎜

⎞ ⎠ ⎟= 0.882 × σ b

2

σ b2 ∞ a − b( )

S0.882xσ2

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

S.L. (km) 13.33 76.76 17.45

p a-b (Pa) 41.9 20.7 40.4

S.L. (km) 98.48 109.11 97.64

CV3 BE I CV3 BE II

Factor Loop 1 Loop 2 Loop1 Loop2

Var2 HL VL Var2 HL VL Var2 HL VL Var2 HL VL

as1 0.063 0.75 1.5 0.016 0.50 1.5 0.063 0.75 1.5 0.003 0.20 1.5

as2 0.063 0.75 1.5 0.016 0.50 1.5 0.063 0.75 1.5 0.003 0.20 1.5

as3 0.220 1.00 1.5 0.063 0.75 1.5 0.220 1.00 1.5 0.950 0.30 1.5

as4 0.230 2.00 1.5 0.063 1.50 1.5 0.230 2.00 1.5 0.230 0.20 1.5

as5 0.270 0.50 1.5 0.068 0.40 1.5 0.270 0.50 1.5 1.300 0.575 1.5

Increments (a-b) and scale-lengths for single ob tests wit different BEs

Tuning factors for CV3 BE with approach I and II

Reults

Single ob tests

Morakot forecasts

South-North Cross-section of increments of u, p, θ, and qfrom single u(148,55,20) tests:

o - b = 1 m/s,

€

σ o2 = 1 m /s

TWRF: CV5 3-outer-loops

U(a – b) = 0.818 m/sS.L. = 71.07 km

CV3 BE I2-outer-loops

U(a – b) = 0.729 m/sS.L. = 124.78 km

CV3 BE II2-outer-loops

U(a – b) = 0.839 m/sS.L. = 77.23 km

South-North Cross-section of increments of u, p, θ, and qfrom single t(148,55,20) tests:

o - b = 1oK,

€

σ o2 = 1oK

TWRF: CV5 3-outer-loopst (a – b) = 0.445o

S.L. = 21.06 km

CV3 BE I2-outer-loops

t (a – b) = 0.129o

S.L. = 98.98 km

CV3 BE II2-outer-loops

t (a – b) = 0.441o

S.L. = 21.61 km

South-North Cross-section of increments of u, p, θ, and qfrom single q(148,55,20) tests:

o - b = 1 g/kg,

€

σ o2 = 1 g /kg

TWRF: CV5 3-outer-loops

q (a – b) = 0.623 g/kgS.L. = 13.33 km

CV3 BE I2-outer-loops

q (a – b) = 0.371 g/kgS.L. = 76.76 km

CV3 BE II2-outer-loops

q (a – b) = 0.678 g/kgS.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,

€

σ o2 = 100 Pa

TWRF: CV5 3-outer-loopsp (a – b) = 41.9 Pa

S.L. = 98.48 km

CV3 BE I2-outer-loops

p (a – b) = 20.7 PaS.L. = 109.11 km

CV3 BE II2-outer-loops

p (a – b) = 40.4 PaS.L. = 97.64 km

CWB/UCAR 2011 project ---- TASK#1Yong-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-loop3, CV3LP2: CV3 BE with 2 outer-loops, tuning approach-I4, 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

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

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.

Remarks

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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