ASRAdvances in Science and ResearchASRAdv. Sci. Res.1992-0636Copernicus PublicationsGöttingen, Germany10.5194/asr-13-121-2016On the temporal variability of the surface solar radiation by means of spectral representationsBengulescuMarcmarc.bengulescu@mines-paristech.frBlancPhilippehttps://orcid.org/0000-0002-6345-0004WaldLucienhttps://orcid.org/0000-0002-2916-2391MINES ParisTech, PSL Research University, Centre for Observation, Impacts, Energy CS 10207, 06904 Sophia Antipolis CEDEX, FranceThese authors contributed equally to this work.Marc Bengulescu (marc.bengulescu@mines-paristech.fr)19July2016131211273November201517June201627June2016This 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/13/121/2016/asr-13-121-2016.htmlThe full text article is available as a PDF file from https://asr.copernicus.org/articles/13/121/2016/asr-13-121-2016.pdf
This work deals with the temporal variability of daily means of the global
broadband surface solar irradiance (SSI) impinging on a horizontal plane by
studying a decennial time-series of high-quality measurements recorded at a
BSRN ground station. Since the data have a non-linear and non-stationary
character, two time-frequency-energy representations of signal processing are
compared in their ability to resolve the temporal variability of the
pyranometric signal. First, the continuous wavelet transform is used to
construct the wavelet power spectrum of the data. Second, the adaptive,
noise-assisted empirical mode decomposition is employed to extract the
intrinsic mode functions of the signal, followed by Hilbert spectral
analysis. In both spectral representations, the temporal variability of the
SSI is portrayed having clearly distinguishable features: a plateau between
scales of two days and two-three months that has decreasing power with
increasing scale, a large spectral peak corresponding to the annual
variability cycle, and a low power regime in between the previous two. It is
shown that the data-driven, noise-assisted method yields a somewhat more
sparse representation and that it is a suitable tool for inspecting the
temporal variability of SSI measurements.
Introduction
The surface solar irradiance has been recognized by the Global Climate
Observing System as one of the Essential Climate Variables, a restrained set
of critical variables extensively used in science and policy circles for
understanding climate evolution and guiding mitigation and adaptation efforts
. As such, not only do long term time-series of the SSI
play a key role in science and policy consulting, but they also strongly
impact the engineering and financial sectors, providing support for, e.g.,
the selection of suitable sites for solar power plants or investment
decisions, respectively . Thus, better
understanding the characteristic time-scales of variability of the solar
radiation at ground-level, as recorded in the aforementioned long term
time-series, is one of the goals of the present work.
However, in order to tackle the topic of the variability of the surface solar
irradiance, the very concept of variability itself must first be properly
addressed. Even though spatial variability is of course also of interest, as
has been shown for example by , in line with the scope of the
study, only the temporal dimension of SSI variability will be considered.
One of the most common ways of quantifying variability is through the use of
simple descriptive statistics, such as the mean and the standard deviation
. Variability is then usually related to the time-scale at
which analysis is carried out, with the fluctuations at the scale of interest
being compared to the mean (or another low-pass filtered approximation) at
the next higher scale. Thus, yearly sums of SSI are compared to their
multiannual mean, daily sums to their monthly mean, minute values are
compared to their hourly mean, etc. When, still, even finer (sub-second)
scales are addressed, the second temporal derivative of irradiance is used to
quantify its instantaneous variability . Another way to
express SSI variability by using statistics is in terms of the probability
density function of the irradiance, or by using a two-sample structure
function, which is the root-mean-square deviation of increments of
irradiance, as described by .
While adhering to the statistical approach and defining the root mean square
of a high-pass filtered version of the data as the fluctuation factor
of the time-series, allude to employing the power spectral
density as a mean of describing variability. Hence, spectral methods provide
a different approach of describing variability in that, by analysing the
frequency contents of the data, a much clearer picture is rendered in terms
of the scales, or the inverse of frequency, at which variations in the SSI
occur. As such, make use of wavelet periodograms to
characterize fluctuations in solar irradiance, while use the
wavelet transform to decompose a time series of SSI for their proposed
Wavelet Variability Model. make use of the power
spectrum of a time-series in order to model the multi-fractal properties of
the SSI by means of, among other spectral methods, Hilbert analysis.
Having seen the variety of techniques used to describe the temporal
variability of the SSI, the present work investigates the characteristic
time-scales of variability embedded in long-term time-series of SSI, as
revealed by two spectral methods in particular. Given the nonlinear and
nonstationary character of solar radiation measurements , the
Hilbert–Huang Transform (HHT) is a natural choice since it has been
specifically designed for the analysis of such data . The
spectral picture rendered by the HHT is compared to the one obtained by the
now classical Continuous Wavelet Transform (CWT) ,
highlighting the commonly shared and contrasting features of the two
representations. Since these methods are applied to long term time-series of
the SSI, the sampling rate of the data has been fixed at one day.
The aims and intents of the study at hand can now be described as
ascertaining what type of information about the temporal variability of the
data the two spectral representations, the HHT and the CWT, may convey. Added
to this, it is well known that the SSI, more specifically a decennial
time-series thereof, exhibits a characteristic time-scale of variability
around a one year period which is easily accounted for by the orbital
parameters of the Earth-Sun system. However, other factors controlling
surface solar irradiance are also at play, such as atmospheric absorption or
scattering, due to e.g. clouds, aerosols, etc. Thus, an ancillary locus of
investigation is whether such processes also induce other characteristic
scales of variability that may be embedded in the scrutinized signal.
In what follows, this study is organized as per the ensuing outline.
Section discusses the data and the analysis methods; the
time-series under investigation is described in Sect. ,
the spectral methods employed are reviewed in Sect. .
The obtained results are then presented in Sect. , while
discussion thereof is deferred to Sect. . Finally,
conclusions and outlook are presented in Sect. .
Data and methodsThe analysed time-series and preprocessing
The time-series under investigation in this study was obtained from
high-quality ground measurements of the SSI, at a station situated in
Carpentras (44.083∘ N, 5.059∘ E), in southeastern France. The
measuring station is part of the Baseline Surface Radiation Network (BSRN, ).
BSRN is a worldwide radiometric network providing accurate readings of the
SSI at 1 min temporal resolution and with an uncertainty requirement at
5 W m-2 for SSI . The original 1 min
measurements for the ten-year period 2001–2010 have been quality checked
following . Next, daily means were computed from this raw
data only if more than 80 % of the samples during daylight were valid.
Isolated missing daily means were completed by linear interpolation applied
to the daily clearness index, which is the ratio between the daily mean of
SSI and the daily mean of the corresponding total solar irradiance received
on a horizontal surface at the top of atmosphere. The resulting time-series
is shown in Fig. .
The ten year time-series of SSI under study, spanning 2001
through 2010. Each point corresponds to a daily mean of irradiance. Time
markers on the abscissa denote the start of the corresponding
year.
Time-frequency-energy representations
For this study, two spectral methods have been employed in order to assess
the variability of the SSI. Most notably, analysis techniques that reveal the
temporal evolution of the scrutinized data have been privileged. Two such
techniques, popular within the geophysics community , are the
Continuous Wavelet Transform (CWT) an the Hilbert–Huang Transform (HHT). The
methods are of interest mostly due to their innate ability to offer insight
into the processes underlying the data by painting so called
time-frequency-energy representations, i.e. graphical representations
depicting the distribution of power, or variance, occurring at specific time
instants and specific scales or periods (assuming period ≃ 1/frequency).
The continuous wavelet transform (CWT)
The continuous wavelet transform is a mean to render a two parameter
representation (time and scale, or frequency) of a signal of interest in
terms of wavelets obtained by time-shifting and dilating a basic wavelet of
fixed shape. Starting from this latter, elementary ”mother” wavelet, any
signal can be decomposed by measuring its degree of similarity to wavelets
that are time-shifted and scaled versions of the original one. Consequently,
this scaling of the basis functions enables the wavelet transform to achieve
high frequency resolution when analysing slowly varying features and high
temporal resolution when investigating bursts and transients.
To formalize this, consider a basic, or mother, wavelet, ψ(t), from
which, by scaling with some factor λ∈R, λ> 0, and
time-shifting by τ∈R, a family of wavelets can be defined:
ψλ,τ(t)=1λψt-τλ.
Then, the CWT can be regarded as measuring the similarity between a signal of
interest, x(t) and the basis functions ψλ,τ(t), as in :
CWTx(λ,τ)=∫-∞∞x(t)ψλ,τ*(t)dt.
From Eq. () the wavelet transform is seen to be linear,
since the analysis consists of computing the inner products between the
wavelets and the signal of interest . Another point to be
made is that the CWT is overly redundant, hence its amenability to
quantitative analysis is arguable .
Since, due to its good trade-off between time and frequency resolution, the
Morlet wavelet is frequently chosen as such a basic wavelet, following
, this study has also settled on the Morlet wavelet as
“mother” wavelet, given in Eq. (), where the parameter ω0≥ 5.
ψ(t)=π-14e-iω0te-t22
The analysing wavelet is in general, as also in particular for
Eq. (), a complex-valued function; it follows that the continuous
wavelet transform is also complex-valued. Hence, the wavelet power spectrum
is defined as the square of the complex amplitude of the wavelet
coefficients, i.e. |CWTx(λ, τ)|2, yielding a
time-frequency-energy representation . Nevertheless this
power spectrum has been found to be biased towards large scales, thus
rectification of the coefficients through division by the scale is undertaken
in order to enable the comparison of the spectral peaks across scales .
The Hilbert–Huang transform (HHT)
In contrast with the CWT, which decomposes data into a set of a priori
chosen basis functions, the Hilbert–Huang transform is completely data-driven
and adaptive. In other words, the transform does not “measure” the similarity
between the data and some predefined “rulers”, such as trigonometric
functions or wavelets, in the case of the Fourier analysis and the CWT,
respectively. Rather, the method lets the signal itself be the driver of the
decomposition. Furthermore, the HHT was designed with the analysis of
non-linear, non-stationary signals in mind , closely
resembling a previous proposal of non-linear analysis of non-stationary
frequency-amplitude structure of long time-scale solar activity
. Thus, owing to its ability of handling data issued
from the non-linear interaction of physical processes, which are often under
the influence of non-stationary external forcings , the method
has seen extensive use in geophysical research .
The HHT consists of two separate steps. First, the oscillations embedded in
the data, called Intrinsic Mode Functions (IMFs) are extracted by means of an
algorithmic procedure, termed the Empirical Mode Decomposition (EMD), whose
outline is sketched in algorithm (). After all the IMFs have been
extracted, the residue or trend, which cannot be mathematically thought of as
an oscillation, is all that is left of the time-series.
EMDx
x(t)∈R {Real-valued signal x(t)}
k← 0 {Initialize IMF counter}
r(t)←x(t) {Initialize residual}
while¬is_monotonic(r(t))do {IMF loop}
k←k+ 1 {Increment IMF counter}
h(t)←r(t) {(Re)process residual}
while¬is_imf(h(t))do {Sifting loop}
[xmin(t), xmax(t)] ←get_extrema(h(t))
L(t)←spline3(xmin(t)) {Lower envelope}
U(t)←spline3(xmax(t)) {Upper envelope}
h(t)←h(t)-L(t)+U(t)2 {Subtract mean of envelopes}
endwhile
ck(t)←h(t) {Store IMF}
r(t)←r(t)-ck(t) {Update residual}
endwhile
returnc1…N(t), r(t) {Return IMFs and residue}
The wavelet power spectrum (left panel) of the ten year time-series
of SSI under study, spanning 2001 through 2010. Pixel colour encodes power
(logarithmic scale colour bar on top) at each time (abscissa) and each scale
(ordinate). Time markers on the abscissa denote the start of the
corresponding year. The white-out area indicates the regions where edge
effects become significant. The global wavelet spectrum in the right panel is
the time-integrated version, i.e. line-by-line sum, of the wavelet spectrum
and indicates the amount of power at each scale, expressed
in dB.
By construction, the EMD extracts oscillatory modes that have a well-behaved
Hilbert transform , by means of which the instantaneous
amplitude and frequency of each mode is computed as follows. Thus, following
the notation from algorithm (), for a real-valued IMF ck(t),
its Hilbert transform can be written as:
σk(t)=Hck(t)=1πP∫-∞∞ck(τ)t-τdτ
where P indicates the Cauchy principal value. Next, each IMF and its
Hilbert transform are used to compute the complex-valued analytic signal:
zk(t)=ck(t)+i⋅σk(t)=ak(t)⋅ei⋅θk(t)
in which
ak(t)=ck2(t)+σk2(t)andθk(t)=tan-1σk(t)ck(t)
are the instantaneous amplitude and instantaneous phase, respectively. The
instantaneous frequency can be then be thought of as the derivative of the
instantaneous phase:
ωk(t)=dθk(t)dt.
The original data can now be written in terms of this amplitude modulation–frequency
modulation (AM–FM) signal model:
x(t)=Re∑k=1Nak(t)⋅ei∫ωk(τ)dτ+r(t)
which is essentially a sum of zero-mean oscillations having symmetric
envelopes that are riding onto the EMD trend.
The second part of the method, Hilbert spectral analysis, is next used to
construct the Hilbert energy spectrum. define this
time-frequency-energy representation as “the energy density
distribution in a time-frequency space divided into equal-sized bins of
Δt×Δω with the value in each bin summed […] at
the proper time, t, and proper instantaneous frequency, ω”.
For the purpose of this study, a modified version of the algorithm, the
Improved Complete Ensemble Empirical Mode Decomposition, introduced by
, is used, as it allows for an exact decomposition of
the data and is more robust with respect to noise. A fast EMD routine,
provided by , has been employed to speed up computations.
The Hilbert spectrum (left panel) of the ten year time-series of SSI
under study, spanning 2001 through 2010. Pixel colour encodes power
(logarithmic scale colour bar on top) at each time (abscissa) and each scale
(ordinate). Time markers on the abscissa denote the start of the
corresponding year. The white-out area indicates the regions where edge
effects become significant. The Hilbert marginal spectrum in the right panel
is the time-integrated version, i.e. line-by-line sum, of the Hilbert
spectrum and indicates the amount of power at each scale, expressed
in dB.
Results
The continuous wavelet transform was applied to the time-series of SSI,
producing the wavelet power spectrum depicted in Fig. .
It is noteworthy that the abscissa is the same as for
the original data, from Fig. , providing the
means to temporally resolve the various features of the spectrum. The
ordinate, having a logarithmic scale this time, indicates the approximate
period, expressed in days. Lastly, colour encoding is used to logarithmically
express the amount of power that the time-series contains at a certain point
in time and at a certain period or scale. The colour bar at the top of the
wavelet spectrum provides a scale assigning numerical values to
the colour encoding. The white-out regions that run along the edges of the
figure indicate the so-called cone of influence (COI), an area that is awash
with edge effects which are a principal limitation of the CWT and that should
be excluded from analysis. The COI edges correspond to the power of an
impulse at the edge dropping by a factor e-2. As
the scale increases, the COI progressively widens until the two edge-bound
regions intersect in the middle of the spectrum, thus imposing a limitation
with respect to the highest period at which the wavelet power spectrum is
able to provide meaningful information. The global wavelet spectrum in the
right panel is the time-integrated version, i.e. line-by-line sum, of the
wavelet spectrum and indicates the amount of power at each scale, expressed in dB.
Secondly, the empirical mode decomposition was used to extract IMFs from the
data. By applying Hilbert spectral analysis to them, another
time-frequency-energy representation of the same data is obtained, as
illustrated in the left panel of Fig. . The axes of
the Hilbert energy spectrum are analogous to the axes of the wavelet spectrum
in Fig. , except for the colour coding, which now
spans a different range. Edge effects are usually contained within a
half-period of a component at data boundaries , an area that
has been whitened out in similar way to wavelet COI from
Fig. . The right panel of Fig.
plots the Hilbert marginal spectrum, a time-integrated version of the
spectrum to its left, representing the total amount of power contained in the
time-series at each scale or period, expressed in dB.
Discussion
The wavelet power spectrum in Fig. reveals that the
most energetic component of the SSI time-series is confined between scales of
approximately 256 and 512 days, as indicated by the thick orange power-band
near the bottom of the plot. In CWT, spectral smearing and leakage occur
because of the finite support of the analysing basic wavelet
. Power “spills over” to neighbouring scales and is assigned
to this wider orange band on the plot instead of a distinct peak. This is
also apparent in the support of this feature in the global wavelet spectrum.
This feature can be interpreted in terms of the yearly seasonal variation of
the SSI with a period of around 365 days; as the amount of sunlight reaching
the ground is markedly different between summer and winter, the apparent
cycle has a strong power contribution at this scale. Nevertheless, this band
also contains a bright yellow strip centred on 365 days, which is thinning
beginning in 2003 and reaching the smallest extent in scale space around 2008,
indicative of a minimum in the eleven year cycle solar cycle
. Is is also interesting to note, that because of the
linear nature of the CWT, non-linearities of the yearly cycle are also
identifiable in the green band at the scale of about 180 days (the second
harmonic of the fundamental yearly cycle), which is dotted with blue patches
that indicate low power during boreal winter. The power of this feature is
such that it also makes a slight impression on the global wavelet spectrum.
Furthermore, the grouping of yellow-orange “filaments” at scales of between
64 and 2 days, that correspond to an increase in SSI during boreal summer,
could also be interpreted as high-order harmonics. However, the same feature
may as well be indicative of amplitude modulation of these bands by the
yearly cycle similar toFig. 1, since this cycle can be
inferred just from the alternation of blue-green (winter lows) and
orange-yellow (summer highs) hues in this band. Power in this high-frequency
band decreases with increasing scale, as can be inferred from the global
wavelet spectrum, where a minima is also noticeable at around 128 days. This
minima, centred in a larger depression, is a consequence of the low-powered
regions occurring in the 150 to 64 days band in the left panel, denoted by
dark blue patches, most notably between the second half of 2003 and 2005, the
second half of 2005 and 2007, and from the second half of 2009 onward. These
dark blue patches correspond to minimal levels of correlation between the
signal and the analysing wavelets at those scales and temporal localizations.
A final remark must be made here by reiterating that owing to the overly
redundant nature of the CWT, even after rectification, the global spectrum is
best interpreted qualitatively and not quantitatively.
The Hilbert spectrum depicted in Fig. shows similar
features. As in Fig. , the most energetic component
is found oscillating around the one year period, in the form of a yellow
trace, with the notable exception that in this representation the frequency
modulation of the component can be clearly distinguished, i.e. there are
slight downward period shifts, with respect to 365 days, during winter and
upward shifts during summer. Also of notice is the fact that the domain of
scale modulation of the yellow trace is somewhat restrained with respect to
the thick power band between 256 and 512 days from
Fig. , as can also be inferred from the support of this
feature on the marginal spectrum where it accounts for the greatest power
peak. The decrease in frequency modulation between 2007 and 2009 corresponds,
once again, to a period of solar activity at its minimum. Furthermore, two
low power components can be identified as having periods between 4 years and
1 year, with two distinct spectral peaks showing on the marginal spectrum at
about 1200 and 800 days, respectively. The physical meaning of these peaks is
ambiguous since most of the power of these components lies in the COI area.
Another feature of the spectrum that stands out is the lack of power in the
2–3 months to one year band, which is best viewed on the marginal spectrum.
This was also hinted at by the depression at about the same scales on the
global wavelet spectrum in Fig. , however since the
EMD does not detect any spectral features in this region, the Hilbert
spectrum, owing to sparsity, assigns a negligible amount of power in this
band. Another feature is the band of quasi-constant power in the
high-frequency region, between 2 days (the Nyquist limit) and 2–3 months. As
already indicated by the CWT global spectrum in
Fig. , here too the power of this plateau decreases with
increasing scale. Modulation in amplitude of this band by the yearly cycle is
apparent here too, and is most visible for the years 2005 and 2007 where the
blue hues occurring during winter turn green-yellow in the high-SSI regime of
summer. A final note regarding the Hilbert spectrum is that binning, both in
the colour and in the scale space (see Sect. ), may yield some
features as continuous lines while others are rendered as point-like,
especially where rapid frequency modulation takes place, such as the
high-frequency plateau.
Conclusion and outlook
To sum up, it has been shown that both the CWT and the HHT are able to
capture the characteristic time-scales of variability of the data. However,
to our opinion, the HHT is able to pick up more subtle features, such as the
yearly period modulation, while also yielding a somewhat more sparse
representation. This makes it a useful tool for the analysis of the temporal
variability of the surface solar radiation in forthcoming studies.
Nevertheless, both transforms provide a time-frequency-energy representation
of the signal where it can be clearly seen that the quasi-annual oscillations
have the highest energetic contribution. A power depression in the spectrum
can also be identified, which is a particular feature of the analysed data
and may prove useful in modelling the SSI for this specific site, i.e. any
such model should avoid injecting power into this frequency band, or filter
it out at the very least. A slightly slanted plateau between 2 days and
2–3 months can also be identified, whose power declines towards lower scales.
Preliminary investigations suggest that this high-frequency region is
noise-like and modulated by the yearly cycle, and is an indication of
cross-scale coupling of phase-amplitude ; further
research is needed for a definitive answer. If this plateau is noise-like, it
would impact numerical modelling of the SSI in the context of forecasting, as
it would imply that, for this specific site at Carpentras, there exists a
deterministic yearly component, a gap in the spectrum at medium scales, and a
stochastic component modulated by the highly energetic yearly cycle.
Data availability
Any user who accepts the BSRN data release guidelines (http://www.bsrn.awi.de/en/data/conditions_of_data_release)
may ask Gert König-Langlo (Gert.Koenig-Langlo@awi.de) to obtain
an account to download these datasets.
Acknowledgements
The authors wish to acknowledge the two anonymous reviewers, whose comments
helped improve the clarity and the readability of the paper.
Edited by: E. Batchvarova
Reviewed by: two anonymous referees
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