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Data assimilation concepts
and methods
March 1999
By F. Bouttier and P.
Courtier
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7 . Optimal interpolation (OI) analysis
The OI is an algebraic simplification of the computation of the weight
in the analysis equations (A1) and (A2).
|
(A1) |
|
(A2) |
The equation (A1) can be regarded as a list of scalar analysis equations, one per model variable in the vector
. For each model variable the analysis increment is given by the
corresponding line of 
times the vector of background departures
. The fundamental hypothesis in OI is: For each
model variable, only a few observations are important in determining the
analysis increment. It is implemented as follows:1) For each model variable  , select a small number  of observations
using empirical selection criteria. |
2) Form the corresponding list
of  background departures  , the
 background error covariances between the model variable
 and the model state interpolated at the  observation points (i.e. the relevant  coefficients of the  -th line
of  ), and the  background
and observation error covariance submatrices formed by the restrictions
of  and  to the selected
observations. |
3) Invert the  positive
definite matrix formed by the restriction of  to the selected observations (e.g. by an LU or Choleski
method), |
4) Multiply it by the  -th line of  to get the necessary line of  . |
It is possible to save some computer time on the matrix inversion by solving directly a symmetric positive linear system, since we know in advance the vector of departures to which the inverse matrix will be applied. Also, if the same set of observations is used to analyse several model variables, then the same matrix inverse (or factorization) can be reused.
In the OI algorithm it is necessary to have the background error covariances
as a model which can easily be applied
to pairs of model and observed variables, and to pairs of observed variables.
This can be difficult to implement if the observation operators are complex.
On the other hand, the 
matrix needs not be specified globally, it can be specified in an
ad hoc way for each model variable, as long as it remains locally positive
definite. The specification of 
usually relies
on the design of empirical autocorrelation functions (e.g. Gaussian or Bessel
functions and their derivatives), and on assumed amounts of balance constraints
like hydrostatic balance or geostrophy.The selection of observations should in principle provide all the observations which would have a significant weight in the optimal analysis, i.e. those which have significant background error covariances
with the variable considered. In practice,
background error covariances are assumed to be small for large separation,
so that only the observations in a limited geometrical domain around the
model variable need to be selected. For computational reasons it may be
desirable to ensure that only a limited number of observations are selected
each time, in order to keep the matrix inversions cheap. Two common strategies
for observation selection are pointwise selection (Fig. 9 ) and box selection (Fig. 10
)
Figure 9 . One OI data selection strategy is to assume that each analysis point is only sensitive to observations located in a small vicinity. Therefore, the observations used to perform the analysis at two neighbouring points
or 
may be different, so that the analysis field will
generally not be continuous in space. The cost of the analysis increases
with the size of the selection domains.
Figure 10 . A slightly more sophisticated and more expensive OI data selection is to use, for all the points in an analysis box (black rectangle), all observations located in a bigger selection box (dashed rectangle), so that most of the observations selected in two neighbouring analysis boxes are identical.
The advantage of OI is its simplicity of implementation and its relatively small cost if the right assumptions can be made on the observation selection.
A drawback of OI is that spurious noise is produced in the analysis fields because different sets of observations (and possibly different background error models) are used on different parts of the model state. Also, it is impossible to guarantee the coherence between small and large scales of the analysis (Lorenc 1981).
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03.12.2001 |
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