96
using direct methods. However, iterative techniques provide an alternative
possibility if the adjoint propagator has been coded. The power method,
whereby a random initial vector is operated on repeatedly by L*L, is
an example. A more sophisticated technique such as the Lanczos algorithm (Strang, 1986) is required if more than the largest singular vector
is required. More recently, calculations with the Jacobi-Davidson method
(Sleijpen and van der Vorst, 1995) has proved both efficient, and allowed
estimation of generalised eigenvectors (see section 2.7 below).
2.7 Singular vectors and eigenvectors of the forecast and
analysis error covariance operator.
We discussed above the notion that there is a natural inner product defined
such that the PDF of the initial state is isotropic with respect to this metric.
In this section we shall show this more explicitly, and give evidence that the
total energy is a reasonable approximation to this preferred inner product.
Consider an initial state of an operational weather forecast, determined
by the operational data assimilation system. We can think of this initial
state as a point X in the phase space of the numerical weather prediction model. Now, as mentioned, a complete operational data assimilation
system should be able to determine not only the initial state, but also
an estimate of the probability that the initial state is in error by a given
amount.
To make this idea more precise, let us consider the (linear) vector space
Tx tangent to X, and let df-L denote the probability that the error lies in a
small volume at the point eiETx(ie f-L is a measure on Tx). Here we shall use
some elementary tensor algebra, with the convention that repeated indices
imply summation. Let us assume that the operational analysis is our best
unbiased estimate of truth, so that at initial time
(2.18)
The covariance of analysis error associated with this measure is given by
the contravariant second-rank tensor
c ij = J eiejdf-L
(2.19)
The linear transformation (2.3) between TX(to) and TX(t) can be written (in
index form) as
using direct methods. However, iterative techniques provide an alternative
possibility if the adjoint propagator has been coded. The power method,
whereby a random initial vector is operated on repeatedly by L*L, is
an example. A more sophisticated technique such as the Lanczos algorithm (Strang, 1986) is required if more than the largest singular vector
is required. More recently, calculations with the Jacobi-Davidson method
(Sleijpen and van der Vorst, 1995) has proved both efficient, and allowed
estimation of generalised eigenvectors (see section 2.7 below).
2.7 Singular vectors and eigenvectors of the forecast and
analysis error covariance operator.
We discussed above the notion that there is a natural inner product defined
such that the PDF of the initial state is isotropic with respect to this metric.
In this section we shall show this more explicitly, and give evidence that the
total energy is a reasonable approximation to this preferred inner product.
Consider an initial state of an operational weather forecast, determined
by the operational data assimilation system. We can think of this initial
state as a point X in the phase space of the numerical weather prediction model. Now, as mentioned, a complete operational data assimilation
system should be able to determine not only the initial state, but also
an estimate of the probability that the initial state is in error by a given
amount.
To make this idea more precise, let us consider the (linear) vector space
Tx tangent to X, and let df-L denote the probability that the error lies in a
small volume at the point eiETx(ie f-L is a measure on Tx). Here we shall use
some elementary tensor algebra, with the convention that repeated indices
imply summation. Let us assume that the operational analysis is our best
unbiased estimate of truth, so that at initial time
(2.18)
The covariance of analysis error associated with this measure is given by
the contravariant second-rank tensor
c ij = J eiejdf-L
(2.19)
The linear transformation (2.3) between TX(to) and TX(t) can be written (in
index form) as