ASRAdvances in Science and ResearchASRAdv. Sci. Res.1992-0636Copernicus PublicationsGöttingen, Germany10.5194/asr-14-35-2017Do modelled or satellite-based estimates of surface solar irradiance accurately describe its temporal variability?BengulescuMarcmarc.bengulescu@mines-paristech.frBlancPhilippehttps://orcid.org/0000-0002-6345-0004BoilleyAlexandreWaldLucienhttps://orcid.org/0000-0002-2916-2391MINES ParisTech, PSL Research University, Centre for Observation, Impacts, Energy, CS 10207, 06904 Sophia Antipolis CEDEX, FranceTransvalor SA, 694 Avenue du Dr. Maurice Daunat, 06255 Mougins CEDEX, FranceMarc Bengulescu (marc.bengulescu@mines-paristech.fr)21February201714354814December20168February201713February2017This 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://asr.copernicus.org/articles/14/35/2017/asr-14-35-2017.htmlThe full text article is available as a PDF file from https://asr.copernicus.org/articles/14/35/2017/asr-14-35-2017.pdf
This study investigates the characteristic time-scales of variability found
in long-term time-series of daily means of estimates of surface solar
irradiance (SSI). The study is performed at various levels to better
understand the causes of variability in the SSI. First, the variability of
the solar irradiance at the top of the atmosphere is scrutinized. Then,
estimates of the SSI in cloud-free conditions as provided by the McClear
model are dealt with, in order to reveal the influence of the clear
atmosphere (aerosols, water vapour, etc.). Lastly, the role of clouds on
variability is inferred by the analysis of in-situ measurements. A
description of how the atmosphere affects SSI variability is thus obtained on
a time-scale basis. The analysis is also performed with estimates of the SSI
provided by the satellite-derived HelioClim-3 database and by two numerical
weather re-analyses: ERA-Interim and MERRA2. It is found that HelioClim-3
estimates render an accurate picture of the variability found in ground
measurements, not only globally, but also with respect to individual
characteristic time-scales. On the contrary, the variability found in
re-analyses correlates poorly with all scales of ground measurements variability.
Introduction
The Sun is of the utmost importance for the planet Earth. Not only does it
play a central role in our solar system, but it also represents the main
power source for the Earth, being the main driver behind weather and climate
. As such, the Global Climate Observing System (GCOS)
has defined the surface solar irradiance (SSI) as an Essential Climate
Variable (ECV), used for the characterization of the state of the global
climate system and for long-term climate monitoring (GCOS2010).
The SSI is known to exhibit variations on a large dynamic range with respect
to both time and geographical position. This is due to a wide array of factors.
Some of these operate at long time-scales, from decades to millennia and
beyond, and are related to the stellar variability of the Sun
, or to changes in the orbital parameters of the Earth,
e.g. obliquity of the ecliptic, eccentricity of the orbit, and axial
precession . The revolution of the Earth around the Sun gives
rise to the yearly cycle. Another well known factor is the rotation of the
Earth around its own axis, that yields variations in the SSI with a period of
1 day. Atmospheric effects such as scattering and absorption, mainly due to
O2 and stratospheric O3, clouds, and aerosols
may exhibit even shorter time-scales. Orographic
factors, such as shadowing effects due to relief, influence the SSI on a
yearly to very short time-scale. The variability of the SSI at different
time-scales is, thus, a leitmotif of this study.
In order to analyse the temporal variability of the SSI, measurements of this
physical quantity are needed. A primary way of obtaining such data is
recording the values of the SSI at ground stations by using pyranometers or
pyrheliometers. Nevertheless, even sources of high quality solar radiation
measurements, such as the Baseline Surface Radiation Network (BSRN), a
worldwide radiometric network providing accurate readings of the SSI at high
temporal resolution , are spatially very sparse, capture
only temporal variability on a very limited set of locations and, in
addition, have been found to have a data gap percentage ranging from 4.4 to
13 % . Furthermore, measurements from point sources can
only reveal information about the temporal variability of the solar radiation
at one particular site.
In practice, information about the SSI is often required at geographical
locations different from any measuring station. But extending the
representativity of ground station measurements to surrounding areas cannot
be applied to regions where the physical and/or climatological distance
between stations is large . Using the nearest-neighbour
approximation, have found that the relative
root-mean-square error (relative RMSE) of the estimates is proportional to
the square root of the distance between sites.
Another option is to make use of satellite based methods, which are a good
supplement in long term solar resource assessment. have
shown that when the distance from a station exceeds 50 km, in the case of
daily aggregates, or 34 km for hourly data, – estimate
this as low as 20 km – satellite-derived values of the SSI are more
accurate than estimating using the nearby measuring point. Thus, given the
scarcity and spatial sparsity of long-term ground measurements of the SSI,
satellite-derived estimates of surface solar radiation remain a good
complement to ground station data .
Yet another possibility for estimating the solar radiation at ground level is
provided by global atmospheric re-analyses from numerical weather models. The
main benefits are the wide, even global, coverage and the spanning of
multi-decennial time periods. However, some authors have found a large
uncertainty relative to satellite-based irradiance estimates and advise
against using data from re-analyses . Others
nevertheless find SSI datasets from re-analyses to be suitable for
photovoltaic applications . Efforts to improve the
adaptation of re-analyses solar radiation datasets to a specific geographical
site are also ongoing .
In this context, we investigate and analyse here the temporal variability in
time-series of daily means of SSI for two geographical locations, at
different time-scales, as found in the outputs of different models, satellite
estimates, re-analyses, and ground measurements. To gain better insight into
the causes of variability of the SSI, we follow the downwelling solar
shortwave irradiance along its path through the atmosphere towards the
surface. The modelled top of the atmosphere (TOA) solar irradiance is first
analysed as a clean input signal, devoid of any atmospheric perturbations, in
order to reveal the natural variability of the exo-atmospheric solar input.
To account for variability owing to atmospheric effects such as scattering or
absorption due to water vapour or aerosols, but excluding any influence of
clouds, the output of a clear-sky (i.e. cloud-free) model of the SSI is
scrutinized. The role of clouds on variability is lastly inferred by
analysing pyranometric ground measurements. The fitness for use of satellite
estimates and re-analyses data is then assessed, by comparison with the measured data.
The novelty of our work stems from the fact that, unlike previous studies
where global statistical indicators are employed , here we
decompose the datasets into their constituent characteristic time-scales
before doing the analysis. To this end, we employ the adaptive, data-driven
Hilbert–Huang Transform (HHT) , which has been shown to be a
good analysis method on solar radiation datasets . In this way, the fitness
for use of the modelled and estimated SSI can be assessed not only at a
global, whole dataset level, but also on an per time-scale basis.
The study develops as follows. Section describes the
datasets and the analysis technique. Results are presented in
Sect. , discussion thereof being deferred to
Sect. . The conclusions and outlook are given in
Sect. . The availability of the software code and the
datasets needed to reproduce our findings is indicated in
Sects. and , respectively.
Data and methodsData
The data consists of multiple time-series of daily means of solar irradiance
corresponding to two geographical locations in Europe: Vienna, Austria
(48.25∘ N; 16.35∘ E; elevation 203 m), and Kishinev,
Moldova (47.00∘ N; 28.82∘ E; elevation 205 m). The temporal
coverage of the data is 9 years, from 1 February 2004 to
31 January 2013. The number of samples in each dataset is
9 × 365 = 3285 days (leap days are omitted). The datasets for
Vienna (VIE) are plotted in Fig. . A similar plot
for Kishinev (KIV) is provided in the Supplement.
The six solar irradiance time-series for VIE investigated in this
study, spanning 1 February 2004 to 31 January 2013. From top to bottom: TOA,
McClear, ERA, MERRA2, HC3v5, and WRDC. Each point corresponds to a daily mean
of irradiance. Time markers on the abscissa indicate the start of the
corresponding year.
Six datasets are used for each location:
modelled exo-atmospheric irradiance;
modelled clear-sky irradiance at ground level;
pyranometric measurements of the SSI;
Meteosat satellite-based SSI estimates;
radiation products from the ERA-Interim re-analysis;
radiation products from the MERRA2 re-analysis.
Data availability and Digital Object Identifiers (DOIs) are indicated in Sect. .
The top of the atmosphere (TOA) solar irradiance time-series has been
generated using the SG2 algorithm . This dataset is built
using the constant value of 1367 W m-2 for total solar irradiance,
though it could have made use of satellite measurements of this quantity.
Recently, the yearly mean of the total solar irradiance has been revised to
1361 W m-2 or 1362 W m-2. This discrepancy does not impact the validity of our analysis.
The dataset of downwelling surface solar irradiance, under clear-sky
conditions (i.e. cloud-free), is generated using the McClear model
. The McClear model is part of the Copernicus Atmosphere
Monitoring Service (CAMS) and its inputs comprise 3 h estimates of the
aerosol properties and total column contents of water vapour and ozone also
provided by CAMS.
Estimates of the SSI derived from Meteosat satellite imagery by the
Heliosat-2 method, as described by and modified by
, are obtained from the HelioClim-3 (HC3v5) database
. The daily means of TOA, McClear, and HC3v5 irradiance were
obtained directly from the SoDa website (http://www.soda-pro.com).
Pyranometric ground measurements of the daily SSI were obtained from the
World Radiation Data Centre (WRDC) for the two stations.
No detailed information on data quality is provided except for a quality
flag. Considering the high quality of maintenance of these stations, we
consider that the data obey the good quality level set by the World
Meteorological Organization (CIMO2014) which specifies a 5 %
uncertainty, expressed as the percentile 95 of the deviations (P95). If a
normal law is assumed for the deviations, then there is a 0.3 % chance that a
deviation exceeds 1.5 times P95, i.e. 7.5 % of the SSI. Given the global
means of SSI of the WRDC datasets, both greater than 135 W m-2,
the 7.5 % uncertainty is 10.2 W m-2 for VIE and 11.3 W m-2 for KIV.
The ERA-Interim product “Surface Solar Radiation Downwards” (ECMWF2009),
from 2004 to 2014, was retrieved using the
ecmwfapi python library on the Meteorological Archival and Retrieval
System (MARS). The raw ERA-Interim data is a forecast of accumulated SSI
expressed in J m-2, and has a spatial resolution of
0.75∘× 0.75∘. Both the H+12 forecast from 00:00 UT re-analysis and
from 12:00 UT re-analysis were summed and then divided by 24 to obtain a daily
SSI value in W m-2 for the 4 nearest points around the location.
These values were then bi-linearly interpolated at the exact location.
The 1 h radiation diagnostics M2T1NXRAD from MERRA2 have been extracted for
the four nearest points and bi-linearly interpolated to generate the time
series at the exact location. The MERRA2 data are in W m-2
directly and have a spatial resolution of 0.5∘× 0.65∘
in latitude and longitude. Values were then summed over each day and divided
by 24 to obtain a daily SSI value in W m-2.
The Hilbert–Huang transform (HHT)
The goal of the study at hand is to first decompose the scrutinized
time-series into uncorrelated sub-constituents that have distinct
characteristic time-scales. Analysis then ensues at each distinct scale of
intrinsic variability. These time-scales, or characteristic periods, are
nothing more than the inverse of the frequency of the various processes from
which the data stems. As such, analysis techniques that depict the changes
with respect to time of the spectral content of a time-series are to be
favoured, since they enable both the identification of periodicities and the
following of the dynamic evolution of the processes generating the data. For a
review of such regularly employed methods in geophysical signal processing
see .
The non-linear and non-stationary characteristics of the SSI
are also worth consideration. Handling such data issued from the non-linear
interaction of physical processes, often also found under the influence of
non-stationary external forcings calls for an adaptive data analysis approach
. The ideal analysis technique must make no a priori
assumptions regarding the character of the data, i.e. neither linearity, nor
stationarity should be presumed, since the nature of the processes that have
generated the data is not usually known before the analysis is carried out.
Adaptivity to the analysed data is also a desirable feature, i.e. letting
the data itself decompose onto a set of basis functions determined by its
local characteristic time scales, instead of a projection onto a predefined
set of patterns. This ensures that the extracted components carry physical
meaning, and that the influence of method-inherent mathematical artefacts on
the rendered picture of temporal variability is kept to a minimum .
As such, this study employs the Hilbert–Huang Transform, an adaptive,
data-driven analysis technique. The HHT is ideally suited for non-linear and
non-stationary data and it adaptively decomposes any time-series into basis
functions derived from the local properties of the data .
The method is used here to extract, depict and analyse the characteristic
time-scales of variability of solar irradiance time-series. The data analysis
method operates in the time domain and makes no beforehand assumptions
regarding the analysed dataset (stationarity or linearity). The method is
also adaptive, letting the data decompose itself onto a finite number of
locally derived data-driven basis functions , in contrast with
the Fourier or wavelet transforms that impose a predefined set of functions
for the decomposition, such as trigonometric functions or wavelets
. Further details on how the HHT compares to other spectral
methods, the Fourier or wavelet transforms included, can be found in
and . A case study comparing the HHT and the
wavelet transform, as applied to surface solar radiation data, is offered in
.
The HHT consists in two steps, the empirical mode decomposition (EMD),
followed by Hilbert spectral analysis (HSA), both detailed hereafter.
Empirical mode decomposition (EMD)
The EMD is algorithmic in nature, and iteratively decomposes data into a
series of oscillations; within a series, oscillations have a common local
time-scale, called Intrinsic Mode Function (IMF). An IMF is a function that
satisfies two criteria: (1) its number of zero crossings and number of
extrema differ at most by one; (2) at any point, the mean value of its upper
and lower envelopes is zero. The theoretical signal model for IMFs is an
amplitude modulation–frequency modulation (AM–FM) one. Given the adaptive
nature of the EMD, the IMFs represent the basis functions onto which the data
is projected during decomposition. Any two IMFs are locally orthogonal for
all practical purposes, however, given the empirical nature of the method no
theoretical guarantee can be provided. In practice, it is found that the
relative difference between the variance of the input signal and the sum of
variances of the IMFs (i.e. the spectral leakage) is typically less than
1 %; only for extremely short data ranges does the leakage increase to 5 %,
comparable to that of a collection of pure trigonometric functions having the
same data length . By design, IMFs have a well behaved
Hilbert transform . The EMD can be sketched as follows:
let r(t) hold the data, initialize IMF counter k=1;
let h(t)←r(t)
find the minima and maxima of h(t);
interpolate minima to find lower envelope: L(t);
interpolate maxima to find upper envelope: U(t);
find mean of envelopes: m(t)←L(t)+U(t)2;
substract the mean: h(t)←h(t)-m(t);
if h(t) is not an IMF, repeat step 3;
store IMF: ck(t)←h(t)
update the residual: r(t)←r(t)-ck(t)
if r(t) is not monotonic, increment k and go to step 2;
return IMFs ck(t), k∈{1, …, N} and residual r(t).
Step 3 is called the sifting loop and it controls the filter character of the
EMD. An infinite number of sifting iterations would asymptotically approach
the result of the Fourier decomposition (i.e. constant amplitude envelopes)
. have shown the wavelet-like dyadic
filter bank character of the EMD and have found that this
dyadic property is enforced by keeping the number of sifting iterations
small, around 10, which also assures maximum component separation and minimum
leakage. The IMFs can also be shown to satisfy the envelope–carrier
relationship, thus guaranteeing the existence of a unique true intrinsic
amplitude function and of a unique phase function .
After all the IMFs are extracted, what is left of the data is called a trend
or residue, which can no longer be considered as an oscillation at the span
of the data. have shown that, for time-series of
daily means of solar irradiance, the trend approximates the yearly mean. More
generally, the trend can be interpreted as low-pass filtered version of the
data , therefore it is excluded from this analysis. To
illustrate the EMD process, the IMFs obtained from the decomposition of the
WRDC time-series for VIE are plotted in Fig. . The modes
IMF1…IMF10 and the residual (Res.) are plotted as YZ slices along the
x axis, with time running on the y axis, and amplitude on the z axis. The
zero-centred oscillatory nature of the IMFs can be clearly seen, as well as
the local time-scale increase with mode number. The IMF1 has a mean time-scale of
3.0 days and exhibits a large variability in time. As the IMF rank increases,
the time-scale increases and the variability decreases. The exception is for
IMF7 at 367 days which exhibits the greatest variability, as discussed later.
Edge effects in the EMD appear because of oscillations of the interpolating
splines and are usually contained within a half-period of a component at data
boundaries .
The IMFs obtained by decomposing the WRDC time-series for VIE,
plotted as YZ slices, with the x axis denoting their number
(IMF1…IMF10) and mean time-scales in days. For reference, the residual
(Res.) is also included. The amplitude, or irradiance, of each IMF is plotted
on the z axis, and time runs on the y axis. Time markers denote 1 January
of the corresponding year.
This study uses a modified version of the original EMD algorithm, the
Improved Complete Ensemble Empirical Mode Decomposition, introduced by
. This enables the exact decomposition of the data,
i.e. the sum of all IMFs, including the trend, reconstructs the original
time series, and is more robust with respect to noise. To decrease
computation time, the fast EMD routine proposed by is
employed. See Sect. for code availability.
Hilbert spectral analyis (HSA)
Once the EMD has decomposed the data into IMFs, the last step of the HHT
consists in the Hilbert spectral analysis. For each IMF ck(t) its Hilbert
transform is computed as given by Eq. (), where k designates
the kth IMF, and P indicates the Cauchy principal value.
σk(t)=Hck(t)=1πP∫-∞∞ck(τ)t-τdτ
The pair can then be used to construct the complex-valued analytic signal
proposed by , described by an amplitude modulation–frequency
modulation (AM–FM) model, as in Eq. ().
zk(t)=ck(t)+i⋅σk(t)=ak(t)⋅ei⋅θk(t)
In the AM–FM model, the instantaneous amplitude is given by Eq. ().
ak(t)=ck2(t)+σk2(t)
The instantaneous phase can be derived from Eq. ().
θk(t)=tan-1σk(t)ck(t)
The instantaneous frequency, i.e. the inverse of the local time-scale, is
then just the first time derivative of the instantaneous phase, as in Eq. ().
ωk(t)=12πdθk(t)dt
The role of HSA is to decompose each IMF into two time-varying components,
namely instantaneous amplitude and instantaneous frequency, in order to
determine, in a time-dependent manner, how much power (amplitude squared)
occurs at which time-scales, as in Eq. (). This
representation, called the Hilbert energy spectrum, plots the data as an
energy density distribution overlaid on the time-frequency space .
S(ω,t)=∑k=1Nak2(t)⋅ei∫ωk(τ)dτ
The time-integrated version of Eq. (), the
Hilbert marginal spectrum SM(ω), is similar, but not identical to,
the traditional Fourier spectrum, and is given in Eq. ().
SM(ω)=∫0TS(ω,t)dt
Results
The 12 datasets (6 datasets per station) have been decomposed by the EMD into
10 IMFs and a residual, as shown in Fig. . Similar
plots are available for the rest of the datasets in the Supplement.
To summarize the results, the mean characteristic scales of
variability for all the IMFs of the VIE datasets have been compiled into
Table , while the corresponding mean amplitudes
are given in Table . Similar summaries for the KIV
datasets are provided in Appendix , as Table
for time-scales and Table for amplitude, respectively.
Mean IMF time-scales in days for the VIE datasets.
Mean IMF amplitudes in W m-2 for the VIE datasets.
IMF1IMF2IMF3IMF4IMF5IMF6IMF7IMF8IMF9IMF10TOA0.10.10.10.13.11865.41.22.00.4McClear6.44.63.73.03.2128101.61.70.4WRDC442419161216968.34.22.8HC3v547262017131491126.52.1ERA332016131019896.95.02.2MERRA2382116131030944.74.53.1Intrinsic time-scales of variability
From Tables and it
can be seen that the most significant time-scale of variability present in
the TOA time-series is around the 1 year mark, as evidenced by IMF6. The mean
period of this component is 366 days and its mean amplitude of 186 W m-2
is two orders of magnitude greater than the other modes.
From this, it can be inferred that this yearly mode, a result of the
revolution of the Earth around the Sun, is the only significant scale of
variability of the TOA. This result is unsurprising, since it can also be
inferred from Fig. top panel, where the TOA series
does not exhibit any other variability apart of this mode, i.e. it is almost a
perfect sine wave with a period of one year. Adjacent to this sixth mode,
IMF5 and IMF7 also display similar time-scales of 344 and 366 days,
however their mean amplitudes are 3.1 and 5.4 W m-2 respectively, thus their origin is attributed to spectral
leakage from the main mode. The first four IMFs have negligible amplitudes
(0.1 W m-2) and are probably the manifestation of residual noise.
IMF8 has a mean time-scale of 1431 days and its low amplitude of
1.2 W m-2 is an indication that it might be a numerical artefact. IMF9
and IMF10 are also assumed to be non-physical, because their large periods of
over 3100 days are indication that these components are entirely immersed in
the time coverage where edge effect are non-negligible. As a reminder, edge
effects are important at the half-period of a component at data boundaries,
which for these two latter IMFs yields 1550 days forward from 1 February 2004
and 1550 backwards from 31 January 2013, spanning almost the entire data range.
For the McClear time-series, as well as for the rest of the VIE datasets,
IMF1…IMF5 display remarkably similar features, such as monotonically
decreasing amplitudes and time-scales that exhibit period doubling, roughly
following the dyadic scale: 3 days → 6.8 days → 13.1 days → 26 days → 51 days. The amplitudes for McClear
are 3 to 7 times less than in other time-series, and less than 6.5 W m-2.
The break in the monotonic decrease of amplitude with scale
for McClear – 3.2 W m-2 for IMF5 vs. 3 W m-2 for IMF4 – is
considered an artefact, since this monotonicity holds for the other
datasets and also for KIV (see Table ). The
presence of these first five dyadic scales in all the datasets, with the
exception of TOA, allude to their possible origin as being cloud-free
atmospheric affects. This can also be observed in
Fig. , where, as opposed to the TOA, the raw McClear
time-series is seen to exhibit slight high-frequency variability, and which
tends to increase during the summer months. For McClear, there is little to
no variability in the 2–3 months to 1 year band (see also
Fig. , discussed later on). As for TOA, IMF6 of the McClear
time-series is clearly associated with the yearly variability, by its mean
period and amplitude of 366 days and 128 W m-2, respectively. Here
too, this is the most energetic spectral component of the dataset. IMF7 of
McClear has a median time-scale of 417 days and a median amplitude of
10 W m-2. IMF8…IMF10 have very low amplitudes of less than
1.8 W m-2, hence their origin cannot be unambiguously determined. This
is especially the case with IMF9 and IMF10, which reside almost entirely in
the edge effect region because of their large periods of about 3000 days and greater.
The Hilbert marginal spectra for the VIE datasets: TOA,
McClear,WRDC, HC3v5, ERA, MERRA2. The abscissa indicates the time-scale on a
binary logarithm, and the ordinate denotes power in dB.
The WRDC dataset unsurprisingly shares many features with HC3v5, ERA and
MERRA2 datasets, since the latter three are intended to be accurate estimates
of the former. The rest of the results will be presented in a lumped form for
these four time-series. For IMF1…IMF5, HC3v5 agrees better with WRDC
than the re-analyses in terms of mean amplitudes and takes on only slightly
higher values (1 to 3 W m-2 or less than 5 % on average,
Table ). This can also be seen in
Fig. , where the HC3v5 data exhibits a slight
overestimation of the low SSI values for WRDC during winter. For the first
five modes, both ERA and MERRA2 have mean amplitudes that are on average 17 %
less than those of WRDC. This is also apparent in
Fig. , where fewer samples of less that 150 W m-2
are occurring between spring and autumn for the re-analyses
time-series; this is mostly visible for the year 2007. For IMF6, HC3v5
closely follows WRDC, both in terms of time-scale: 156 days vs. 159 days,
and amplitude: 14 W m-2 vs. 16 W m-2. For the same
IMF6, both ERA and MERRA2 exhibit significantly greater time-scales: 180 and
217 days, respectively, and amplitudes: 19 and 30 W m-2.
The first six modes are responsible for the high-frequency
variability, which is also manifest in Fig. ,
i.e. the estimates and ground measurements of the SSI are significantly more
variable than the McClear time-series. For all these datasets the yearly
variability cycle is captured by IMF7, with generally good agreement across
datasets in terms of both time-scale (367 to 371 days) and amplitude (89 to
96 W m-2). The greatest modes, IMF8…IMF10, have amplitudes
less than the uncertainty threshold, with the exception of IMF8 for HC3v5.
Generally similar results are also obtained for KIV, as summarized in
Tables and . Apart minor
differences in numerical values, the only notable exception in the KIV
datasets are the significantly greater time-scales for IMF6 of 280 and 298 days
respectively for ERA and MERRA2, as opposed to 199 and 176 days for WRDC and
HC3v5, respectively.
The Hilbert marginal spectra for the KIV datasets: TOA,
McClear,WRDC, HC3v5, ERA, MERRA2. The abscissa indicates the time-scale on a
binary logarithm, and the ordinate denotes power in dB.
Results of Hilbert spectral analysis
The previous summary of the results, although informative, is static in the
sense that only two features are used to characterize, in an approximative
manner, each time-evolving IMF: the long-term average amplitude and time
period. To make use of the full potential of the HHT, which can follow both
the temporal and the spectral evolution of the data, Hilbert spectra were
also computed for all the datasets (not shown) and are provided in the
Supplement. From these Hilbert spectra the marginal,
time-integrated, versions were computed and are presented in
Fig. for VIE and in Fig. for KIV.
The TOA spectrum for VIE in Fig. confirms the previous
findings. For this dataset the only significant mode of variability is found
at the one year mark, and has a power of about 63 dB. A slight peak of 17 dB
is also present around 1400 days, corresponding to IMF8. The end region of
the spectrum, from 2500 days onwards also contains some power, but the
respective IMFs have been shown to be heavily affected by edge effects, thus
the origin of this feature is ambiguous at best.
The McClear dataset is seen to introduce variability in the high-frequency
regime, whose power decreases almost monotonically from 30 dB at 2 days, to
about 2 dB at roughly 300 days. Most of this variability occurs during
summer, as observed in Fig. (see also the full
Hilbert spectrum of McClear in the Supplement). The yearly
variability component stands out again, this time with just less than 60 dB
in power. From here, power decreases to a minor spectral shoulder of 22 dB at
800 days (IMF8), after which it fades out towards larger frequencies.
As previously shown, the high frequency variability (IMF1…IMF5) of
HC3v5 matches more closely that of WRDC, while the re-analyses have slightly
(2–5 dB) less power. The power of these features is 15 dB greater in the
estimates and ground measurements of SSI than that found in the clear-sky
regime. From 170 days to 256 days, WRDC and HC3v5 overpower the re-analyses.
After 256 days, the power in the re-analyses overcomes that of WRDC and
HC3v5, until approximately one year. This can also be seen from
Tables and , where IMF6 for
the re-analyses is seen to be greater than the other two time-series, both in
terms of amplitude and of time-scale. Again, the yearly variability component
is the largest spectral characteristic, with WRDC peaking at 57 dB. Here, the
other datasets closely agree with the ground measurements as per the VIE
results summary tables. After the 500 days mark the interpretation of the
different spectral features is ambiguous, both because of their mean
amplitudes failing to rise above the uncertainty level, and also because of
the progressively large impact of the edge effects, especially towards the
end of the spectrum.
The spectra for the KIV datasets in Fig. are very similar
to their VIE counterparts. For KIV too, TOA exhibits only a sharp yearly
variability component (IMF6), while McClear introduces an almost
monotonically decreasing band of high-frequency variability. Unlike for VIE,
the McClear does not drop abruptly up to one year, but exhibits a rebound of
17 dB around 150 days. This is interpreted as an artefact, induced by
energetic oscillations in and near the left edge effect boundary in the full
Hilbert spectrum (see Supplement). From 2 days to 2 years, the
HC3v5 spectrum follows the WRDC one more closely that the re-analyses. The
latter two exhibit large downward excursions of 5 to 10 dB, from 70 days to
roughly 150 days. For the yearly variability component, there is better
agreement than for VIE between the estimates and the ground measured SSI
data, all four of them peaking simultaneously at 57 dB. The large peak of 42 dB
found in ERA at 530 day should be ignored, since it is caused by an
energetic oscillation in the edge effect region (see Supplement).
As for the VIE datasets, after the 500 days mark the interpretation spectral
features becomes ambiguous.
Time-scale comparison of SSI estimates and ground measurements
Still another possibility of investigating the data is to make use of the
adaptive, data-driven, time-domain filter character of the EMD. Looking at
pairs of IMFs in the time domain, it is possible to construct 2-D histograms
of the satellite and re-analyses estimates of SSI compared to the concomitant
ground measurements. This gives a good overview of the similarity, at each
characteristic time-scale of variability, between satellite estimates of the
SSI or re-analyses radiation products and the WRDC measurements, which serve
as ground truth.
The 2-D histogram for IMF2 of HC3v5 and WRDC for VIE. Each pixel
encodes relative frequency according to the colour-bar on the right. The
solid black line denotes the identity line and the dash-dotted red line
represents the best fit line. The linear regression equation is indicated in
the legend. The time-scale, root-mean-square error and coefficient of
determination are indicated in the panel above the
legend.
Figure illustrates the 2-D histogram for the
second IMF of the HC3v5 and WRDC datasets for the VIE station. The colour of
each pixel denotes the relative frequency of the WRDC–HC3v5 irradiance
pairs, as encoded on the colour-bar on the right. In
Fig. , the pattern of dots has a positive slope
which indicates a positive correlation between the two variables. A robust
best-fit linear regression has been performed, with the resulting line shown
in dash-dotted red. Plotted in solid black is the identity line; were the two
datasets identical the scatters would fall exactly onto it. The equations for
the two lines are indicated in the legend. The line describing the best-fit
can be seen to deviate very little from the identity line, with a slope of 1.021
indicating that for IMF2 at VIE the satellite-derived SSI slightly
over-estimates the SSI measured at ground level, and an intercept of -0.016,
which is the expected mean irradiance value of HC3v5 when then WRDC
irradiance is zero. Indeed, the linear regression model manages to explain
92.4 % of the total variability, as indicated by the coefficient of
determination (R2= 0.924). The points do not fall exactly
onto the best-fit line and exhibit a small scattering, with a
root-mean-square error (RMSE) of 6.2 W m-2. The characteristic
time-scale of variability is also indicated on the plot, as 6.8 days.
Figure presents a similar graph, but for ERA
and WRDC datasets. Here the best-fit line deviates significantly from the
identity line, with a slope of 0.647 and an intercept of 0.119. This is also
reflected in the coefficient of determination of R2= 0.636. The
scatter is also significantly larger, with a RMSE of 10.3 W m-2.
Statistical indicators for correlations at different time-scales
between SSI estimates and ground measurements for VIE.
The 2-D histogram for IMF2 of ERA and WRDC for VIE. Each pixel
encodes relative frequency according to the colour-bar on the right. The
solid black line denotes the identity line and the dash-dotted red line
represents the best fit line. The linear regression equation is indicated in
the legend. The time-scale, root-mean-square error and coefficient of
determination are indicated in the panel above the
legend.
Similar plots to those in Figs.
and have been computed for IMF1…IMF5, IMF7
and globally, for the whole time-series, for the ground measurements and for
the satellite and re-analysis estimates. Graphs are available in the
Supplement. Results, in terms of root-mean-square error and
coefficient of determination, are indicated in Tables
and for VIE and KIV, respectively. IMF6 has been excluded,
because Tables and
show that for the re-analyses the time-scale of this mode deviates
significantly from the ground measurements, thus the comparison is
meaningless. IMF8…IMF10 have also been excluded because the mean
amplitudes are below the uncertainty level (see Tables and ).
Table shows that for VIE, on a per time-scale basis as
well as globally, the closest estimate of ground measurements of the SSI is
the HC3v5 dataset, both in terms of explained variance, and in terms of
scatter. The lowest coefficient of determination for HC3v5 is 0.924 for the
weekly variability (IMF2), meaning that Fig.
represents the worst case scenario for this
particular dataset. The largest percentage of variance explained, 99.2 %, is
attained for the yearly variability (IMF7). Globally, the HC3v5 accounts for
97.9 % of the observed variability (RMSE = 14.1 W m-2),
outclassing ERA with 92.1 % (RMSE = 24.8 W m-2) and MERRA2 with
92.8 % (RMSE = 26.4 W m-2). For IMF1…IMF5, MERRA2
outperforms ERA in terms of R2 except for the monthly
variability (IMF4), as also reflected in the range of this coefficient of
[0.684; 0.762] for MERRA2 as opposed to a range of [0.636; 0.755] for ERA.
The yearly variability of the ground measurements is better expressed by ERA
than by MERRA2, both in terms of coefficient of determination and RMSE.
Generally, all the datasets exhibit monotonically decreasing RMSE for the
first five modes and very good agreement for the yearly variability
(R2> 0.985).
Similar statements can be made about the results for KIV, presented in
Table . Here too, the HC3v5 dataset outperforms ERA and MERRA2
both in terms of coefficient of determination and RMSE, across all
time-scales and also at the whole time-series level. The minimum R2
for HC3v5 is 0.885 for IMF5, while the minimum values of R2
for ERA end MERRA occur for IMF1 and are 0.611 and 0.621, respectively.
As for VIE, all the datasets exhibit monotonically
decreasing RMSE for the first five modes and very good agreement for the
yearly variability (R2> 0.982).
Discussion
The results in the previous section have highlighted some features of the
data that will be expanded upon here.
It has been inferred from the mean time-scales and mean amplitudes of the
decomposed data (Tables and )
as as well as from the Hilbert marginal spectra
(Figs. and ) that the yearly mode of
variability is the most prominent feature of all the datasets. This result is
unsurprising since both stations are situated at mid-latitude and it can be
explained in terms of orbital geometry through the yearly cycle of seasons;
it can also be inferred by visually inspecting the raw data from
Fig. which shows large variability with periodicity of one
year. For WRDC, HC3v5, ERA and MERRA2, there is good agreement with respect
to this mode (IMF7) both in terms of amplitude and of time-scale (see also
Tables and ).
Apart from this yearly component, the TOA exhibits no other form of
significant variability, also in good agreement with its trace from
Fig. , which registers as an undisturbed sinusoidal waveform.
High-frequency variability, from 2 days up to 2–3 months, is manifest in
McClear through its first five IMFs. This feature is also present in the rest
of the datasets with greater power when compared with McClear. Hence, this
feature can be attributed to clear-sky (no cloud) atmospheric effects
(scattering and absorption by ozone, water vapour, aerosols, etc.). Looking
at the McClear graph in Fig. , it becomes evident
that this high-frequency variability manifests itself more strongly during
the summer than during the winter months. In other words, the yearly cycle
modulates the power of this high-frequency feature through a non-linear
cross-scale amplitude-phase coupling. This feature is also apparent in the
HC3v5, WRDC, ERA and MERRA2 datasets (see Supplement) and is in
agreement with the findings of who underlined its
stochastic nature. The time-scales for the individual modes composing this
high-frequency feature agree well across these latter four datasets. In terms
of amplitudes, however, for VIE HC3v5 slightly overestimates the WRDC
measurements while ERA and MERRA2 underestimate more severely, while for KIV
all the SSI estimates yield lower values than the ground data, although less
so for HC3v5. This is also readily apparent in Tables
and , where for the first five IMFs, HC3v5 outperforms ERA and
MERRA2 by a large margin. This result is a major contribution of the study,
since have proposed that the accuracy of SSI forecasts
crucially depends on the ability to forecast the stochastic component. As
such, practitioners interested in modelling and forecasting the daily SSI are
better off using satellite estimates that radiation products from
re-analyses, at least for the two sites studied herein. This finding can be
explained by the fact that re-analyses assimilate state variables such as
temperature, moisture and wind, while the SSI is diagnostic. Stated
differently, in re-analyses, radiation and cloud properties are derived from
a model and, as such, they include the uncertainty of this model. Re-analyses
often predict clear sky conditions while the actual conditions are cloudy
. HC3v5 is based on Meteosat imagery (Sect. )
and as such directly takes clouds into account.
Another significant result of this study is the fact that, for both VIE and
KIV, the McClear datasets do not have a variability component in between this
high-frequency feature and the yearly cycle. In other words, IMF6 for the
McClear data represents the yearly cycle, unlike the ground measurements or
the SSI estimates, where IMF6 is an intermediate component before the yearly
component represented by IMF7. This has first been discussed as a
“variability gap” by , when analysing a decennial
dataset of daily means of SSI measured by BSRN ground station at Carpentras,
France, that experiences clear-sky conditions for most of the year.
Subsequently, have shown that this “variability gap”
is also manifest for a similar dataset of ground measurements taken at
Boulder, Colorado, USA, a location that also experiences a high number of
days with clear skies. Hence, this study confirms the fact that, indeed, a
clear-sky atmosphere does not introduce any spectral features in between
2–3 months and 1 year. Since the ground data for both VIE and KIV feature such
spectral components, it can be concluded that these two locations do not
experience so many cloud-free days and/or that they experience a lot of
broken clouds conditions. Here too, ERA and MERRA2 are outperformed by HC3v5,
since the time-scale of this IMF6 in the re-analyses datasets is greatly
different form the true time-scale found in ground measurements, and which is
accurately reflected by the satellite estimates.
Larger scales of variability (IMF8…IMF10) have been discarded from
this analysis because of their failing to stand above the uncertainty threshold.
Conclusions and outlook
In this study we have investigated the characteristic time-scales of
variability found in long-term time-series of daily means of SSI. We have
also studied the fitness for use of satellite estimates of the SSI and
radiation products re-analyses as alternatives to pyranometric ground
measurements. The novelty of our work is the use of the adaptive, data-driven
Hilbert–Huang Transform (HHT) to decompose the datasets into their distinct
characteristic time-scales of variability before undergoing analysis.
We have shown that the TOA only presents variability at the one year
time-scale. The clear-sky atmosphere introduces stochastic high frequency
variability, from 2 days to 2–3 months, which exhibits non-linear
cross-scale phase-amplitude coupling with the yearly cycle. This feature is
also present, and amplified, in ground measurements, satellite estimates and
re-analysis products. The fact that the cloud-free atmosphere does not
introduce variability from 2–3 months to one year, i.e. the “variability
gap” alluded to in previous studies, has been confirmed. It has also been
shown that, HC3v5 outperforms ERA and MERRA2 by a large margin in terms of
estimating the measured SSI, not only at a global, whole dataset level, but
also on an per time-scale basis, and especially with respect to the
stochastic variability component. This has implications on the forecast and
modelling of the SSI, where satellite estimates should be preferred instead
of re-analysis products. Our study, hence, refines the existing methodology
to assess the fitness for use of surrogate SSI products, through an improved
in-depth comparison of their local time-scales of variability.
A limitation of our study needs to be pointed out. Before carrying out the
analysis, we have used the EMD on each time-series and have only compared
modes with similar time-scales. That is, we have used the mono-variate
version of the EMD, where mode alignment (identical time-scales for the IMFs
across datasets) is not enforced. Nevertheless the non-alignment of modes is
not to be considered a weakness of our approach. Because identical
time-series will be decomposed into identical modes, by not enforcing similar
time-scales across the modes of different datasets, changes in the
time-scales of the modes (e.g. IMF6 of HC3v5 matches WRDC unlike ERA or
MERRA2) also provide supplementary clues as to the fitness for use of the
surrogate SSI datasets in lieu of ground measurements. Mode alignment can be
enforced by more advanced, multi-variate versions of the EMD. Two such
techniques are the noise-assisted multi-variate empirical mode decomposition (NA-MEMD)
introduced by or the adaptive-projection
intrinsically transformed multivariate empirical mode decomposition (APIT-MEMD)
proposed by . The latter method is
particularly of interest since it is also able to deal with power imbalances
and inter-channel correlations found in multichannel data. With NA-MEMD or
APIT-MEMD, all the datasets would be treated in a unitary manner as a single
multi-variate signal, thus mode alignment would be enforced. This would also
enable the use of of more advanced descriptors, such as multi-scale measures
suitable for multi-variate datasets and inter-component measures,
e.g. intrinsic correlation, intrinsic sample entropy or intrinsic phase synchrony
. However, the exercise is significantly more technical and
is proposed as a future study.
Lastly we recognize the restrained geographical character of the study and,
as a future exercise, we propose its extension to many more geographical
locations and possibly including several different satellite estimates and
re-analyses radiation products, in order to determine whether the findings
reported herein also hold for different regions and for different SSI surrogates.
Code availability
The software used for this study, comprising general EMD and HSA routines is
publicly available online, as follows.
The fast EMD routine used in this study is provided by and
can be downloaded at: http://rcada.ncu.edu.tw/FEEMD.rar.
Methods pertaining to Hilbert spectral analysis are part of a general HHT
toolkit provided by and can be downloaded at:
http://rcada.ncu.edu.tw/Matlab runcode.zip.
The code for the ICEEMD(AN) algorithm is provided by
María Eugenia Torres on her personal webpage, and can be downloaded at:
http://bioingenieria.edu.ar/grupos/ldnlys/metorres/metorres_files/ceemdan_v2014.m
Data availability
The data can be accessed as follows:
The ERA-Interim data set (ECMWF2009) can be accessed at:
https://doi.org/10.5065/D6CR5RD9.
The MERRA2 radiation diagnostics M2T1NXRAD timeseries (GMAO2015)
is available at: 10.5067/Q9QMY5PBNV1T.
TOA and McClear data from Copernicus Atmosphere Monitoring Service
(Copernicus2015, ) can be retrieved at:
http://www.soda-pro.com/web-services/radiation/mcclear.
The WRDC global radiation daily sums for Europe (WRDC2014) can be
accessed at: http://wrdc.mgo.rssi.ru/wrdc_en.htm.
The HelioClim-3v5 dataset was downloaded from the SoDa Service web site
(http://www.soda-pro.com) managed by the company Transvalor. Data are
available to anyone for free for years 2004–2006 as a GEOSS Data-CORE (GEOSS
Data Collection of Open Resources for Everyone) and for-pay for the most
recent years with charge depending on requests and requester.
Mean IMF time-scales and amplitudes for KIV
Tables and present
the mean time-scales and respectively the mean amplitudes of the IMFs for the
KIV datasets.
Mean IMF time-scales in days for the KIV datasets.
The Supplement related to this article is available online at doi:10.5194/asr-11-35-2017-supplement.
All authors contributed equally to this work.
The authors declare no competing interests. HC3v5 solar
radiation products are commercialized by Transvalor SA through its online SoDa
Service at http://www.soda-pro.com, except for years 2004–2006 which are
available to anyone for free as a GEOSS Data-CORE.
Acknowledgements
The authors thank the World Radiation Data Centre for maintaining the
radiation archives and hosting the website for downloading data. They thank
the ground station operators of the WMO network for their valuable
measurements. The authors are indebted to the company Transvalor SA which is
taking care of the SoDa Service for the common good, therefore permitting an
efficient access to the HelioClim databases.
Edited by: S.-E. Gryning
Reviewed by: two anonymous referees
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