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FLORENCE RABIER
Hx a =(HK)
y
T .
(10)
DFS=Tr( y Hx a )=Tr(HK) characterizes how the assimilation system uses
the observations to pull the signal from the background. In the optimal case
(i.e K = K true ), this is also the relative reduction of variance (Tr (KH) = Tr
((B-A)*B -1 ) = Tr(I-AB
-1 )). It is only an upper bound in non-optimal cases. It
indicates what the system does. One would need other information to give
insight about what should be done to get the best analysis.
How to estimate Tr (HK) ? This is not straightforward for largedimension systems where the matrices are often implicitly known and not
explicitly computed. Cardinali et al (2003, 2005) compute an estimate using
the singular vectors of the hessian of the cost function provided by the
Lanczos/Conjugate gradient minimizer. Another method was introduced in
Desroziers and Ivanov (2001) and is used in Chapnik et al (2005), based on
Girard (1987) for the evaluation of the trace of a matrix only known as an
operator. Let us present the basis of this method, which is relatively easy to
implement.
Let us consider a vector H following a normal (Gaussian) distribution
with mean 0 and covariance matrix the identity I. Girard (1987) proposed to
use the following mathematical identity in order to evaluate the trace of a
matrix A only known as an operator
E(H
T
AH)= Tr(A).
(11)
The evaluation of the trace of KH, or equivalently the trace of HK can be
perfomed based on this equation. The basic idea is to produce two analyses,
one being deduced from the other by perturbing the observations. The
difference between these two analyses will then be equal to the operator K
applied to the perturbation, and applying H to this difference of analyses will
give access to the HK operator needed. This method, as presented in
Chapnik et al (2005) involves several steps
1. Perform a normal analysis from the information (x b , y) producing the
analysis vector x a
2. Perform a perturbed analysis from the information (x b , y*) with
perturbed observations y*=y+R 1/2 H leading to x a *. One notes that x a *- x a =
K(y* - y).
3. Then (y*- y) T R -1 H(x a *-x a ) provides an approximation to Tr(HK).
If the number of observations is large, one sequence of such steps might
be enough to get a reasonable estimation of the trace. Otherwise, one can use
several realizations of the analysis, and concatenate the results.
FLORENCE RABIER
Hx a =(HK)
y
T .
(10)
DFS=Tr( y Hx a )=Tr(HK) characterizes how the assimilation system uses
the observations to pull the signal from the background. In the optimal case
(i.e K = K true ), this is also the relative reduction of variance (Tr (KH) = Tr
((B-A)*B -1 ) = Tr(I-AB
-1 )). It is only an upper bound in non-optimal cases. It
indicates what the system does. One would need other information to give
insight about what should be done to get the best analysis.
How to estimate Tr (HK) ? This is not straightforward for largedimension systems where the matrices are often implicitly known and not
explicitly computed. Cardinali et al (2003, 2005) compute an estimate using
the singular vectors of the hessian of the cost function provided by the
Lanczos/Conjugate gradient minimizer. Another method was introduced in
Desroziers and Ivanov (2001) and is used in Chapnik et al (2005), based on
Girard (1987) for the evaluation of the trace of a matrix only known as an
operator. Let us present the basis of this method, which is relatively easy to
implement.
Let us consider a vector H following a normal (Gaussian) distribution
with mean 0 and covariance matrix the identity I. Girard (1987) proposed to
use the following mathematical identity in order to evaluate the trace of a
matrix A only known as an operator
E(H
T
AH)= Tr(A).
(11)
The evaluation of the trace of KH, or equivalently the trace of HK can be
perfomed based on this equation. The basic idea is to produce two analyses,
one being deduced from the other by perturbing the observations. The
difference between these two analyses will then be equal to the operator K
applied to the perturbation, and applying H to this difference of analyses will
give access to the HK operator needed. This method, as presented in
Chapnik et al (2005) involves several steps
1. Perform a normal analysis from the information (x b , y) producing the
analysis vector x a
2. Perform a perturbed analysis from the information (x b , y*) with
perturbed observations y*=y+R 1/2 H leading to x a *. One notes that x a *- x a =
K(y* - y).
3. Then (y*- y) T R -1 H(x a *-x a ) provides an approximation to Tr(HK).
If the number of observations is large, one sequence of such steps might
be enough to get a reasonable estimation of the trace. Otherwise, one can use
several realizations of the analysis, and concatenate the results.