99
wavenumbers). Only the energy metric shows some consistency between
singular vector and analysis error structure.
A more accurate estimate of inner product is, in principle, available
from the 3DVAR data assimilation system. In 3DVAR a cost function
J based on both data and a first guess error, is minimised. The second
derivative, or Hessian, of the cost function gives a measure of the analysis
error covariance. In terms of the Hessian, the singular vector computation
becomes equivalent to a generalised eigenvector problem. In 3DVAR the
Hessian is known in operator form (Fisher and Courtier, 1995). Tests are
in progress, using the Jacobi-Davidson scheme, to estimate the corresponding singular vectors using this Hessian. Further generalisation, in which
the background error covariance in the Hessian is flow dependent, will be
possible with the development of Kalman filter schemes (Bouttier, 1993).
2.8 Correspondence with breeding vectors
Toth and Kalnay (1993, 1995) have discussed a technique which they use
to generate initial perturbations for the NMC model. The technique is
referred to as 'breeding', and is simply described. A random initial perturbation is generated with a specified amplitude, characteristic of a typical
uncertainty in the initial state. Two integrations are run over a specified
cycle time (eg 12 hours). The first is an integration of the operational
weather prediction model from the operational initial state. The second
integration is made using the same model, but is initialised by adding the
perturbation to the operational analysis. At the end of the integration period, the difference between the two integrations is renormalised using the
specified amplitude. The process is repeated for the next cycle time using
the renormalised perturbation to generate the perturbed initial state. The
process is repeated ad infinitum. Toth and Kalnay (1995) call the breeding vectors, local Lyapunov vectors. Clearly they are not entirely local,
depending on the history of evolution of the breeding vector. A more accurate description would be in terms of the local orientation of the global
Lyapunov vectors (L. Smith, personal communication).
It is claimed that the breeding method mimics the analysis cycle that is
used to generate operational initial conditions. As a result, it is argued that
analysis errors will tend to rotate into the direction of the breeding vector.
As discussed in section 3.3 below, the singular vectors form the basis for the
calculation of initial perturbations for the ECMWF ensemble prediction
wavenumbers). Only the energy metric shows some consistency between
singular vector and analysis error structure.
A more accurate estimate of inner product is, in principle, available
from the 3DVAR data assimilation system. In 3DVAR a cost function
J based on both data and a first guess error, is minimised. The second
derivative, or Hessian, of the cost function gives a measure of the analysis
error covariance. In terms of the Hessian, the singular vector computation
becomes equivalent to a generalised eigenvector problem. In 3DVAR the
Hessian is known in operator form (Fisher and Courtier, 1995). Tests are
in progress, using the Jacobi-Davidson scheme, to estimate the corresponding singular vectors using this Hessian. Further generalisation, in which
the background error covariance in the Hessian is flow dependent, will be
possible with the development of Kalman filter schemes (Bouttier, 1993).
2.8 Correspondence with breeding vectors
Toth and Kalnay (1993, 1995) have discussed a technique which they use
to generate initial perturbations for the NMC model. The technique is
referred to as 'breeding', and is simply described. A random initial perturbation is generated with a specified amplitude, characteristic of a typical
uncertainty in the initial state. Two integrations are run over a specified
cycle time (eg 12 hours). The first is an integration of the operational
weather prediction model from the operational initial state. The second
integration is made using the same model, but is initialised by adding the
perturbation to the operational analysis. At the end of the integration period, the difference between the two integrations is renormalised using the
specified amplitude. The process is repeated for the next cycle time using
the renormalised perturbation to generate the perturbed initial state. The
process is repeated ad infinitum. Toth and Kalnay (1995) call the breeding vectors, local Lyapunov vectors. Clearly they are not entirely local,
depending on the history of evolution of the breeding vector. A more accurate description would be in terms of the local orientation of the global
Lyapunov vectors (L. Smith, personal communication).
It is claimed that the breeding method mimics the analysis cycle that is
used to generate operational initial conditions. As a result, it is argued that
analysis errors will tend to rotate into the direction of the breeding vector.
As discussed in section 3.3 below, the singular vectors form the basis for the
calculation of initial perturbations for the ECMWF ensemble prediction