The potential for measurement redundancy to reduce uncertainty in atmospheric
variables has not been investigated comprehensively for climate observations.
We evaluated the usefulness of entropy and mutual correlation concepts, as
defined in information theory, for quantifying random uncertainty and
redundancy in time series of the integrated water vapour (IWV) and water vapour
mixing ratio profiles provided by five highly instrumented GRUAN (GCOS,
Global Climate Observing System, Reference Upper-Air Network) stations in
2010–2012. Results show that the random uncertainties on the IWV measured
with radiosondes, global positioning system, microwave and infrared
radiometers, and Raman lidar measurements differed by less than 8 %.
Comparisons of time series of IWV content from ground-based remote sensing
instruments with in situ soundings showed that microwave radiometers have the
highest redundancy with the IWV time series measured by radiosondes and
therefore the highest potential to reduce the random uncertainty of the
radiosondes time series. Moreover, the random uncertainty of a time series
from one instrument can be reduced by

The use of redundant measurements is considered the best approach to reduce the uncertainty of an atmospheric variable. For this reason, several atmospheric observatories have extended their observing capabilities and have acquired multiple instruments that measure the same atmospheric variables with different measurement techniques and retrieval algorithms.

Redundancy can be defined as the duplication or the multiplication of the
estimation of an atmospheric variable with the aim of increasing reliability
in the study of the same variable over the time. Without doubt, redundant
measurements provide added value towards the full exploitation of the synergy
among different measurements techniques: the main advantages are related to

filling gaps and improving measurement continuity over time and vertical range;

increasing the sampling rate by merging measurements from different instruments;

addressing instrument noise and identifying possible biases or retrieval problems by comparing different techniques and instruments.

Comprehensive studies to quantify the effective value of redundant measurements and their ability to reduce uncertainty of essential climate variables (ECVs), as retrieved by multiple ground-based techniques and in situ active and passive remote sensing, are missing. To this end, GRUAN (GCOS, Global Climate Observing System, Reference Upper-Air Network) aims at providing long-term, highly accurate measurements of atmospheric profiles, complemented by surface-based state-of-the-art instrumentation, for full characterization of ECVs and their changes in the complete atmospheric column (Seidel et al., 2009; Thorne et al., 2013). GRUAN, which is now being implemented, is aimed at supporting a network of 30–40 high-quality, long-term upper-air observing stations, building on existing observational networks.Cross-checking of redundant measurements for consistency is an essential part of the GRUAN quality assurance procedures. A fully equipped GRUAN site should make at least three redundant measurements of all GCOS ECVs (Seidel et al., 2008). As a consequence, the GRUAN community has fostered GATNDOR (GRUAN Analysis Team for Network Design and Operations Research), a scientific team charged with addressing key scientific questions of major interest to GRUAN and identifying reliable metrics for quantifying the value of redundant measurements.

The present study used observations of the vertical-profile of water vapour mixing ratio and the integrated water vapour (IWV) content from a few GRUAN sites equipped with radiosondes, global positioning system (GPS), lidars, radiometers, spectrometers, and radars. Studies of redundant measurements should be based on the preliminary identification of a reliable metric. Linear correlation (Pearson's or Spearman's) has typically been used to study redundant measurements and their reliability. More recently, Fassò et al. (2014) presented a new approach for an advanced statistical modelling based on functional data analysis of the relationships among collocation uncertainty and a set of environmental factors (e.g. wind speed and wind direction). The approach, which can decompose the total collocation uncertainty, could be adapted to evaluate the measurement redundancy. In this paper, we present the results of the GATNDOR study of redundant measurements at GRUAN sites. The present study identifies mutual correlation (MC), which is related to the concept of entropy, as a suitable metric for quantifying the value of measurement redundancy. In information theory, entropy is a measure of the probabilistic uncertainty associated with a random variable. The approach presented here represents a fast, efficient way to quantify the value of redundant measurements and to correlate the value with factors such as number of instruments, as reported in this work, type of measurement techniques, and retrieval algorithms.

The aims of the paper are

to show the potential of entropy and MC as metrics for quantifying uncertainty (in a probabilistic sense) and the value of redundancy in climate time series;

to study, according to GRUAN standards, the uncertainty and the value of redundancy of in situ and ground-based remote sensing techniques for estimating ECVS;

to provide the GRUAN community and others interested in the observation of atmospheric thermodynamics with recommendations for the establishment of an observation protocol to reduce the uncertainty of a measurement time series through measurement redundancy;

to aid site scientists, managers, and funders in making informed decisions on new instrument procurements to maximize the scientific return on the capital expenditure.

Section 2 outlines information theory concepts used for the study of redundancy and presents the data sets considered in this work. The data sets were provided by five candidate GRUAN sites: the Atmospheric Radiation Measurement (ARM) Program Southern Great Plains in Oklahoma, USA (Miller et al., 2003); CIAO (Consiglio Nazionale delle Ricerche, Istituto di Metodologie per l'Analisi Ambientale (CNR-IMAA) Atmospheric Observatory) in Potenza, Italy (Madonna et al., 2011); Lindenberg in Germany (Adam et al., 2005); Payerne in Switzerland (Calpini et al., 2011); and Sodankylä in Finland (Hirsikko et al., 2014). Section 3 provides results and preliminary remarks on the value of redundant measurements in reducing uncertainty and introduces a possible criterion for addressing redundancy in the frame of GRUAN. Section 4 summarizes the conclusions.

Comparisons among time series of in situ and ground-based remote sensing
measurements have been performed mostly by using the concept of variance and
root-mean-square difference, less frequently in terms of “information”
content (e.g. Majda and Gershgorin, 2010). In information theory, as in
thermodynamics, entropy is a measure of the number of specific ways a system
can be arranged. Entropy is often considered a measure of disorder or
uncertainty in the outcome or the prediction of an event. Commonly used in
time series analysis is the Shannon–Wiener entropy measure (Cover and Thomas,
1991). Given

In information theory, MC is a measure of the statistical dependence of two
random variables or, equivalently, the amount of information that one
variable contains about the other (Cover and Thomas, 1991). The MC value can
be considered a qualitative indication of how well one measurement explains
the other. This means that MC quantifies the reduction of uncertainty in a
variable

The MC of two discrete random variables

The MC can be also linearized; differences between non-linear and linear
redundancy provide a qualitative test for the non-linearity of the
investigated problem. The linear MC is defined as (Cover and Thomas, 1991)

Many applications require a metric – a distance measure not only between
points but also between data clusters (or time series of data). Different
distances are defined in the literature (Arkhangel'skii and
Pontryagin, 1990). Here,

Finally, the conditional entropy is defined as

The data sets considered in this study include radiosonde, Raman lidar,
infrared and microwave radiometry (MWR) observations from the GRUAN sites
(Lindenberg, Payerne, Potenza, Sodankyla and ARM Southern Great Plains). The
data sets are collected by each station according to their quality assurance
criteria. More information about the selected sites can be found at

Water vapour measurements from sensors not considered in this study are also available for the considered sites (as noted in Table 1); they are a subject for future study. The current water vapour measurements were selected according to data availability for each site. A similar investigation could be performed for other ECVs. For coherency, we used sonde data processed at each site rather than GRUAN products, which are still not available at all sites and for all radiosonde types. Moreover, retrieval algorithms for passive instruments usually take advantage of historical radiosonde data sets as a statistical constraint.

Simultaneous data from all available instruments were selected according to
the conditions of clear sky (per lidar measurements or radiosonde humidity),
nighttime, and, if lidar data are available, a relative error of lidar water
vapour mixing ratio at 7

Data from different sites are currently processed with different algorithms; this could affect the comparison. However, the study of entropy is also a good check for the effect of retrieval inconsistencies. A linear regression on the entire time series (3 years) of IWV data and vertical profiles of water vapour mixing ratio at the altitude levels removed natural or artificial trends (e.g. calibration drifts). This was done to suppress the bias component of the time series uncertainty and to ensure that the reported entropies are related only to the random uncertainty.

The two crucial issues that need to be considered for entropy calculation using the histogram of a variable are the minimal quantity of data required to reduce inaccuracies in the calculation and the choice of the optimal binning to represent the actual probability density functions (PDFs) of the variable.

To make our histogram representative of the real underlying PDF of the
variable and to calculate the related entropy, a minimal number of data
points is needed. The data sets considered here include

To determine the optimal binning, several statistical methods have been proposed (Knuth, 2013). In Fig. 1, entropy is shown as a function of the number of bins used to build the histogram for the Payerne radiosonde data sets. The value of entropy increases up to 0.81 for a histogram with 100 bins. Between 25 and 100 bins, entropy tends to assume asymptotic behaviour. In this work, in view of the behaviour shown in Fig. 1 and the number of data points available, 50 bins per histogram are used.

Instruments available (and model when applicable) at the GRUAN sites generating data sets considered in this study of uncertainty and redundancy. Symbol ! indicates that the instrument is available at the site, but the data were not used in the study. Abbreviations: CFH, cryogenic frost-point hygrometer; MWR, microwave radiometer; MWP, microwave profiler; GPS, global positioning system; FTIR, Fourier transform infrared radiometer; AERI, atmospheric emitted radiance interferometer.

In this section, normalized entropy, MC, and conditional entropy are calculated for the data sets (and instruments) identified in Table 1. Both quantities were calculated to quantify uncertainty and redundancy in the IWV time series, as well as in the times series of the vertical profile of water vapour mixing ratio. In this investigation of time series of atmospheric water vapour measurements, entropy includes all contributions affecting the uncertainty of a measurements time series – sampling uncertainty, uncertainty due to the time and vertical average, atmospheric variability, and all other relevant environmental factors (Kitchen, 1989; Fassò et al., 2014), such as solar radiation affecting daytime in situ soundings.

Entropy as a function of the number of bins used to build the histogram for the Payerne radiosonde.

Figure 2 (left) is an example of a series of samples of the IWV for the Lindenberg instruments (Table 1), while Fig. 2 (right) shows the corresponding histograms of the time series. After linear detrending of the time series described above, the histograms were used to calculate entropy and MC. The shape of the histograms in Fig. 2 clarifies both how outliers can occur by chance in any distribution, often indicating either measurement errors or a heavy-tailed distribution in the population, and also the absence of any guarantee that the distribution will be a normal one. The discrepancies between the time series reported in Fig. 2 (left) translate into a sort of bi-modal distribution characterized by a high kurtosis (Fig. 2, right). Thus, caution is needed in assuming a normal distribution; statistics, like entropy, that are robust to outliers and independent on the underlying distribution are more reliable for characterizing the uncertainty of a time series.

To show the reader the advantages of using entropy and MC instead of using
standard deviation (

Following previous studies, the Taylor diagram described above is compared in
Fig. 3 with a modified Taylor diagram (Correa and Lindstrom, 2012) obtained
by replacing the standard deviation with the entropy and

Figure 4 compares the normalized entropies

Example of the time series (left) of integrated water vapour obtained with the instruments available at the Lindenberg site (reported in Table 1) and histograms (right) of the time series shown in the left panel. After detrending of the time series, the histograms were used to calculate entropy and mutual correlation.

The left panel shows a Taylor diagram obtained for the GPS IWV time
series collected at Lindenberg, and the same time series but adding to the
IWV probability density function 5, 10, 20, 30, and 40 outliers respectively.
The correlation has been calculated with respect to an underlying Gaussian
distribution fitted to 637 the data; in the right panel, a modified Taylor
diagram is obtained by replacing the standard deviation with the entropy and

Comparison of the normalized entropy retrieved for the instruments measuring integrated water vapour at the Lindenberg (LIN), Payerne (PAY), Potenza (POT), and Southern Great Plains (SGP) sites. The data set considered includes all available measurements in 2010–2012. The numbers above the bars represent the number of cases selected, according to the quality assurance criteria for each station.

For the available measurements and in the range of atmospheric variability over the analysed stations there are no large differences in the uncertainties of IWV measurements. With the exception of Payerne, lidar entropy is the closest to radiosonde entropy, whether calibrated by using the sonde itself or the MWR. Moreover, at Payerne the lidar offers the lowest entropy of the instrument ensemble. At the SGP, GPS has the lowest entropy, though the values for all considered instruments are quite close. Similarly, at Lindenberg, where the MWR has the lowest entropy, all values are similar. At Potenza, the lowest entropy value is for the microwave profiler. As a whole, differences in the entropy of the time series between the different instruments are within 8 %. Obviously, the different atmospheric variability of each site can also result in large deviations between entropy values. This deviation could be smoothed if a longer temporal data set was investigated. Moreover, differences in the observation techniques and their experimental implementation (e.g. different measurement angles and fields of view) might also contribute to differences in the calculated entropies and to non-linear calibration drifts.

Comparison of the statistical distances between pairs of times series data retrieved for the instruments measuring integrated water vapour with respect to the time series obtained from the radiosondes at the Lindenberg (LIN), Payerne (PAY), Potenza (POT), and Southern Great Plains (SGP) sites. The data set considered includes all available measurements in 2010–2012.

The statistical distance

Normalized entropy and MC are compared for the available measurements of the
water vapour or relative humidity (RH) vertical profiles. In Fig. 6, the
distance of the water vapour vertical profiles obtained with the Raman lidar
(RL) and atmospheric emitted radiance interferometer (AERI) with respect to
radiosonde (RS92) profiles are compared. Lidar profiles were retrieved by
integrating signals over 10

Comparison of the statistical distances of the Raman lidar (RL) and the atmospheric emitted radiance interferometer (AERI) time series from the RS92 radiosonde time series of water vapour vertical profile at the Southern Great Plains (SGP) site. The data set considered includes all measurements available at the SGP in 2010–2012, in 144 profiles.

Figure 7 (left panel) compares the entropies computed for the RH profiles
provided by RS92 radiosondes, Intermet radiosondes (I-Met), and CFH measuring
in situ water vapour vertical profiles at SOD. Figure 7 (right panel) shows
the profiles for one case on 15 March 2010. Because only 20 simultaneous
profiles are included in this comparison, the calculated entropy values might
underestimate the real uncertainty for the sensors. A larger data set should
be considered for a full assessment of the differences in entropy for the
various in situ measurements; this will be considered in future work, taking
advantage of the data set available in the GRUAN network at the Boulder and
Lindenberg sites. The comparison in the left panel of Fig. 7 reveals that the
entropy values for all sensors differ by more than 0.2 from the ground to
2

Comparison of the normalized entropy values (left) for the RS92 and Intermet radiosondes (I-Met) and the cryogenic frost-point hygrometer (CPH) measuring the in situ water vapour vertical profile at the Sodankyla site in 2010 (left panel); comparison of the relative humidity profiles in one case on 15 March 2010 (right panel).

The conditional entropy quantifies the amount of information needed to
describe the outcome of a random variable

These results also show that the residual uncertainties obtained with two conditioning constraints (two instruments) can be better than or similar to the value with only one instrument as a constraint. This is relevant when synergetic products must be defined and retrieved by using algorithms that can integrate information from ground-based or satellite sensors. This is the case for all optimal estimation algorithms based on the Bayes' theorem, which is frequently adopted to improve atmospheric profiling. To quantify the effective advantages of integration, the presented analysis can be performed in advance of the elaboration of algorithms integrating measurements from different sensors. Moreover, conditional entropy can be applied similarly for directly measured quantities, like radiances, as well as for data products such as water vapour ground-based remote sensing. This is the case for algorithms making use of satellite measurements from polar and geostationary satellites to improve the resolution or reduce the uncertainty affecting the estimation of ECVs, but it is also true for algorithms merging satellite and ground-based passive sensor data to improve atmospheric profiling.

Comparison of the normalized conditional entropy values retrieved for most of the possible combinations of instruments measuring integrated water vapour at the Potenza site (upper panel) and the Southern Great Plains site (bottom panel).

Comparison of the normalized mutual correlation for the linear (LMC) and non-linear cases (MC), calculated for the lidar and radiosonde data sets from the Potenza site.

A comparison between MC and linear mutual correlation (LMC) provides a
qualitative test for the non-linearity of the investigated problem. The plot
in Fig. 9 shows a comparison between MC and LMC for the lidar and radiosonde
at POT. In this case, both MC and LMC are normalized by the maximum entropy
(the total number of entries in the joint histogram). The LMC underestimates
the correlation between the two variables by about 10 %. Above
5.5

The analysis above shows how to approach the problem of quantifying measurement redundancy by using the concepts of information theory. However, the usefulness of this approach can be clarified only if some criteria are identified to classify when two data sets are redundant. This obviously depends on the investigated variable and on the uncertainty limits assumed to be minimum requirements for studying a certain atmospheric process or climate trend.

Here, we present an example showing the relationship between distance values
and the random uncertainty affecting IWV measurements. The aim is to clarify
the use of MC and the related distance for quantifying redundant IWV
measurements at GRUAN sites. The plot of Fig. 10 shows the distance between
the radiosonde IWV time series at Lindenberg and the corresponding time
series obtained by adding variable random noise to the radiosonde time
series. The random noise is added to reproduce the effect of an additional
random uncertainty, with relative values of 0–100 % affecting an IWV
time series with respect to the reference series. For example, a distance
value lower than about 0.2 corresponds to a random uncertainty 20 %
larger than that of the original time series assumed as the reference. This
example indicates a very simple way to approach data sets from different
instruments or techniques, fixing a threshold consistent with the desired
redundancy requirements. According to the GCOS requirements for the
state-of-art capability, also reported in the GRUAN manual
(

Statistical distance between the integrated water vapour time series retrieved from the radiosonde at the Lindenberg site and the corresponding time series obtained by adding random noise to the radiosonde time series to simulate the effect of increasing relative random uncertainty.

The ultimate aim of this study is to recommend the best combination of instruments for monitoring atmospheric water vapour. Though entropy and MC are robust concepts provided in information theory, representing appropriate metrics to quantify the uncertainty and redundancy of atmospheric measurements, they have never been applied extensively to climate data. In this paper, we show how entropy and MC can be used to evaluate the random probabilistic uncertainty in the ECV by analysing measurement redundancy.

The following conclusions can be drawn from the results of this study of data
sets of water vapour from five GRUAN observation stations in 2010–2012:

The random uncertainty in the IWV time series obtained with the different
instruments considered in this study (Raman lidar, GPS, MWR, microwave
profiler, sondes) differs by

In terms of the best performances for each instrument at the different sites, the comparison of IWV time series showed that MWR have the highest redundancy and therefore the highest potential to reduce the random uncertainty of IWV time series as measured by radiosondes.

The distance between the time series of water vapour profiles at each altitude level has been also performed to show how to evaluate the redundancy of collocated in situ, active and passive profiling instruments, though for passive instruments this also depends on the retrieval algorithms and on which first-guess prior covariance is used.

Both RS92 and I-Met radiosondes can measure in situ atmospheric water vapour with the same random uncertainty as the CFH, though the sondes are affected by a bias error that cannot be evaluated with the present approach.

A conditional entropy analysis showed that conditioning of the time series with more than one instrument, assumed as constraints, can decrease the residual entropy by at least 60 % versus the use of one conditioning instrument. Moreover, the use of two conditioning instruments versus one results in similar or slightly better residual uncertainty.

An analysis of the relationship between distance and the random
uncertainty showed that a maximum random error

As a whole the concepts of entropy and mutual correlation demonstrate their potential if used as metrics for quantifying random uncertainty and redundancy in time series of atmospheric observations. The examples discussed in this work support the use of the mutual correlation as a more general concept than other linear metrics for the study of redundant measurements. Moreover, the analysis based on the entropy, MC and conditional entropy can be used for a preliminary feasibility study of the effective advantages obtained in using retrieval algorithms integrating measurements provided by different observation platforms, ground-based or satellite, both for direct measurements (e.g. radiances) and retrieved products (e.g. temperature, water vapour content, aerosol optical depth). For example, this is the case of those algorithms integrating measurements from different sensors using the Bayes' theorem (that is based on the concept of conditional probability) as well as for those algorithms integrating radiances measured by different sensor in different spectral ranges (e.g. Romano et al., 2007).

Data sources were as follows: ARM SGP data through the US Department of
Energy (