98
is the adjoint propagator. Hence with the specific choice of metric (2.24),
the eigenvectors of C are precisely the evolved singular vectors of L. The
vectors at initial time which evolve into these directions at the forecast
time are given by the corresponding initial singular vectors of L. This
choice of metric is sometimes known as the Mahalanobis metric (eg Mardia
et al, 1979).
How do we define a metric in practice? Three simple choices are based
on the enstrophy, energy and streamfunction squared. The spherical harmonic spectrum of typical 48-hour enstrophy and energy-norm singular
vectors are shown in Fig 3 at initial (dashed) and at final (solid) time. The
enstrophy spectrum (Fig 3a) of the enstrophy SVs are red at initial time,
blue at final time. Because of the large enstrophy amplification, values at
initial time are multiplied by 40 to plot them on the same scale as final
values. (However, Fig 3b, which has no rescaling, shows that there is little
energy amplification associated with these enstrophy SVs.) By contrast,
the energy spectrum (Fig 3c) of the energy SV s peaks at sub-synoptic
scales at initial time and at synoptic scales at optimisation time.
We now want to compare these spectra with spectra of analysis error.
As a surrogate for a set of analysis error fields, let us take the differences between analyses made during periods when the ECMWF operational system
and some (potentially operational) experimental system were being run in
parallel. We have chosen two different periods when such parallel tests were
being made. The first corresponded to the testing of an experimental model
formulation, the second to the testing of an experimental analysis methodology (3-dimensional variational data assimilation, 3DVAR). Twenty one
analysis difference fields were taken from each period. The total horizontal
wavenumber spectra of the energy, enstrophy, and streamfunction squared
difference fields associated with the northern extratropical component of
these difference fields are shown in Fig 4a-c respectively. It is clear that in
terms of energy, the difference field is indeed almost white. By contrast,
the enstrophy spectrum is blue and the streamfunction squared spectrum
is red. (Molteni et al, 1996, have demonstrated that these results can be
replicated using difference fields from operational analyses from different
operational centres.) Comparing 2.4 with 2.3 we can rule out enstrophy as
a suitable metric (the initial enstrophy singular vectors have no amplitude
on small scales, whilst the analysis errors have all their enstrophy on small
scales). A streamfunction metric would also be ruled out (although not
shown, an initial streamfunction singular vector is strongly peaked at high
is the adjoint propagator. Hence with the specific choice of metric (2.24),
the eigenvectors of C are precisely the evolved singular vectors of L. The
vectors at initial time which evolve into these directions at the forecast
time are given by the corresponding initial singular vectors of L. This
choice of metric is sometimes known as the Mahalanobis metric (eg Mardia
et al, 1979).
How do we define a metric in practice? Three simple choices are based
on the enstrophy, energy and streamfunction squared. The spherical harmonic spectrum of typical 48-hour enstrophy and energy-norm singular
vectors are shown in Fig 3 at initial (dashed) and at final (solid) time. The
enstrophy spectrum (Fig 3a) of the enstrophy SVs are red at initial time,
blue at final time. Because of the large enstrophy amplification, values at
initial time are multiplied by 40 to plot them on the same scale as final
values. (However, Fig 3b, which has no rescaling, shows that there is little
energy amplification associated with these enstrophy SVs.) By contrast,
the energy spectrum (Fig 3c) of the energy SV s peaks at sub-synoptic
scales at initial time and at synoptic scales at optimisation time.
We now want to compare these spectra with spectra of analysis error.
As a surrogate for a set of analysis error fields, let us take the differences between analyses made during periods when the ECMWF operational system
and some (potentially operational) experimental system were being run in
parallel. We have chosen two different periods when such parallel tests were
being made. The first corresponded to the testing of an experimental model
formulation, the second to the testing of an experimental analysis methodology (3-dimensional variational data assimilation, 3DVAR). Twenty one
analysis difference fields were taken from each period. The total horizontal
wavenumber spectra of the energy, enstrophy, and streamfunction squared
difference fields associated with the northern extratropical component of
these difference fields are shown in Fig 4a-c respectively. It is clear that in
terms of energy, the difference field is indeed almost white. By contrast,
the enstrophy spectrum is blue and the streamfunction squared spectrum
is red. (Molteni et al, 1996, have demonstrated that these results can be
replicated using difference fields from operational analyses from different
operational centres.) Comparing 2.4 with 2.3 we can rule out enstrophy as
a suitable metric (the initial enstrophy singular vectors have no amplitude
on small scales, whilst the analysis errors have all their enstrophy on small
scales). A streamfunction metric would also be ruled out (although not
shown, an initial streamfunction singular vector is strongly peaked at high