Advances in LAM 3D-VAR formulation

Vincent GUIDARD

Claude FISCHER

Météo-France, CNRM/GMAP

Introduction

• Through various experiments, a drawback of biperiodic increments has arisen : « wrapping around » analysis increments

Introduction

Introduction

• Through various experiments, a drawback of biperiodic increments has arisen: « wrapping around » analysis increments Controlling the lengthscale of the correlation functions is necessary: compact support

• Introduction of a large-scale information in the LAM analysis to let the increments due to the observations be of mesoscale

1.1 Compact support - definition

• Aim : Reducing the lengthscale of structure functions

• The COmpactly SUpported (COSU) correlation functions are obtained through a gridpoint multiplication by a cosine-shape mask function.

The mask should be applied to the square root of the gridpoint correlations (Gaspari and Cohn, 1999)

)²()²(),(),( ,, jyixmaskyxqyxq jijiCOSU

1.1 Compact support - definition

Steps to modify the power spectrum:

1. Power spectrum modal variances

2. Fill a 2D spectral array from the 1D square root of the modal variances

3. Inverse bi-Fourier transform – mask the gridpoint structure – direct bi-Fourier transform

4. Collect isotropically and square to obtain modified modal variances

5. Modal variances power spectrum

1.2 Compact support – 1D model

GridpointAuto-CorrelationsT 22

1.2 Compact support – 1D model

PowerSpectrumT 22

1.2 Compact support – 1D model

Analysis

1.3 Compact support – ALADIN

Univariate approach:Original B

Horizontal covariances COSU 100km-300km

1.3 Compact support – ALADIN

Multivariate approach:

The multivariate formulation (Berre, 2000):

*u is the umbalanced part of the * errorH is the horizontal balance operator

uuu

uu

u

PsT

PsTPsT

q),(q

),(),(

SRQH

PNH

MH

1.3 Compact support – ALADIN

• COSU Horizontal autocorrelations; Vertical cross-correlations and horizontal balance operator not modified

Whatever the distance of zeroing, results are « worse » than with the original B.

Explanation : the main part of the (temperature) increment is balanced, while only the horizontal correlations for are COSU, but not for H.

1.3 Compact support – ALADIN

– Cure 1: a modification of the power spectrum in order to have COSU correlations for H same results as the original B.– Cure 2: another solution is to compactly support the horizontal balance operator

1.3 Compact support – ALADIN

Original B All COSU 300km-500km

1.3 Compact support – conclusion

• Single observation:– Very efficient technique in univariate case– Needs drastic measures (COSU horizontal balance) to be efficient in multivariate case

• Full observation set:– No impact, even with drastic measures– Further research is necessary – Problems possibly due to a large scale error which this mesoscale analysis tries to reduce use of another source of information for large scales

2.1 « Large scale » cost-function

• Aim : input a large scale information in the LAM 3D-VAR.

• The large scale information is the analysis of the global model (ARPEGE) put to a LAM low resolution geometry

• Thanks to classical hypotheses, plus assuming that the global analysis error and the LAM background error are NOT correlated, we simply add a new term to the cost function

2.1 « Large scale » cost-function

• J(x) = Jb(x) + Jo(x) + Jk(x), where

H1 : global LAM low resolutionH2 : LAM high resolution LAM low res. V : « large scale » error covariancesxAA : global analysis

)()()()()( 2AA

11

2AA

1 xxxxxJT

k HHHH V

2.2 Large scale update - evaluation

•1D Shallow Water « global » model (I. Gospodinov) LAM version with Davies coupling (P. Termonia)Both spectral models

•1D gridpoint analyses implemented:– Using LAM background and observation (Jb+Jo) BO

– Using LAM background and global analysis (Jb+Jk) BK

– Using all information (Jb +Jo+Jk) BOKPlus dynamical adaptation DA

•Aim: comparing DA and BKcomparing BO and BOK

2.2 Large scale update - evaluation

• Dynamical Adaptation versus BK

LAM background

BK analysis

DA

global analysis

truth

Statistically(Fisher and Student tests on bias and RMS):No difference between DA and BK

2.2 Large scale update - evaluation

• BO versus BOK: observation over all the domain

LAM background

BOK analysis

BO analysis

global analysis

truth

Statistically:No difference between BO and BOK

+ observation

2.2 Large scale update - evaluation

• BO versus BOK: obs. over a part of the domain

LAM background

BOK analysis

BO analysis

global analysis

truth

Statistically:BOK better than BO

+ observation

2.3 Large scale update - conclusion

• The large scale information seems useful only in border-line cases, in the Shallow Water model

• Next steps :– Evaluation in a Burger model– Ensemble evaluation of the statistics in ARPEGE-ALADIN (based on the work of Loïk Berre, Margarida Belo-Pereira and Simona Stefanescu)– Implementation in ALADIN