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Chapter 1. Theory
Chapter 2. Computational
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1.2 Singular vectors
1.2.1 Formulation of the singular vector computation
One way to define singular vectors is by means of a maximization problem. The scalar which has to be maximized can be written as:
|
(1.1) |
where
denotes the Euclidean inner product, 
, and 
and 
are positive definite operators. The operator 
is the propagator of the tangent model. It assigns to a particular
vector 
the linearly evolved vector 
for a given forecast time and with respect to a reference trajectory.
Hence, the scalar defined by (1.1) is the ratio between the 
-norm of the evolved vector 
and the 
-norm of 
at initial time. Notice that the norm at initial and final time may
differ. The leading singular vector has the property that it maximizes the
scalar, the second singular vector maxizes the scalar in the space 
-orthogonal to the leading singular
vector, and so forth. In this way, one obtains a set of singular vectors
which are 
-orthogonal at initial time and 
-orthogonal at final time. The actual computation of the singular
vectors in the IFS is done by solving an equivalent eigenvalue problem.
Observe that the solutions of the maximization problem (1.1)
also satisfy the following generalized eigenvalue problem (1.2).
|
(1.2) |
where
is the adjoint of 
. The defining equation (1.2) can be generalized by activating
operators in the singular vector computation, see Section 2.1. It is, for instance, possible
to set the state vector to zero outside a pre-scribed area at optimization
time, by using a projection operator 
. Consequently, the growth of singular vectors outside
the target area is not taken into account in the actual computation. In
using this projection operator, the eigenvalue problem
(1.2) becomes 
. To keep the notation as simple as possible,
these additional operators will be left from the basic eigenvalue problem
(1.2).The operator
determines the properties by which the singular
vectors are constrained at initial time. As such, it can be interpreted
as an approximation of the inverse of the analysis error covariance matrix
. Currently, there are two methods to compute singular vectors, depending
on the form of 
. Both methods will be discussed in Sections 1.2.2 and 1.2.3.1.2.2 Use of the total energy norm
When using the total energy norm, or any other simple operator, at initial time, the generalized eigenvalue problem (1.2) can be simplified to an ordinary eigenvalue problem. In this case the
-norm of 
reads as
|
(1.3) |
where
and 
stands for the vorticity, divergence, temperature, specific humidity
and logarithm of the surface pressure component of the state vector 
, and 
is the specific heat of dry air at constant pressure, 
is the latent heat of condensation at 
, 
is the gas constant for dry air, 
is a reference temperature and 
= 800 hPa is a reference pressure. The parameter 
defines the relative weight given to the specific humidity term.
Since the operator
is a diagonal matrix, one can easily define
a matrix 
so that 
. Multiplying both sides of (1.2) to the left and right with 
, yields the equation
|
(1.4) |
which can be solved using the Lanczos algorithm, see Appendix Section A.1. The energy metric is believed to be a first-order approximation to the analysis error covariance metric (Palmer et al. 1998).
1.2.3 Use of the Hessian of the 3D-Var objective function
In the incremental formulation of 3D-Var, the Hessian of the objective function
can be used as an approximation of the inverse
of the analysis error covariance matrix. The objective function has the
form
|
(1.5) |
and the increment
where 
attains its minimum, provides the analysis 
which is defined by adding 
to the background 
|
(1.6) |
The operators
and 
are covariance matrices of the background and observation error respectively
and 
is the innovation vector
|
(1.7) |
where
is the observation vector and 
is a linear approximation of the observation operator in the vicinity
of 
.The Hessian
of the objective function is given by
|
(1.8) |
Provided that the background error
and the observation error 
are uncorrelated, with 
the true state of the atmosphere, the Hessian 
is equal to the inverse of the analysis error covariance matrix 
. This follows by noting that the objective
function is quadratic and attains its unique minimum at 
and consequently, by using (1.6),
|
(1.9) |
Rewriting (1.9) gives
|
(1.10) |
Using the assumption that the background and observation error are uncorrelated the above equation implies that
|
(1.11) |
By now multiplying each side of (1.11) to the right with its transpose, the desired result follows.
The defining eigenvalue problem for the singular vectors becomes
|
(1.12) |
Since the objective function is quadratic in the incremental formulation, the Hessian
in (1.12) can be evaluated by computing
the difference between two gradients: 
. The generalized eigenvalue problem (1.12) is solved by using the Jacobi-Davidson
algorithm, see Appendix Section A.2.Next Section
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10.04.2002 |
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