2Observation Representativeness Error

NWP models simulate the atmospheric state on a given model grid, thereby principally
limiting the representation of atmospheric phenomena to scales larger than the grid point
distance. However, in addition such models generally filter away small-scale phenomena (of
several grid lengths) in order to avoid numerical instability of the discrete numerical model
equations, which also prevents upscale error growth originating from the relatively uncertain
small scales (section 3 of [R5] and [R22]). Moreover, data assimilation systems act as socalled
low pass filters on the information provided by the observations, thereby essentially
rejecting observed information on scales smaller than the typical error structure of the NWP
model (background error covariance). So, observations do generally not affect the spectrum
of NWP model scales, but rather replace error variance with observed (true) variance. As a
consequence the simulated atmospheric state by NWP models is a smooth representation of
the true atmospheric state, lacking atmospheric variance in particular on the smallest
(turbulent) scales. Observing systems sample the true atmospheric state and thus do
measure small-scale atmospheric phenomena. This variance in the measured atmospheric
scales which is not part of the NWP model atmospheric state is called the spatial observation
spatial representativeness error [R3]. It is not an error in the sense of a deficiency of the
observing system such as e.g. instrument noise, but it merely describes that part of the
observation that cannot be well represented by the model and should therefore be treated as
an observation error when compared to the NWP model state, e.g., in data assimilation.
The smallest spatial scale of atmospheric phenomena that a NWP model can reasonably
represent is called the effective model resolution here. This should not be confused with the
sampling or grid distance which the NWP community usually refers to as “resolution”. The
discrete numerical equations actually cannot generally resolve atmospheric variance on the
sampling scale, but use higher-order closures that allow realistic atmospheric variance only
on scales 5-10 times the grid size [R5]. The effective model resolution differs for different
NWP models and is in fact not well defined in the literature. [R5] defines effective model
resolution as the wavelength (i.e., the inverse of the wave number) where the NWP model
spectra first deviates from the atmospheric spectrum, the latter obtained from high-resolution
observations. [R15] on the other hand uses a statistical description of atmospheric
turbulence. Structure functions and wind energy spectra are derived from these and a 2-
dimensional box-car averaging technique is applied to the turbulence data. The length of the
box-car for which the averaged structure function (or spectrum) compares best with the NWP
model structure function (or spectrum) defines the effective resolution. From the structure
functions, typical values found for the effective model resolution are 117 km for a global
model with 35 km grid size and 80 km for a mesoscale model with 13 km model grid size.
When using wind energy spectra 8 times larger values of about 650 km for the mesoscale
model are found.
Results from the recently issued survey [A3] among NWP centers show a large spread and
uncertainty of reported numbers for the effective model resolution, indicating that NWP
centers worldwide do not have a clear picture of this quantity. It is very relevant in data
assimilation though, for the quantification of the observation representativeness error
variance that substantially contributes to the total observation error variance. It is clear that a
correct specification of the observation representativeness error is needed to subscribe a
correct weight of the observation in the analysis.
From the above it is clear that the observation representation error depends on how the
observation is spatially accumulated. The change from Aeolus burst-mode (BM) to
continuous-pulsed mode (CM) laser operation offers an increased flexibility for the
accumulation of measured data. For BM operation the horizontal accumulation was limited to
50 km along track. Continuously measured satellite information (both horizontally and
vertically) may be processed and accumulated to optimally represent the effective resolution
of the NWP model as this would increase the accumulated signal and reduce the total
observation error, both the measurement error and the spatial representation error parts. It
is, thus, of interest to study the effects of accumulation, both horizontally and vertically, for
the Aeolus CM.
This report describes the use of high-resolution observations of various parts of the
atmosphere to quantify i) the effective model resolution of a global (ECMWF) and regional
(HiRLAM) model and ii) the expected observation representativeness error for Aeolus-CM
measurements as a function of horizontal accumulation length and vertical accumulation
depth.