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Assimilation techniques: 3dVar
April 2001
By Mike Fisher
European Centre for Medium-Range Weather
Forecasts.
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2 . The incremental method
Ideally, the analysis cost function should be specified in terms of fields which have the same resolution as the forecast model. However, this makes the cost function computationally expensive to minimize (especially in 4dVar, where the much of the computational expense is in the tangent-linear and adjoint models.)
The incremental method reduces computational expense by minimizing a cost function which has a lower resolution than is used by the forecast model. This is reasonable because analysis increments are generally rather smooth, at least with current methods for specifying background error correlations.
The incremental method is an iterative procedure. For each iteration
, an approximate analysis 
is generated from an initial approximation 
by:
|
(1) |
denotes the pseudo-inverse of an operator
which reduces the resolution of the model fields to the resolution
used for the cost function. (For example, 
might correspond to spectral truncation.) The pseudo-inverse of 
is an operator which increases resolution
(for example by padding the spectrum with zeros). In this case, 
increases the resolution of the increment 
to match that of the analysis. The increment,
, is calculated by minimizing the cost function:
|
(2) |
acts on the high resolution fields 
whereas the operator 
acts on the low resolution increment. The incremental method is described above as an iterative procedure. However, there is generally no guarantee that the iterations will converge. For this reason, a typical implementation of the method performs only a few iterations.
2.1 Implementation of the Incremental method in the ECMWF 3dVar
In the case of the ECMWF 3dVar system, a single iteration of the incremental method is performed. The initial approximation to the analysis,
, is simply the background. The analysis consists of 3 steps:
2.1 (a) Comparison with the observations at high resolution
This step calculates
. The background fields are transformed to gridpoint space and then
interpolated to observation locations. 12-point bi-cubic interpolation is
used for upper-air fields. Bi-linear interpolation is used for surface fields
to avoid problems near surface discontinuities (e.g. coastlines) and steep
orography. Observation operators are applied to the interpolated model fields to calculate the model equivalents of the observations. The observations are compared with their model equivalents and the differences between the two (i.e. the departures) are written to the observation file, for use in the minimization. Obviously-bad observations are screened out at this stage by removing observations with very large departures.
2.1 (b) The minimization
This step calculates the low-resolution increment,
. The high-resolution upper-air background fields are truncated to
T63. The gridpoint surface fields are transformed to spectral space, truncated
to T63, and transformed back to gridpoint space. This ensures consistency
between the spectral and gridpoint fields used in the minimization. In the
case of the 31-level model, non-linear normal mode initialization (NNMI)
is applied to the fields to adjust them to the low resolution orography.
(NNMI has been found to give unsatisfactorary results in the 50-level and
60-level versions of the model, so no initialization of the low-resolution
background is performed for these vertical resolutions.) The low resolution cost function for the increments is minimized using a quasi-Newton method (Gilbert and Lemaréchal, 1989). Variational quality control of observation departures is applied during the minimization (Andersson and Järvinen, 1998).
2.1 (c) Updating at high resolution
This step calculates the high-resolution analysis,
. Since the low resolution analysis is balanced with respect
to the low resolution orography, it is desirable to initialize the analysis
increment to adjust it to the high-resolution orography. However, NNMI is
a non-linear procedure. It must be applied to whole fields rather than to
increments. For the 31-level model, the high resolution analysis is defined
as:
|
(3) |
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