Research of short- -term forecast errors term forecast errors - - PowerPoint PPT Presentation

Research of short Research of short-

• term forecast errors

term forecast errors structure in the atmospheric boundary layer structure in the atmospheric boundary layer

Klimova E.G. Institute of Computational Technologies Russian Academy of Sciences, Siberian Branch Siberian Hydrometeorological Research Institute (SibHRI) klimova@ict.nsc.ru

Work is in part supported by the Project on development of Work is in part supported by the Project on development of variational variational data assimilation system data assimilation system (Russian (Russian Hydrometeorological Hydrometeorological Center, Moscow). Center, Moscow).

Klimova E.G. , Kilanova N.V., Dubrovskaya O.A. - Institute of Computational Technologies Russian Academy of Sciences, Siberian Branch Siberian Hydrometeorological Research Institute (SibHRI) Zaripov R. - Russian Hydrometeorological Center, Moskow

The purpose of work: an estimation of the short The purpose of work: an estimation of the short-

• term forecast

term forecast errors in atmospheric errors in atmospheric boundary layer boundary layer The basic problems:

• a large volume of the surface observations;
• dependence of correlation of surface forecast errors and

forecast errors at the top levels from stability of an atmospheric PBL;

• influence of orography.

Data assimilation Data assimilation a algorithms lgorithms: : Kalman Kalman filter filter

1 1 1 1 1 1; 1

; ( ) ; ( ) ; ( ); 0,..., . ( )( ) ; ( )( ) .

f a k k k f a T k k k k k f T f T k k k k k k k a f k k k k a f f k k k k k k f f t f t T a a t a t T k k k k k k k k k k

x A x P A P A Q K P M M P M R P I K M P x x K y M x k K P E x x x x P E x x x x

− − − − − − −

= = + = + = − = + − = = − − = − −

Data assimilation a Data assimilation algorithms lgorithms: : 4DVAR 4DVAR

) ( ) ( 2 1 ))) ( ( ( ))) ( ( ( 2 1 ) ( min

1 1

x x P x x x M H y O x M H y x J

b b T b i i i i T i i n i i i

− − + + − − =

− − = =

∑

′ × ″ − =

−

b b

x x b a

J J x x

1

1 0,

( ), ( ) ,

a fg fg T T fg

x x K y Mx K PM MPM O y Mx ε

−

= + − = + = +

ε O E E

T =

= ) )( ( , ε ε ε

• random m-vector of the
• bservational errors,

) ( 12

2 12

r F P

f

σ =

P – covariance matrix of the forecast errors.

Observational data analyses: Observational data analyses:

If to assume homogeneity and isotropy of the forecast errors, covariance approximately it is possible to describe as:

Observational data analyses Observational data analyses:

:

) x ~ y ( w x x

f i

• i

i ik f k a k

− + =

∑

) r ) i , i ( Cov ( ) k , i ( Cov w

2 ik

+ = ) ( F ) k , i ( Cov

ik 2

ρ σ =

• interpolation to the grid point
• the weight function

ps p1 p2 pn Lf

Methods of the forecast error covariance estimation : Methods of the forecast error covariance estimation :

• NMC-method;
• innovations;
• ensemble of forecasts.

Forecast error covariance estimation : Forecast error covariance estimation :

) (

fg

Mx y r − =

• The forecast errors are estimated by the “innovation” vector :
• The “innovations” covariance:

P – forecast error covariance matrix, R – observations error covariance matrix. Usually assume, that

R MPM rr

T T

+ =

} r , , r , r { diag R

2 n 2 2 2 1

L =

Forecast errors covariance estimation : Forecast errors covariance estimation : For forecast error covariance estimation we use: For forecast error covariance estimation we use:

• Observations for June - August 2006 (SYNOP, TEMP,

TEMPDW, TEMPDT).

• 6-hour forecasts on model WRF.
• The FNL data for the period June - August 2006.

Forecast error covariance estimation : Forecast error covariance estimation :

• The estimation of the forecast errors is made for WRF model:

http://www.mmm.ucar.ed/wrf

• Horizontal grid size is 18 km, 21 vertical levels.
• Parameterization of turbulence in the PBL: Mellor-Yamada-

Janjic.

• Surface Layer: similarity theory (Monin and Obukhov).

6 6-

• hour forecast

hour forecast on the

• n the WRF:

WRF: Т Т ( (° °K) K) on

• n 1000

1000 mb mb

Black points designate surface stations (SINOP), yellow – radiosonde

• bservations (TEMP).

Horizontal covariance of the forecast Horizontal covariance of the forecast errors: errors:

• The forecast errors are estimated by the “innovation”

vector:

• The covariances are calculated for each pair of
• bservation stations and binned for each interval of 20

km over the range from 0 to 1500 km (in supposition of homogeneity and isotropy of the forecast errors).

• «Innovations» are calculated on observations: T on 2m

SINOP and TEMP and 6-hour forecasts: T on 2m and

• п model levels.

) (

fg

Mx y −

Estimation of horizontal forecast error covariance: Estimation of horizontal forecast error covariance:

The covariances are calculated for each pair of observation stations and binned for each interval of 20 km over the range from 0 to 1500 km (in supposition of homogeneity and isotropy of the forecast errors). s ) k , i ( 1 s

ρ ≤ ρ < ρ −

) r r cov( N 1 ) cov(

k ) k , i ( i s

∑

= ρ

Variance for s-th bin:

2 s 2 k ) k , i ( i s 2

) cov( ) r r cov( N 1 ) ( ρ − = ρ σ

∑

Horizontal Horizontal covariance and correlation functions of forecast covariance and correlation functions of forecast errors: errors: Т Т, , June June -

• August

August 2006 2006: :

Covariance T

1 2 3 4 5 6 7 500 1000 1500 Distance (km) cov_T dispersiya

Correlation T

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 500 1000 1500 Distance (km) corr_T dispersiya

Covariances are calculated for the bins of fixed interval for the range (0,1500 km) km 20 r , r ) 1 s (

s

= ∆ ∆ × − = ρ

Number of available station pairs within each bin Number of available station pairs within each bin

• f fixed interval
• f fixed interval (

(ND ND) )

ND 50 100 150 200 250 300 350 400 450 250 500 750 1000 1250 1500 Distance (km) ND

ND 50 100 150 200 250 300 20 40 60 80 100 120 140 160 180 200 220 Distance (km) ND

Covariances are calculated for the bins of fixed interval for the range (0,1500 km) km 20 r , r ) 1 s (

s

= ∆ ∆ × − = ρ

Correlation and covariance Correlation and covariance functions for different time of day functions for different time of day

• 0,1

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 500 1000 1500 Distance (km) Corr06 Disp06 Corr18 Disp18

• 1

1 2 3 4 5 6 500 1000 1500 Distance (km) Cov06 Disp06 Cov18 Disp18

Covariance Correlation Cov06, Corr06, Cov18, Corr18 – functions, estimated in 06 and 18 hours UTC (local times 12h and 00h )

Correlation and covariance Correlation and covariance functions for different time of day functions for different time of day (The estimation is made on sample from The estimation is made on sample from N independent groups of N independent groups of

• bservations)
• bservations)
• 1

1 2 3 4 5 6 500 1000 1500 Distance (km) cov_06 disp_o6 cov_18 disp_18

• 0,1

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 500 1000 1500 Distance (km) corr_06 disp_06 corr_18 disp_18

Covariance Correlation Cov06, Corr06, Cov18, Corr18 – functions, estimated in 06 and 18 hours UTC (local times 12h and 00h )

Estimation of vertical covariance of forecast errors in PBL Estimation of vertical covariance of forecast errors in PBL : :

• Vertical covariance matrix is calculated on “innovations“ by the data for

June - August 2006. The matrix is calculated at isobaric levels from 1000 up to 700 mb with step 25 mb (13 on 13).

• Data TEMPDT with special points on temperature were interpolated

linearly on ln (p) on the given isobaric levels.

• Forecasts on model WRF were considered on the same isobaric levels.
• Cases - "stable" and "unstable" stratification were considered depending
• n the Richardson number value calculated on data TEMP (radiosonde):

2

dz dV dz d g Ri       θ θ =

Vertical Vertical covariances covariances and correlations of forecast and correlations of forecast errors errors : : Т Т, , June June -

• August

August 2006 2006

1 2 3 4 5 6 50 100 150 200 250 300 Rf>0 Rf<0 Rf>0.25 Rf<0.25 Rf>0.505 Rf<0.505 0,2 0,4 0,6 0,8 1 1,2 100 200 300 Rf>0 Rf<0 Rl>0.25 Rl<0.25 Rl>0.505 Rl<0.505

Covariance Correlation Ri>0 Ri>0.25 Ri>0.505

At all levels (1000 mb, 925 mb, 850 mb) >

Ri<0 Ri<0.25 Ri<0.505

Even at one level <

Vertical Vertical covariances covariances and correlations of forecast and correlations of forecast errors errors : : Т Т, , June June -

• August

August 2006 2006

100 200 300 400 500 600 Rl>0 Rl<0 Rl>0.25 Rl<0.25 Rl>0.505 Rl<0.505 Lf Nd

Rl>0 Rl<0 Rl>0.25 Rl<0.25 Rl>0.505 Rl<0.505 Nd 532 71 513 90 485 118 Lf 80,5 148,9 78,1 164,3 78,2 145,1

Nd – the size of sample Lf – «integrated» scale of covariance function (mb):

∫

∞

ρ ρ =

f

d ) ( Cov ) ( Cov 1 L

Vertical Vertical covariances covariances and correlations of forecast and correlations of forecast errors for different time of day errors for different time of day

0h

• 1

1 2 3 4 5 6 50 100 150 200 250 300 Ri>0 Ri<0 Ri>0.25 Ri<0.25 Ri>0.505 Ri<0.505

12h

1 2 3 4 5 6 50 100 150 200 250 300 Ri<0 Ri>0 Ri>0.25 Ri<0.25 Ri>0.505 Ri<0.505

Covariance 0h Covariance 12h Covariance in 0 and 12 hours UTC (local times 6h and 18h )

Vertical Vertical covariances covariances and correlations of forecast and correlations of forecast errors for different time of day errors for different time of day

00 h

100 200 300 400 Rf>0 Rf<0 Rf>0.25 Rf<0.25 Rf>0.505 Rf<0.505 Lf Nd

12 H

50 100 150 200 250 Rf>0 Rf<0 Rf>0.25 Rf<0.25 Rf>0.505 Rf<0.505 Lf Nd

Nd – the size of sample Lf – «integrated» scale of covariance function (mb):

∫

∞

ρ ρ =

f

d ) ( Cov ) ( Cov 1 L

Three Three-

• dimensional forecast error covariance

dimensional forecast error covariance : : Т Т, , June June -

• August

August 2006 2006

Ri>0

• 1

1 2 3 4 5 500 1000 1500 Distance(km) p1000 p925 p850 p700

Ri<0

• 0,6
• 0,4
• 0,2

0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 500 1000 1500 Distance (km) p1000 p925 p850 p700

) y , x , pn ( r ) y , x , ps ( r ) ; pn , ps ( F

l k j i

= ρ

ps – surface pressure, pn – pressure on standard isobaric levels (1000, 925, 850, 700 mb)

Three Three-

• dimensional forecast error covariance

dimensional forecast error covariance : : Т Т, , June June -

• August

August 2006 2006

Ri>0.25

• 1

1 2 3 4 5 500 1000 1500 Distance (km) p1000 p925 p850 p700

Ri<0.25

• 1

1 2 3 4 5 6 500 1000 1500 Distance (km) p1000 p925 p850 p700

) y , x , pn ( r ) y , x , ps ( r ) ; pn , ps ( F

l k j i

= ρ

ps – surface pressure, pn – pressure on standard isobaric levels (1000, 925, 850, 700 mb)

Three Three-

• dimensional forecast error covariance

dimensional forecast error covariance : : Т Т, , June June -

• August

August 2006 2006

Ri>0.505

• 1

1 2 3 4 5 500 1000 1500 Distance (km) p1000 p925 p850 p700

Ri<0.505

• 1
• 0,5

0,5 1 1,5 2 2,5 3 3,5 4 4,5 500 1000 1500 Distance (km) p1000 p925 p850 p700

) y , x , pn ( r ) y , x , ps ( r ) ; pn , ps ( F

l k j i

= ρ

ps – surface pressure, pn – pressure on standard isobaric levels (1000, 925, 850, 700 mb)

Three Three-

• dimensional forecast error covariance

dimensional forecast error covariance : : Т Т, , June June -

• August

August 2006 2006

200 400 600 800 1000 Ri>0 Ri<0 Ri>0.25 Ri<0.25 Ri>0.505 Ri<0.505 Lf (km) p850 p925 p1000

Lf (km) p1000 p925 p850 Ri>0 402 383 436 Ri<0 494 581 941 Ri>0.25 394 374 415 Ri<0.25 554 521 830 Ri>0.505 384 359 396 Ri<0.505 577 609 754

Lf – “integrated” scale (km):

∫

∞

ρ ρ =

f

d ) ( Cov ) ( Cov 1 L

Conclusions Conclusions

• The condition of homogeneity and isotropy for horizontal

functions is better carried out on distances up to 500 km and for correlations.

• The behaviour of a variance and radius of correlation

essentially depends on time of day.

• Variance and horizontal and vertical scales of forecast error

covariance in PBL depends on a degree of stability of proceeding processes.

Plans Plans

• Estimation of error covariance for U,V, Q.
• Research of influence of orography on forecast

error covariance in PBL.