A Multivariate Reduced-order Optimal Interpolation Method and its Application
283
where only the background error variances in the diagonal matrix ni' are predicted,
using an external scheme, from the previous analysis error variances. The correlations in matrix C are assumed to be constant in time. Since OI does not evolve
dynamically the errors, the modelling and parameterization of W contains most of
the physics of the estimation problem and should be treated very carefully.
Equations (1-4) hold for OI as well as for the EKE The OI gain has the same
structure as the EKF gain:
(6)
where R is the observational error covariance matrix, and an approximation is used
instead of the predicted EKF forecast error. As in the EKF gain, the full observation operator H() is replaced in (6) by the tangent linear observation operator H.
Finally, let us turn to the analysis errors. In most OI schemes, the forecast error
variances are derived from the analysis error variances at the previous filter step
(e.g. Ide et al., 1997) using an ad hoc scheme. Considering any linear (optimal or
suboptimal) filter, the analysis error covariance matrix can be written:
(7)
This is valid for all forms of the gain K and therefore for the form (6), with the
best available estimates of the forecast (background) and observational error covariance matrices. This assumes that both types of errors are uncorrelated to each
other, which is a reasonable good assumption for error of a purely observational
nature, but might fail if e.g. observations are mapped prior to assimilation.
15.2.2 3-D EOFs
Numerical weather forecasting, for the purposes of which OI schemes were
brought in and continuously improved over the years, has had to face the question
of how to model realistic, three-dimensional, multivariate error correlations, and
how to predict error variances, using expressions such as (5). Historically, many
"recipes" chiefly aimed at splitting the problem into simpler ones have been used
by the various meteorological agencies and research groups (see e.g. Gustavsson,
1981; Lorenc, 1986). In the atmosphere, many attempts have revolved around the
idea of using (5) for a basic set of increments (mass field, wind field), and implementing adjustments or balance relationships such as geostrophy for the other
increments.
In the ocean, and in particular in the Mediterranean, such a "splitting" approach
is more complicated to set up, because of the lack of experience on what works
best, spatial inhomogeneities such as coasts, islands and straits, and relatively poor,
irregular data coverage. Here we will review methods to introduce empirical, spatially coherent multivariate structure functions in the modelling of errors.
Let us write the spectral factorization of the background error covariance matrix
W:
283
where only the background error variances in the diagonal matrix ni' are predicted,
using an external scheme, from the previous analysis error variances. The correlations in matrix C are assumed to be constant in time. Since OI does not evolve
dynamically the errors, the modelling and parameterization of W contains most of
the physics of the estimation problem and should be treated very carefully.
Equations (1-4) hold for OI as well as for the EKE The OI gain has the same
structure as the EKF gain:
(6)
where R is the observational error covariance matrix, and an approximation is used
instead of the predicted EKF forecast error. As in the EKF gain, the full observation operator H() is replaced in (6) by the tangent linear observation operator H.
Finally, let us turn to the analysis errors. In most OI schemes, the forecast error
variances are derived from the analysis error variances at the previous filter step
(e.g. Ide et al., 1997) using an ad hoc scheme. Considering any linear (optimal or
suboptimal) filter, the analysis error covariance matrix can be written:
(7)
This is valid for all forms of the gain K and therefore for the form (6), with the
best available estimates of the forecast (background) and observational error covariance matrices. This assumes that both types of errors are uncorrelated to each
other, which is a reasonable good assumption for error of a purely observational
nature, but might fail if e.g. observations are mapped prior to assimilation.
15.2.2 3-D EOFs
Numerical weather forecasting, for the purposes of which OI schemes were
brought in and continuously improved over the years, has had to face the question
of how to model realistic, three-dimensional, multivariate error correlations, and
how to predict error variances, using expressions such as (5). Historically, many
"recipes" chiefly aimed at splitting the problem into simpler ones have been used
by the various meteorological agencies and research groups (see e.g. Gustavsson,
1981; Lorenc, 1986). In the atmosphere, many attempts have revolved around the
idea of using (5) for a basic set of increments (mass field, wind field), and implementing adjustments or balance relationships such as geostrophy for the other
increments.
In the ocean, and in particular in the Mediterranean, such a "splitting" approach
is more complicated to set up, because of the lack of experience on what works
best, spatial inhomogeneities such as coasts, islands and straits, and relatively poor,
irregular data coverage. Here we will review methods to introduce empirical, spatially coherent multivariate structure functions in the modelling of errors.
Let us write the spectral factorization of the background error covariance matrix
W: