| Abstract |
In variational data assimilation, an optimal anal-
ysis is derived from the knowledge of background
and observation error statistics. According to Da-
ley (1993), the observation error can be divided
into two components: the instrumental error of a
measuring device and the representativeness error
of an observation operator. The first one is often
considered to be a white Gaussian noise, whereas
the second is thought to be responsible for spatial
correlations in the observational error. The repre-
sentativeness error depends on a resolution func-
tion of the measuring instrument, observation den-
sity, model grid resolution, and specification of an
observation operator. This dependency was inves-
tigated by Liu and Rabier (2002) in a simple one-
dimensional (1D) framework. They have found
an approximate relation of the aforementioned pa-
rameters corresponding to an optimal analysis.
However in operational practice, the optimal
analysis is usually not achievable, since the
observation-error correlations are difficult to es-
timate and expensive to specify in the assimila-
tion procedure. Therefore, a suboptimal assimi-
lation scheme is often used, in which the obser-
vation errors are assumed to be uncorrelated. In
this scheme, the observations with strongly corre-
lated errors must be filtered out prior to assimila-
tion, in order to achieve a good analysis quality.
This error-decorelation operation is called obser-
vation thinning. Although it is commonly used in
operational practice by most of the weather pre-
diction centers nowadays, the question of optimal
thinning that provides the best balance between
the observation-error correlation and the forecast
error is still not well understood. Liu and Rabier
have investigated the thinning of observations with
respect to the analysis error and found that small
observation-error correlation-coefficients of ≤ 0.15
can be anticipated in the suboptimal assimilation
scheme. They considered a simple non-adaptive
thinning strategy, in which the observation posi-
tions were constrained to a regular grid and the
sets of equidistant observations only were used in
the assimilation. However, recent studies of Joly
et al. (1997), Langland et al. (1999) and Daescu
and Navon (2004) demonstrate that the forecast
quality may benefit from adaptive observations,
the spatial distribution of which is not regular, but
exhibits a higher observation density in the regions
where the forecast errors are expected to grow
most rapidly. The identification of these regions
of interest is a complicated and expensive task in
general, since it deals with an investigation of the
forecast model dynamics at the time of assimila-
tion (see, e.g., Berliner et al. (1999)). An alterna-
tive much simpler and cheaper approach is based
on the assumption, that such critical for the fore-
cast error areas may be approximately identified
from the information distributed among the obser-
vations available for the assimilation. For exam-
ple, these regions may be the areas of atmospheric
fronts, storms, etc., characterized by a large vari-
ation of some atmospheric variables. In this ap-
proach that got the name ”adaptive observation
thinning”, the spatial distribution of the thinned,
i.e., filtered, observations depends on (adapts to)
their values.
In the context of adaptive thinning, the problem
of optimal thinning becomes more difficult. Sev-
eral adaptive observation thinning methods, based
on different heuristics, were proposed recently by
Ochotta et al. (2005) and Ramachandran et al.
(2005), giving rise to the question about a thinning
strategy, that provides a spatial distribution of ob-
servations to be assimilated that is optimal in the
sense of the analysis or forecast error. To answer
this question, a one-dimensional setting is not suf-
ficient, since most of the thinning schemes operate
on data sets of higher dimensionality. This mo-
tivated us to generalize the one-dimensional anal-
ysis scheme proposed by Liu and Rabier (2002)
to a two-dimensional framework for assimilation
of scalar-valued observation sets distributed on a
2-sphere, in order to be able to perform a compar-
ative analysis of different thinning methods with
respect to the analysis error.
Although being similar in their approaches,
the one- and two-dimensional frameworks differ
in the following important aspect. Due to the
regular spatial observation distribution, the 1D-
investigation represents a purely statistical study,
in which no realizations of observations or true
signal values are necessary. In the present work
in contrary, the assimilation of more realistic ran-
domly distributed observations is of interest. For
this reason, a purely statistical consideration of the
analysis error is not possible and its quality is to
be estimated experimentally over an ensemble of
analysis errors resulting from assimilation of sim-
ulated observations. This paper describes the sim-
ulation framework and presents the experimental
results on the analysis error as a function of model
resolution, observation resolution and observation
density. |