Statistical approach

To recover the information on the past SCs from the decadally-averaged SN, we first search for the potential relationships between various SC properties and the decadally-averaged values using the directly observed SN.

Following the paper I, we use the updated version (v2.0) of the annual international sunspot number series by WDC-SILSO¹ (SNv2, Clette et al. 2014). We also employ the group sunspot number series (GSN) constructed by Chatzistergos et al. (2017, R_CH he-reafter) to estimate the uncertainty of the results (Sect. 5.4.1) due to the uncertainty in the SN record. RCHis a new GSN series, which is based on a non-linear non-parametric method for cross-calibration of individual historical GSN records.

Using the directly observed SN we quantify four main SC properties: (1) annual SN at the cycle maximum,Smax, (2) annual SN at the preceding minimum,Smin, (3) cycle rise time,L_rise, and (4) cycle length,L_cyc. Together with the SC shape function (which will be introduced in Sect.5.3.2), these give a relatively complete description of a cycle.

Earlier studies have compared various SC properties with each other (e.g.,Waldmeier 1935,1939;Hathaway et al. 2002;Solanki et al. 2002a;Karak and Choudhuri 2010). We here compare each property directly with the decadally-averaged SN,hSNi. Moreover, even though a potential correlation betweenSmax and Smin has been suggested (Wilson et al. 1998), we compare these two characteristics with the hSNi individually. This is because thehSNiis the only information available from the isotope data, and thus deter-mining one parameter from another could easily propagate the errors.

The levels of both the SN maxima and minima affect the decadal SN average. The-refore, we compare thehSNiwith two parameters, S_max and S_min that are closest to it.

Then we compare thehSNiwith theLrise of the cycle (k) who possesses that foundSmax. However, due to the anti-correlation between the strength of a given cycle and the length of the previous cycle, we compare thehSNiwith the length of the previous cycle (Lcyc,k−1).

Figure5.1visualizes how the decadally-averaged SN values are compared with the SC parameters. The annual S_Nv2series over the period 1700 – 2017 is shown in panel a, where the numbers of the cycles (C) are indicated above the curve. The SN series is segmented into calendar decades beginning from 1700, as shown by the vertical dashed lines, where the numbers of the decades (D) are denoted in the upper x-axis. The decadally-averaged SN,hS_Nv2i, is shown by a dot in each decadal segment. Since the solar cycle lengths are not exact 10 years, there are in total 32 decades corresponding to 29 cycles. For instance, the decadal values of both D4 and D5 are compared with theSmax andLrise of cycle C0, and theL_cycof cycle C-1.

1http://www.sidc.be/silso/datafiles

D0 D5 D10 D15 D20 D25 D30

C-1 C0

C10

C20

Figure 5.1: Observed SN series, S_Nv2, segmented into calendar decades (vertical dashed lines). The decadal SN value, hSNv2i, calculated in each decade is shown by a dot. The numbering of the decadal segments is denoted as D in the upper x-axis, and the numbering of solar cycle as C. Note that in Sect. 5.3.5we will test different segmentation dates and their influence on the results.

Figure5.2shows the relationships between the four SC parameters and the decadally-averaged SN, hS_Nv2i. The solid lines represent the best fits in four panels, and the 3-σ uncertainties are shown by the dashed lines. Note that since both,hS_Nv2iand the SC para-meters contain uncertainties, a commonly-used ordinary linear squares regression (OLS) is not suitable for finding the best fit, as it only takes the uncertainties in one variable into account. We hence apply the ordinary least squares bisector regression (OLS-bisector), which is defined as the fit that bisects the angles of two OLS regressions (i.e., standard OLS and inverse OLS,Babu and Feigelson 1992). By doing so, the best fit accounts for the uncertainties in both, independent and dependent variables. The Pearson correlation coefficient,Rc, between thehSNv2iand theSmax(panel a) is 0.86, whereas theRc between thehS_Nv2iand the other three characteristic values (panel b – d) are 0.61, -0.66, and -0.61, respectively. TheR_cvalues are also given in the corresponding panels.

We note that the relationships obtained so far are based on the decadal SN values averaging from 1700, namely: 1700 – 1709, 1710 – 1719, ..., and so on. Later in Sect.

5.3.5, we will shift these decadal segments by 1 – 9 years to examine the sensitivity of these linear relationships to the segmentation periods.

Using the set of relationships obtained above, we are now able to simulate (postdict) the SC parameters back in time from thehS_Nv2i. Nonetheless, based on our simulation approach, we can only determine the length of the previous cycle, whereas the length and the end timing of the last cycle of the series are unknown. These two values, L_cyc,f and Tend,f, are, hence, needed as initial conditions in this approach. According to the current decaying SC, we take 2019 as the end of the C24, whose cycle length would then be 11 years.

5.3 Methodology

D D

D D

Figure 5.2: Four relationships of four studied SC parameters vs.hS_Nv2i: (a)S_max. (b)S_min. (c) Lcyc. (d) Lrise. The solid lines are the best OLS-bisector fits, with 3-σ uncertainties shown by the dashed lines. Each data point represents one decade (D), the values are obtained by comparing with the SC parameters of the corresponding cycle (C), see the text for the details.

Figure 5.3: Comparisons of fitted SC shape (red) and directly measured SN, S_Nv2(annual, black) after 1700 with Pearson correlation coefficientRc=0.96.