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

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)

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 variancespower spectrum

1.2 Compact support – 1D model Gridpoint Auto- Correlations T 22

1.2 Compact support – 1D model Power Spectrum T 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 * error H is the horizontal balance operator

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) = J b (x) + J o (x) + J k (x), where H 1 :global  LAM low resolution H 2 :LAM high resolution  LAM low res. V :« large scale » error covariances x AA :global analysis

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 (J b +J o )  BO – Using LAM background and global analysis (J b +J k )  BK – Using all information (J b +J o +J k )  BOK Plus dynamical adaptation  DA Aim:comparing DA and BK comparing 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