Advances in LAM 3D-VAR formulation

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Advances in LAM 3D-VAR formulation

• Vincent GUIDARD
• Claude FISCHER
• Météo-France, CNRM/GMAP

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Introduction

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

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Introduction
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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

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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)

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1.1 Compact support - definition

• Steps to modify the power spectrum
• Power spectrum modal variances
• Fill a 2D spectral array from the 1D square root
  of the modal variances
• Inverse bi-Fourier transform mask the gridpoint
  structure direct bi-Fourier transform
• Collect isotropically and square to obtain
  modified modal variances
• Modal variances power spectrum

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1.2 Compact support 1D model
Gridpoint Auto- Correlations T 22
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1.2 Compact support 1D model
Power Spectrum T 22
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1.2 Compact support 1D model
Analysis
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1.3 Compact support ALADIN

• Univariate approach

Horizontal covariances COSU 100km-300km
Original B
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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

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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 z are COSU, but not for Hz.

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1.3 Compact support ALADIN

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

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1.3 Compact support ALADIN
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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

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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

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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

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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 (JbJo)? BO
• Using LAM background and global analysis (JbJk)
  ? BK
• Using all information (Jb JoJk) ? BOKPlus
  dynamical adaptation ? DA
• Aim comparing DA and BK comparing BO and BOK

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2.2 Large scale update - evaluation

• Dynamical Adaptation versus BK

Statistically (Fisher and Student tests on bias
and RMS) No difference between DA and BK
LAM background
BK analysis
DA
global analysis
truth
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2.2 Large scale update - evaluation

• BO versus BOK observation over all the domain

Statistically No difference between BO and BOK
LAM background
BOK analysis
BO analysis
global analysis
truth
observation

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2.2 Large scale update - evaluation

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

Statistically BOK better than BO
LAM background
BOK analysis
BO analysis
global analysis
truth
observation

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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