Reconstruction and prediction of the total solar irradiance: From the Medieval Warm Period to the 21st century

Abstract

h i g h l i g h t s:
We present a new method to estimating TSI between the years 1000 and 2100.
We found a grand minimum for the 21st century. We found conspicuous periodicities of 11 and 120 years.
The solar activity grand minima periodicity is of 120 years.
To decide when the solar activity is high or low, we using the power of the TSI.

a b s t r a c t Total solar irradiance is the primary energy source of the Earth's climate system and therefore its varia-tions can contribute to natural climate change. This variability is characterized by, among other manifes-tations, decadal and secular oscillations, which has led to several attempts to estimate future solar activity. Of particular interest now is the fact that the behavior of the solar cycle 23 minimum has shown an activity decline not previously seen in past cycles for which spatial observations exist: this could be signaling the start of a new grand solar minimum. The estimation of solar activity for the next hundred years is one of the current problems in solar physics because the possible occurrence of a future grand solar minimum will probably have an impact on the Earth's climate. In this study, using the PMOD and ACRIM TSI composites, we have attempted to estimate the TSI index from year 1000 AD to 2100 AD based on the Least Squares Support Vector Machines, which is applied here for the first time to esti-mate a solar index. Using the wavelet transform, we analyzed the behavior of the total solar irradiance time series before and after the solar grand minima. Depending on the composite used, PMOD (or ACRIM), we found a grand minimum for the 21st century, starting in $ 2004 (or 2002) and ending in $ 2075 (or 2063), with an average irradiance of 1365.5 (or 1360.5) Wm À2 AE 1r ¼ 0:3 (or 0.9) Wm À2 . Moreover, we calculated an average radiative forcing between the present and the 21st century minima of $ À0:1 (or À0.2) Wm À2 , with an uncertainty range of À0:04 to À0:14 (or À0:12 to À0:33) Wm À2 . As an indicator of the TSI level, we calculated its annual power anomalies; in particular, future solar cycles from 24 to 29 have lower power anomalies compared to the present, for both models. We also found that the solar activity grand minima periodicity is of 120 years; this periodicity could possibly be one of the prin-cipal periodicities of the magnetic solar activity not so previously well recognized. The negative (positive) 120-year phase coincides with the grand minima (maxima) of the 11-year periodicity. Ó 2014 Elsevier B.V. All rights reserved.

Full text

Reconstruction and prediction of the total solar irradiance: From
the Medieval Warm Period to the 21st century
V.M. Velasco Herrera
a,
⇑
, B. Mendoza
a
, G. Velasco Herrera
b
a
Departamento de Ciencias Espaciales, Instituto de Geofísica, Universidad Nacional Autónoma de México, Ciudad Universitaria, Coyoacán, 04510 México D.F., Mexico
b
Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Ciudad Universitaria, Coyoacán, 04510 México D.F., Mexico
highlights

We present a new method to estimating TSI between the years 1000 and 2100.

We found a grand minimum for the 21st century.

We found conspicuous periodicities of 11 and 120 years.

The solar activity grand minima periodicity is of 120 years.

To decide when the solar activity is high or low, we using the power of the TSI.
article info
Article history:
Received 2 July 2013
Received in revised form 4 July 2014
Accepted 11 July 2014
Available online 23 July 2014
Communicated by W. Soon
Keywords:
Total solar irradiance
Grand Minima
Estimation of the 21st century minimum
ACRIM
PMOD
Least Square Support Vector Machine
abstract
Total solar irradiance is the primary energy source of the Earth’s climate system and therefore its varia-
tions can contribute to natural climate change. This variability is characterized by, among other manifes-
tations, decadal and secular oscillations, which has led to several attempts to estimate future solar
activity. Of particular interest now is the fact that the behavior of the solar cycle 23 minimum has shown
an activity decline not previously seen in past cycles for which spatial observations exist: this could be
signaling the start of a new grand solar minimum. The estimation of solar activity for the next hundred
years is one of the current problems in solar physics because the possible occurrence of a future grand
solar minimum will probably have an impact on the Earth’s climate. In this study, using the PMOD
and ACRIM TSI composites, we have attempted to estimate the TSI index from year 1000 AD to 2100
AD based on the Least Squares Support Vector Machines, which is applied here for the first time to esti-
mate a solar index. Using the wavelet transform, we analyzed the behavior of the total solar irradiance
time series before and after the solar grand minima. Depending on the composite used, PMOD (or
ACRIM), we found a grand minimum for the 21st century, starting in 2004 (or 2002) and ending in
2075 (or 2063), with an average irradiance of 1365.5 (or 1360.5) Wm
2
1
r
¼0:3 (or 0.9) Wm
2
.
Moreover, we calculated an average radiative forcing between the present and the 21st century minima
of 0:1 (or 0.2) Wm
2
, with an uncertainty range of 0:04 to 0:14 (or 0:12 to 0:33) Wm
2
.Asan
indicator of the TSI level, we calculated its annual power anomalies; in particular, future solar cycles from
24 to 29 have lower power anomalies compared to the present, for both models. We also found that the
solar activity grand minima periodicity is of 120 years; this periodicity could possibly be one of the prin-
cipal periodicities of the magnetic solar activity not so previously well recognized. The negative (positive)
120-year phase coincides with the grand minima (maxima) of the 11-year periodicity.
Ó2014 Elsevier B.V. All rights reserved.
1. Introduction
Solar radiation is one of the main influences on the Earth’s
climate. Over the 11-year solar cycle, total solar irradiance (TSI)
variations of 0:1%have been observed between the solar minima
and maxima of cycles 22 and 23 (Kopp and Lean, 2012). This mod-
ulation is mainly due to the interplay between dark sunspots and
bright faculae and network elements (Foukal and Lean, 1988).
Long-term reconstructions of TSI (e.g. Steinhilber et al. (2009),
Wang et al. (2005), Krivova et al. (2007), Muscheler et al. (2007),
Bard et al. (2000), Fröhlich (2006), Vieira et al. (2011)) showed
epochs of maxima and minima when substantial changes in the
http://dx.doi.org/10.1016/j.newast.2014.07.009
1384-1076/Ó2014 Elsevier B.V. All rights reserved.
⇑
Corresponding author. Tel.: +55 56224113.
E-mail address: vmv@geofisica.unam.mx (V.M. Velasco Herrera).
New Astronomy 34 (2015) 221–233
Contents lists available at ScienceDirect
New Astronomy
journal homepage: www.elsevier.com/locate/newast

TSI occur. These changes can contribute to climate variability (e.g.
Gray et al. (2010)).
These TSI reconstructions are based on the evolution of the
sun’s total and/or open magnetic flux. For instance Wang et al.
(2005) used a flux transport model to simulate the evolution of
the solar total and open magnetic flux by means of the group
sunspot number (GSN). This model obtained two annual TSI
reconstructions, one with and one without a secularly varying
background. From 1850 onwards these reconstructions are recom-
mended as the solar forcing for the fifth Coupled Model Intercom-
parison Project 20th century simulations (Taylor, 2009). The
Krivova et al. (2010) TSI model used the GSN to reconstruct the
evolution of the solar surface magnetic field, relying on time con-
stants representing the decay and conversion of the different
photospheric magnetic flux components.
Other reconstructions explicitly used the solar modulation
potential (
U
). This potential quantifies the galactic cosmic ray
deceleration produced by the solar activity (Gleeson and Axford,
1968):
U
is obtained from cosmogenic isotope time series cor-
rected for geomagnetic field variations. The Steinhilber et al.
(2009) reconstruction is based on an observationally-derived rela-
tionship between TSI and open magnetic flux (Fröhlich, 2009); the
authors obtained the open flux from
U
. Also, Delaygue and Bard
(2011) use
U
to reconstruct the TSI.
The Vieira et al. (2011) model derived the relationship between
solar-cycle averaged open magnetic flux and TSI from the Krivova
et al. (2010) model, and this is used to reconstruct the TSI through-
out the Holocene, based on
14
C record.
The TSI reconstructions are used in climate models for among
other things, to assess the effect of solar radiative forcing on the
climate. Because the physical constraints on the amplitude and
timing of TSI histories remain a very difficult challenge, we propose
that using as many different approaches for TSI reconstructions
will lead to a better and objective sensitivity experiments regard-
ing how Earth climate can response to changing TSI.
Studies using cosmogenic isotope data and sunspot data (e.g.
Solanki et al. (2004), Abreu et al. (2008), Velasco et al. (2008)) indi-
cate that we are currently within a grand activity maximum, which
began after 1930.
However, the behavior of the solar cycle 23 minimum has
shown an activity decline not previously seen in the past solar
cycles for which spatial observations exist (e.g. Kirk et al. (2009),
Lee et al. (2009), Smith and Balogh (2008), McComas et al.
(2008)). The descending phase and minimum measurements show
that the TSI has fallen below the previous two solar minima values:
the mean PMOD composite TSI for September 2008 was
1365:26 0:16 Wm
2
, compared to 1365:45 0:1Wm
2
in 1996
or 1365:57 0:01 Wm
2
in 1986 (Fröhlich, 2009).
Studying the solar wind, the interplanetary magnetic field
strength, and the open solar flux over the past century,
Lockwood et al. (2009) found that all three parameters showed a
long-term rise peaking around 1955 and 1986 and then declining,
yielding predictions that the grand maximum will end in the years
2013, 2014 or 2027, depending on the parameter used. Other stud-
ies have indicated that the current maximum will not last longer
than two or three more solar cycles more (Abreu et al., 2008). It
has been suggested that a Dalton-type minimum already began
in the preceding minimum solar cycle 24 reaching to solar cycles
24 and 25 (e.g. Russell et al. (2010), Rigozo et al. (2010)). In addi-
tion, a summary and analysis of maximum sunspot number predic-
tions for solar cycle 24 indicate an average of 115 40 (Pesnell,
2012). Moreover, up to now the observed smoothed sunspot max-
imum number of solar cycle 24 shows less than 70 sunspots
(www.sidc.be), consistent with the lower estimate.
Frequency analysis of solar activity series (e.g. Tobias et al.
(2004)) showed several significant long-term periodicities whose
existence has inspired attempts to predict trends in solar activity.
The future estimation of solar activity for the next hundred years
is one of the current problems in solar physics because the possible
occurrence of a future grand solar minimum will probably have a
significant impact on the Earth’s climate.
In this paper we modeled the TSI between 1000 and 2100 AD
using the Least Squares Support Vector Machines, which is being
applied for the first time to estimate a solar activity index. Using
the PMOD and ACRIM TSI composites as the original calibration
and training set, we produced two TSI reconstructions and future
TSI estimates, compared them with previous results, and calcu-
lated their main periodicities, power index and radiative forcings
for Earth’s climate.
2. Data
2.1. Statistical characteristics of the TSI-composites
Since 1978, several independent space-based instruments have
measured the TSI. Three main composite series were constructed:
the Active Cavity Radiometer Irradiance Monitor (Willson and
Mordvinov, 2003; Scafetta and Willson, 2014), the Royal Meteoro-
logical Institute of Belgium (RMIB) (Dewitte et al., 2004) and the
Physikalisch-Meteorologisches Observatorium Davos (PMOD)
(Fröhlich, 2006). The composites employed different calibration
techniques and mathematical algorithms.
Here we used the annual TSI mean obtained from the daily
PMOD (www.pmodwrc.ch) and ACRIM (www.acrim.com) compos-
ites (Fig. 1) between 1979 and 2013.
Figs. 1a and c show the Probability Density Function (PDF) of
the PMOD and ACRIM composites respectively, which yields a
binomial distribution with the characteristic that the first maxi-
mum is greater than the second. In the case of the PMOD compos-
ite, these maxima are 1365:7Wm
2
and 1366:3Wm
2
and for
the ACRIM-composite: 1361:2Wm
2
and 1362:1Wm
2
,
respectively.
We note that the difference between the first and the second
peak in the PDF is larger for the ACRIM composite than for the
PMOD data (i.e., compare Fig. 1a and c).
2.2. Power index
As an indicator of the level of TSI activity, we defined the fol-
lowing TSI annual power index (P
i
):
P
i
¼X
N
i
k¼1
ðTSI
k
Þ
2
N
i
where N
i
is the total number of data for each year i. The normalized
annual power index anomalies can be used to ponder and compare,
which then allows us to decide when periods of higher or lower TSI
occur; it is defined as:
^
P
i
¼P
i
hP
TSI
i
MAXðkPkÞ
where hP
TSI
iis the average PMOD or ACRIM average power index,
MAXðkPkÞ is the maximum in absolute value of the power index
anomalies and (
^
) denotes normalization.
According to Fig. 1b and d, the power anomaly signs of both
composites have similar time behavior for cycles 21–23: around
the minimum of each cycle, the normalized power anomalies
are negative; for the rest of the cycle phases, the normalized
power anomalies are positive. But we highlight that during the
ascending phase of solar cycle 24, the values are negative in both
TSI-composites. This empirical evidence would indicate that dur-
ing the current activity phase, solar cycle 24 has a very low level
222 V.M. Velasco Herrera et al. / New Astronomy 34 (2015) 221–233

compared to the previously observed cycles. We do not expect
the power intensities to be similar because the composite
calibrations are different around the minimum of each cycle;
the normalized power anomalies are negative. Around the
maxima of solar cycles 21–23, the normalized power anomalies
are positive, but in solar cycle 24 they are negative in both
TSI-composites.
2.3. Sunspot
We also used information from sunspots: the group sunspot
number (GSN) between 1610 and 1995 (Hoyt and Schatten,
1998) and the sunspot numbers (SSN) from 1996 to 2013 (Royal
Observatory of Belgium, http://sidc.oma.be/) for this study. As
the GSN series ended in 1995, there are 17 years missing values.
Fig. 1. TSI composites between 1979 and 2013 cycles 21–24. (a) Probability Density Function of PMOD composite. (b) Daily PMOD composite data (black line) and TSI
normalized annual power anomalies (gray bars). (c) Probability Density Function of ACRIM composite. (d) Daily ACRIM composite data (black line) and TSI normalized annual
power anomalies (gray bars).
V.M. Velasco Herrera et al. / New Astronomy 34 (2015) 221–233 223

Taking advantage of the fact that there is a great similarity
between the GSN and the sunspot numbers from 1878 to 1995
(solar cycles 12–22), we calculated the GSN from 1996 to 2013
with the following expressions:
^
SSN ¼SSN hSSNi
r
SSN
ð1Þ
where ^
SSN is the standardized annual sunspot number, SSN is the
annual sunspot number, hSSNiis the mean sunspot number value
for the period 1996–2013 and
r
SSN
is the standard deviation.
GSN
new
¼
r
GSN
^
SSN þhGSNið2Þ
Here GSN
new
is the annual group sunspot number between 1996
and 2013;
r
GSN
and hGSNiare the group sunspot number standard
deviation and the group sunspot number mean value for solar
cycles 12–22 respectively (Fig. 2).
3. Method
3.1. Wavelet transform
We applied the wavelet transform to the TSI models described
in Section 3.2 between the years 1000 AD and 2100 AD, in order
to study local variations of spectral power at multiple periodicities
(Torrence and Compo, 1998). The spectral power is the periodicity
amplitudes of the wavelet transform. The wavelet transform of a
discrete sequence Y
n
is defined as:
W
n
ðsÞ¼X
N1
n
0
¼0
Y
n
w

o
ðn
0
nÞ
sdt
 ð3Þ
where sis the scale, nis the translation parameter (slide in time)
and the (⁄) denotes complex conjugation.
Here we chose the Morlet function as the mother wavelet
because it provides a higher periodicity resolution and it is com-
plex, allowing us to filter the series in bandwidths (Soon et al.,
2011).
The decomposition of a signal (Y
n
) can be obtained from a time-
scale filter (Leal-Silva and Velasco, 2012). The time-scale filter (the
inverse wavelet transform) is defined as (Torrence and Compo,
1998):
Y
n
¼d
j
dt
1=2
C
d
w
o
ð0ÞX
j
2
j¼j
1
ReðW
n
ðs
j
ÞÞ
s
1=2
j
ð4Þ
where j
1
and j
2
define the scale range of the specified spectral
bands, d
j
is the scale averaging factor, C
d
is a constant (d
j
¼0:6
and C
d
¼0:776, for Morlet wavelet), and w
0
is an energy normaliza-
tion factor (Torrence and Compo, 1998).
To calculate the confidence level, we normalized the time ser-
ies; in this way the series have a Gaussian distribution. Meaningful
wavelet meaningful periodicities (confidence level greater than
95%) must be inside the cone of influence (COI). COI is the region
of the wavelet spectrum outside of which the edge effects become
important.
We also included the global spectra in the wavelet plots to show
the power contribution of each periodicity inside the COI. We
established our significance levels in the global wavelet spectra
with a simple red noise model, i.e. increasing power with decreas-
ing frequency, (Gilman et al., 1963). The uncertainty of the peaks is
obtained from the full width at half maximum.
3.2. Least Squares Support Vector Machines
To regress the TSI between 1610 and 1978, we applied a non-
parametric and nonlinear method based on the Least Squares
Support Vector Machines Nonlinear Regression (LS-SVM) with
radial basis function (RBF) kernel. To estimate the TSI from the
Medieval Warm Period (beginning in the year 1000 AD) to 1610
and extrapolate forward to the full 21st century, we used the
LS-SVM together with the Nonlinear Autoregressive Exogenous
Model (NARX) (Vapnik, 1998; Suykens et al., 2005). The LS-SVM–NARX
is originally applied to calibrate the model between the years 1979
and 2013.
The LS-SVM is a generalization of the algorithms developed in
the 1960’s (Vapnik and Lerner, 1963; Vapnik and Chervonenkis,
1964). In its beginnings the LS-SVM was applied to classification
problems; subsequently, it was proposed as a regression method
(Weigend and Gershenfeld, 1994) and nowadays it has trans-
formed into an estimation method (Drucker et al., 1997;
Weigend and Gershenfeld, 1994). The LS-SVM is defined as
(Vapnik, 1998; Suykens et al., 2005):
W
¼X
k
X
k
I
k
þbð5Þ
X¼XðIÞ
where
W
is the estimated output (TSI), I
k
denotes the input data
(GSN) at time k(discrete time index), X
k
is the weight, and bis
the ‘‘bias’’ term. As Xdepends on I, the Eq. (5) is nonlinear.
For the NARX models (Vapnik, 1998; Suykens et al., 2005), we
used the function:
U
k
¼fðY
k1
;Y
k2
;...;Y
kp
;U
k1
;U
k2
;...;U
kq
Þð6Þ
where U
k
is the TSI value, Y
k1
;Y
k2
;...;Y
kp
denotes the output,
and U
k1;k2;...;kq
is the input data at time k. The order of the system
is determined by the input and output values pand q, representing
the number of lags. The function fin Eq. (6) is non-analytic and has
been modeled with the LS-SVM. At each step ‘‘k’’, the input data in
LS-SVM is the value of the GSN at time ‘‘k’’. The output at time ‘‘k’’ is
the estimated value of the TSI (Eq. (5)).
In all artificial intelligence models, the goodness of the estima-
tion has a limit (Cherkassky and Ma, 2004): the cost function must
be small enough (close to zero); this is subjectively selected or
determined by the user. To compute a cost function, we used the
mean squared error (MSE).
Additionally, the LS-SVM uses the ‘‘

-insensitive’’ loss function
for TSI estimation. The region of uncertainty is the ‘‘Cone of

-accu-
racy’’. In our case it is one standard deviation for each solar cycle.
We referred interested readers to the papers cited here for more
formal and detailed descriptions of the method and we focused
1650 1700 1750 1800 1850 1900 1950 2000
0
50
100
150
0
50
100
150
TIME (YEARS)
Fig. 2. Sunspot Number time series from 1700 to 2013 (black line). Group sunspot
number time series between 1610 and 2013 (gray area). From 1610 to 1995 we
used the Hoyt time series. To calculate the group sunspot number from 1996 to
2013 we used Eqs. (1) and (2).
224 V.M. Velasco Herrera et al. / New Astronomy 34 (2015) 221–233

in this paper strictly on the application of the method for TSI
estimates.
3.3. Wavelet-LS-SVM-algorithm for multi-channel system
Different studies suggest that solar mid-term (i.e., 1–2 years) as
well as secular periodicities originate from chaotic quasi-periodic
processes and not from stochastic or intermittent processes (e.g.
Bouwer (1992), Mendoza and Velasco (2011)). Chaotic quasi-peri-
odic processes can be decomposed in simple signals of modulated
amplitude, which can be used to regress and estimate. Then, the
periodicities of the solar activity can be decomposed in multi-
channel or bandwidth. Here each periodicity has been analyzed
as a channel as follows:
(I) Using Eq. (3) we found the periodicities of the TSI-
composites.
(II) With Eq. (4) we found the time series for each of the period-
icities of (I).
In this paper we used steps (I) and (II) with the following two
LS-SVM algorithms to estimate the TSI:
3.3.1. LS-SVM calibration and regression per channel
In our case
W
obtained from Eq. (5) is the estimated TSI. To cal-
culate the TSI using the LS-SVM method, we proceeded as follows
for each channel (periodicity):
(i) Calibration of the LS-SVM–NARX with TSI composites
(PMOD and ACRIM): we determined the weights (
X
k
) and
‘‘bias’’ term (b) from Eq. (5). We used as input the 34 GSN
data points from 1979 to 2013; they correlated to 80% of
the total data distributed arbitrarily along this time span.
In Fig. 3a we calibrated the LS-SVM–NARX reconstruction
with the PMOD TSI composite. The model reproduces the
composite very well; in fact, the linear correlation coefficient
is 0.98 between 1979 and 2013. The red dotted lines are the
‘‘Cone of

-accuracy’’ and we obtaining a MSE of 0.01; the
testing of the remaining 20% presented an MSE of 0.01.
(ii) Regression of TSI between 1610 and 1978: once we cali-
brated the model and determined the weights (
X
) and ‘‘bias’’
term from step (i), we carried out the TSI estimate by enter-
ing the GSN data points from 1610 to 1978 in the calibrated
LS-SVM.
3.3.2. LS-SVM estimation per channel
To estimate the TSI from the Medieval Warm Period to 1610 and
throughout the 21st century, we further followed with steps:
(iii) Selection of the model lags pand q.
(iv) Determination of the weight and bias.
(v) Estimation of a TSI model using Eq. (6).
(vi) Computation of a cost function.
(vii) Test of the estimation, by comparing the estimated TSI-
model with the measured TSI-composite (PMOD or ACRIM)
value.
(viii) Test of the cost function: if this function was small enough
we stopped; otherwise we changed one of the parameters
and repeated from (iii) onwards.
As an example we used as input data the TSI model obtained
from the LS-SVM regression over the 1610–1978 interval to test
the reproduction of the actual PMOD composite. Fig. 3b shows
the comparison. This result has an MSE of 0.06 and a linear corre-
lation coefficient of 0.90.
We note that the LS-SVM–NARX model reproduces quite well
the evolution in amplitude of the TSI-PMOD and that the model
presents a maximum lag of 10–15% with respect to the PMOD
TSI composite minima.
We further note that if the LS-SVM–NARX model could repro-
duce exactly the values of the TSI-PMOD composite, then the
model is oversaturated (Cherkassky and Ma, 2004) and the future
TSI estimations will have much larger errors.
After analysing all of our results, we noticed that in general all
the reconstructions are affected by the model variables. We sug-
gest that the accuracy of the estimates may be limited by an uncer-
tainty principle:
r
ðAÞ
r
ðTÞ
r
ð/Þ
r
ðwÞ
>0
where
r
ðAÞ
;
r
ðTÞ
;
r
ð/Þ
and
r
ðwÞ
are the standard deviations of
amplitude (A), period (T), phase ð/Þand other parameters ðwÞ
respectively; thus, we will never know the ‘‘exact’’ values of the
estimations.
4. Results
4.1. TSI estimations
In Fig. 4 we showed the LS-SVM reconstructions (black line)
between 1000 and 2013 AD. The red lines are the ‘‘Cone’’ of

-accuracy, i.e., the uncertainty of the reconstruction, which is
one standard deviation for each solar cycle. When using GSN as
input data, the LS-SVM model composites show low secular
amplitudes.
1980 1985 1990 1995 2000 2005 2010
1365
1365.5
1366
1366.5
TIME (YEARS)
TSI (W/m2)
1980 1985 1990 1995 2000 2005 2010
1365
1365.5
1366
1366.5
1367
TIME (YEARS)
TSI (Wm−2)
(a)
(b)
Fig. 3. (a) Calibration of the LS-SVM–NARX model (blue line) with the PMOD TSI
composite (red line). (b) Test of the estimation. Comparison of PMOD TSI composite
(red line) with the LS-SVM–NARX model (blue line). The red and black dotted lines
are the ‘‘Cone

-accuracy’’. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)
V.M. Velasco Herrera et al. / New Astronomy 34 (2015) 221–233 225

In Fig. 5 we compared the LS-SVM PMOD-based reconstruction
(blue line) with other reconstructions. Panel 5apresents high tem-
poral-frequency resolution reconstructions: The reconstructions of
Vieira et al. (2011) and Wang et al. (2005), with and without a
back-ground term, are presented as brown, pink and red lines
respectively. The actual PMOD TSI composite is the black line.
From Fig. 5a we noticed that whereas these TSI reconstructions
remain almost constant during the Maunder minimum, the
LS-SVM reconstruction (blue line) changes. The values of our TSI
reconstruction during the Maunder minimum are as an upper
limit, similar to the reconstruction of Vieira et al. (2011). In addi-
tion, our model has a time evolution similar to that of Vieira
et al. (2011), particularly before and after the Maunder minimum.
Panel 5b shows the comparisons of different TSI low temporal-
frequency resolution reconstructions with no information on the
11-year cycles. The reconstructions of Shapiro et al. (2011),
Steinhilber et al. (2009), Bard et al. (2000) and our filtered model
are presented as red, pink, green and blue lines respectively. We
filtered our model using Eq. (4) for periodicities >30 years. The
LS-SVM filtered model lies between the reconstructions of Bard
et al. (2000) and Steinhilber et al. (2009).
To better show the differences and similarities between the
LS-SVM reconstructions and the reconstructions obtained by
different physical models, we calibrated them with the filtered
PMOD composite for periodicities>30 years using the algorithms
of Eqs. (1) and (2).
In Fig. 5c we compared the LS-SVM filtered reconstruction (blue
lines) with the calibrated and filtered reconstructions of Bard et al.
(2000) (green lines), Shapiro et al. (2011) (red line), and the
Steinhilber et al. (2009) (pink line). It is clear that the LS-SVM
reconstruction gives results which are very similar to those of each
of the different TSI reconstructions. We noticed that with this
calibration, the time evolution of the Shapiro et al. (2011) and
Steinhilber et al. (2009) reconstructions look very similar; probably
because both reconstructions were based on the
10
Be records.
This calibration allowed us to differentiate clearly the last eight
grand solar minima (Fig. 5c) in all the reconstructions. Table 1 pre-
sents some of their characteristics. The mean duration of the grand
minima is about 60 years.
We have thus shown the capacity of the LS-SVM to reproduce
the characteristics reported by different reconstructions and TSI
composites (PMOD and ACRIM) since the Medieval Warm Period.
In Fig. 6a we compared the LS-SVM estimation based on the
PMOD-composite (the smoothed model is the blue line) with a
very recent TSI prediction (Steinhilber and Beer, 2013) that used
the WTAR Method (red line) for the 21st century. Fig. 6b shows
the LS-SVM estimation based on the ACRIM composite.
Steinhilber and Beer (2013) applied a Constant Amplitudes-Fast
Fourier Transformation (FFT Method) and Modulating Amplitudes,
that is a combination of the wavelet decomposition method with
an Autoregressive (AR) model (WTAR Method), to estimate the
TSI for the next 500 years. One of the limitations of the use of
the AR model to estimate values (e.g., for TSI) is that the output
variable depends linearly on its own previous values; in addition,
the WTAR method assumes that the input data (e.g., TSI recon-
struction) are stationary.
However, as can be easily shown in the present paper, the value
of the PMOD-TSI shows that the amplitudes are not stationary and
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
1360
1361
1362
1363
TIME (YEARS)
TSI (W/m
2
)
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
1363.5
1364.5
1365.5
1366.5
1367.5
TIME (YEARS)
TSI (W/m
2
)
(a)
(b)
Fig. 4. LS-SVM reconstruction from 1000 to 2013 AD. (a) PMOD-based model (black line). (b) ACRIM-based model (black line). The red lines are the ‘‘cone

-accuracy’’. (For
interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
226 V.M. Velasco Herrera et al. / New Astronomy 34 (2015) 221–233

do not depend linearly on their own previous values. The authors
found a grand minimum starting in cycle 26 with the lowest irra-
diance around cycle 29.
In another recent estimation of future TSI values, Jones et al.
(2012) derived relationships by regressing 25-year means of the
modulation parameter
U
against a variety of historic reconstruc-
tions of TSI, similarly averaged, over the period of the TSI recon-
struction. The relationship between the solar cycle amplitude and
its 25-year mean is also found, which is used to add variations over
an assumed 11-year cycle to the TSI estimates based on the
U
.In
their study they used three different historic TSI reconstructions
to scale the modulation parameter, covering the last three
centuries or so (Lean, 2000; Krivova et al., 2007; Lean and Rind,
2009). For the average estimation of future TSI, the authors found
a grand minimum starting in solar cycle 26, with the lowest irradi-
ance starting around solar cycle 28. Our estimation of the grand
minimum starts in cycle 24, with the deepest part in solar cycle 27.
In Fig. 7 we showed the wavelet analysis of the modeled TSI
from 1000 to 2100 AD. The global wavelets present periodicities
of 11 3;60 20;120 30 and 240 40-years.
The periodicity of 11 years corresponds to the Schwabe cycle.
The periodicity of 60 years (Yoshimura–Gleissberg cycle) could
be associated with solar barycentric motion (Velasco Herrera,
2013) and has been reported using the cosmogenic isotopes
14
C,
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
1364.4
1365
1366
TIME (YEARS)
TSI (W/m2)
LS−SVM
Bard
Shapiro
Steinhilber
Late Medieval
Wolf
Oort
Maunder
Medieval Modern
Sporer
Dalton
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
1360
1364
1368
TIME (YEARS)
TSI (W/m2)
Steinhilber
Bard
Shapiro
LS−SVM
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
1364
1365
1366
1367
TIME (YEARS)
TSI (W/m2)
Wang
Vieira
Wang
LS−SVM
PMOD
(a)
(b)
(c)
Fig. 5. Comparison of the LS-SVM PMOD-based model: (a) High spatial-frequency resolution. (b) Low spatial-frequency resolution. (c) Calibration of TSI reconstructions with
the low frequency TSI-PMOD.
V.M. Velasco Herrera et al. / New Astronomy 34 (2015) 221–233 227

10
Be and SSN. This periodicity has also been observed in sunspot
and cosmic rays (Yoshimura, 1979; Gleissberg, 1967; Velasco and
Mendoza, 2008; Frick et al., 1997). Although the first candidate
proposed for causing this periodicity is magnetic field changes in
the solar surface, other sources of TSI variability have been pro-
posed, such as changes in the photospheric temperature (Kuhn
and Libbrecht, 1991) and long-term changes in the solar diameter
(Sofia and Unruh, 1994) or in its convective strength (Hoyt and
Schatten, 1998). This periodicity could also be associated with
solar barycentric motion (Leal-Silva and Velasco, 2012).
The 120-year and 240-year (de Vries or Suess cycle) periodici-
ties have been reported using the cosmogenic isotopes
14
C and
10
Be (Velasco and Mendoza, 2008; Stuiver and Braziunas, 1993).
The periodicity of 120-years can be associated with solar magnetic
activity (Velasco Herrera, 2013) and the periodicity of 240 years,
with solar barycentric motion (Jose, 1965). Concerning the
120-year periodicity, Abreu et al. (2012) studying solar activity
for the past 9400 years using the modulation potential function
determined by the cosmogenic radionuclides
10
Be and
14
C, found
the periodicities of 88, 104, 150 and 506 years using the Fourier
Table 1
TSI grand minima between the years 1000 and 2100. PMOD = PMOD-based model, ACRIM = ACRIM-based model, Steinhilber= Steinhilber et al. (2009), Bard = Bard et al. (2000)
and Shapiro= Shapiro et al. (2011). S–E = Grand minima start and end years. hDMiis the mean duration of grand minima.
Solar Minimum PMOD ACRIM Steinhilber Bard Shapiro 120-Cycle
S–E DM S–E DM S–E DM S–E DM S–E DM S-E
Oort 10261085 59 1013–1074 61 990–1058 68 1000–1092 92 1000–1064 64 1050–1110
Medieval 1142–1201 59 1137–1200 63 1163–1208 45 1138–1197 59 1136–1205 69 1170–1230
Wolf 1263–1326 63 1263 –1326 63 1263–1333 70 1280–1334 54 1271–1335 64 1290–1350
LM 1389–1450 61 1389–1450 61 1413 –1483 70 1414–1481 67 1399–1461 62 1410–1470
Sporer 1509–1570 61 1510–1571 61 1518–1558 40 1526–1576 50 1518–1576 58 1530–1590
Maunder 1635–1702 67 1636–1706 70 1643–1723 80 1652–1720 68 1642–1711 69 1650–1710
Dalton 1767–1826 59 1773–1833 60 1798–1843 45 1787–1835 48 1775–1833 58 1770–1830
Modern 1885–1945 60 1887–1943 56 1883–1928 45 1878–1928 50 1886–1940 54 1890–1950
21st C 2004–2075 71 2002–2063 61 2010–2070
hDMi62.2 61.7 57.8 61 62.5
1100 1300 1500 1700 1900 2100
1360
1361
1362
TIME (YEARS)
TSI (W/m2)
LS−SVM
LS−SVM−Smoothed
1100 1300 1500 1700 1900 2100
1364
1365
1366
1367
TIME (YEARS)
TSI (W/m2)
LS−SVM
LS−SVM−Smoothed
Steinhilber−Smoothed
(a)
(b)
Fig. 6. LS-SVM reconstruction. (a) Using the PMOD composite and showing a comparison between our model and the Steinhilber and Beer (2013). (b) The LS-SVM ACRIM-
based model.
228 V.M. Velasco Herrera et al. / New Astronomy 34 (2015) 221–233

Transform method. Scafetta and Willson (2013) and Scafetta
(2012) found the 120-year periodicity, but the period seemed to
be a combination of the 115 and 130 harmonics. Taking into
account these uncertainties, our 120-year periodicity coincides
with those reported by these authors.
Velasco Herrera (2013), using the solar modulation potential
function for a total of 11,000 years and applying wavelet trans-
form, found periodicities of 60, 128, 240, 480, 1000 and
2100 years. The 120-year, 480-year (unnamed), 1000-year (Eddy
cycle) and 2100-year (Hallstattzeit cycle) periodicities have been
reported using different solar activity proxies (see e.g., Soon
et al. (2014)).
In the central panels of Fig. 7a and b, we noted that the 11-year
and 120-year periodicities appear well above the red noise level
in the global spectra (left panels). Several studies have reported
that during the Maunder minimum, the Schwabe cycle increased
to 14 years (e.g., Nagaya et al. (2012)). The 11-year periodicity
has an uncertainty of 3 years, (see left panels), with the upper
limit consistent with 14 years. The 240-year periodicity appears
near or above the red noise level.
Fig. 7. Wavelet analysis of the LS-SVM models. The time series between 1000 and 2100 is shown in the top panels. The wavelet spectrum is shown in the central panels, the
curved black line is the COI. The global wavelet is show in the left panels, the dotted lines are the red-noise level. (a) PMOD-based model. (b) ACRIM-based model. (For
interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
V.M. Velasco Herrera et al. / New Astronomy 34 (2015) 221–233 229

4.2. Bandwidth analysis
In Fig. 8 we showed the model decomposed by bandwidths.
Fig. 8a–c show the 11-year, 120-year and 240-year periodicities
and their phases. These results were obtained using Eq. (4).
As GSN were so scarce during the Maunder minimum (Fig. 2), it
has been suggested that the solar dynamo stopped or ceased oper-
ating (e.g., Charbonneau (2010), Ossendrijver (2003)). However
Fig. 8a clearly shows that the 11-year amplitude is attenuated
but not absent during the grand minima.
From Fig. 8a it is difficult to see the beginning and the ending
of the grand minima. However we can solve this problem by
studying the nature of variation of the 120-year periodicity. The
negative (positive) 120-year phase in Fig. 8b coincides with the
grand minima (maxima) of solar activity. Based on a 120-year
periodicity, we find the duration of the grand minima presented
in Tables 1 and 2.
Usoskin et al. (2007) have calculated the duration of grand min-
ima. The only grand minimum that coincides with our work is the
Maunder minimum, with a duration of 80 years, while we found a
duration of 70 years. However, we would like to point out that
these authors calculated the minima from the amplitude of the
sunspot numbers, while in the present work, as we stated above,
we used the 120-year periodicity to decide when a grand mini-
mum (maximum) starts and ends. In Table 2 we showed the calcu-
lated beginning, ending and duration of solar cycles 24–31; we also
showed the years of the cycle maxima and their corresponding TSI.
The results of the models based on the PMOD and ACRIM compos-
ites show differences in the duration of the cycles and in the year of
the cycle maxima. There are several definitions of grand minima,
all based on the activity levels of the solar indices time series.
For instance, Usoskin et al. (2007) defined a grand minimum as a
period when the sunspot level is less than 15 sunspots during at
least two consecutive decades. It is difficult to determine the
beginning and termination of the grand minima. In fact, we cannot
define a unique period to a group of few solar cycles (a short wave
packet, Feynman et al. (1963)), and therefore there is an indetermi-
nacy of the periodicity of the grand minima.
In the present paper we use the 120-year periodicity and the
wavelet spectral power to decide whether we have found a grand
minimum or not. According to the equivalent of Parseval’s theorem
for wavelet analysis, the total energy is conserved under the wave-
let transform, and the power in a time series is equal to the spectral
power (Torrence and Compo, 1998). This means that the defini-
tions of grand minima based on the activity levels of the solar
activity indices time series and the definition of grand minima
based on the spectral power are equivalent.
Fig. 8c shows the 240-year cycle. It seems that according to the
lag between the 120-year and the 240-year cycles, the grand solar
minima should have different amplitudes. For instance, when its
negative phase coincides with the negative phase of the 120-year
cycle, we have the deepest minima, as in the case of the Maunder
minimum. This indicates that in order to study the grand minima
or maxima of the solar activity, we must decompose the time ser-
ies in their various periodicities and focus on both, their ampli-
tudes and phases (Velasco et al., 2011).
4.3. Power index
As an indicator of the level of TSI activity for each solar cycle i
between 1000 and 2100 AD, we proposed the following TSI power
index (P
i
) as similarly noted earlier in Section 2.2:
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100
−1
0
1
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100
−1
0
1
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100
−0.5
0
0.5
TIME (YEARS)
(a)
(b)
(c)
Fig. 8. Analysis of the TSI-models in bandwidths for the time series between 1000 and 2100. (a) The 11-years component. (b) The 120-years component. (c) The 240-years
component. Black lines are the PMOD-based model and red lines are the ACRIM-based model. (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
Table 2
Estimations of several characteristics of future solar cycles.
Solar cycle PMOD ACRIM
Start–End Duration Year of the TSI Start–End Duration Year of the TSI
(years) (years) maximum maximum (years) (years) maximum maximum
24 2008–2019 11 ± 1.3 2013 ± 1.3 1365:89 0:26 2008–2020 12 1:5 2013 1:5 1361:29 0:48
25 2019–2030 11 ± 1.3 2024 ± 1.3 1365:91 0:33 2020–2033 13 1:6 2027 1:6 1360:85 0:32
26 2030–2041 11 ± 1.3 2036 ± 1.3 1365:78 0:29 2033–2042 9 1:1 2037 1:1 1360:46 0:21
27 2041–2052 11 ± 1.3 2046 ± 1.3 1365:90 0:35 2042–2055 13 1:6 20491:6 1360:89 0:39
28 2052–2064 12 1:5 2059 1:5 1365:84 0:33 2055–2065 101:2 2060 1:2 1360:85 0:29
29 2064–2075 11 ± 1.3 2069 ± 1.3 1365:98 0:36 2065–2075 10 1:2 20711:2 1361:13 0:28
30 2075–2085 10 1:2 2080 1:2 1365:97 0:35 2075–2090 15 1:9 2084 1:9 1362:35 0:59
31 2085–2096 10 1:2 2090 1:2 1366:39 0:45 2090–2101 11 1:3 2095 1:3 1362:48 0:71
230 V.M. Velasco Herrera et al. / New Astronomy 34 (2015) 221–233

P
i
¼X
N
i
k¼1
ðTSI
k
Þ
2
T
i
where N
i
is the number of years and T
i
is the duration in years of
solar cycle i.
We used the PMOD and ACRIM composites averaged from 1979
to 2012, to calculate the normalized power index anomalies for
each cycle (^
P) as:
^
P
i
¼P
i
hPMOD=ACRIMi
MAXðkPkÞ
where hPMOD/ACRIMiis the average of the PMOD or ACRIM power
index and MAXðkPkÞ is the maximum in absolute value of the power
index anomalies. The power index anomalies are used to decide
when a cycle presents higher or lower TSI.
Fig. 9 shows these TSI power index anomalies from 1000 to
2100 AD. Negative (positive) anomalies correspond to lower
(higher) TSI with respect to the PMOD/ACRIM composites.
The PMOD-based model (Fig. 9a) shows multi-secular periodic-
ities while the ACRIM-based model (Fig. 9b) shows the secular
120-year periodicity.
According to the power anomalies, the PMOD-based model for
the 21st century grand minimum goes between cycles 28 and 31,
while for the ACRIM-based model, this minimum goes between
cycles 29 and 31. Fig. 9a also indicates that solar cycles 25, and
27 present lower activity than solar cycles 26 and 28, regardless
of the fact that the latter present the lowest TSI (see Table 2).
Fig. 9b and Table 2 indicated that cycles 25, 26 and 28 have the
lowest TSI, although cycles 25 and 26 have the lowest activity,
cycle 28 presents higher activity while having the same TSI as cycle
25.
The lowest power anomalies are around the grand minima.
According to Fig. 9a, the lowest anomalies occur during the
Maunder minimum, but in Fig. 9b they occur during the Oort
minimum.
4.4. Radiative forcings of grand minima
The calculation of the average TSI for the grand minima appears
in Table 3 together with the associated radiative forcings (RF).
Table 3 indicates that between the Maunder minimum and the
present, represented by the PMOD (ACRIM) TSI average between
1979 and 2003, the RF is 0.25 (0.19) Wm
2
.Schmidt et al. (2011)
have considered several recent TSI reconstructions (Wang et al.,
2005; Krivova et al., 2007; Muscheler et al., 2007; Bard et al.,
2000; Fröhlich, 2006; Steinhilber et al., 2009; Vieira et al., 2011),
and based on them, the authors presented an RF range of
0.1–0.23 Wm
2
. Thus, our results are consistent with this range.
For the deepest part of the 21st century, Jones et al. (2012) found
an RF that is between the Dalton and the Modern minima. Between
the PMOD (ACRIM) time span and the 21st century minimum,
there is a radiative forcing of 0:09ð0:18ÞWm
2
with an
uncertainty range of 0:04 to 0:14ð0:12 to 0:33ÞWm
2
. The
21st century forcing corresponding to the PMOD-based model is
the third least severe of the grand minima since the year 1000
AD, with the Oort minimum being the least deep. The RF
1000 1200 1400 1600 1800 2000
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
TIME (YEARS)
1000 1200 1400 1600 1800 2000
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
TIME (YEARS)
(a)
(b)
Fig. 9. TSI power annual anomalies normalized. For years of high (low) solar activity the power anomalies are positive (negative). (a) PMOD- based model. (b) ACRIM-based
model.
V.M. Velasco Herrera et al. / New Astronomy 34 (2015) 221–233 231

ACRIM-based forcing will be the fourth most severe since 1000 AD,
with the Medieval being the deepest. The great differences
between the PMOD and ACRIM-based forcings reflect the different
long-term trends of the composites.
In addition, Table 3 demonstrates the great differences in RF
between the PMPD and ACRIM composites.
5. Conclusions
We modeled the TSI from 1000 to 2100 AD to obtain a TSI index,
using for the first time the non-parametric and non-linear method
based on the Least Squares Support Vector Machines (LS-SVM)
with Nonlinear Autoregressive Exogenous model and using a radial
basis function kernel. We, however, leave open the uniqueness and
correctness of the two different time histories of our reconstructed
TSI records using either the PMOD or ACRIM calibrated baselines.
With the LS-SVM method, we found results that have time evo-
lutions similar to those produced by Vieira et al. (2011) through
physical modeling.
We found a grand minimum for the 21st century, starting in
2004 (2002) and ending in 2075 (2063), with an average
PMOD (ACRIM) irradiance of 1365.5 (1360.5) Wm
2
1
r
¼0:30
(0.9) Wm
2
.
We found a PMOD (ACRIM) Maunder minimum to a present RF
of 0.24 (0.19) Wm
2
, which is consistent with other estimations.
Between the PMOD (ACRIM) time span and the 21st century
minimum, there is a radiative forcing of 0:09 ð0:18ÞWm
2
with an uncertainty range of 0:04 to 0:14 ð0:12 to
0:33ÞWm
2
.
Applying the wavelet analysis we found periodicities at 11
and 120 years.
During the grand minima, the 11-year periodicity has dimin-
ished spectral power, while the 120-year periodicity maintains
more or less the same spectral power.
The 120-year periodicity has not been widely reported before.
The solar activity grand minima periodicity is of 120 years; this
periodicity could possibly be one of the principal periodicities of
the magnetic solar activity.
The negative (positive) 120-year phase coincides with the grand
minima (maxima) of the 11-year periodicity and, according to the
Table 3
Averages of the TSI grand minima and associated radiative forcings (RF). The RF was
obtained from the difference between the indicated TSI average minimum and the
PMOD (ACRIM) TSI composite average along 1979–2003: 1366 Wm2
(1361:5Wm
2).
Minimum PMOD RF-PMOD ACRIM RF-ACRIM
(Wm
2
) (Wm
2
) (Wm
2
) (Wm
2
)
Oort 1365.6 0.07 1360.2 0.23
Medieval 1366.0 0.00 1361.2 0.05
Wolf 1365.5 0.09 1360.9 0.11
L Medieval 1364.8 0.21 1360.3 0.21
Spoerer 1364.8 0.21 1360.8 0.12
Maunder 1364.6 0.25 1360.4 0.19
Dalton 1365.0 0.18 1360.9 0.11
Modern 1365.4 0.11 1361.1 0.07
21st C 1365.5 0.09 1360.5 0.18
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
TIME (YEARS)
TSI (W/m2)
LS−SVM
LS−SVM−Smoothed
Hoyt
0 200 400 600 800 1000 1200 1400 1600 1800 2000
1356
1360
1364
1368
1372
TIME (YEARS)
TSI (W/m2)
(a)
(b)
Fig. A.1. Shapiro and Hoyt TSI reconstruction types. (a) Comparison between LS-SVM (black line) and the Shapiro et al. (2011) reconstruction (red line) since the year 0 to
2005. (b) Calibration of ACRIM-based model with the Hoyt and Schatten (1993) reconstruction (red line). The smoothed model is the blue line. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of this article.)
232 V.M. Velasco Herrera et al. / New Astronomy 34 (2015) 221–233

lag between the 120-year and 240-year, we would have different
amplitudes for the grand solar minima. This indicates that in order
to study the grand minima or maxima of the solar activity, we
must decompose the time series in their various periodicities and
focus on both their amplitudes and their phases.
Future solar cycles 26 and 28 will have the lowest TSI power-
index anomalies, even though they do not have the lowest TSI
amplitudes.
Acknowledgments
This work was partially supported by CONACyT-180148 and
PAPIIT-IN103112-3 grants. We would like to thank two anony-
mous referees for valuable and constructive suggestions. The
authors would like to thank Shapiro A.I. and Scafetta N. for provid-
ing the time series of TSI reconstructions.
Appendix A. Shapiro and Hoyt TSI reconstruction records
There are diverse TSI reconstructions based on different calibra-
tion and physical models. Using the algorithms described in
Sections 3.2 and 3.3, we showed the capacity of the LS-SVM (black
line) to reproduce the Shapiro et al. (2011) reconstruction (red
line) from the year 0 to 2005 AD in Fig. A.1a, with the main differ-
ence that the LS-SVM method also reconstructed the 11-year solar
cycle. Moreover, this figure showed the LS-SVM estimation of a
grand minimum for the 21st century, with a TSI level that lies
between the Wolf and Dalton minima.
Figure Fig. A.1b showed the calibration of the ACRIM-based
model (black lines) with the Hoyt and Schatten (1993) reconstruc-
tion (red line, with updates by N. Scafetta) from 1850 to 2010. We
noticed that our method demonstrates higher amplitudes during
the grand minima, as compared to Fig. 6b. Also, the estimated
TSI grand minimum during the 21st century lies between the Wolf
and Dalton minima. Thus, when the LS-SVM method is supplied
with additional information, it can reproduce any TSI reconstruc-
tion histories.
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