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Targetting and Assimilation a dynamically
consistent approach Anna Trevisan, Alberto
Carrassi and Francesco Uboldi ISAC-CNR Bologna,
Italy, Dept. of Physics University of Ferrara,
LEGOS/SHOM, Toulouse, France
Summary
Results

• A new assimilation method and results of its
  application in an adaptive observations
  experiment are presented.
• Targetting and assimiliation are considered as
  two faces of the same problem and are addressed
  with a dynamically consistent approach.
• Assimilation increment is confined within the
  unstable subspace where the most important
  components of the error are expected to grow.
• The analysis cycle is considered as an
  observationally forced dynamical system whose
  stability is studied.
• Unstable vectors of the Data Assimilation System
  are estimated by a modified Breeding technique.
• Mathematical Formulation
• The analysis increment is confined to the
  N-dimensional unstable subspace spanned by a set
  of N vectors en
• (1)
• where xa is the analysis, xf is the forecast
  (background) state, E is the I x N matrix (I
  being the total number of degrees of freedom)
  whose columns are the en vectors and the vector
  of coefficients a, the analysis increment in the
  subspace, is the control variable . This is
  equivalent to estimate the forecast error
  covariance as
• (2)
• where ? represents the forecast error covariance
  matrix in the N-dimensional subspace spanned by
  the columns of E. The solution is
• (3)
• here H is the (Jacobian of the) observation
  operator, R is the observation error covariance
  matrix, and yo is M-dimensional observation
  vector. It is intended here that N ? M, so that
  for each vector en there is at least one
  observation. If, in particular, we consider the
  case of a single vector and a single observation,
  NM1, E consists of the single column vector e
  and G of the scalar g2 . The solution is then
• (4)
• Where are scalars, too. If the observation is
  perfect
• (5)

• Comparison between two different experiments
• Experiment type I all observations, fixed and
  adaptive, are assimilated by means of 3DVAR
• Experiment type II the fixed observations are
  assimilated by means of 3DVAR the adaptive
  observations by means of the proposed method

The adaptive observation in both experiments is
placed at the location where the current bred
vector attains its maximum amplitude
Noisy Observations
Perfect Observations
FIG (1) Normalized RMS analysis error as a
function of time. The error is expressed in
potential enstrophy norm and it is normalized by
natural variability. Dotted line Experiment I-P
Continous Line Experiment II-P
FIG (4) Same as fig (1) Dotted line
Experiment I-N Continous Line Experiment II-N
FIG (5) Same as fig (3) Blue line Experiment
I-N Red Line Experiment II-N
FIG (2) Normalized time and vertical average RMS
analysis error.From Experiment II-P
Stability Analysis
FIG (3) Normalized time vertical and latidudinal
average RMS analysis error. Blu line Experiment
I-P Red line Experiment II-P. Red line results
from the same fields as fig (2), but averaged
also in latitude
FIG (6) Leading Lyapunov exponent of the
assimilation system as a function of time
(perfect observations). Values are averaged over
all previous instants. The growth rate is
expressed in units of days-1. Dotted line
Experiment type I-P (3DVar) Continuous Line
Experiment type II-P.
Conclusion The main conclusion of this work is
that the benefit of adaptive observations is
greatly enhanced if their assimilation is based
on the same dynamical principles as targetting.
Results show that a few carefully selected and
properly assimilated observations are sufficient
to control the instabilities of the Data
Assimilation System and obtain a drastic
reduction of analysis error. The success obtained
in the QG-Model makes us very hopeful that, even
in an operational environment with real adaptive
observations, this method can yield significant
improvement of the assimilation performance.

• References
• detailed results and theoretical discussion
  Carrassi A., A. Trevisan and F. Uboldi, 2004.
  Deterministic Data Assimilation and Targeting by
  Breeding on the Data Assimilation System.
  Submitted to J. Atmos. Sci..
• on the 3DVar scheme Morss R., 1999. Adaptive
  Observations Idealized sampling strategy for
  improving numerical weather prediction. PhD
  thesis, Massachussets of Technology, Cambridge,
  MA, 225 pp.
• on the QG model Rotunno R., and J. W. Bao,
  1996. A case study of cyclogenesis using a model
  hierarchy. Mon. Wea. Rev., 124, 1051-1066
• a previous study with a small nonlinear model
  Trevisan A., and F. Uboldi, 2004. Assimilation of
  standard and targeted observation within the
  unstable subspace of the observation-analysis-fore
  cast cycle system. J. Atmos. Sci., 61, 103-113
• update of applications in progress Uboldi F.,
  A. Trevisan and A. Carrassi, 2004. Developing a
  Dynamically Based Assimilation Method for
  Targeted and Standard Observations. In review for
  Nonlinear Process in Geophysics.

A.Trevisan_at_isac.cnr.it
CONTACTS
carrassi_at_fe.infn.it
uboldi_at_thor.cst.cne
s.fr
Acknowledgments The authors thank Rebecca Morss
and Matteo Corazza for providing the code of
QG-Model and 3D-Var and for getting us started
with its use. We also thank Eugenia Kalnay and
Zoltan Toth for their useful comments.