The time series of global radiation observed by a dense
network of 99 autonomous pyranometers during the HOPE
campaign around Jülich, Germany, are investigated with
a multiresolution analysis based on the maximum overlap
discrete wavelet transform and the Haar wavelet. For different
sky conditions, typical wavelet power spectra are calculated
to quantify the timescale dependence of variability in global
transmittance. Distinctly higher variability is observed at all
frequencies in the power spectra of global transmittance under
broken-cloud conditions compared to clear, cirrus, or overcast skies.
The spatial autocorrelation function including its frequency dependence
is determined to quantify the degree of similarity of two time
series measurements as a function of their spatial separation.
Distances ranging from 100

The Sun is the primary source of energy for the Earth's climate
system. Clouds strongly modulate the radiation budget through
reflection of solar radiation back to space, and by trapping
terrestrial radiation within the atmosphere

This can be attributed to our currently still very limited
understanding of cloud processes and the resulting cloud–radiation
interactions, due to their complexity and the wide range of scales
involved. Small-scale processes such as updraughts and downdraughts,
turbulent mixing, as well as the availability and composition of
cloud condensation nuclei and large-scale dynamics, influence the
formation and life cycle of clouds, which subsequently determine their
optical properties and thus their interaction with radiation

Satellite observations are one very important source of information
for investigating clouds and their radiative effects. Current
operational retrievals of cloud properties from passive satellite
sensors do however invoke the assumption of plane-parallel,
horizontally homogeneous clouds. While these retrievals have been
extensively evaluated with ground-based measurements over the past
years

These complications can be mainly attributed to horizontal photon
transport, radiative smoothing, and sub-pixel inhomogeneity

The attribution of deviations between ground-based observations,
satellite observations, and model results is also complicated by the
effects of spatial collocation and the limited representativeness of a
point measurement for domain averages implicitly assumed in any such
comparisons

Focusing on solar radiation,

Comparing satellite-based solar radiation retrievals from the Advanced
Very High-Resolution Radiometer to pyranometer observations,

Over the past decades, several ground-based surface radiation networks
have been established

Spatial and temporal scaling properties of the time series of observed
global radiation can be derived using a wavelet-based multiresolution
analysis. Wavelet-based estimators of variance, covariance, and
cross-correlation decompose their scale-independent counterparts on a
scale-by-scale basis. Multiple studies have adapted similar
wavelet-based methods to explore a wide range of subjects involving
the atmospheric time series applications

In our study, the statistical properties inferred from a multiresolution
analysis (MRA) of the time series of global radiation are subsequently
used to quantify the representativeness of a point measurement for
a surrounding domain considering typical domain sizes and different
sky conditions. Instead of directly considering the global radiation,
its transmission by the atmosphere, denoted as
global transmittance is considered in this paper, because
the changes in incoming solar radiation are removed at least to
first order. The present study is focused at addressing the following
research questions:

How do the power spectra of global transmittance differ for different sky conditions?

How representative is the time series observed at one station for other nearby stations?

How representative is the single station observation for domain averages considering different spatial- and temporal-averaging scales?

This paper is organized as follows: in Sect. 2 details of the observational data used in this study are presented. An overview of our methods is given in Sect. 3, with more details on the theory given in the Appendix. Section 4 discusses the results of the multiresolution analysis, the behavior of the power spectra, and the spatial correlation under different prevailing sky conditions. These results are further used to investigate the spatial representativeness of a point measurement for spatial averages over typical domain sizes, and to quantify the expected deviations. Finally, the summary and conclusions with an outlook are presented in Sect. 5.

As part of the HOPE campaign, a high-density network of 99 autonomous
pyranometer stations was operated across a spatial domain covering
50.85–50.95

The limited spectral range (0.3–1.1

Classification of days into clear, cirrus, overcast, and broken cloudy-sky condition during the HOPE Jülich campaign.

The real-time sky conditions were assessed using hemispheric images
from a Total Sky Imager (TSI) operated at the Research Center
Jülich (FZJ) during the HOPE campaign. Time–azimuth (

Time representation of the Haar

A multiresolution analysis (MRA) based on the maximum overlap
discrete wavelet transform (MODWT;

Averaging time periods and wavenumber range corresponding to each wavelet detail (

In the MRA, the day is chosen as fundamental frequency

Multiresolution analysis of global radiation (red) and
corresponding transmittance (blue) showing smooths (left panel) and
details (right panel) as a function of local time (in hours,

Figure

Horizon graphs for MRA of global transmittance from a
pyranometer station at FZJ during the HOPE campaign represented
as a function of local time (in

An effective graphical technique to the MRA is the horizon graph

From the MRA, the wavelet power spectrum of transmittance can be
calculated (Sec. 3.1), which describes the partitioning of
signal power into frequency ranges, and reflects the characteristics
of the prevailing sky conditions. Additionally, the spatial
autocorrelation function describes the similarity of variations in
the time series measured at two stations as a function of their
distance. By determining both the power spectrum and the
frequency-dependent spatial autocorrelation function across the
observation domain under different sky conditions, the
representativeness of a point measurement for an area-averaged value
can be quantified, including the expected deviation. Various
statistical parameters, namely the variance, covariance, and explained
variance linking the time series of a point measurement to that of an
area-averaged value, are derived in Appendix A. In this study, we
consider three typical spatial areas (

Further, the estimated representativeness error of the transmittance
(

Wavelet-based spectral power density characterize the variability contained in specific frequency intervals for both stationary and non-stationary processes. As the time series of global transmittance results from a non-stationary process (i.e., its statistical properties are not time invariant), the wavelet power spectrum is a suitable tool for the analysis of the variability contained within specific frequency intervals, and to study the effect of temporal and spatial averaging on the variability of the time series.

In Fig.

On days with cirrus clouds, the spectral power density is lower than for broken clouds and higher than for clear skies. Due to the changes in solar elevation and thus air mass over the day, a pronounced diurnal cycle in global transmittance is observed in clear-sky situations, which introduces significant variance at longer time periods.

In the case of overcast sky, the variance of transmittance is found
to be the lowest at high frequencies (i.e., 10.5–5.25

Considering the cumulative explained variance, it can be seen that for
broken clouds, high-frequency variability contributes most strongly to
the total variance of the global transmittance (Fig.

Summary of the spectral power density of scalar variables with observed scale regimes and
spectral exponents (

Wavelet variance of global transmittance (as shown in
Fig.

Various studies have described the properties of stratocumulus/cumulus
clouds using power spectra (or spectral density,

In Fig.

It should be noted that the global irradiance is a hemispherically integrated property and thus there cannot be an exact one-to-one relation to the cloud variability or to (directional) radiance variability. However, the irradiance variability should show a correlation to a smoothed cloud structure. Finding an appropriate smoothing kernel requires intensive investigations of the interaction of clouds and radiation including 3-D radiative effects, and is beyond the scope of this study.

Spatial autocorrelation

An important aspect for assessing the density of a measurement network
is the representativeness of observations at one station for other
close by network stations as a function of their distance. To
investigate this aspect for the network operated during the HOPE
campaign, the spatial autocorrelation

Summary of various parameterizations used for modeling the behavior
of spatial autocorrelation

The behavior of the spatial autocorrelation (

Decorrelation lengths

An overview of various parameterizations used for modeling the
behavior of the spatial autocorrelation function as a function of
station distance is presented in Table

In our study, the spatial correlation of transmittance variations
decays faster than linear at small distances as is indicated by the
exponent (

The spatial representativeness of a point measurement at the center of
a domain of interest depends on the size of the domain, the temporal
averaging applied, and the spatiotemporal variability present in the
observations. Generally, higher variability leads to a reduction of
representativeness. Statistically optimum methods for spatial
averaging have been developed to provide spatial means including
uncertainty estimates when using data from a number of stations

Power spectrum of spatially averaged transmittance as a
function of frequency under different sky conditions:

In Fig.

At 10

The

Explained variance (

The level of similarity between two time series is often expressed by
metrics such as the explained variance or the RMSE,
and suitable averaging timescales are often determined by studying
the sensitivity of these metrics to the choice of averaging scale. The
explained variance (

Mean deviation between point measurement and spatial averages of global transmittance
(

Area-averaging error in the global transmittance (

Finally, the deviation between point observations and spatial averages
is determined for different domain sizes and temporal-averaging
periods, combining the two effects discussed before. The magnitude of
the expected deviation as a function of domain size and temporal-averaging period is shown in Fig.

Table

On clear days, the representativeness error of a point measurement for
an area-averaged mean value increases only slightly as the averaging
period decreases, and ranges from 2.1 to 3.3 %. The difference
between the maximum and minimum deviations resulting from the choice
of averaging period is found to be around 0.6 % (

Power spectrum of the

The range of deviations of a point observation under cirrus clouds is
found to be around 1.6 % (

During overcast skies, the representativeness error again increases
substantially with increasing domain size, doubling and tripling its
magnitude when going from 1

As expected, the magnitude of deviations in global transmittance and
corresponding radiation due to the limited representativeness of a
point observation is found to be distinctly higher for all considered
domain sizes and frequency intervals under broken cloudy
situations. It varies from 4.5 to 11.5 % (

Based on our findings for different sky conditions, the comparison of
time series corresponding to spatial averages of global radiation on
the one hand, and point measurements on the other hand, can result in
large deviations due to the limited representativeness of the point
measurement. Similar effects are expected to occur for other observables
such as liquid water path. To address this issue of representativeness,
we recommend here to apply a low-pass filter, which removes variability
at higher frequencies without significant correlation. Even for lower
frequencies, a low-pass filter should be applied to adjust the power
spectrum of a point time series towards that of the spatially averaged
time series, at least if the reduction factor of the amplitude of variations
shown in Fig.

Variability in global radiation results from the combined variability
of the direct and diffuse radiation components. During the
HOPE Melpitz campaign (May–July 2015;

Power spectra of the global, direct, and diffuse transmittance are
shown in Fig.

This behavior highlights that the strong influence of the direct
radiation on the power spectra of global radiation. A plausible
explanation is the hemispherical field of view of the diffuse radiation
observations, which is less sensitive to small-scale variations in
cloud properties than the direct beam of sunlight.

A unique data set of global radiation observations has been collected
using a dense network of pyranometer stations

The most significant findings of this study are summarized as follows:

The power spectra of global transmittance exhibit unique
characteristics for different prevailing sky conditions associated
with the dominant cloud type. For days with broken clouds, the
variability of global transmittance is significantly and distinctly
higher for all considered frequencies than for other situations,
and contains remarkable contributions (1 %

The spatial autocorrelation between stations decreases
strongly with increasing frequency. Variations at different points
separated by more than 1

While the time series of spatially averaged irradiance
fields generally resemble the behavior of a point measurement, its
power spectrum is strongly attenuated (96–98 % for
10

As a consequence of the previous conclusions, point
measurements can deviate strongly from the spatial mean of a
surrounding domain. This effect can reach as much as 80

The methods presented in this paper allow for an explicit treatment of the effects of temporal and spatial averaging on the spatiotemporal variability of global radiation, and can easily be adapted to other geophysical fields. We have applied this methodology to estimate the inherent uncertainty arising from a comparison of two time series with fundamentally different spatial- and temporal-averaging scales, as is commonly done in radiation closure studies, the evaluation of atmospheric models, or satellite products with point measurements. The findings contribute towards a better understanding of the uncertainties in such comparisons.

In future work, we plan to apply these findings towards an assessment
of the level of accuracy of satellite-based estimates of shortwave
irradiance from Meteosat SEVIRI with ground-based measurements

Due to the spatially distributed nature of the pyranometer network,
the present work can also be extended to estimate Lagrangian instead
of the Eulerian decorrelation scales, by considering the maxima in the
time-lagged cross-correlation of transmittance time series observed at
different sites. This time shift can be converted into an estimate of
wind speed and direction, and will allow for a separation of changes in
radiation resulting from advective changes in clouds, which depend on
the wind flow, and from temporal changes in cloud properties, which
are independent of current wind speed and direction. Such an analysis
will also enable a comparison of spatial and temporal decorrelation
scales obtained from geostationary satellite observations

Finally, this work can serve as reference for evaluating the
representation of clouds including their radiative effects and spatial
variability in high-resolution atmospheric models

The pyranometer network data used are available upon request to Andreas Macke
(macke@tropos.de) or Hartwig Deneke (deneke@tropos.de). The data are archived at
the Integrated Climate Data Center. This archive is also referred as the “Standardized
Atmospheric Measurement Data” and is accessible at

Let

The mean of the time series at

The variance of the time series at

The covariance of any two time series at

The autocorrelation

We now assume that the measurement field within the observation domain is
statistically homogeneous (i.e., invariant under translation due to the
shift in the origin of the coordinate system) and isotropic (i.e., invariant
under rotations and reflections of the coordinate system). Consequently, the
following properties hold:

homogeneity:

isotropy:

By adopting the above assumptions in Eq. (A4), the autocorrelation

Therefore, the autocorrelation

For a spatial area

The mean of the area-averaged time series is given by

The variance of the area-averaged time series is given
by

Therefore, the variance of area-averaged time series is directly proportional to
the variance of the time series centered in the observation domain and the domain-weighted autocorrelation function

Now, the statistical parameters between the time series centered in the domain and the area-averaged time series for the
domain area

The covariance of the time series

The square of the cross-correlation

Now, the variance of the difference between the point time series

Alternately, we define a damped time series

The variance of the difference between the point time series

Expressing the Eq. (A16) in terms of the standard deviation, the area-averaging error

Comparing Eqs. (A13) and (A17), we find

The authors declare that they have no conflict of interest.

The authors acknowledge essential technical support from the TROPOS mechanics and electronics workshops in designing
and building the autonomous pyranometer, especially Cornelia Kurze and Hartmut Haudek. Many thanks to all the private landowners
for their support. We are grateful to the Research Center Jülich (FZJ) for their valuable logistic support in setting up and
maintaining the instruments. The first author acknowledges the funding support of the Federal Ministry of Education and Research
(BMBF), Germany, as part of the HD(CP)