Climatologies from satellite measurements: the impact of orbital sampling on the standard error of the mean
and Physics
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Atmospheric
Measurement
Techniques
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Biogeosciences
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Atmos. Meas. Tech., 6, 937–948, 2013
www.atmos-meas-tech.net/6/937/2013/
doi:10.5194/amt-6-937-2013
© Author(s) 2013. CC Attribution 3.0 License.
of the Past
M. Toohey1 and T. von Clarmann2
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Climatologies from satellite measurements: the impact of orbital
sampling on the standard error of the mean
Climate
1 GEOMAR Helmholtz-Centre for Ocean Research Kiel, Kiel, Germany
Earth System
Dynamics
Correspondence to: M. Toohey ()
Received: 31 October 2012 – Published in Atmos. Meas. Tech. Discuss.: 9 November 2012
Revised: 19 February 2013 – Accepted: 10 March 2013 – Published: 10 April 2013
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Geoscientific
variations in the sampledInstrumentation
field, leading to maximum sensitivity to arbitrary phase shifts between the sample distribution
Methods and
and sampled field. The occurrence of such instances is thus
DatainSystems
very sensitive to slight changes
the sampling distribution,
and to the variations in the measured field. This study highlights the need for caution in the interpretation of the oft-used
classically computed SEM,
and outlines a relatively simple
Geoscientific
methodology that can be used to assess one component of the
Modelmean
Development
uncertainty in monthly
zonal mean climatologies produced from measurements from satellite-borne instruments.
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1
Introduction
Hydrology and
Earth System
Sciences
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Atmospheric observations are often averaged within timespace intervals, such as calendar months and latitude
bands, producing so-called “climatologies” (e.g., Grooß
and Russell III, 2005; Hegglin and Tegtmeier, 2011; von
Ocean
Clarmann et al., 2012).
While theScience
motives behind the construction of such climatologies can be simply pragmatic –
for instance to simplify comparison with similarly averaged
model fields – averaging does have the advantageous effect
of reducing the impact of random variations present in individual measurements due to measurement errors and natural
Earth
variability. The standard errorSolid
of the mean
(SEM) is a statistical quantity which quantifies the random error in the calculated mean value.
In general terms, the standard error describes the random
error of an estimate based on limited sampling of a population. For example, the SEM describes the potential variation
The Cryosphere
of a sample mean of n samples if other, equally probably sets
of n samples were drawn instead. The “classic” and oft-used
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Published by Copernicus Publications on behalf of the European Geosciences Union.
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Abstract. Climatologies of atmospheric observations are often produced by binning measurements according to latitude
and calculating zonal means. The uncertainty in these climatological means is characterised by the standard error of
the mean (SEM). However, the usual estimator of the SEM,
i.e., the sample standard deviation divided by the square
root of the sample size, holds only for uncorrelated randomly sampled measurements. Measurements of the atmospheric state along a satellite orbit cannot always be considered as independent because (a) the time-space interval
between two nearest observations is often smaller than the
typical scale of variations in the atmospheric state, and (b)
the regular time-space sampling pattern of a satellite instrument strongly deviates from random sampling. We have developed a numerical experiment where global chemical fields
from a chemistry climate model are sampled according to
real sampling patterns of satellite-borne instruments. As case
studies, the model fields are sampled using sampling patterns
of the Michelson Interferometer for Passive Atmospheric
Sounding (MIPAS) and Atmospheric Chemistry Experiment
Fourier-Transform Spectrometer (ACE-FTS) satellite instruments. Through an iterative subsampling technique, and by
incorporating information on the random errors of the MIPAS and ACE-FTS measurements, we produce empirical estimates of the standard error of monthly mean zonal mean
model O3 in 5◦ latitude bins. We find that generally the classic SEM estimator is a conservative estimate of the SEM, i.e.,
the empirical SEM is often less than or approximately equal
to the classic estimate. Exceptions occur only when natural
variability is larger than the random measurement error, and
specifically in instances where the zonal sampling distribution shows non-uniformity with a similar zonal structure as
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2 Karlsruhe Institute of Technology, Institute for Meteorology and Climate Research, Karlsruhe, Germany
M
938
M. Toohey and T. von Clarmann: Sampling and the SEM
SEM estimator is given by the standard deviation of the sample divided by the square root of the sample size, however,
this estimator is only valid when the measurements are uncorrelated. Consideration of correlations of measured data
is a standard in various applications of statistical estimators
inferred from atmospheric measurements, e.g., Jones et al.
(1997) consider inter-site correlations in the estimation of
global mean temperatures; Weatherhead et al. (1998) present
a scheme to consider autocorrelations in estimating uncertainties in trends; and von Clarmann et al. (2010) propose
a generic approach to consider arbitrary correlations in trend
estimation. For the SEM, one of the most fundamental estimators of a finite sample of atmospheric data, little literature
is available.
Correlations in atmospheric measurement sets depend
upon the underlying time-space correlations of the atmosphere, and the time-space sampling patterns of the measurements themselves. Observational datasets from satellite instruments have distinct sampling patterns which depend on
the orbit and measurement technique of the instrument. Different sampling patterns can lead to differences in the means
of two datasets: in this case, the difference is referred to as
a sampling bias. For example, Aghedo et al. (2011) have examined the role of sampling in biasing monthly mean values of satellite-based measurements of tropospheric chemical species and temperature. However, the potential impact
that sampling may have on the SEM of atmospheric climatologies has not, to our knowledge, been formally addressed.
The goals of the present study are (1) to raise awareness
of the potential impact of sampling on the SEM of climatologies built from satellite-based atmospheric measurement
sets, (2) to develop a strategy for estimating the magnitude
of its impact, and finally (3) to estimate the impact that sampling considerations have on the SEM for some sample cases.
In order to assess the impact of time-space sampling patterns
on the SEM, we present a numerical recipe which makes use
of model fields from a coupled chemistry climate model. Assuming that the model accurately reproduces, in a statistical
sense, the correlations of the true atmosphere on scales larger
than the horizontal footprint (...truncated)
