Introduction
The power of the electromagnetic radiation from the Sun that
reaches the surface of the Earth is estimated at around 1017 W. Thus,
solar irradiance is the main driver behind the weather and climate systems on
the planet. As such, the Global Climate Observing System (GCOS) program has
identified the surface solar irradiance (SSI) as an Essential Climate
Variable that helps understand climate evolution and guides adaptation and
mitigation efforts . Long-term time series of the SSI are
instrumental in engineering and finance by enabling, for example, the optimal
determination of geographical sites for solar power plants and guiding
investment decisions, respectively . Thus,
better knowledge of the SSI and of its temporal variability, as recorded in
long-term time series, is one of the intents of this work.
Temporally, the SSI exhibits a very wide dynamic range. Its short-term
timescales of variability, such as clouds briefly obscuring the Sun, are
observed over seconds. At the opposite scale, thousands or even millions of
years are to be used, as related to the change of the orbital parameters of
the Earth–Sun system or to stellar evolution . In spite of
this large span of characteristic scales of temporal variability, most of the
studies dealing with this physical quantity have focused primarily on a few
selected timescales of interest. As such, reports have either dealt with
global averages and long-term trends
, have only scrutinized
the short-term, high-frequency variability ,
or have focused exclusively on a few intermediate scales
. Although considerably differing in methods,
taken together the previously cited studies add valuable contributions to our
knowledge of the SSI. But is it possible to analyze the variability in the
SSI across multiple timescales in a unitary way?
To do so, first a decomposition of the time series into uncorrelated
sub-constituents with distinct characteristic timescales should be
preferred. Analysis would then ensue in a like manner for each scale. The
timescales, or characteristic periods of a time series, can be identified
with the inverse of the frequency at which the processes that generate the
data occur. It then follows that methods portraying the changes of the
spectral content of a time series with respect to time are potentially good
candidates. This would enable both the identification of the periodicities
and of the dynamic evolution of the processes generating the data. A general
class of useful signal processing techniques can thus be identified in the
so-called time-frequency distributions that depict the intensity (or energy)
of a signal in the time and the frequency domains simultaneously
. Such methods are commonly employed for geophysical signal
processing .
Another factor to be taken into account is the nonlinear and non-stationary
characteristics of the measured solar radiation data .
Handling such data issued from the nonlinear interaction of physical
processes, often also found under the influence of non-stationary external
forcings, calls for an adaptive data analysis approach .
The study at hand will make use of the Hilbert–Huang transform (HHT), an
adaptive, data-driven analysis technique designed specifically for
investigating nonlinear and non-stationary data . The HHT
adaptively decomposes any dataset into basis functions that are derived
solely from the local properties of the time series. A time-frequency-energy
representation of the data is then constructed from these basis functions.
The HHT has seen extensive use in geophysical signal analysis and spectral
estimation
.
The HHT has also been previously employed on SSI datasets
. A similar
technique has been independently proposed by for the
analysis of the nonlinear, non-stationary, long-range solar activity. In
this light, the use of the HHT for the study of the temporal variability in
the SSI appears to be appropriate. The inner workings of this data processing
method are detailed in a dedicated subsection.
Regardless of the methods used, when analyzing data there is always the need
to discriminate between deterministic signals and what are assumed to be
background stochastic realizations . The classical way to
solve this when employing the HHT on geophysical signals, such as the SSI, is
to presume some model for the background power spectrum, against which the
identified features are then compared
. In contrast, the present study
parts with the traditional approach, by adopting a novel, adaptive null
hypothesis introduced by that requires no a priori knowledge of the nature of the
background processes; further discussion thereof will be
presented in due course. A somewhat similar objective can be found in the
work of , though their method of discrimination between
stochastic and deterministic components is fundamentally different.
also propose a method for discriminating
frequency-dependent stochastic components by empirically estimating their
power law spectral energy distribution and respective confidence bounds.
Approaches for discriminating high-frequency fluctuations from large
timescale modulations are also described by and
.
At this point, the general outline of our study can be summarized as follows.
We analyze measurements of daily means of SSI at different geographical
locations. We focus on identifying and analyzing the intrinsic modes of the
temporal variability in the SSI, as revealed by the HHT. We also investigate
the physical and statistical significance of these modes. We show that the
HHT is able to discriminate between a deterministic yearly cycle and multiple
high-frequency (quasi-)stochastic components. We also find a non-null,
statistically significant rank correlation between the amplitude envelopes of
the high-frequency scales and the yearly cycle. We then discuss the possible
implications of our findings on the modeling and forecast of the SSI.
The study is organized as follows. Section discusses
the data sources and the preprocessing. In Sect. the adaptive
data analysis approach is described. Section will present
the results obtained, with the discussion thereof being deferred to
Sect. . Conclusions and outlook are presented in
Sect. . Code and data availabilities are indicated in the
Code and Data availability sections, respectively. Lastly, acknowledgements
and a bibliographical list conclude the study.
Ground measurement stations listing.
Code
Locationa
Latitudeb
Longitudeb
Climatec
BOU
Boulder
(US)
40.0500
-105.0070
BSk
CAR
Carpentras
(FR)
44.0830
5.0590
Csa
PAY
Payerne
(CH)
46.8150
6.9440
Cfb
TAT
Tateno
(JP)
36.0581
140.1258
Cfa
a Country codes according to ISO 3166-1
alpha-2.
b Positive north for latitude and positive east for longitude, following ISO 19115.
c Köppen–Geiger climate classification according to
.
Adaptive data analysis
Ideally, data analysis methods should require that no assumptions be made
about the nature of the scrutinized time series, i.e., neither linearity nor
stationarity should be presumed. This is because the true character of the
underlying processes that have generated the data is usually not known
beforehand. Adaptivity to the analyzed data would also be a sought-after
feature, in the sense of not imposing a set of patterns against which data
would be decomposed, but rather letting the data themselves drive the
decomposition. This latter criterion ensures both that the extracted
components carry physical meaning and that the influence of the mathematical artifacts inherent to the method on the
rendered picture of temporal variability is kept to a minimum .
Since such a decomposition is only determined by the local characteristic
timescales of the data, its appropriateness to nonlinear and non-stationary
time-series analysis is immediate .
The Hilbert–Huang transform
The Hilbert–Huang transform (HHT) is an adaptive data analysis technique
built with the previous consideration in mind. It involves two distinct steps –
the empirical mode decomposition (EMD) followed by Hilbert spectral analysis.
In-depth discussion of each step is carried out within the dedicated
subsections that follow.
The empirical mode decomposition
The first step of the HHT is the empirical mode decomposition (EMD), an
algorithmic procedure in essence, by which oscillations that present a common
local timescale are iteratively extracted from the data. These oscillatory
components of the data are called intrinsic mode functions (IMFs). An IMF is
any function that satisfies two criteria: (1) its number of extrema and zero
crossings differs at most by one and (2) at any data point the mean value of
its upper and lower envelopes is zero. These two properties ensure that IMFs
have a well-behaved Hilbert transform . Owing to the
adaptive nature of the EMD, the IMFs represent the basis functions onto which
the data are projected during decomposition. This is in contrast with the
Fourier or wavelet transforms where the basis functions are fixed in advance
. Once all the IMFs have been extracted, all that is left of
the time series is a residue, or trend, which cannot be mathematically
thought of as an oscillation at the span of the data. A sketch of the EMD
algorithm is provided in Algorithm 1.
Lines 6–12 of the EMD algorithm represent the so-called “sifting loop”
which has a two-fold purpose – to discard any riding waves and to render the
IMFs more symmetric. The stoppage criterion for the sifting loop is closely
related to how the latter controls the filter character of the EMD. On the
one hand, an infinite number of sifting iterations would asymptotically
approach the result of the Fourier decomposition (i.e., constant amplitude
envelopes) . On the other hand, several studies performed on
time series of pure noise have
shown the decomposition behaves like an adaptive “wavelet-like” dyadic
filter if the number of sifting iterations is kept small, around 10, which
also assures maximum component separation and minimum leakage .
This stoppage criterion of 10 sifting iterations is currently the recommended
one for practical applications and is also the one
employed in the study.
It also worth noting that the preferred interpolation method in the EMD,
i.e., lines 8 and 9 of Algorithm 1, are cubic splines .
Because of oscillations of these interpolating splines edge effects may appear in the EMD but are
usually contained within a half-period of a component at data boundaries
.
The eight IMFs obtained by decomposing the BOU time series; from top
to bottom: IMF1–IMF8. The panels plot SSI (ordinate) versus time
(abscissa). Time markers on the horizontal axes indicate 1 January of the
corresponding year. The zero-centered oscillatory nature of the modes can be
clearly seen. Also apparent is the local timescale increase with mode
number.
![]()
One of the drawbacks of the original EMD is that it may introduce a
phenomenon known as “mode mixing”. This is the manifestation of
oscillations with dissimilar timescales in the same IMF or the presence of
oscillations with similar timescales in different IMFs. A workaround was
proposed by with ensemble empirical mode decomposition (EEMD).
The idea was to run the decomposition over an ensemble of copies of the
original signal to which white Gaussian noise has been added, with the final
result obtained by averaging. Although the EEMD improved the mode-mixing
problem, the different sums of signal and noise produced different numbers of
modes, making the final averaging somewhat difficult. Added to this, the
reconstructed signal still contained some residual noise and thus was not
identical to the original. To overcome this situation,
have proposed another iteration of the EMD, the complete EEMD with adaptive
noise (CEEMDAN). This method also decomposes the white noise into modes,
along with the signal, such that at each stage of the decomposition a
particular noise is added and a unique residue is computed to obtain each
mode. However, the modes of CEEMDAN still contain some residual noise and
sometimes spurious modes appear in the early stages of the decomposition. The
next iteration of the method, the improved complete ensemble EMD (ICEEMD or
ICEEMDAN), overcomes these issues by fixing the signal-to-noise ratio for all
stages of the decomposition process . The ICEEMD method
is that used in this study. In addition, a fast EMD routine provided by
has been used to decrease the computation time.
The power spectral density (PSD) of the eight IMFs for BOU (solid
line) on a logarithmic scale normalized with respect to the power of the
highest spectral peak. The period, or inverse frequency, runs on the abscissa
in a base-2 logarithm. The individual spectra are shown in the same colors
as the IMFs from Fig. ; from left to right:
IMF1–IMF8. The Fourier estimates of the median periods, marked along
the dash-dotted lines, are seen to increase with mode number. Notable
features are the prominent spectral peak of IMF6 at ∼ 365 days
corresponding to the yearly cycle and the apparent dyadic repartition of the
timescales for IMF1–IMF5.
![]()
To illustrate the workings of the EMD, the eight IMFs of the BOU time series
are presented in Fig. in the order they were obtained,
from top to bottom. As EMD operates in the time domain, the IMFs have the same
temporal support as the original data and, by construction, upper and lower
amplitude envelopes that are symmetrical with respect to zero. It can be
observed in Fig. that, as the decomposition progresses, the
timescale of the IMFs increases; i.e., the intrinsic oscillations are getting
further spaced apart with increasing IMF number. Another view of this is
brought by Fig. , where the power spectral density (PSD) and
a Fourier estimate of the mean period of each IMF are plotted. To aid the
reader, the colors used to portray the individual IMF spectra are the same
as for the time-domain representation from Fig. . The
spectral shapes of the IMF1–IMF5 are similar in form, i.e., bell
curves, and their median periods roughly follow a dyadic scale, i.e., doubling
with increasing IMF number as 3.1 days → 7.3 days →
13.9 days → 30.5 days → 54.0 days. This doubling of
the timescale for these first five IMFs is the hallmark output of an
efficient dyadic filter. Subsequently, it is shown that this dyadic
repartition is pertinent to identifying deterministic signals from random
realizations of quasi-stochastic background processes. This finding is even
more interesting, since the median periods have been estimated with a
Fourier-based method, which measures the period globally over the whole
time range of the IMFs. In contrast, a measure of the local period in
the Hilbert sense is a much better estimate, since it has an accuracy as low
as a quarter wavelength of temporal resolution with respect to the average
timescale of the IMF .
Hilbert spectral analysis
Once the empirical mode decomposition is completed, the second and last step
of the HHT consists in the Hilbert spectral analysis of the previously
obtained IMFs. Each IMF and its Hilbert transform are used to construct a
complex analytic signal, described by an amplitude-modulation–frequency-modulation (AM–FM) model. This decomposition into two time-varying parts
corresponding, respectively, to instantaneous amplitude and instantaneous
frequency is very useful for the purpose of this study. It enables the
identification, in a time-varying sense, of how much power (i.e., the square
of amplitude) occurs at which timescale (i.e., the inverse of frequency).
The Hilbert transform of each real-valued IMF ck(t) can be written as
σkt=Hckt=1πP∫-∞∞ck(τ)t-τdτ,
where subscript k designates the kth IMF and P indicates the Cauchy
principal value. From each IMF and its Hilbert-transformed version, a unique
complex-valued analytic signal can be obtained :
zk(t)=ck(t)+i⋅σkt=ak(t)⋅ei⋅θk(t),
in which
ak(t)=ck2(t)+σk2(t)
is the instantaneous amplitude and
θk(t)=tan-1σk(t)ck(t)
is the instantaneous phase. The instantaneous frequency is the first time
derivative of the instantaneous phase:
ωk(t)=12πdθk(t)dt.
Hilbert spectral analysis of the fifth IMF of the BOU time series.
The intrinsic mode function (IMF5 panel) is the product of its constituent
slowly varying amplitude-modulation part (AM panel) and of its
rapidly changing frequency-modulation component (FM panel). The time-varying
local timescale, extracted from the FM component, is also depicted
(timescale panel). Time markers on the abscissa denote the beginning of the
corresponding year.
![]()
Figure provides a visual guide to this concept by
illustrating the AM–FM decomposition of IMF5 for the BOU time series. The
top panel (IMF5) of the figure reproduces the mode function, which is also
the real part of the analytic signal from Eq. (). The
amplitude of the latter (AM), given in Eq. (), which is the
envelope of the original signal, is then extracted and plotted in the second
panel. This amplitude is not a constant, but rather a time-dependent
function. Next, by removing the AM component from the signal through simple
division, the frequency modulation component is obtained, i.e., the complex
exponential in Eq. (); the real part of this component
(FM) is plotted in the third panel. The FM is a trigonometric function with a
phase argument that is a time-dependent function, as seen from
Eq. (). The local frequency is then just the first
temporal derivative of this phase, as defined in Eq. ().
The inverse of the local frequency, i.e., the local timescale of the signal,
is depicted in the bottom panel (timescale), where its temporal variability
can be clearly distinguished. Owing to their time-varying character, the
amplitude and frequency are usually encountered in the literature under the
terms instantaneous amplitude and instantaneous frequency, respectively.
The Hilbert spectrum (a) of the 10-year time series of SSI
for BOU, spanning 2001 through 2010. Pixel color encodes power (logarithmic-scale color bar on top)
at each instant (abscissa) and each scale
(ordinate). Time markers on the horizontal axis denote the start of the
corresponding year. The whited-out area indicates the regions where edge
effects become significant. The Hilbert marginal spectrum in the
panel (b) is the time-integrated version, i.e., line-by-line sum, of
the Hilbert spectrum and indicates the amount of power at each
scale.
![]()
The original time-series x(t) can then be expressed as a sum of AM–FM
signals riding onto the EMD trend, r(t), as follows:
x(t)=Re∑k=1Nak(t)⋅ei∫ωkτdτ+r(t).
The square of the instantaneous amplitude and the instantaneous frequency of
the IMFs can then be used to represent the data as an energy density
distribution overlaid on the time-frequency space, as in
Eq. (). This representation, called the Hilbert
energy spectrum, is defined by 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 and
designated as a2(t) at the proper time, t, and proper instantaneous
frequency, ω”.
S(ω,t)=∑k=1Nak2(t)⋅ei∑ωkt
The time-integrated version of Eq. (), the Hilbert
marginal spectrum SM(ω), is similar, but not identical, to
the traditional Fourier spectrum:
SM(ω)=∫0TS(ω,t)dt.
An example of Hilbert spectral representation is given in Fig. a
where the BOU time series is shown as an energy density distribution
over-imposed on a time-frequency space as in Eq. ().
Each pixel in the Hilbert spectrum is identified by three attributes –
color, abscissa, and ordinate – through which it denotes the local power
(color, log scale) of the corresponding time series, at a certain time
(abscissa) and at a certain timescale (ordinate, log scale). For the sake
of readability, the spectrum is binned in time, scale, and color space
and has been smoothed. Hence, some aliasing may occur. Some features may be
represented as continuous lines while others are rendered as point-like,
especially where rapid frequency modulation takes place, such as in the
high-frequency bands.
Interpretation of Hilbert spectral features at data boundaries must be done
with care due to possible oscillations of the spline interpolants used in the
EMD (see Algorithm 1). This effect is similar to the “cone of influence” in
the popular wavelet transform . With the EMD, edge effects
are usually contained within a half-period of a component at data boundaries
. In Fig. this region has been whited out.
The plot in Fig. b is the Hilbert marginal spectrum, or the
time-integrated variant of the image at its left, indicating the amount of
power at each timescale. This time-agnostic representation is comparable,
but not identical, to the Fourier spectrum of the same time series. It should
be once again emphasized that the Hilbert marginal spectrum is obtained from
local features of the data, with its components having instantaneous
amplitude and instantaneous frequency, as opposed to the global
outlook of the Fourier spectrum whose constituents have constant amplitude
and constant frequency throughout the whole domain.
Adaptive background null hypothesis
Which confidence can be attributed to the information extracted by the EMD?
More specifically, how can one ascertain that a certain IMF is the result of
a real physical process as opposed to it possibly being a stochastic
manifestation of background processes?
In the past, several investigations have been carried out in order to
identify the effects of the EMD when applied to time series issued from
various models, such as white, red, or fractional Gaussian noise,
.
As a result, it has been consistently shown that, irrespective of the assumed
noise model, the EMD acts as an efficient wavelet-like dyadic filter,
decomposing the stochastic inputs into IMFs having the same spectral shape
but that are shifted in the frequency domain.
Nevertheless, the rejection of a null hypothesis based on an a priori assumed
model of the background does not preclude the probability that the now
statistically significant deemed signals originate from a stochastic process
of a different kind. Furthermore, as the EMD is an adaptive, data-driven
decomposition, it would be desirable to also employ a null hypothesis that
shares the same characteristics, making no beforehand assumptions about the
character of the background processes.
Following P. Flandrin (personal communication, 2015) and
this study will make use of the robust statistical properties of the EMD with
respect to a wide class of background models in order to adaptively contrast
potential signals against presumed stochastic realizations, as detailed
hereafter. Owing to its dyadic filter character, the EMD decomposes noise
time series into IMFs having similar spectral shape but that are translated
to roughly the next lower octave in the spectral domain. When the sampling
step is increased, i.e., the sampling frequency is reduced by fractionally
resampling the input, these components cannot preserve their original
locations in the spectral domain and are instead shifted towards lower
frequencies. Hence, significance testing of IMFs is done by verifying if the
IMF remains unchanged in the time-frequency representation of the signal
during fractional resampling of the latter.
A Hilbert marginal spectrum SMk(ω) is first constructed
for each IMF from its instantaneous amplitude ak(t) and instantaneous
frequency ωk(t). Next the spectrum-weighted mean frequency (SWMF)
ω‾k of each IMF is computed :
ω‾k=∫SMk(ω)ωdω∫SMk(ω)dω.
Then, the time series is fractionally resampled by making the original
sampling rate Δt progressively larger, i.e., the time spacing of the
data points becomes
Δtl=Δt⋅l,l∈1.1,1.2,…,1.9.
For each sampling rate l and for each IMF k, the SWMFs are then
recomputed, obtaining a set ω‾k,l. To enhance the
visibility of the evolution of frequency as a function of the resampling
rate, normalization is performed as in
ω^k,l=ω‾k,lω‾k,1,
with ω‾k,1 being the SWMFs of the modes of the data
having the original sampling rate. Therefore, the normalized SWMFs for the
IMFs of the original data will be unity, i.e., ω^k,1=1,∀k.
Since the EMD is an efficient dyadic filter, frequency deviation from the
unity line will occur for IMFs generated by stochastic processes. It follows
that when ω^k,l≃1,∀l, the null hypothesis
that mode k is the realization of stochastic processes can be rejected.
Box plot of the instantaneous timescales of the IMFs for the four
stations. The top and the bottom edges of the boxes represent the first (Q1)
and, respectively, the third (Q3) quartiles. The bars inside boxes denote the
second quartile (Q2), i.e., the median. The whisker length is set at at most
1.5 times the interquartile range, i.e., 1.5×(Q3-Q1);
hence, the whiskers roughly correspond to ±2.7 standard deviations, or
equivalently ∽ 99 % of the data, assuming normal distribution.
The median for each box is expressed numerically above the lower whiskers.
Outliers are omitted. Numeric values for all the statistical descriptors are
shown in Table .
![]()
Results
The IMFs obtained from the BOU time series from Fig. have
already served as an illustrative example on the operation of the EMD. The
IMFs for the other datasets (not shown) are very similar and are
discussed in the following. It must be noted that, like BOU, the CAR time series
is decomposed into 8 IMFs, while the PAY data have 9, and 10 IMFs are obtained
for TAT. Besides the IMFs, for each time series the decomposition also yields
a residual, or trend (also not shown). With respect to the decennial time
span of the analysis (10 years), the trend can be thought of as a low-pass
approximation of the data , but not as an oscillation.
Since this work focuses mostly on the characteristic scales of temporal
variability, the EMD trends along with their statistical significance and
physical meaning do not fall within the scope of the study; for such
discussion, see for example .
From the Fourier spectra of the IMFs in Fig. it can be seen
that, owing to its median period of 364.8 days, IMF6 can be unambiguously
associated with the yearly cycle, as dictated by the orbital parameters of
the Earth–Sun system. IMF6 also accounts for the most prominent visual
feature in the original data (Fig. a: BOU), with its maxima
and minima denoting summer and winter, respectively. Further evidence is
brought by the spectral shape of IMF6, distinguished by a sharp peak that has
the largest power in Fig. . Also noteworthy is that IMF6
seems to modulate the previous five IMFs, as these latter seem to exhibit
amplitude excursions that are approximately in phase with the amplitude of
IMF6, a phenomenon that is most visually distinguishable in
Fig. for the first three IMFs during the year 2005.
Finally, the last two components, IMF7 and IMF8, having median periods of
783.3 and 1457.4 days (Fig. ), respectively, are seen to
exhibit only slight amplitude deviation from zero in their temporal
representation. Moreover, these fluctuations in amplitude occur at the end of
the signal for IMF7 and at the front edge for IMF8. Interpretation of these
components should, thus, be done with care, since edge effects for the EMD
are known to be usually contained within a half-period of a component at data
boundaries , i.e., approximately 1 year for IMF7 and 2 years for
IMF8.
With the FM components obtained, it becomes possible to illustrate the
frequency contents of each time series in terms of its individual IMFs, as
shown in Fig. , where by means of box plots the
distribution of the local timescale of each mode is conveyed. This box plot
representation is somehow incomplete, as it only accounts for the period
distribution of the modes and does not take into account either the amplitude
or the temporal localization of the events. For readability, the
characteristic period of each IMF with its range of variability is also shown
numerically in Table . The box plots of the instantaneous
amplitude of each IMF are given in Fig. .
Statistical descriptors of the instantaneous timescales of the
IMFs, expressed in days.
Station
Descriptor*
IMF1
IMF2
IMF3
IMF4
IMF5
IMF6
IMF7
IMF8
IMF9
IMF10
BOU
Lower whisker
2.0
2.9
4.3
6.9
18.1
288.1
319.6
1435.5
First quartile
2.5
5.5
11.2
21.6
42.5
342.9
405.4
1476.0
Second quartile
3.1
6.6
13.8
27.4
51.6
368.2
724.7
1531.5
Third quartile
3.9
8.7
18.1
36.0
68.1
393.7
807.6
1611.1
Upper whisker
6.0
13.4
28.3
57.3
106.3
469.5
1192.1
1710.9
CAR
Lower whisker
2.1
2.9
3.7
8.2
12.4
299.7
328.6
916.4
First quartile
2.7
5.7
11.6
21.8
46.6
337.0
367.6
1165.4
Second quartile
3.2
6.9
14.1
26.8
55.7
364.3
469.5
1305.1
Third quartile
4.1
9.0
17.7
33.7
69.8
388.9
716.2
2062.3
Upper whisker
6.3
13.8
26.7
51.4
104.4
443.0
1031.8
2212.8
PAY
Lower whisker
2.0
3.1
4.1
7.9
10.2
28.9
328.8
573.7
1493.0
First quartile
2.5
5.7
11.6
22.7
44.4
231.5
378.1
658.0
1637.0
Second quartile
3.1
7.0
14.5
27.4
53.5
356.6
413.6
707.5
1668.6
Third quartile
3.9
9.0
19.4
36.7
67.3
447.4
477.2
772.9
1733.5
Upper whisker
6.0
14.0
31.0
57.7
101.5
755.0
625.3
918.9
1757.2
TAT
Lower whisker
2.1
2.4
5.0
7.5
16.4
48.3
215.4
378.8
742.8
1908.4
First quartile
2.6
5.9
11.8
23.7
48.9
105.6
328.7
522.8
1196.3
2169.2
Second quartile
3.3
7.1
14.3
28.3
61.8
143.2
366.6
609.0
1440.3
2402.6
Third quartile
4.2
9.4
18.5
34.6
81.1
181.3
404.7
795.0
1687.8
2831.9
Upper whisker
6.5
14.7
28.5
51.1
129.0
294.6
513.9
1145.5
1837.4
3029.0
* A box plot illustration of the statistical descriptors is
shown in Fig. .
For all time series, IMF1–IMF5 have very similar median periods
(Fig. ) that approximate the dyadic sequence:
3.5 days → 7 days → 14 days → 28 days → 56 days.
This dyadic repartition of their median timescales is worthy of attention
since, as it is apparent in Sects.
and , it plays a major role in discriminating
which IMFs can be attributed to deterministic phenomena as opposed to being
the output of random realizations of background processes. Moreover, besides
the notable similarity among the medians of these modes, for all the datasets
both the interquartile ranges and the total ranges of these first five modes
exhibit approximately the same variability. Added to this, IMF6 for BOU, CAR,
and PAY, as well as IMF7 for TAT, whose median periods are, respectively,
368.2, 364.3, and 356.6, as well as 366.6 days, can clearly be associated
with the yearly cycle given by the revolution of the Earth around the Sun.
This yearly component is very similar for BOU, CAR, and to a lesser extent
TAT, with an interquartile range that is concentrated around almost the same
median value and with the only minor difference being the slightly extended
range for TAT of 300 days as opposed to 200 days for the other two. The PAY
yearly mode differs from those of the other stations, with its interquartile
range and foremost its range being much larger, the latter even overlapping
the interquartile ranges of IMF5 and IMF4. This is a result of the
mode-mixing phenomenon described in Sect. that may arise
with the EMD, i.e., the coexistence or mixing of different timescales in the
same IMF, mainly related to the intermittence of signal and to contamination
with noise . Nevertheless, the spectral part of IMF6 which
overlaps IMF5 and IMF4 has very low power ; thus, this
phenomenon does not influence the validity of the analysis. With this in
mind, one notes that for BOU and CAR no spectral components are present in
the 100- to 300-day band. Furthermore, TAT is the only dataset that has a
transitional mode of 143.2 days, with the median period in between the first
five IMFs common to all stations and the yearly cycle.
At this point, the Hilbert frequency distribution of the IMFs for BOU may be
compared to the Fourier one from the PSD in Fig. . As
previously mentioned, the Hilbert estimates are based on local
features of the data and thus are more accurate than the Fourier ones when
applied to non-stationary signals. This can be seen especially when comparing
the range of the first five high-frequency IMFs, which is upper bounded to
about 100 days in Fig. , whereas in the PSD from
Fig. the spectra of the same components are seen to span the
whole timescale range. This also holds for IMF6, which has very narrow
Hilbert period range, whose Fourier analogue is the sharp peak in the PSD of
the same mode. Similar statements can be made for IMF7 and IMF8. To sum up,
it is found that, while the Hilbert period distributions of the modes have
compact supports, the Fourier representations of the same components span the
whole frequency range. Nevertheless, most of the power in the Fourier PSD is
assigned to a frequency band that closely corresponds to the Hilbert range.
Owing to the global nature of the Fourier transform, however, additional
spectral coefficients are needed to provide a complete mathematical
description of the data.
Box plot of the instantaneous amplitudes of the IMFs for the four
stations. The bottom and the top edges of the boxes represent the first (Q1)
and, respectively, the third (Q3) quartiles. The bars inside boxes denote the
second quartile (Q2), i.e., the median. The whisker length is set at at most
1.5 times the interquartile range, i.e., 1.5×(Q3-Q1);
hence, the whiskers roughly correspond to ±2.7 standard deviations, or
equivalently ∽ 99 % of the data, assuming normal distribution.
The median for each box is expressed numerically below the lower whiskers.
Outliers are omitted.
![]()
Resuming the discussion of the IMF timescales from
Fig. , it can be observed that the low-frequency, i.e.,
greater than 1 year, variability in the data, trend notwithstanding, is
assigned into slightly overlapping (within the same time series) IMFs that
span the spectrum starting from the 1-year mark. For BOU and CAR time series,
there are only two modes extending beyond 1 year. First, IMF7 can be seen to
span approximately the same range for both these stations, from about 1 year
to slightly more than 3 years. For BOU, however, the interquartile range and
especially the median period is shifted towards higher periods, i.e.,
724.7 days vs. 469.5 days for CAR. The last modes of IMF8 of these stations are
very different, with a very narrow range around the median of 1531.5 days for
BOU, as well as a range of 900 to over 2000 days and a median of 1305.1 days for CAR.
For the PAY data, the low-frequency components have narrower spectral
support, with two IMFs (IMF7 and 8) that cover the band from 1 to 2.5 years
and median periods of 413.6 and 707.5 days, as well as the IMF9 around 4.5 years
(∽ 1668 days) with a very narrow range. It must also be noted that
IMF7 for BOU, CAR, and PAY has the same lower-end support, and that the
couple (IMF7, IMF8) of PAY taken together somehow emulates IMF7 for BOU and
CAR. Lastly, TAT is the only dataset whose the low-frequency variability is
expressed by three components, IMF8–IMF10, with mean periods of 609,
1440.3, and 2402.6 days. While the first quartile of IMF9 coincides with the
upper range of IMF8, the upper range of IMF9 is slightly below the lower
range of IMF10; hence, the last two modes do not overlap at all. By its range,
IMF8 of TAT approximates IMF7 for BOU and CAR, but there is no proximity in
terms of median or interquartile range. Similarly, the IMF9 of TAT resembles the IMF8
of CAR in terms of range, but their medians are not in close agreement and
their interquartile ranges even less so.
With the scrutiny of these low-frequency components, the discussion of the timescale distribution of the
IMFs from Fig. can now be concluded. However, as
previously mentioned, this particular illustration, although instructive, is
incomplete. First, the box plot representation does not take the
instantaneous variations in frequency into account, but renders global
aggregates instead – much like the traditional Fourier methods, with the
interquartile range spread in addition. This is done on purpose, with the
intent of making it easier for the readership not accustomed to the HHT to
create analogies with the more familiar methods (e.g., Fourier analysis,
wavelets). Second and last, this particular representation is totally devoid
of any information pertaining to the local amplitude, or power, or variance,
of the data. With these considerations in mind, the Hilbert spectrum, a true
time-frequency representation for nonlinear and non-stationary data, is
discussed next. Since the goal of this exercise is to lay the groundwork for
the forthcoming discussion, only the spectrum for the BOU data is provided as
an example.
The BOU Hilbert spectrum from Fig. exhibits a high-frequency
feature between 2 and ∽ 100 days, which corresponds to first five
IMFs of the time series. The instantaneous timescales of these modes overlap
(Fig. ), hence the appearance on the Hilbert spectrum
of a continuum instead of distinct bands. This spectral feature has
relatively low power, which decreases with increasing period, as can be
inferred from the sloped dent in the marginal Hilbert spectrum corresponding
to this region. In the 2- to 32-day band, amplitude modulation by the yearly
cycle can be inferred from the periodic change in color, with yellow–green
tones, occurring mostly during the high irradiance regime of summer, that
turn blue during the winterly minima. Next, in the band between 100 and
300 days, a gap in the spectrum is apparent, as can also be inferred from the
lack of support in this region for any of the BOU IMFs in
Fig. . The yellow trace, corresponding to IMF6,
exhibits frequency modulation around the 1-year period, seen as oscillations
in the range of 300 to 450 days, which is also the support of this mode in
Table . The color of
this IMF indicates that it has the highest power of all the components, as
can also be inferred from the large peak on the marginal spectrum. The
corresponding timescale fluctuations are centered in 365 days, and frequency
modulation is greatest during 2003 through 2005. From 2006 onwards, however,
frequency modulation is less pronounced – perhaps capturing the low solar
activity around the 2008 minimum in the 11-year cycle solar cycle
. The final two low-frequency, blue–green traces on the
spectrum correspond to IMF7 and IMF8. For IMF7, mode mixing is apparent
through the occasional sharing of the yearly timescale band with IMF6,
between mid-2003 and 2005. IMF7 has such low power that it fails to leave an
imprint on the marginal spectrum and it seems to suddenly spring into
existence during summer 2003, which is in perfect agreement with its temporal
representation from Fig. (panel IMF7), whence it can be
seen to have negligible amplitude during the first 2.5 years. Also in
agreement with its temporal depiction from Fig. (panel
IMF8), IMF8 starts out in light-green hues and slowly vanishes during
mid-2007. Although this last BOU mode manages to register on the marginal
spectrum through two very slight indentations around 1500 days (which is
about the median period of this mode from Figs.
and ), most of its power lies within edge effect
territory, hence interpretation of these slight bumps is ambiguous at best.
Thus far, all time series have been shown to share a high-frequency
constituent between 2 and 100 days composed of five IMFs with mean periods
following a dyadic sequence, and an IMF around 365 days that captures the
yearly variability. For BOU, CAR, and PAY, a low-power region can be found in
the 100- to 300-day band. Beyond the 1 year timescale, the low-frequency
variability in the 1.5- to 6-year band is captured by another two (BOU and
CAR) or three (PAY and TAT) components. The TAT data are the only time series
that has an IMF in the low-power band between the high-frequency feature and
the yearly cycle (median period 143.2 days).
Discussion
The previously identified features of the SSI time series will now be
discussed in terms of their intrinsic temporal scales of variability and
physical statistical meaning.
Discriminating deterministic signals from stochastic components in the IMFs
At this point, having identified the spectral characteristics of the SSI
time series by means of the HHT, a question arises with regard to their
physical and statistical significance, namely how can one ascertain which
features represent the expression of real, deterministic physical phenomena
and which ones can be attributed to random realizations of background
processes. Such a method, proposed by , is described in
Sect. . The procedure was applied to the first eight
IMFs of all the time series and the results are presented in
Fig. (from top to bottom: BOU, CAR, PAY, and TAT). First,
each time series was resampled with a fractional sampling rate up to a
factor of 2; i.e., the original uniform time spacing of the data,
Δt, was progressively made larger and larger, as described in
Eq. (): Δtl=Δt⋅l, where l∈1.1,1.2,…,1.9 is the resampling rate and runs
along the horizontal axis. Next, the HHT was used to decompose the resulting
time series into IMFs and to compute their spectrum-weighted mean
frequencies, following Eq. (). In order to emphasize the effects
of the fractional resampling on the spectral contents of the IMFs, these
latter frequencies were then normalized by the SWMF of the original, non-resampled data as per Eq. (). For each dataset, this ratio
is indicated on the y axis, as ω^k,l, with k∈1…8 indicating the IMF number. It then becomes possible
to follow the evolution of the normalized SWMF of each individual IMF as a
function of the fractional resampling rate (Fig. ). As
the EMD is an efficient wavelet-like dyadic filter, it follows that the
IMFs of time series of random processes undergo a translation towards lower
frequencies under fractional resampling. Therefore, for those IMFs whose
SWMFs are not downshifted during resampling, the null hypothesis that they
are purely stochastic can be rejected, i.e., they represent meaningful
signals. Stated otherwise, an IMF k is deemed not to be stochastic in
nature if its normalized SWMFs ω^k,l stay close to the
unity line for all l. From Fig. it can be observed that
for all the stations the only component that maintains a quasi-constant
frequency under fractional resampling is the mode representative of the
yearly variability, i.e., IMF6 for BOU, CAR, and PAY, as well as IMF7 for TAT. All the
other IMFs experience the previously described frequency downshifting; hence,
for them the null hypothesis that they are purely stochastic in nature cannot
be rejected. Since the normalized SWMFs of the yearly components clearly
stray from the black dashed line in Fig. , the result that
they are not stochastic in nature is unambiguous. This also indicates that
the signal-to-noise ratio of these components is well above the minimum value of
0.2–0.3 required to reveal potential signals .
The drift of normalized SWMF ω^k,l (ordinate)
of IMF k, with k∈1…8, as a function of the
resampling rate l (abscissa) for the four time series. From top to bottom:
BOU, CAR, PAY, and TAT. The black dashed diagonal depicts the behavior of a
pure noise time series under an ideal dyadic filter. For all datasets, the
only mode that maintains a quasi-constant frequency under fractional
resampling is the IMF associated with the yearly cycle, i.e., IMF6 for
BOU, CAR, and PAY and IMF7 for TAT. In all the other IMFs quasi-stochastic
behavior is apparent through frequency downshifting towards the next lower
octave, approximately following the dashed line.
![]()
At this point, several precautionary notes are compulsory. First, the rule of
inference used here is modus tollens; i.e., the results from
Fig. do not imply that the modes which experience
downshift in their SWMFs are made up of pure noise. It is subsequently shown
that, for the first five IMFs at least, this is indeed the case; although
(quasi-)stochastic in nature, they are not completely devoid of information.
Second, the result is mostly qualitative, since it is difficult to define a
confidence interval owing to the adaptive nature of the null hypothesis that
can account for different types of models of the stochastic background. Third
and last, the approach is best applied only to the high-frequency modes, with
respect to the data length and sampling, since by resampling spurious
low-frequency oscillation may inadvertently be introduced .
This is further supported by the fact that, as the IMF number progresses, the
region where the influence of edge effects becomes important gets larger and
larger, hence only adding uncertainty to the interpretation of the results.
This is also the reason why this type of analysis was only carried out on the
first eight IMFs of each dataset.
Amplitude modulation through nonlinear cross-scale coupling
This section investigates whether the first five IMFs can be modeled as
purely uncorrelated, random noise or whether they also contain any other form of
information. To test this, the rank correlation between the yearly and
sub-yearly IMFs and their envelopes, e.g., the AM part in the middle panel of
Fig. , has been computed for each SSI time series.
Kendall's rank correlation coefficient, τ, a statistical measure of
ordinal association describing how similar the orderings of the data are when
ranked , is employed here to establish whether each pair
of the two variables, AMx and IMFy with x,y∈1…7, may be regarded or not as independent. τ=1
indicates perfect agreement between rankings, while τ= -1 denotes
perfect disagreement – i.e., one ranking is the reverse of the other; for
τ=0 the two variables are statistically independent.
The resulting rank correlation coefficients and the associated p values,
are presented in Fig. . For each panel, the
columns denote the EMD modes (IMFx) and the rows the amplitudes of their envelopes (AMy).
The background color of each cell AMx,IMFy indicates the rank correlation τ between
IMFy and the AM part of IMFx within the same dataset. The
legend of the color encoding is found on the color bar at the bottom of the
figure. The associated p values are presented numerically in each cell for
the sake of completeness and transparency . For BOU,
CAR, and PAY, IMF6 accounts for the yearly variability in the time series;
hence, the correlation matrices are 6×6 in size. For TAT, the yearly
mode is IMF7; thus, in this case the correlation matrix has a size of
7×7. Two conclusions can be drawn from
Fig. .
Values of τ significantly different from zero, shown in red, are
recorded in the last column for all stations. These demonstrate a modulation
of the amplitude of the components having sub-year timescales, i.e.,
AM1 to AM5 (and AM6 for TAT), by the yearly IMF, at a statistically
significant level (p∼0). The effect is
most pronounced for PAY, as inferred from the darker red shades (larger rank
correlation coefficients).
For the BOU and CAR datasets the first row (AM1) exhibits blue and dark blue
cells for IMF3–IMF5 at the statistically significant level. This
indicates a negative rank correlation. Similar, but lighter, amplitude
modulation is observed on the second row (AM2), but only by IMF4 and IMF5.
For the PAY series, this negative rank correlation is greatly reduced for the
first row (light blue tones) and is absent in the second row. For TAT no such
correlation can be observed. At this point it is interesting to note that, in
a similar way to the discussion from Sect. , the
different features of the datasets from Fig.
also enable a classification of the local climate experienced by the
measuring stations.
It should be mentioned that the amplitude modulation of high-frequency
components by lower frequency ones is also found in the sunspots number time
series and in multiple solar proxies .
The short-term intrinsic periodicities in the solar proxies appear to be
indicative of “randomly distributed dynamical processes in the solar
atmosphere” that are closely related to the 11-year solar activity and
therefore, unsurprisingly, the high-frequency modes are found to be modulated
by this latter cycle . But this phenomenon is not limited
to solar activity signals and has also been identified in surface air
temperature records , and time series of the sea level
, and may indicate cross-scale nonlinear couplings
.
The intrinsic timescales of variability in the SSI
Firstly, the median periods of the IMFs composing the high-frequency band are
revisited. It is shown in Fig. that they follow a
dyadic repartition that approximates the series dyadic sequence: 3.5, 7, 14,
28, and 56 days. Such a doubling in frequency in IMFs has been previously
reported in astrophysical and geophysical signals. When investigating three
independent datasets of satellite observations of the (extraterrestrial)
total solar irradiance, consistently find a similar
dyadic-scale progression of modes at 13.5, 27, and 54 days, statistically
significant within the 95 % level, that correspond to the 27-day solar
rotation period and its (sub-)harmonics. find intrinsic
periodicities having an average of 25-2+7 and 44-5+10 days in
five different solar proxy signals. The mean periods and the associated error
bars in (sub-)superscript, estimated at the half-level width of the
corresponding probability histogram, were obtained by analyzing the sunspot
area for the whole Sun, and for the northern and southern solar hemispheres
taken separately, the 10.7 cm radio flux intensity, and the helioseismic
frequency shift. also find periodicities of 5, 7, 9, 13.5,
and 27 days in different radiation belt, solar wind, geomagnetic, and auroral
parameters. Compelling as it may seem, nevertheless, the imprint of a solar
rotation signature on ground measurements of the SSI is highly unlikely, as
it would imply the existence of hitherto unknown physical mechanisms in
Earth's atmosphere (Gerard Thuiller, personal communication, 2015). The
amplitudes of the IMFs of the TSI time series and those of the IMFs in the
SSI data differ at times by 2 orders of magnitude (e.g., compare
Fig. with Fig. 1 in ). If the solar
rotation signature were to be seen in the IMFs of the SSI this would require
the existence of amplifying processes. and
have studied the possibility of such a mechanism and
have concluded that, irrespective of the mechanisms invoked and of the
amplification of the solar variability, for the past decades solar forcing is
only a minor contributor and thus not able to account for most of the global
warming observed in the second half of the 20th century, which could be
better explained by an increase in greenhouse gases. Further proof is
provided subsequently, this time from a signal theoretical point of view, in
support of the view that it is unlikely that the solar rotation signature is
captured in measurements of the SSI.
Rank correlations between IMFs and their AM components for
BOU (a), CAR (b), PAY (c), and TAT (d).
Kendall's rank correlation coefficient τ is color-coded according to
the color bar on the bottom. IMFs run vertically, along the columns, and
their AM components run horizontally, along the rows. The numeric values
within the cells are the associated
p values.
![]()
Secondly, in the 100- to 300-day band, two of the stations, BOU and CAR, do
not exhibit any variability. For PAY, the support of yearly IMF6 protrudes in
this region, although its first quartile rests well below the 200-day mark.
As mentioned before, the power of the portion of this IMF that extends into
the high-frequency range is very small (not shown). Hence, while not totally
devoid of spectral features, this band contains negligible power. A distinct
mode is present at TAT in this band, whose median period of 143.2 days
somehow seems to continue the dyadic sequence of the previous five modes.
Since a similar transitional mode has also been found for two locations in
Europe , presently no explanation in terms of physical
processes, such as monsoon rainy seasonality, can be proposed for IMF6 of
TAT. These findings are important for the modeling and forecasting of the
SSI, as follows. On the one hand, models for BOU and CAR should not contain
any power in this band, or should at least filter it out. For TAT, on the
other hand, any model attempting to reconstruct the SSI should ensure that
the 100- to 300-day region is not a spectral void. In
Sect. , evidence is
presented that the spectral band spanning from 2 to 300 days seems to be
composed mostly of random realizations of stochastic background processes,
which can be modeled following, e.g., ,
, , and .
Thirdly, the median periods detected around the 1-year mark in all the
datasets can be explained by the revolution of the Earth around the Sun and the
associated orbital parameters. The interpretation of these components is
unambiguous, with one notable exception for the PAY time series, whose IMF6
exhibits mode mixing; i.e., it has a total range that overlaps some of the
modes in the high-frequency band. Nevertheless, it will be subsequently shown
that it is indeed these components that account for variability at the 1-year
timescale.
Lastly, the components indicative of low-frequency variability on timescales
greater than 1 year are discussed. The intrinsic timescales found in these
IMFs seem to match once more those pertaining to the so-called quasi-biennial
oscillations (QBOs) that have been
observed in solar activities and proxies with periodicities between 0.6 and
4 years , as well as in
meteorological data like who identifies a 1.68-year peak
in cloud cover or high-latitude stratospheric temperatures and geopotential
heights . Nevertheless, within the scope of the current
analysis, the interpretation of these low-frequency variability components as
a real, possibly QBO-like, signal is uncertain.
The local climate imprint in the IMFs
It is shown in Sect. that the four measuring
stations experience different climates and exhibit differences in terms of
KT. Figure shows that the high-frequency
band composed of the first five IMFs is very much alike for all stations.
This section investigates the possible relationship between local climate and
dissimilarities in terms of the repartition of the IMFs with mode number 6
and higher.
It can be noted that the IMF6 for both BOU and CAR has a well-defined period
(Fig. ), with a median of, respectively, 368.2 and
364.3 days and very narrow interquartile range. In addition, for both
stations, the IMF6 is the mode having the greatest amplitude, and by far,
compared to the other modes (Fig. ). The IMFs 7 and
8 for CAR have less marked periods, i.e., the interquartile ranges are greater
than for IMF6, and the amplitude of each IMF is very small. These
observations may be related to the high frequency of cloud-free days seen in
Fig. because, in absence of clouds, the variability in the
daily mean of SSI is predominantly driven by the variability in the solar
irradiance received at the top of the atmosphere during the year.
PAY and TAT need four IMFs to account for the low-frequency variability,
i.e., one IMF more than BOU and CAR. IMF6 in PAY has a median period of 356.6 days,
close to 1 year (Fig. ) with a large interquartile
range. The median amplitude of the IMF6 is approximately half of that of BOU
or CAR (Fig. ) and the amplitude exhibits large
variations. The median amplitude of the IMF7 is similar to that of IMF6 while
the period of the IMF7 is well marked with a narrow interquartile range. This
may be related to the abundance of the presence of broken clouds that render
the SSI signal highly intermittent. This intermittence of the signal could,
in turn, explain the mode mixing observed in IMF6 .
Similar to PAY, TAT also has a low median clearness index
K̃TTAT=0.51, which helps explain the
presence of a sixth IMF (median period: 143.2 days) between the
high-frequency components and the yearly IMF7 (median period: 366.6 days). In
other words, the amplitudes of the stochastic components in the sub-year band
are higher at TAT than at PAY, or, conversely, there is a lower
signal-to-noise ratio of the yearly cycle. Hence, this high power of the
background drives the EMD to assign a dedicated intrinsic mode for this
region, as opposed to PAY, where the signal in this spectral band is assigned
to the yearly IMF through mode mixing.
Conclusion and outlook
To sum up, the HHT
analysis of decennial time series of daily means of measurements of the SSI
from distinct BSRN stations has revealed the following: the presence of a
high-frequency band (2–100 days) consisting of quasi-stochastic IMFs that
have been shown to be amplitude modulated by the yearly cycle; a low-power
spectral band in the 100- to 300-day region; a well-defined spectral
peak at the 1 year mark accounting for the yearly variability; and multiple
QBO-like components whose character has been, inconclusively, attributed to
quasi-stochastic random processes.
This separation of the (quasi-)periodic components of the signal from the
apparently random realizations of a stochastic background has been shown to
significantly augment accuracy in time-series modeling . Our
findings can be thus directly used to improve models for estimating SSI from
satellite images or forecasts of the SSI.
We have shown that the adaptive Hilbert–Huang transform is a versatile
tool in analyzing SSI datasets, exhibiting significant nonlinearity and
non-stationarity. First, we have employed it to extract the intrinsic modes
of variability in the SSI at distinct timescales. Second, the HHT has been
used to discriminate between the deterministic yearly cycle and the
quasi-stochastic high-frequency components. The same methodology could also
be employed on different geophysical signals, such as wind speed time series and
river discharge datasets.
When modeling climate processes as dynamical systems with low-frequency
oscillations and noise effects, have shown that “even
the `approximately right' noise can help, rather than hinder”. Here, we have
provided a recipe not only for extracting, but also for characterizing the
stochastic high-frequency constituents of long-term time series of the SSI.
Indications with respect to modeling these quasi-stochastic components have
also been provided. With respect to SSI forecast models, it is exactly this
spectral region that is the focus of attention
. venture as far as
stating that “the accuracy of the solar irradiance forecasting models
depends almost exclusively on the ability to forecast the stochastic
component”. In this light, the recipe for discriminating the realizations of
random background processes that we have put forth can be seen as one of the
more significant contributions of our paper.
We have also proposed that a classification of the measuring stations
according to climate and/or solar insolation conditions may be possible,
based on the Hilbert spectral features of the data. Thus, one future research
pathway could consist in creating a catalogue of the variability in the solar
resource, at different timescales, on a global scale via satellite estimates
of the SSI. Current meteorological reanalyses are too noisy in their
estimates of the SSI to form the basis for such a catalogue
. In terms of solar power production, the low-frequency
variability data would aid with policy and investment decisions, while
short-term variability would be of interest from a monitoring, operations, and
engineering perspective.