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Data assimilation concepts
and methods
March 1999
By F. Bouttier and P.
Courtier
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2 . The state vector, control space and observations
2.1 State vector
The first step in the mathematical formalisation of the analysis problem is the definition of the work space. As in a forecast model, the collection of numbers needed to represent the atmospheric state of the model is collected as a column matrix called the state vector x. How the vector components relate to the real state depend on the choice of discretization, which is mathematically equivalent to a choice of basis.
As explained earlier, one must distinguish between reality itself (which is more complex than what can be represented as a state vector) and the best possible representation of reality as a state vector, which we shall denote
, the
true state at the time of the analysis. Another important value
of the state vector is 
, the
a priori or background estimate of the true state before the analysis
is carried out, valid at the same time1. Finally, the analysis is
denoted 
, which is what we are looking
for.2.2 Control variable
In practice it is often convenient not to solve the analysis problem for all components of the model state. Perhaps we do not know how to perform a consistent analysis of all components2, or we have to reduce the resolution or domain of analysis because of insufficient computer power. This is difficult to avoid as the resolution and sophistication of forecast models tend to be as high as the computing power allows, i.e. too high for the analysis which is more expensive because the observations have to be processed on top of the management of the model state itself. In these cases the work space of the analysis is not the model space, but the space allowed for the corrections to the background, called control variable space. Then the analysis problem is to find a correction
(or analysis increment) such that
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is as close as possible to
. Formally
the analysis problem can be presented exactly like before by a simple translation:
instead of looking for 
, we look for 
in a suitable subspace3.2.3 Observations
For a given analysis we use a number of observed values. They are gathered into an observation vector
. To use them in the analysis procedure it is necessary to be able
to compare them with the state vector. It would be nice if each degree of
freedom were observed directly, so 
could be regarded as a particular value of the state vector. In practice
there are fewer observations than variables in the model and they are irregularly
disposed, so that the only correct way to compare observations with the
state vector is through the use of a function from model state space to
observation space called an observation operator4 that we will denote by 
. This operator generates the values 
that the observations
would take if both they and the state vector were perfect, in the absence
of any modelling error5. In practice 
is a collection of interpolation operators from the
model discretization to the observation points, and conversions from model
variables to the observed parameters. For each scalar observation there
is a corresponding line of 
. The number of observations, i.e. the dimension of vector 
and the number of lines in 
, is varying
if the observing network is not exactly periodic in time. There are usually
many fewer observations than variables in the model.2.4 Departures
The key to data analysis is the use of the discrepancies between observations and state vector. According to the previous paragraph, this is given by the vector of departures at the observation points:
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When calculated with the background
it is
called innovations, and with the analysis 
, analysis residuals. Their study provides important
information about the quality of the assimilation procedure.Training Course Notes Front Page >>
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1 It is sometimes called the first guess, but the recommended word is background, for reasons explained later.
2 This is often the case with surface or cloud-related variables, or the boundary conditions in limited-area models.
3 Mathematically speaking, we constrain
to belong to the affine manifold spanned by 
plus the control variable vector subspace.4 also called forward operator
5 the values
are also called
model equivalents of the observations. |
03.12.2001 |
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© ECMWF |
