88
P. Courtier
dealing with the full matrix is intractable and one has to introduce simplifications. In both the
ECMWF (Courtier et al., 1993) and the
NMC (Parrish and Derber, 1992) implementations of Jb the basic ideas are similar and already
expressed in Phillips (1986). 3D isotropy and geostrophic coupling over the sphere using Hough
mode separation (ECMWF) or a balance equation (NMC) are practically feasible while providing a theoretical improvement on top of the 01 implementation. We now describe the key
points of the ECMWF implementation while referring to Courtier et al. (1993) for a detailed
presentation.
4.3.3 Structure function specification in 3D-Var
2D univariate homogeneous covariances
The basic property we are using is that, as homogeneous covariances over the sphere are by definition invariant with rotations, it becomes diagonal while expressed in the spherical harmonics
basis (Boer, 1983). More precisely, the covariance between two points P and Q of the sphere
is a function of only J.l = cos () where () is the angle between P and Q. 1 may be expressed as
a Legendre polynomial series
n
(The square root is here only for normalisation).
We then have the cost function expression Jb for a field x( >., J.l) = L~=o L::'=-n x:;' ynm(,X, J.l) and
a background field
N
n
L L xb;.ynm(,X,J.l)
n=Om=-n
N
n
Jb(x)
L 1;:1 L Ix:;' - xb;.12
n=O
2D univariate, homogeneous correlations
Knowing the field CTb('x,J.l) of the standard deviations of error, assuming the correlations homogeneous means that the covariances of X«~'''» are homogeneous; we are back to the previous
CTb A,P,
case 1 being a correlation which verifies 1(0) = 1 and then L~=o In(2n + 1) = 1.
The algorithm to compute Jb then becomes
1. from the spectral component x:;' and xbn of the control variable and the background
respectively, computes the difference 8x:;' = x:;' - Xbn
2. transforms 8x:;' to grid point space and obtains the field 8x( >., J.l)
3. divides 8x('x, J.l) by the standard deviations CTb('x, J.l)
4. transforms back to spectral space and obtains the spectral coefficients (§.E.)m 0". n
P. Courtier
dealing with the full matrix is intractable and one has to introduce simplifications. In both the
ECMWF (Courtier et al., 1993) and the
NMC (Parrish and Derber, 1992) implementations of Jb the basic ideas are similar and already
expressed in Phillips (1986). 3D isotropy and geostrophic coupling over the sphere using Hough
mode separation (ECMWF) or a balance equation (NMC) are practically feasible while providing a theoretical improvement on top of the 01 implementation. We now describe the key
points of the ECMWF implementation while referring to Courtier et al. (1993) for a detailed
presentation.
4.3.3 Structure function specification in 3D-Var
2D univariate homogeneous covariances
The basic property we are using is that, as homogeneous covariances over the sphere are by definition invariant with rotations, it becomes diagonal while expressed in the spherical harmonics
basis (Boer, 1983). More precisely, the covariance between two points P and Q of the sphere
is a function of only J.l = cos () where () is the angle between P and Q. 1 may be expressed as
a Legendre polynomial series
n
(The square root is here only for normalisation).
We then have the cost function expression Jb for a field x( >., J.l) = L~=o L::'=-n x:;' ynm(,X, J.l) and
a background field
N
n
L L xb;.ynm(,X,J.l)
n=Om=-n
N
n
Jb(x)
L 1;:1 L Ix:;' - xb;.12
n=O
2D univariate, homogeneous correlations
Knowing the field CTb('x,J.l) of the standard deviations of error, assuming the correlations homogeneous means that the covariances of X«~'''» are homogeneous; we are back to the previous
CTb A,P,
case 1 being a correlation which verifies 1(0) = 1 and then L~=o In(2n + 1) = 1.
The algorithm to compute Jb then becomes
1. from the spectral component x:;' and xbn of the control variable and the background
respectively, computes the difference 8x:;' = x:;' - Xbn
2. transforms 8x:;' to grid point space and obtains the field 8x( >., J.l)
3. divides 8x('x, J.l) by the standard deviations CTb('x, J.l)
4. transforms back to spectral space and obtains the spectral coefficients (§.E.)m 0". n