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Data assimilation concepts
and methods
March 1999
By F. Bouttier and P.
Courtier
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11 . Estimating the quality of the analysis
It is usually an important property of an analysis algorithm that it should be able to provide an estimate of the quality of its output. If there is no observation the quality is obviously that of the background. In a sequential analysis system the knowledge of the analysis quality is useful because it helps in the specification of the background error covariances for the next analysis, a problem called cycling the analysis. If the background is a forecast, then its errors are a combination of analysis and model errors, evolved in time according to the model dynamics. This is explicitly represented in the Kalman filter algorithm.
If the analysis gain
has been calculated,
e.g. in an OI analysis, then the analysis error covariance matrix is provided
by Eq.
(A3)
|
which reduces to
(A4) in the unlikely case where 
has been computed exactly.In a variational analysis procedure, the error covariances of the analysis can be inferred from the matrix of second derivatives, or Hessian, of the cost function thanks to the following result:
11.1 Theorem: use of Hessian information
The Hessian of the cost function of the variational analysis is equal to twice the inverse of the analysis error covariance matrix:
|
| Proof: |
The Hessian is obtained by differentiating
 twice with respect to the control variable
 : |
|
Now we express the fact that
 and we insert the true model state  into the
equation: |
|
| Hence |
|
| When it is multiplied on the right by its transpose, and the expectation of the result is taken, the right-hand side then contains two terms that multiply |
|
| which is zero because we assume background and observation errors are uncorrelated. The remaining terms lead to, successively: |
|
| which proves the result. |
11.2 Remarks
Figure 14 . Illustration in a one-dimensional problem of the relationship between the Hessian and the quality of the analysis. In one dimension, the Hessian is the second derivative, or convexity, of the cost-function of the variational analysis: two examples of cost-functions are depicted in the upper panel, one with a strong convexity (on the left), the other with a weaker one (on the right). If the cost-function is consistent with the pdfs involved in the analysis problem, the Hessian is a measure of the sharpness of the pdf of the analysis (depicted in the lower panel). A sharper pdf (on the left) means that the analysis is more reliable, and that the probability of the estimated state to be the true one is higher.
A simple, geometrical illustration of the relationship between the Hessian and the quality of the analysis is provided in Fig. 14 . In a multidimensional problem, the same interpretation is valid along cross-sections of the cost-function.
If the linearization of the observation operator
can be performed exactly, the cost function 
is exactly quadratic and 
does not depend
on the value of the analysis: 
can be determined
as soon as 
is defined, even before the analysis
is actually carried out1. If the linearization is not exact,
is not constant. It may depend a lot on 
, even if 
itself does not look very different from
a quadratic function. For instance, if 
is continuously
differentiable but not strictly convex, there are points at which 
. If 
is not
continuous, then there are points at which 
is not defined
at all. It means that 
must be
exactly linear in order to be able to calculate 
using the Hessian. In practice 
must be
modified to use the tangent linear of 
, which
can be acceptable in a close vicinity of 
.The identity
shows clearly how the observed data
increases the inverse error covariances, also called information matrices.Ref: Rabier and Courtier 1992.
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1 Actually, neither
nor 
depend on the values of the background or of the observations. |
03.12.2001 |
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