ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-17-3317-2017Multiresolution analysis of the spatiotemporal variability in global radiation observed by a dense network of 99 pyranometersMadhavanBomidi Lakshmimadhavan.bomidi@tropos.deblmadhavan@gmail.comhttps://orcid.org/0000-0001-8782-9249DenekeHartwighttps://orcid.org/0000-0001-8595-533XWitthuhnJonashttps://orcid.org/0000-0002-4818-5011MackeAndreashttps://orcid.org/0000-0003-2550-6641Leibniz-Institute for Tropospheric Research (TROPOS), Permoserstraße 15, 04318 Leipzig, Germanynow at: Department of Marine Sciences, Goa University, Goa 403 206, IndiaBomidi Lakshmi Madhavan (madhavan.bomidi@tropos.de, blmadhavan@gmail.com)8March2017175331733381August20168November201614February201714February2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://acp.copernicus.org/articles/17/3317/2017/acp-17-3317-2017.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/17/3317/2017/acp-17-3317-2017.pdf
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 m to 10 km are considered,
and a rapid decrease of the autocorrelation function is found with
increasing frequency and distance. For frequencies above
1/3min-1 and points separated by more than 1 km,
variations in transmittance become completely uncorrelated.
A method is introduced to estimate the deviation between a point
measurement and a spatially averaged value for a surrounding
domain, which takes into account domain size and averaging
period, and is used to explore the representativeness of a single
pyranometer observation for its surrounding region. Two distinct
mechanisms are identified, which limit the representativeness; on
the one hand, spatial averaging reduces variability and thus
modifies the shape of the power spectrum. On the other hand,
the correlation of variations of the spatially averaged field and a
point measurement decreases rapidly with increasing temporal
frequency. For a grid box of 10 km× 10 km and
averaging periods of 1.5–3 h, the deviation of global
transmittance between a point measurement and an area-averaged
value depends on the prevailing sky conditions: 2.8 (clear),
1.8 (cirrus), 1.5 (overcast), and 4.2 % (broken clouds). The
solar global radiation observed at a single station is found to
deviate from the spatial average by as much as
14–23 (clear), 8–26 (cirrus),
4–23 (overcast), and 31–79 Wm-2
(broken clouds) from domain averages ranging from
1 km × 1 km to 10 km× 10 km in area.
Introduction
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 .
A better understanding of the small-scale variability in the radiation
field at the surface resulting from clouds will have numerous
practical applications, ranging from climate-related research focused
on cloud radiative effects and cloud–aerosol interactions, representing of radiative transfer in numerical weather prediction
and to solar energy forecasting
. According to the latest Intergovernmental Panel
on Climate Change report, the impact of various cloud types on the net
radiation budget is not fully understood to the extent that for some
cloud types neither the magnitude nor even the sign is known
.
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
e.g.,. Consequently,
clouds induce the largest amount of uncertainty in climate projections
and weather prediction .
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 e.g.,, significant biases and uncertainties
remain due to the limitations of this assumption e.g.,.
These complications can be mainly attributed to horizontal photon
transport, radiative smoothing, and sub-pixel inhomogeneity
e.g.,. To address these issues,
proposed a parameterization to account for
unresolved sub-grid-scale variability, which does however depend on a
priori information about typical variability for different cloud
types. They also identified an increase in optical thickness and a
decrease in relative variability in the transition from cumuliform to
stratiform clouds. studied power spectra
obtained from high-resolution Landsat observations, and identified
different behavior for scales below 1 km and within the
interval from 1 to 5 km as a consequence of both cloud
morphology and three-dimensional (3-D) cloud radiative effects. Based on a large ensemble
of 3-D cloud fields as input for 3-D radiative transfer models,
reported that spatially transmitted solar
radiation and domain average cloud properties are highly correlated.
In related research, and showed
that a stochastic cloud generator together with a 3-D radiative transfer
model can be used to link the statistical properties of cloud
observations to those of the resulting solar radiation field with
satisfactory accuracy.
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 e.g.,.
Large inconsistencies are expected to occur in particular for short
time periods (< 1 h) and broken-cloud fields, if point
measurements are compared to large satellite pixels or
coarse-resolution model output (> 1 km).
Focusing on solar radiation, concluded that for
stratocumulus clouds, a high frequency of observations is required to
estimate the hourly averaged global radiation from satellites with
acceptable accuracy (∼ 5 % error for six scans per hour). To
estimate the representativeness of a point measurement for a larger
domain, used data from a network of pyranometers
during the First International Satellite Cloud Climatology
Project Regional Experiment, and showed that a spatial
separation between the measurement sites of up to 150 km can
be allowed for daily averages. This is, however, mainly attributable
to the fact that correlation is dominated by the diurnal cycle of
solar radiation at the top of atmosphere. In another study,
found a characteristic timescale of 60 min
for solar radiation on cloudless days, and twice that long for cloudy
days, after removal of the diurnal cycle component. They also
concluded that to achieve a correlation of 0.9 between measurements at
a point and averaged over a surrounding area on cloudy days, the
central site can be considered representative for a region with a
radius of 30 km. Both and
reported that the representativeness of a point measurement for area
averages depends on the considered averaging time and the prevailing
cloud type.
Comparing satellite-based solar radiation retrievals from the Advanced
Very High-Resolution Radiometer to pyranometer observations,
reported a large root mean square error (RMSE) of
86 Wm-2 for individual station records even when
averaging over 10 × 10 satellite pixels and over
40 min. In contrast, a much better accuracy (RMSE ∼ 33 Wm-2) is achieved if the average of 30 stations is
considered. They interpret this finding as evidence that a significant
fraction of the RMSE in the comparison results from the variability
of the global radiation field due to the limited representativeness of
the pyranometer measurements for the satellite-retrieved values.
Over the past decades, several ground-based surface radiation networks
have been established e.g.,.
However, a dense network of solar radiation measurements at the surface with station distances
smaller than a typical satellite pixel or model grid has to our knowledge not been realized before.
Such a network has been developed and operated during the High
Definition Clouds and Precipitation for advancing Climate
Prediction (HD(CP)2) Observation Prototype
Experiment (HOPE) conducted around Jülich,
Germany . This unique data set can
provide insights into the small-scale variability of global radiation
due to various cloud types, and possibly enable the development of
parameterizations of the unresolved spatiotemporal variability in the
radiation field. Using this data set, explored the
fluctuations of the clear-sky index (i.e., the ratio of instantaneous
global radiation to the radiation on the Earth with a cloud-free
atmosphere) on clear, overcast, and mixed sky conditions with a simple
increment statistics to study the smoothing effects of distributed
photovoltaic power production.
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 , solar
radiation , fluctuation analysis of the power
generated by photovoltaic plants , geophysical
seismic signal analysis , signal and image
processing, and vegetation monitoring.
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.
Data sets
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∘ N and 6.36–6.50∘ E (∼ 10 km× 12 km area)
around Jülich, Germany, from 2 April to 24
July 2013. Each of these stations continuously recorded the global
radiation (G in Wm-2) using a silicon photodiode
pyranometer (model: EKO ML-020VM) with 10 Hz resolution. A
Global Positioning System module embedded on the data acquisition
board of each station provides an accurate time reference. The global
radiation measurements have been averaged into 1 s time
periods during the conversion of the ASCII log files into NetCDF data
files following the Climate and Forecast Metadata Conventions
version 1.6 . From these measurements we have
derived the global transmittance (T), which is calculated by
normalizing the global radiation (G) under all sky conditions by the
extraterrestrial radiation at the top of atmosphere assuming a value
of the solar constant of 1360.8 Wm-2 from
and accounting for the cosine of the solar zenith angle and Sun–Earth
distance. The solar zenith angle and the Sun–Earth distance have been
calculated following the guidelines of .
The limited spectral range (0.3–1.1 µm) of silicon
photodiodes is a well-known limitation of this type of
pyranometer . Changes in the spectral
distribution of downward irradiance compared to the conditions
during calibration can lead to errors of up to 5 %, particulary
at higher solar zenith angles. While the derived global
transmittance is sensitive to aerosols and cloud optical thickness,
information on cloud thermodynamics phase and cloud droplet
effective radius is beyond the spectral range of these silicon
photodiode pyranometers. Detailed information about the
pyranometer network setup during the HOPE campaign, data
processing, quality control, and uncertainty assessment due to
various potential sources of error are presented in .
Classification of days into clear, cirrus, overcast, and
broken cloudy-sky condition during the HOPE Jülich campaign.
Sky conditionObservation days (day/month)Clear4 May, 8 June, 9 July, 21 JulyCirrus22 April, 24 April, 16 JulyOvercast9 June, 28 JuneBroken clouds13 April, 25 April, 1 May, 2 May,24 May, 4 June, 19 July
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 (t–azi)
plots were generated from the TSI images. Every line in these t–azi
plots contains pixels from the azimuth angle range from 0 to
360∘, sampled at an elevation angle of 45∘. These
plots capture both spatial and temporal variability of clouds, and
help to identify the dominating advection direction of clouds, which
shows up in sine-like patterns . Since the
ground-based observations have a field of view, which does not exceed
50 km in radius , Meteosat SEVIRI
(Spinning Enhanced Visible and Infrared Imager) images based on the
day-natural RGB color composites were additionally
used for the physical interpretation and thermodynamic phase
identification of the cloud types present over the observation
domain. The 0.6, 0.8, and 1.6 µm spectral channels where
enhanced in resolution using the high-frequency component of the
broadband HRV (high-resolution visible)
channel (0.4–1.1 µm; ). Based
on the predominant sky conditions during the daylight period
(06:00–18:00 h local time), we have classified selected days as
clear, cirrus, overcast, or broken cloudy conditions (see
Table ).
Time representation of the Haar (a) scaling and (b) wavelet
function, and the frequency response of the associate (c) low-pass and
(d) bandpass filters. Adopted from .
MethodsMultiresolution analysis
A multiresolution analysis (MRA) based on the maximum overlap
discrete wavelet transform (MODWT; ) and the Haar
wavelet is applied to the time series of global flux
transmittance measurements of the pyranometer network. The Haar
wavelet filters correspond to rectangular scaling and wavelet
functions, which act as low-pass and bandpass filters, respectively
(see Fig. ). Maximum time localization is achieved
through the minimal support of the filters. This also minimizes the
range of edge effects. The choice of a rectangular function as
low-pass filter also has the advantage that it corresponds to an
arithmetic average for a specific period and is thus simpler to
interpret than the weighted averages obtained by other wavelets.
The drawback of the rectangular function as a low-pass filter is its
sub-optimal frequency separation, which could result in lower
correlations found between time series than those obtained by
Gaussian averaging. A summary of the methodology is given
here, while a more formal mathematical treatment with relevant
references to the literature can be found in the
Appendix of .
Averaging time periods and wavenumber range corresponding to each wavelet detail (DJ).
In the MRA, the day is chosen as fundamental frequency
f=1day-1, and the frequency domain is partitioned
into bands delimited by the harmonics fJ given by
fJ=2J×f. For obtaining this partitioning, the
original data set has been resampled from 86 400 to 216
(= 65 536) samples per day before subjecting it to the MRA.
To avoid aliasing effects caused by the resampling step, a
3 s running mean has been applied as low-pass filter
prior to subsequent decimation of samples by a factor
of 512675. Only harmonics from J=3 to J=14 with
corresponding averaging time periods of 3 h (=213
samples) and 5.25 s (=22 samples) are considered
in the following analyses, avoiding the influence of changes in
solar zenith angle below 75∘ and the anti-aliasing filter
above this frequency range, respectively. The running means
of the original time series for the different harmonics J
(corresponding to an averaging time period of
2-J×86400s) are referred to as
smooths denoted by SJ. Further, the differences
between two subsequent smooths are called the details,
DJ (=SJ+1-SJ), and contain the variability
(or fluctuation behavior) within a frequency band delimited
by two harmonics. The averaging time periods and wavenumber
ranges (km-1) corresponding to each wavelet
detail DJ from J=3 to J=13 are given in
Table . Transformation from time
or frequency domain to wavenumber space is accomplished
using the frozen turbulence hypothesis .
It states that the variability in the surface radiation is mainly
dominated by the advection of the spatial structures of the
cloud fields across the point of observation rather than
local changes of the fields. Hence, the frequency
domain (f) is converted to wavenumber scale
k (=2π/λ, where λ is the
wavelength in m or km) by assuming a
mean advection velocity v (=λf).
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, h) for
a pyranometer station at FZJ on 25 April 2013. Shaded gray region
on both panels correspond to the region with solar zenith angle
> 75∘. The smoothing time is given in the left panels,
while the correlation of the details in global radiation and
transmittance is listed in the right panels.
Figure shows the results of the MRA applied
to the global radiation (G) and to the global transmittance
(T) for measurements from the pyranometer station located
at FZJ (hereafter, referred as PYR76) on 25 April 2013 for
scales J=3 to J=12 (i.e., 3 h to 21 s). On
this day, light fog prevailed in the morning with some cirrus
clouds. Thereafter, broken cumulus mediocris clouds were
observed until late afternoon, followed by rapidly increasing
low stratus clouds leading to an overcast sky by evening.
The left panel of Fig. contains 10 smoothed
versions (SJ,J∈[3,12]), the smooths of
the original time series corresponding to averaging timescales from 3 h to 21 s. The right panels of
Fig. show the corresponding details
(DJ,J∈[3,11]). As the scale J decreases,
the time series of transmittance details exhibit significant
variability. Large fluctuations are observed in details
D3, D4, and D5, which can be related
to variability in transmission resulting from longer-term
changes in dominant cloud structures and composition
(S12 in Fig. ). A higher number
details do not show this as they examine local-scale
variability in cloud features (D9 to D11).
Note that the MRA results were limited to solar zenith
angle below 75∘ to exclude edge effects. Based
on , the maximum overlap discrete
wavelet transform decomposes the variance of a time
series on a scale-by-scale basis and can be estimated
from the variance (var) of the MODWT coefficients
as given below:
var(TJ)=var(SJ)+∑j=1Jvar(Dj).
This result can be generalized to the calculation of the
correlation, where the wavelet coefficients of two time
series can be used to provide an estimate of the
correlation at a given scale .
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 h) for different sky
conditions: (a) clear – 4 May 2013, (b) cirrus – 16 July 2013,
(c) overcast – 9 June 2013, and (d) broken clouds – 25 April 2013. The
top panels represent the horizon plots of transmittance details.
Middle panels represent the original time series of transmittance.
The t–azi plots of the sky imager at 45∘ elevation angle
are shown in the bottom panels. Shaded gray color in the top and
middle panels of (a) and (d) corresponds to the region with solar
zenith angles > 75∘. A horizon graph is constructed
by dividing a normal line plot into bands defined by uniform value
ranges. The bands are then layered to reduce the chart height.
Negative values (red bands) can be mirrored or offset onto the same
space as positive values (blue bands) such that the colors are
differentiated. These layered bands are nested together. Such a
visualization allows us to identify extraordinary behaviors or
predominant patterns, view changes, interpret each of
the time series independently from the others, and perform
comparisons between the different temporal periods
.
An effective graphical technique to the MRA is the horizon graph
. As illustrative examples, the horizon graphs of the global
transmittance details for different scales (see Table ) from
the PYR76 station are shown in Fig. for days with different sky
conditions: clear (4 May 2013), cirrus (16 July 2013), overcast (9 June
2013),
and broken clouds (25 April 2013). Each row in the top panel of
Fig. includes a different detail of the MRA, while the
middle panel shows the original time series of global transmittance. In
addition, the t–azi plots at 45∘ solar elevation angle are included
as lower panels to illustrate the sky conditions during each observation day.
While the fluctuations in the transmittance at different lower frequencies
can be perceived from the contrasting color bands, significant variability
can be observed in the situations with broken clouds even at high frequencies
corresponding to periods of 1 min or shorter.
Spatial representativeness of point measurements
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 (A) of interest with
1 km × 1 km, 3.2 km × 3.2 km
and 10 km× 10 km. The expected deviation (δ)
between a point measurement and an area-averaged value for a
surrounding domain is calculated as
δJ=(1-γS,J2)⋅var(SJ)+∑j=1J(1-γD,j2)⋅αA,j⋅var(Dj),
where the variance of the transmittance smooths (SJ) and
details (Dj) are obtained from the power spectrum of the point
measurement, and αA (from Eq. A11) is a linear reduction
factor relating the variance of the point measurement (from Eq. A2) to
the variance of an area-averaged time series (from Eq. A8). The
explained variance (i.e., γS,J2 and γD,J2
from Eq. A10) between the point and area-averaged values are
obtained separately for transmittance smooths (SJ) and
details (DJ) for the different spatial and temporal scales.
Then, the expected deviation δJ for each wavelet
detail is calculated based on the explained variance and summed
to yield an estimate of the total variance, accounting for a reduced
temporal variability of the spatially averaged transmittance
by the reduction factor.
Further, the estimated representativeness error of the transmittance
(δT) time series can be converted into a deviation in global
radiation (δG) by multiplication with a fixed value of the
top-of-atmosphere solar irradiance, which avoids the known influence
of changes in solar zenith angle. A fixed value of 680.4 Wm-2
is used here, which is half the solar constant and is taken as an estimate
of the daytime mean value during summer months for the considered
region. This procedure can be adopted to improve photovoltaic power
forecasting models under different sky conditions, especially with broken
clouds, which require absolute values of radiation instead of transmittance.
(a) Wavelet variance, and the (b) cumulative variance (from
Eq. ) of global transmittance from all the pyranometer
stations in the observation domain as a function of considered
frequency/averaging period for cases during the HOPE campaign. As
the time period is inversely proportional to the frequency, the time
periods (on x axis) are represented in ascending order of frequency
scales. The vertical bars around the mean value represent the
observed minimum and maximum variances. The dashed horizontal
line in (a) corresponds to the measurement uncertainty of our
pyranometer. The dashed horizontal lines in (b) denote the total
variance of the original time series averaged across all stations
within the observation domain.
Results and discussionPower spectra of global radiation
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. , the wavelet power spectrum of the global
transmittance is shown together with the cumulative variance (or
standard deviation, from Eq. ) for different sky conditions.
The average power spectrum is obtained by averaging the power
spectra of all the pyranometer stations. The cumulative variance
quantifies the fraction of variance resolved by an observation, which
has been smoothed with a specific averaging period, and is determined
using the spectral power density decomposition given in
Eq. (). It thus gives an indication for how much variability
is lost if averaging is applied to the time series. As the frequency
increases, the variability in global transmittance decreases, irrespective
of the prevailing sky conditions. However, there are clear differences
in the shapes of the power spectra for the different sky conditions.
During situations with broken clouds, the variability of transmittance
is distinctly higher than for all other cases, irrespective of the considered
frequency interval. It is well-known that in the presence of broken clouds,
multiple reflections and scattering events off the sides of clouds and
at the surface lead to significant horizontal photon transport and
strong 3-D radiative effects. For the associated types of low-level
clouds, such as fair weather cumulus or towering cumulus, a high
global transmittance can frequently be observed at the surface
exceeding that of a clear sky, which is usually referred to as
“enhancement effect” . Similar effects also
occur when patches of cirrus or altocumulus clouds are present
in the field of view, but do not obscure the sun.
also demonstrated that the global radiation is very sensitive to cloud
inhomogeneities, in particular for broken-cloud fields due to
contributions from the direct radiation. Overall, for broken-cloud
fields, strong spatial and temporal variations are present over a wide
range of frequencies.
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 s), with a
steep increase up to a time period of 11.25–22.5 min.
Thereafter, the variability is slightly higher and comparable to
that observed for cirrus cloud situations. Under a homogeneous
overcast sky with optically thick clouds, the global radiation is
contained completely in the diffuse component, and the radiance
at the cloud base observed from the ground will be relatively
uniform over time. However, under partly overcast skies, the
global transmittance of clouds is also influenced by multiple
reflections of solar radiation between the surface and the cloud
base, which causes an increased variance. Note that variations
of the transmittance lower than the measurement uncertainty
(±0.0013) of our pyranometer stations are neglected here,
which is the case for higher frequencies corresponding to time
periods below 42 s.
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. b).
For other sky conditions (overcast, cirrus and clear), only a small
decrease in variability (∼ 10 Wm-2) is observed,
if the averaging period is increased from 1 min up to 3 h.
In case of broken clouds, the corresponding decrease is about 3
times (∼ 34 Wm-2) the value observed for other sky
conditions.
Summary of the spectral power density of scalar variables with observed scale regimes and
spectral exponents (β) as obtained by using Eq. () in different studies.
Wavelet variance of global transmittance (as shown in
Fig. a) represented as a function of horizontal scales
denoted by 1/k (in m) for different sky conditions during the HOPE campaign. Dashed lines represent the least-square fits at different scale
regimes using Eq. (). The gray color horizontal line correspond
to the measurement uncertainty of our pyranometer.
Various studies have described the properties of stratocumulus/cumulus
clouds using power spectra (or spectral density, E) of cloud top
height fluctuations, liquid water content (LWC), liquid water path
(LWP), or solar radiation transmission as a function of the horizontal
spectral scale (1/k) with a power-law relationship of the
form :
E(k)≈k-β,
where k is the wavenumber and β is the power-law exponent. The power spectrum of global transmittance
details as a function of horizontal scale (in m) is
shown in Fig. (same as Fig. a
but with a change of x axis). Here, we examine a least-square fit
for single point measurements within the larger context of power
spectrum in Eq. () for the scale covering 5 m to
10 km. The dashed lines correspond to least-square fits
for different scaling regimes under the prevailing sky conditions.
For clear and cirrus cloudy skies, the power-law exponent is
obtained as 0.61 and 0.55, respectively, for the horizontal scales
covering 5 m to 10 km. This indicates that clear
and cirrus cloudy skies resemble a flat and nearly
wavenumber-independent spectrum. In case of overcast
sky, two distinct regimes with scaling exponents of 0.52
(1.5–10 km) and 1.68 (∼5/3; 50 m–1.5 km)
are observed. The occurrence of a scale break suggests
that different physical processes dominate in the two
regimes. While the scaling regime above 1.5 km is
much flatter, the lower scaling regime (< 1.5 km
is indicative of the fluctuations in global transmittance as a
result of advection. For the cases with broken clouds,
two distinctly linear regimes are identified separated
by a wavelength of 200 m. The first regime has a
flat scaling exponent of 0.11 (0.2–10 km), where
as the smaller scaling regime has an exponent of
1.1 (5–100 m), which indicates a linearly dependent
stationary power spectrum. also reported
a “scale-break” at scales of 2–5 m in the spectral
density of LWC indicating a transition of the scaling
regime from β=53 (5 m≤1/k≤ 2 km)
to the one that showed larger variance than expected at
smaller scales corresponding to β≤1
(8–12 cm≤1/k≤ 2–5 m) for the
stratocumulus/cumulus clouds. At smaller scales,
entrainment of environmental air into the clouds changes
the cloud microphysics resulting in an enhancement of
LWC variance. An overview of various scalar fields
with their associated scale regimes and spectral
exponents obtained in different studies is given
in Table .
In Fig. , the wavelet variance for all cloud conditions
is largest at long time periods implying that large-scale cloud structures
with their associated global transmission are important at this scale.
The size distribution of broken cumulus/stratocumulus clouds has
been studied by , which
describes the typical distribution of cloud sizes in terms of their
number density N (=A-C1, where A is the cloud area
and C1 is an exponent determined by a least-square fit). These
studies also point to the importance of low wavenumbers or large
cloud sizes in dominating the variance of the time series of liquid
water and solar radiation transmission, with partly cloudy
skies characterized by few large clouds and many smaller ones.
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 ρ as a function of station distance d
for days with different sky conditions: (a) clear – 4 May 2013 (top row),
(b) cirrus – 16 July 2013 (second row), (c) overcast – 9 June 2013
(third row), and (d) broken clouds – 25 April 2013
(last row). Here, S3 corresponds to the wavelet smooth of global transmittance at 3 h averaging
timescale, while D3 to D9 represent the wavelet details of global transmittance.
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 ρ has been determined as
a function of station distance for the wavelet smooth S3 and the
wavelet details D3 to D9 of global transmittance, and are shown
in Fig. . In this plot, points represent the correlation
coefficient obtained for the individual station pairs. Results are
again shown separately for different sky conditions. The
autocorrelation is generally found to decrease as station distance
and frequency increases, with significant differences notable
depending on sky conditions.
Summary of various parameterizations used for modeling the behavior
of spatial autocorrelation ρ as a function of the station distance d.
Literature referenceAveraging periodsParameterizationRemarks1, 15, 30, 60 min, and dailyρ=a-b⋅dca, b and c are fit coefficients10, 20, 40 and 80 min1, 2, 3 and 4 hρ=(1+dΔt⋅Δv)-1Δt is the time interval, and15 minΔv is the relative cloud speed10×2Js,J∈[0,9]ρ=a+b⋅exp(-dc)a, b and c are fit coefficients15 minρ=1-dbaa and b are fit coefficientsPresent study2-J×86400s,J∈[3,14]ρ=exp[-dab]a and b are fit coefficients
The behavior of the spatial autocorrelation (ρ) as a function of
distance between stations (d in km) is shown as a blue line in
Fig. , and has been modeled by an exponential decay
function as given below:
ρ(d)=exp[-dab],
where a (in km) and b (dimensionless exponent)
represent fit coefficients. If the station distance is negligible
(d→0), then ρ→1 (perfect correlation). Similarly,
if the station distance is infinite (i.e., d→∞), then
ρ→0 (no correlation). We have applied the
Levenberg–Marquardt least-squares fitting technique to
determine the fit coefficients. When the correlation drops
below the e-folding value (i.e., ρ≤1e), the
associated distance between stations is defined as the
decorrelation length. This occurs when the fit coefficient
a equals to the station distance d and thus a is referred
as the decorrelation length. Figure confirms
our expectation that the decorrelation length decreases for
increasing frequencies, following an approximately linear
trend with slightly different slopes and offsets depending
on sky conditions. The RMSE,
which measures the quality of fit has been found to
decrease linearly with decreasing frequency.
Decorrelation lengths a (in km), determined as e-folding
time of the spatial correlation
function, and its dependence on the time period of variations.
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 .
used a linear model to parameterize the
dependence of correlation of the time series of global radiation
measurements at different sites based on their distance of
separation (≲ 100 km). Subsequently, the
same linear function was used to fit the correlation of wavelet
smooths corresponding to the transmittance (from Multi-Filter
Rotating Shadowband Radiometer) and reflectance (from
Meteosat SEVIRI pixels) as a function of distance in the
study by . They observed that the
correlation falls off faster than linear at small distances
due to the exponent (c<1). In a study on the
correlation between the solar power generation of solar cell
inverters, it was shown that the correlation was dependent on the
distance between the inverters, the wavelet timescales, and the
amplitude of daily fluctuations . They used an
exponential decay model with some constraints on the fit coefficients.
The spatial decorrelation of the time series of SEVIRI pixels for its
solar and infrared channels was studied for different cloud amounts as
a function of distance (≲ 200 km) at different
locations over Europe . In the most recent
study, used the model of spatial
correlation using a range of cloud speeds from 2 to 10 ms-1,
and demonstrated that the model is not able to capture
the correlation structure for mixed sky conditions.
In our study, the spatial correlation of transmittance variations
decays faster than linear at small distances as is indicated by the
exponent (b) in Eq. (), and depends strongly on the
type and/or amount of clouds. Small-scale cloud features significantly
decrease the correlation on days with broken clouds. The side reflections
from clouds is strongly enhanced in broken-cloud conditions and could
be important for lowering the correlation . We point
out a likely influence of the cloud speed on the decorrelation length.
Additionally, anisotropy in the decorrelation relative to the
direction of cloud motion is expected, and might influence the
observed relationship . In the following parts of
the paper, the empirically fitted autocorrelation functions are used
to represent the spatial variability at a given temporal frequency
across the observation domain.
Spatial representativeness of a point measurement
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
, and allow us to provide an estimate of
the representativeness error, defined here as deviation of point
measurement from the spatial mean for a considered domain. These
techniques have been previously applied to global surface air
temperature and precipitation measurements, as well as surface networks of
soil moisture observations .
Similarly, the observations from our high-density pyranometer
network can be used to evaluate or quantify the uncertainties
due to small-scale cloud inhomogeneity during validation
studies. In this paper, we utilize the spatial autocorrelation
functions determined in the previous section to calculate the
power spectral density of spatial averages, and the deviation of
spatial averages from point measurements. Thereby, we avoid
averaging of multiple stations to obtain an approximation of a
spatial average, but rely on the assumption that the global
transmittance field within the observation domain is statistically
homogeneous and isotropic, and that its autocorrelation
function follows Eq. (). A concise mathematical
treatment for quantifying the effects of spatial and temporal
averaging is given in Appendix A.
Power spectrum of spatially averaged transmittance as a
function of frequency under different sky conditions: (a) clear,
(b) cirrus, (c) overcast, and (d) broken clouds; var(TD)
denotes the power spectrum of a point measurement of global
transmittance as shown in Fig. a.
In Fig. , the power spectra of area-averaged
transmittances for different domain sizes are compared to those of a
point measurement. They generally follow a similar trend compared to
the point measurement irrespective of sky conditions, but show a
stronger decrease of variability with increasing frequency and area,
which illustrates that spatial averaging acts as low-pass filter. At
lower frequencies corresponding to time periods of 1.5–3 h,
only minor differences in variability are notable for time series for
spatially extended domains and point observations, at least for the
range of domain sizes considered here. However, at higher frequencies,
the variability of the spatially averaged global transmittance is
significantly reduced compared to that of the point observation,
irrespective of the prevailing sky condition. Again, the variance of
spatially averaged transmittance is observed to be higher under broken
clouds at all frequencies and spatial resolutions, compared to that
for clear, cirrus, and overcast conditions.
At 10 km× 10 km and for variations corresponding to
time periods of 1.5–3 h, the spatially averaged variance is
lower by 10 (clear), 16 (cirrus), 18 (overcast), and 38 % (broken
clouds) than the variability observed by a single station. Even for a
domain size of 1 km× 1 km, the spatially averaged
variance is 2–4 % lower than the variability obtained by a point
measurement for the considered sky conditions.
The e-folding times (min) for the explained variance
between the point measurement and area-averaged values under
different sky conditions.
Explained variance (γD2, the square of the
cross-correlation) between the wavelet details of the point
measurement and the area-averaged values of global transmittance as
a function of their time period and for different domain sizes and
sky conditions: (a) clear, (b) cirrus, (c) overcast, and (d) broken
clouds. The black symbols respectively denote the explained
variance (γS32) of wavelet smooths S3
(3 h) corresponding to the domain size. The black dashed
horizontal line in each of the sub-figures represent the e-folding
time of e-1=0.368 and the dashed vertical lines corresponds
to the decorrelation period for the selected domain
sizes.
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 (γD2), given by the square of the
cross-correlation between the time series at a single station and its
area-averaged counterpart, is used here and shown in
Fig. for different sky conditions, including its
dependence on the temporal scales of averaging for the three domain
sizes. The explained variance is thus used here as a measure to
quantify the synchronicity of variations, while the power spectrum
quantifies their mean amplitude. An exponential decay of the explained
variance is observed as the temporal frequency increases for all
spatial domain sizes. As expected, the deviation between a single
station and an area average becomes larger at higher frequencies and
for larger spatial areas. Consequently, the variations observed at
a single station should no longer be used to predict the variations of
the area-averaged transmittance at higher frequencies and for larger
domains. Further, the explained variance between the wavelet smooth
S3 (3 h) of the point measurement and the area-averaged
values of global transmittance is insensitive to the domain sizes.
The decorrelation times for which the point and area-averaged
variations become essentially uncorrelated is defined here by the
e-folding value of e-1 (= 0.368) for the correlation, and are
listed in Table for the different sky conditions. The
e-folding time of 6 min indicates that variations with
frequencies higher than 1/6min-1 are more or less
uncorrelated between the point measurement and a spatial area
of 1 km× 1 km. It should also be noted that the spatial
average has a significantly lower power spectral density at these
frequencies. We thus think these variations associated with small-scale fluctuations in clear-sky
turbidity and only evident in the point measurements are possibly induced by small-scale
structure in water vapor and/or aerosols. However, we cannot rule out that
such variability corresponds to undetected small clouds or even
measurement artifacts such as shading of the instruments by birds.
Mean deviation between point measurement and spatial averages of global transmittance
(δT) and corresponding global radiation (δG, in Wm-2)
for different averaging time periods and domain sizes.
Area-averaging error in the global transmittance (δT) and
corresponding global radiation (δG, in W m-2) with different sky conditions: (a) clear, (b) cirrus,
(c) overcast, and (d) broken clouds
for different domain sizes represented as a function of averaging
time periods. The dashed horizontal lines correspond to the maximum
deviation observed for the different domain sizes. Dashed vertical lines
represent the minimum-averaging time above which the area-averaging
errors are less sensitive at different spatial resolutions.
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. for different sky
conditions. It is generally observed that the representativeness error
increases with the size of the spatial domain, and decreases for
longer averaging periods. Also, the error converges against a limit
value at high frequencies, indicating that the contribution of
high-frequency variability beyond the frequency range considered here
only causes a negligible further increase of the representativeness
error.
Table provides a quantitative estimate of the
deviations of both global transmittance and corresponding global
radiation for three different domain sizes, three averaging periods, and for
different sky conditions. As the averaging frequency interval and
domain size increases, the deviation between point measurement and
corresponding area averages increases, irrespective of the prevailing
sky condition. As expected, the range of deviations for both long
(3 h, S3) and short (5.25 s, including D13)
averaging periods is the largest for broken clouds.
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 % (∼ 4 Wm-2) regardless of spatial domain.
Power spectrum of the (a) direct, (b) diffuse, and
(c) global transmittance as a function of temporal frequency for
a clear-sky (4 July 2015), overcast (21 June 2015), and a broken
cloudy-sky (17 June 2015) conditions observed during the
HOPE Melpitz experiment. The black dashed
horizontal line in (c) represents the combined measurement
uncertainty of the pyranometer system.
The range of deviations of a point observation under cirrus clouds is
found to be around 1.6 % (∼ 11 Wm-2 for a
1 km× 1 km) domain, and 2 % (∼ 14 Wm-2 for both 3.2 km × 3.2 km and
10 km× 10 km) domains, and for 3 h temporal averaging. A
strongly increasing linear trend of the deviations is found from a
reduction of averaging period, indicating that small-scale changes of
cirrus cloud properties resulting from microphysical, dynamical, and
radiative processes can be removed effectively by sufficiently long
temporal averaging. investigated the structure and
lifetime of cirrus clouds using model simulations, and concluded that
radiation together with latent heating leads to much more dynamic and
inhomogeneous clouds.
During overcast skies, the representativeness error again increases
substantially with increasing domain size, doubling and tripling its
magnitude when going from 1 km× 1 km to domain sizes of
3.2 km × 3.2 km and 10 km× 10 km for
short averaging periods. While the small deviations for a
1 km× 1 km domain below 1 % indicate that the
hemispheric nature of a pyranometer measurement is able to resolve
variability at the kilometer scale well, large-scale variations in
cloud optical properties lead to deviations up to 3.3 % in
transmittance or 22.6 Wm-2 for a 10 km× 10 km
domain.
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 % (∼ 31.1–78.3 Wm-2)
over spatial areas ranging from 1km × 1 km to 10 km× 10 km.
Again, deviations decrease strongly with increasing averaging period
by more than 50 % for 3 h averaging. An interesting observation
is that the representativeness error at different spatial resolutions
seems to converge for 1 h or longer averaging periods.
reported that cumulus cloud inhomogeneity
gave rise to an instantaneous error in global radiation of up to
40 Wm-2 or even higher at different solar zenith angles.
This well-known “broken-cloud effect” arises from variability in the
direct and diffuse radiation (based on solar position), and can lead
to an enhancement of global radiation above clear-sky conditions.
As a result, large inconsistencies can occur for collocated satellite
and surface measurements during broken cloudy conditions.
Similarly, reported significant deviations from
1-D radiative transfer due to horizontal photon transport if the
horizontal dimensions of a considered atmospheric
column are decreased. provided further evidence for
the relevance of 3-D radiative effects through the observed anisotropy
in the reflected solar radiation, which increasingly deviates from 1-D
radiative transfer if the spatial resolution of the satellite is
increased. reported that at 1 km scale, the
errors associated with horizontal photon transfer and the plane-parallel approximation cancel at least to some degree for stratiform
boundary layer clouds.
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. can be estimated. Nevertheless, significant
deviations cannot always be avoided, but should be quantified, for
example using the methodology introduced in this paper.
Power spectra of direct and diffuse irradiance
Variability in global radiation results from the combined variability
of the direct and diffuse radiation components. During the
HOPE Melpitz campaign (May–July 2015; ),
two EKO ML-020VM pyranometers were operated in close proximity,
using a sun tracker and shading to obtain the diffuse radiation from
one of the instruments, and to study this aspect in more detail.
Power spectra of the global, direct, and diffuse transmittance are
shown in Fig. , and allow for an assessment of the individual
contributions. It is evident from the plots that the spectra for the direct
and global components are very similar for all sky conditions. The
spectral power density resulting from variations of the diffuse transmittance
is lower, and only contains significant variations at low frequencies,
again a conclusion valid for all sky conditions. Please note that the
large variability in direct horizontal transmittance also in overcast
conditions is due to our classification. In particular, even on the days
classified as overcast, some periods with significant direct irradiance
due to cloud gaps were observed and evidently dominate the power
spectrum of the global transmittance.
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.
also demonstrated that the global radiation is very sensitive to cloud
inhomogeneities, in particular for broken-cloud fields due to contributions
from the direct radiation. A more thorough investigation of the differences
of the power spectra of the direct, diffuse, and global radiation components
is planned for the future.
Summary and conclusions
A unique data set of global radiation observations has been collected
using a dense network of pyranometer stations
during the HOPE Jülich campaign , and is
analyzed in this paper to characterize the small-scale spatiotemporal
variability of the global radiation field. The individual time series have
been subjected to a multiresolution analysis based on the Haar wavelet
following the methodology of . Characteristic
properties have been identified from this analysis for clear-sky, cirrus,
overcast, and broken-cloud conditions. Power spectra for the individual
time series and the spatial autocorrelation function are presented. A
method has been introduced to assess the representativeness of the
time series of a point measurement compared to results for a larger
area centered around the measurement location. This method allows one
to determine the optimal accuracy that can be achieved for the validation
of satellite products for a given pixel footprint, or the evaluation of an
atmospheric model with a given grid cell resolution. The present study
is representative for mid-latitude summer conditions and the results may
not be applicable to other regions such as the tropics characterized by
local convection, large cumulonimbus clouds, and weaker regional winds.
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 % ∼ 7 Wm-2)
even at high frequencies below 1 min-1. This finding is
noteworthy as a recommendation for the operation of Baseline Surface Radiation Network (BSRN stations, which only require one to store 1 min
averages , thereby missing significant
amounts of variability.
The spatial autocorrelation between stations decreases
strongly with increasing frequency. Variations at different points
separated by more than 1 km are completely uncorrelated
for higher frequencies above 1/3min-1. The decorrelation
lengths decrease linearly with increasing frequency (on a log–log scale)
and a distinct dependence on cloud and sky conditions was not observed
(see Fig. ).
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 km× 10 km, 80–90 % for 3.2 km× 3.2 km,
and 55–80 % for 1 km× 1 km) at higher frequencies
(∼ 1 min-1) and for larger domains. Variations between
the spatial average and the point measurement are not correlated
at high frequencies. As a consequence, only a small fraction of the
high-frequency variability observed in a point measurement can be
found in area-averaged (e.g., satellite, model, reanalysis) data.
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 Wm-2
for a grid box of 10 km× 10 km corresponding to an
averaging time period of 5.25–10.5 s (D13) during broken-cloud conditions.
For a comparison of time series of a point measurement with that of a spatially averaged value,
the power spectrum of the point measurement should be adjusted to match that of the spatial average to ensure best
correspondence. A low-pass filter should be applied to remove high frequencies for which the correlation drops below
a certain threshold. To determine this threshold, the autocorrelation function
has to be known, which, however, depends on the prevailing sky condition.
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
e.g.,, to separate retrieval
uncertainties from the inherent uncertainty arising from the limited
representativeness of one data set for the other. Based on the results
presented here, it is important to explicitly take into account the
sky condition including their occurrence frequencies in the
validation, as the representativeness error is situation dependent and
will therefore influence the validation statistics. A classification
of sky conditions based on the observed power spectrum seems promising
given the distinct features described above. However, the dependence
of power spectra on cloud cover and solar elevation warrants further
investigation. Finally, the pyranometer network observations include
temperature measurements, allowing one to study the correlation of
variability in irradiance and temperature.
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 ,
and thereby can contribute towards improved climate predictions. The
spatial- and temporal-scaling properties of atmospheric transmittance
are closely linked to those of the cloud fields. Significant
deviations of modeled and observed values can thus be attributed to
deficiencies in the simulation of clouds and their interaction with
solar radiation e.g.,. Towards
this goal, it is also important to clarify to what extent 3-D radiative
effects contribute to such deviations. A radiation closure study using
reconstructed 3-D cloud distributions based on observations (see,
e.g., ) as input to a radiative transfer code
e.g., could be an essential step
towards this, and is planned for the future.
Data availability
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
https://icdc.cen.uni-hamburg.de/index.php?id=samd.E2809D.
Spatial representativeness of a point time series
Let Ψ(x,t) represent the time series of a point measurement at point x in the observation domain of
interest. The following statistical parameters are defined for this time series:
The mean of the time series at x is given by
Ψ(x,t)‾=E[Ψ(x,t)].
The variance of the time series at x is given by
var(Ψ(x,t))=E[(Ψ(x,t)-Ψ(x,t)‾)2].
The covariance of any two time series at xi
and xj is given bycov(Ψ(xi,t),Ψ(xj,t))=E[(Ψ(xi,t)-Ψ(xi,t)‾)⋅(Ψ(xj,t)-Ψ(xj,t)‾)].
The autocorrelation ρ between any two time series at xi and xj is given byρ(Ψ(xi,t),Ψ(xj,t))=cov(Ψ(xi,t),Ψ(xj,t))var(Ψ(xi,t))⋅var(Ψ(xj,t)).
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:
Ψ(x,t)‾=Ψ‾ for all x and t, and
var(Ψ(x,t))=C (i.e., with C constant for all x and
t);
isotropy:
cov(Ψ(xi,t),Ψ(xj,t))=f(d(xi,xj)),
where d is the distance between the stations, and f is a positive function defined for d>0.
By adopting the above assumptions in Eq. (A4), the autocorrelation ρ becomesρ(Ψ(xi,t),Ψ(xj,t))=E[(Ψ(xi,t)-Ψ‾)⋅(Ψ(xj,t)-Ψ‾)]E[(Ψ-Ψ‾)2]=cov(Ψ(xi,t),Ψ(xj,t))var(Ψ)=ρ(d(xi,xj)).
Therefore, the autocorrelation ρ is a function of the distance d between xi and xj.
For a spatial area A, the area-averaged time series is obtained as
ΨA(t)=1A∬AΨ(x,t)dx.
The following statistical parameters are found for the area-averaged time series:
The mean of the area-averaged time series is given by
ΨA(t)‾=E[ΨA(t)]=Ψ‾.
The variance of the area-averaged time series is given
byvar(ΨA)=E[(ΨA-Ψ‾)2]=E[1A(∬A(Ψ(xi,t)-Ψ‾)dxi)⋅1A(∬A(Ψ(xj,t)-Ψ‾)dxj)]=1A2⋅∬A∬AE[(Ψ(xi,t)-Ψ‾)⋅(Ψ(xj,t)-Ψ‾)]dxidxj=1A2⋅∬A∬Acov(Ψ(xi,t),Ψ(xj,t))dxidxj=var(Ψ)⋅[1A2⋅∬A∬Aρ(d(xi,xj))dxidxj].
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 ρ.
Now, the statistical parameters between the time series centered in the domain and the area-averaged time series for the
domain area A are given below:
The covariance of the time series Ψ and the area-averaged value ΨA is given by
cov(Ψ,ΨA)=E[(Ψ-Ψ‾)⋅(ΨA-Ψ‾)]=E[(Ψ(x,t)-Ψ‾)⋅1A(∬A(Ψ(xi,t)-Ψ‾)dxi)]=1A⋅∬AE[(Ψ(x,t)-Ψ‾)⋅(Ψ(xi,t)-Ψ‾)]dxi=1A⋅∬Acov(Ψ(x,t),Ψ(xi,t))dxi=var(Ψ)⋅[1A⋅∬Aρ(d(x,xi))dxi].
The square of the cross-correlation γA (or explained variance) of the time series
centered in the observation domain Ψ and the area-averaged value ΨA is obtained as
the ratio of the square of the corresponding covariance to the product of their individual variances (using Eqs. A8 and A9).
In order to quantify the variance of the difference between the time series Ψ and the area-averaged value ΨA,
we assume that the variance of the area-averaged time series is linearly related
to the variance of the point time series with an optimal filter, αA (see Eq. A8) defined as follows:
var(ΨA)=αA⋅var(Ψ).
Now, the variance of the difference between the point time series Ψ and the area-averaged time series ΨA
(i.e., the unexplained variance) is given by
var(Ψ-ΨA)=var(ΨA)+var(Ψ)-2⋅cov(ΨA,Ψ)=var(ΨA)+var(Ψ)-2⋅γA⋅var(ΨA)⋅var(Ψ)=(αA+1)⋅var(Ψ)-2⋅γA⋅αA⋅var(Ψ)=[αA+1-2⋅γA⋅αA]⋅var(Ψ).
Expressing Eq. (A12) in terms of the standard deviation, the area-averaging error δ between the point time
series Ψ and area-averaged time series ΨA can be obtained as follows:
δ(Ψ-ΨA)=(αA+1-2⋅γA⋅αA)⋅δ(Ψ).
Alternately, we define a damped time series Ψ′ as the representative variability at a single station and as given below:
Ψ′=αA⋅(Ψ-Ψ‾)+Ψ‾.
The above Eq. (A14) implies that
var(Ψ′)=αA⋅var(Ψ).
The variance of the difference between the point time series Ψ′ and the area-averaged time series ΨA is then given by
var(Ψ′-ΨA)=var(ΨA)+var(Ψ′)-2⋅cov(ΨA,Ψ′)=αA⋅var(Ψ)+αA⋅var(Ψ)-2⋅γAvar(ΨA)⋅var(Ψ′)=2⋅αA⋅var(Ψ)-2⋅γAαA⋅var(Ψ)⋅var(Ψ′)=2⋅αA⋅(1-γA)⋅var(Ψ).
Expressing the Eq. (A16) in terms of the standard deviation, the area-averaging error δ between the point time
series Ψ and area-averaged time series ΨA can be obtained as follows:
δ(Ψ′-ΨA)=2⋅αA-2⋅αA⋅γA⋅δ(Ψ).
Comparing Eqs. (A13) and (A17), we find δ(Ψ′-ΨA)<δ(Ψ-ΨA) as γA≤1.
The authors declare that they have no conflict of interest.
Acknowledgements
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)2 project (grant no. 01LK1212C). We thank our colleague Fabian Senf, and Piet Stammes
from KNMI, the Netherlands, for their valuable comments and suggestions during various stages of this manuscript. We also thank
John Kalisch, Sebastian Bley, Daniel Merk, Felix Dietzsch, Timo Hanschmann, Michael Eickmeier, Anja Hünerbein, Felix Peintnet,
Alexander Graf (FZJ), and Ronny Badeke for extending their support during the HOPE campaigns. Daily TSI time–azimuth data
were provided by Jan H. Schween from the group of Susanne Crewell at the University of Cologne, Köln.
We appreciate the support provided by Alexander Los and Kees Hogendijk from EKO Instruments, the Netherlands.
Edited by: S. Buehler
Reviewed by: M. Nunez and one anonymous referee
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