Horizontal and vertical background error correlations

Numerical Weather Prediction (NWP) model background error correlations play a decisive
role in data assimilation. Their spatial structures directly influence the spatial scales on which
observations can impact the NWP analysis state. Closely spaced observations will impact
the same model state variable making their information content redundant to some extent.
Conventional observing systems usually undersample the background error structures, while
satellite observations generally oversample horizontally (but not vertically). The spatial
background error scales represent those scales where the NWP model is in error. Errors on
the large scales are relatively small, whereas errors on the small scales are relatively large,
this is, the variance of large-scale waves is generally well known, whereas the variance in
small-scale (mesoscale) systems is generally unknown. Since atmospheric spectra are
decaying with wave number, the absolute amount of variance in the smaller scales is rather
limited as compared to the larger scales. Therefore, both the atmospheric spectrum and the
NWP model error spectrum are decaying with wave number on the mesoscales. At the
smallest scales the NWP model spectrum and the specified NWP background error spectrum
both decay faster than the true atmospheric spectrum, i.e., the smallest scales are not
represented in a NWP model state, but are rather parameterized.
NWP models generally filter away small-scale phenomena (of a few grid lengths) in order to
avoid numerical instability of the discrete numerical model equations, but it also prevents
upscale error growth originating from the relatively uncertain small scales, mentioned earlier.
The smallest spatial scale of atmospheric phenomena that a NWP model can reasonably
represent, called effective resolution here, is thus not obviously determined and subject of
study in this report1. It is very relevant in data assimilation though, since it determines the socalled
spatial representativeness observation error. This part of the “observation error” is
constituted of the atmospheric variance measured by the observation, but not part of the
NWP model atmospheric state (due to lack of effective resolution). The observation
representativeness error thus depends on how the observation is spatially accumulated.
As such, the specified spatial background error structures are at interplay with the spatial
sampling and spatial representation of the observing systems in different ways:
1) Satellite information at relatively high spatial sampling would oversample the
background spatial error structures and therefore multiple observations would affect
the same atmospheric state variable in the NWP model in the analysis step. As a
consequence, the observations would be weighted (averaged) in order to update the
particular state variable and the difference in the observations would essentially be
lost. This is, a data assimilation system acts basically as a low pass filter and
essentially largely rejects observed information on scales smaller than the
background error correlations ([R8, page 123 below Eq. (4.4.22) and Eq. (3.3.7) on
page 73).
2) Continuously measured satellite information, both horizontally and vertically, may thus
be processed and accumulated to optimally represent the effective resolution of the NWP model as this would increase the accumulated signal and reduce the
observation error, both the measurement error and the spatial representativeness
error parts. When the observation quality would depend non-linearly on the signal-tonoise
ratio, accumulation would obviously be preferred. It is thus of interest to study
the effects of accumulation, both horizontally and vertically, for the Aeolus CM.
3) Observations made at different, but nearby locations and that represent the same
NWP model state variable(s) may have correlated observation representativeness
error. This is, they both resolve the same particular atmospheric phenomena within a
NWP model resolution cell that are part of their respective observation
representativeness error which are thus probably correlated.
In order to investigate the optimized Aeolus observation size and spacing w.r.t. the NWP
model background, spatial NWP model background errors and error correlations will be
investigated. This will be done in two ways: (i) by extracting model background error
structures as currently used in state of art global and regional models and (ii) by extracting
background error (and observation error) structures from observation minus background (ob)
and observation minus analysis (o-a) statistics of high resolution datasets of radiosonde,
scatterometer and aircraft data. The motivation for the latter is that recent observation-model
intercomparison studies have revealed that nowadays models tend to overestimate both the
background error variance, the observation error variance and observation error correlation
length scales [R14,R15,R16] for various observing systems. The second method is therefore
meant to further validate the currently used model background error structures that have
been obtained from ensemble and NMC methods, as described in [R2].
The models used in both analysis include the operational ECMWF global model and mesoscale
HirLAM/HARMONIE model. Results shall be compared with the outcome of [R9] and
the differences (if any) shall be discussed.