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.

GJ Marseille, H Schyberg, A Stoffelen. Horizontal and vertical background error correlations

published, 2012

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