Estimation of background error statistics in ARPEGE 4D-var

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Estimation of background error statistics Estimation of background error statistics in ARPEGE 4D-varin ARPEGE 4D-var

Margarida Belo Pereira(Instituto de Meteorologia, Lisboa)

Loïk Berre(Météo-France, Toulouse)

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Importance of background error Importance of background error estimativeestimative

- The analysis field results from a combination of observations and background (short range forecast)- The weights given to the observations and to the background depend on error statistics- The background errors statistics determines the way as the information from observations is spread spatially- How to estimate the background error statistics?

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

NMC method NMC method (operational in ARPEGE 4D-VAR)(operational in ARPEGE 4D-VAR)

1236tX

024tX

3636tX24

24tX0tX

ObsObsObsObs

024

2424 tta XXx 12

3636

36 ttf XXx

Analysis error

Forecast error

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Alternative to NMC method?Alternative to NMC method?

Ensemble Analysis MethodEnsemble Analysis Method

Perturbed observations (5)

Perturbed analysis (5)6h forecast

Data assimilation

Random numbers (5)+

observations

Perturbed Perturbed background (5)background (5)

Experiments Ensemble with five 4D-VAREnsemble with five 4D-VAR

cycles of the non-stretchedcycles of the non-stretched

version of ARPEGE modelversion of ARPEGE model

with T299 and 41 levelswith T299 and 41 levels

PeriodPeriod

1 of February to 24 of March1 of February to 24 of March

of 2002of 2002

Background Background differences->Bdifferences->B

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Level 21 (500hPa)

Standard deviation (normalized) of vorticity background error

Ensemble Method

Truncation T42

Level 32 (850hPa)

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Impact of geographical variation of standard deviation of background errors

Forecast against ECMWF analysis Forecast against observations

Geopotential (anomaly correlation)

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Impact of geographical variation of standard deviation of background errors

Forecast against ECMWF analysis Forecast against observations

Wind speed (anomaly correlation)

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Ensemble Method versus NMC MethodEnsemble Method versus NMC Method

Spectral Space

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Spectra of vorticity background errorSpectra of vorticity background error

Ensemble Method: the errors for the wind field have a bigger contribution from the mesoscale and subsynoptic scales than with the NMC method

Ensemble method

This result is valid also for the other variables, except for divergence

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

How to estimate the length scale of autocorrelation How to estimate the length scale of autocorrelation function?function?

Which assumptions are made?

Background error covariances are assumed to be

0~

2

22

~

x

x xd

dL

- stationary

- isotropic (horizontal length scale doesn’t depend on direction)

- spatially homogeneous

Definition of length scale (L) of the autocorrelation function

(Daley, 1991) for the one-dimensional case

L is a measure of the inverse curvature of the autocorrelation

function at the origin

For a sharp autocorrelation function, the curvature is large, so L is small. So, L gives an idea about

how the autocorrelation function decays with distance, from its initial value. In pratice, L is a measure

of the influence radius of one observation.

autocorrelation function

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Surface pressure

Autocorrelation of background errors

Length scale of autocorrelation

The autocorrelation tends to zero faster in

Ensemble method than in NMC method

The length scale of vorticity is smaller than

the one of temperature, this difference is smaller

in Ensemble method than in NMC method

Length scale of background errors

are smaller in Ensemble method

than in NMC method

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Ensemble Method NMC Method

North-South variation of vertical correlation of background error

NMC Method

Vorticity

Ensemble Method Temperature

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Vertical profile of standard deviation of background error

On operational ARPEGE 4D-VAR, the vertical profiles of total standard deviation of the background errors are rescaled by a factor of 0.9

To use the statistics from Ensemble Method it is need to rescale the vertical profile?

to account for mismatch between the magnitudes of the 12/36-hours forecast differences and the 6-hour forecast errors

what is the best factor?

1.5 (green curve)

Standard deviation

Mo

del

leve

ls

NMC x 0.9

Vorticity

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Impact of background covariances estimated by Ensemble method (against NMC method)

Geopotential Wind

Forecast against ECMWF analysis

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Autocorrelation function in gridpoint spaceAutocorrelation function in gridpoint space

E

N

Isotropic: Lx= Ly Lx < Ly Lx > Ly

oLx

Ly

OR

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

2

2

1

121 xxvv

)()( 2121

2

2

22

)(

)(

xx

Lx

xv

Covariance of v between 2 points =>

Length scale of autocorrelationLength scale of autocorrelation

Helmholtz’s theorem => Rotational component of meridional wind

Covariance of stream function =>

standard deviation of background error

autocorrelation

Zonal length scale of autocorrelation =>

Meridional length scale of autocorrelation => 2

2

22

)(

)(

yy

Ly

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Ensemble Method versus NMC MethodEnsemble Method versus NMC Method

Gridpoint Space

Length scale of autocorrelation

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Horizontal length scale of vorticity

Ensemble LAND

Ensemble SEA

NMC SEA

Lx

Lx

Ly

Ly

Ensemble EURATL

Ensemble GLOBAL

NMC EURATL

NMC SEA

Ly is larger than Lx,

this difference is larger in EURATL region

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Horizontal length scale of geopotential

Horizontal length scale of temperature

Lx

LxLy

Ly

Both Lx and Ly in Ensemble method

are smaller than in NMC method

Lx is smaller over

land than over sea,

mainly in Ensemble method

Ly is larger than Lx,

also for temperature

and geopotential

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

Ensemble Method

Horizontal length scale of wind

(Zonal and meridional length scale of zonal and meridional wind)

LAND SEA

EURATL

Lx (u) > Ly(u), except in PBL

Ly (v) > Lx(v), mainly in EURATL region

=>

v is more anisotropic in this region

u is more isotropic

in EURATL region

u is more anisotropic over

sea than over land, except near surface

Lx (v)

Ly (v)

Lx (u)

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

North-South variation of horizontal length scale of wind background error

Lx(u) and Lx (v) in Ensemble method

are smaller than in NMC method and both

are larger in the tropics

Lx(u) is larger than Lx (v) in both method

6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings

ConclusionsConclusions

• In Ensemble method the length scale of autocorrelation is shorter than in NMC method

• This difference has a positive impact on forecasts• The meridional length scale is larger than the

zonal length scale for all variables, except zonal wind

• The meridional length scale is more homogeneous than the zonal length scale

• The zonal length scale is larger over the tropics