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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

N1

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

k1

;Y

k2

;...;Y

kp

;U

k1

;U

k2

;...;U

kq

Þð6Þ

where U

k

is the TSI value, Y

k1

;Y

k2

;...;Y

kp

denotes the output,

and U

k1;k2;...;kq

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 10261085 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 20491:6 1360:89 0:39

28 2052–2064 12 1:5 2059 1:5 1365:84 0:33 2055–2065 101:2 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 20711:2 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 0:09ð0:18ÞWm

2

with an

uncertainty range of 0:04 to 0:14ð0:12 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 ð0:18ÞWm

2

with an uncertainty range of 0:04 to 0:14 ð0:12 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 Wm2

(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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