Introduction to Numerical Weather Prediction Data Assimilation
89
3D univariate separable structure functions
As already said, assuming separable structure functions separable means that the correlation
between two points can be expressed as a product of a function of the horizontal distance only
and a function of the vertical distance only.
In 3D-Var, one may consider the vertical distance as a function of the hybrid vertical coordinate.
After projection on the eigenvectors of the vertical correlation matrix, we are back to the 2D
case. In optimal interpolation, the separability is assumed with pressure as vertical coordinate;
this is different from 3D-Var where the vertical coordinate is terrain-following close to the
surface. This leads to differences in the analysis increments near orography.
3D univariate, non separable
The idea is to have the vertical correlation matrix dependent on the total wavenumber n. More
precisely, having obtained (§.!' .)m as in step iv of the 2D univariate case for all levels k (it is
0. n
a vector of length k), one may consider a vertical covariance matrix A( n) as a function of the
total wavenumber n.
The implication of this formulation on the structure function in grid point space is discussed in
Phillips (1986) and in the 3D-Var framework in Courtier et al. (1993). This implementation of
non separability can be seen as a kind of 3D isotropy with sharper vertical structures associated
to small horizontal scales. The specification of the matrices A( n) is achieved using statistics
of the departure between a 24 h forecast and a 48 h forecast valid for the same time. The
approach proposed by Parrish and Derber ( 1992) is described and validated by Rabier and
McNally (1993).
Multivariate
(x - Xb) is separated into a Rossby and a Gravity contribution. The latter is penalized ensuring
10% of the flow ageostrophic. For more details, we refer to Courtier et al. (1993), in particular
for some difficulties introduced by the vertical discretization while going back from geopotential
to temperature and surface pressure. In the current implementation, it requires separability of
the Ub variation.
4.4 Dealing with the Time Dimension
We said in the introduction that the time dimension is a critical aspect for the specification
of the initial condition of a numerical weather prediction. In this section we introduce an
algorithm, the Kalman filter (Kalman, 1960; Ghil et al., 1981) which provides a comprehensive
and rigorous framework in the linear case. It may easily be extended to the quasi linear case
and is then called extended Kalman filter (Jaswinski, 1970).
4.4.1 The extended Kalman filter
Here we only present the algorithm; the theoretical results can be found in the above references
and, in particular, in Jazwinski (1970). Denoting by x(t) the state of the atmosphere at time
t, we are able to propagate forward in time the information using the forecasting model M,
x(t + T) = M(t + T, t)x(t).
M is non-linear but the tangent linear model R remains valid, to a large extent, for the propagation of the forecast errors (Lacarra and Talagrand, 1988; Vukicevic, 1991; Rabier and Courtier,
1992)
c5x(t + T) = R(t + T, t)c5x(t) + random noise (model errors).
89
3D univariate separable structure functions
As already said, assuming separable structure functions separable means that the correlation
between two points can be expressed as a product of a function of the horizontal distance only
and a function of the vertical distance only.
In 3D-Var, one may consider the vertical distance as a function of the hybrid vertical coordinate.
After projection on the eigenvectors of the vertical correlation matrix, we are back to the 2D
case. In optimal interpolation, the separability is assumed with pressure as vertical coordinate;
this is different from 3D-Var where the vertical coordinate is terrain-following close to the
surface. This leads to differences in the analysis increments near orography.
3D univariate, non separable
The idea is to have the vertical correlation matrix dependent on the total wavenumber n. More
precisely, having obtained (§.!' .)m as in step iv of the 2D univariate case for all levels k (it is
0. n
a vector of length k), one may consider a vertical covariance matrix A( n) as a function of the
total wavenumber n.
The implication of this formulation on the structure function in grid point space is discussed in
Phillips (1986) and in the 3D-Var framework in Courtier et al. (1993). This implementation of
non separability can be seen as a kind of 3D isotropy with sharper vertical structures associated
to small horizontal scales. The specification of the matrices A( n) is achieved using statistics
of the departure between a 24 h forecast and a 48 h forecast valid for the same time. The
approach proposed by Parrish and Derber ( 1992) is described and validated by Rabier and
McNally (1993).
Multivariate
(x - Xb) is separated into a Rossby and a Gravity contribution. The latter is penalized ensuring
10% of the flow ageostrophic. For more details, we refer to Courtier et al. (1993), in particular
for some difficulties introduced by the vertical discretization while going back from geopotential
to temperature and surface pressure. In the current implementation, it requires separability of
the Ub variation.
4.4 Dealing with the Time Dimension
We said in the introduction that the time dimension is a critical aspect for the specification
of the initial condition of a numerical weather prediction. In this section we introduce an
algorithm, the Kalman filter (Kalman, 1960; Ghil et al., 1981) which provides a comprehensive
and rigorous framework in the linear case. It may easily be extended to the quasi linear case
and is then called extended Kalman filter (Jaswinski, 1970).
4.4.1 The extended Kalman filter
Here we only present the algorithm; the theoretical results can be found in the above references
and, in particular, in Jazwinski (1970). Denoting by x(t) the state of the atmosphere at time
t, we are able to propagate forward in time the information using the forecasting model M,
x(t + T) = M(t + T, t)x(t).
M is non-linear but the tangent linear model R remains valid, to a large extent, for the propagation of the forecast errors (Lacarra and Talagrand, 1988; Vukicevic, 1991; Rabier and Courtier,
1992)
c5x(t + T) = R(t + T, t)c5x(t) + random noise (model errors).