Predicting the Maximum Amplitude of Solar Cycle 25 Using the Early Value of the Rising Phase

Abstract

The rising rate [$\beta _{\mathrm{a}}$] of a solar cycle is a good indicator for the subsequent maximum amplitude [$S_{\mathrm{m}}$] of sunspot numbers. We compared the correlation between $S_{\mathrm{m}}$ and $\beta _{\mathrm{a}}$ and that between $S_{\mathrm{m}}$ and the early value of the smoothed monthly mean sunspot number [$S_{\mathrm{N}}$] $\Delta m$ months after the solar minimum. Our main conclusions are as follows: i) The correlation coefficient [$r$] between $S_{\mathrm{m}}$ and $S_{\mathrm{N}}$ is slightly higher than that between $S_{\mathrm{m}}$ and $\beta _{\mathrm{a}}$, and both increase with $\Delta m$ as the cycle progresses; ii) In the first year of the cycle, the correlation is weak [$r\sim$ 0.56]. At the inflection point [$\Delta m=21$], the correlation is stronger [$r = 0.83$]. After the inflection point, $r$ increases slowly with $\Delta m$. Three years after the solar minimum, $r\geqslant 0.90$. Around the average rise time [52 months], $r=0.95$; iii) The correlation between $S_{\mathrm{m}}$ and $S_{\mathrm{N}}$ (or $\beta _{\mathrm{a}}$) in even-numbered cycles is stronger than that in odd-numbered ones, and the latter is slightly weaker than that for all the cycles; iv) The mean relative error [$\eta$] of $S_{\mathrm{m}}$ decreases and the MSE (Mean Square Error) skill score [$S_{\mathrm{c}}$] increases with $\Delta m$. One, two, three, and four years after the solar minimum: $\eta \leqslant 19\%$, $14\%$, $10\%$, and $6.5\%$, $S_{\mathrm{c}}\geqslant 0.24$, 0.68, 0.86, and 0.97, respectively; v) Currently [$\Delta m=20$], the maximum amplitude of Cycle 25 is predicted to be $135.5\pm 33.2$ and to occur around December 2024 (± 11 months).

Appendices

Appendix A: Predicting $S_{\mathrm{m}}$ of Cycle 24 Using the Sunspot Numbers of Version 1.0

As an example, we examine the predictive ability of the methods in this work and Du (2020b) on the maximum amplitude [$S_{\mathrm{m}}$] of Cycle 24 using the sunspot number series [$S_{\mathrm{N}}$] of Version V1.0. Du and Wang (2012) predicted $S_{\mathrm{m}}$ for Cycle 24 [$S_{\mathrm{p}}=S_{\mathrm{p}}(24)$] by the rising rate [$\beta _{\mathrm{a}}$, Equation 1]. The result is shown in Figure 6a (for $\Delta m>0$). In the original work, data were available only until February 2011 [$\Delta m=27$], and the result at $\Delta m=27$ was $r(\Delta m)=0.88$ and $S_{\mathrm{p}}=84.0$ (marked with an asterisk in Figure 6a), slightly higher than the observed value [$S_{\mathrm{m}}(24)=81.9$, the horizontal dash-dotted line]. Now, we extended the result to $\Delta m=60$ months (four months before the peak timing of Cycle 24, April 2014). We see that with the increase of $\Delta m$, the correlation coefficient [$r( \Delta m)$, dotted] between $S_{\mathrm{m}}$ and $\beta _{\mathrm{a}}$ for Cycles 1 – 23 increases (if $\Delta m\leqslant 50$), and the predicted value [$S_{\mathrm{p}}$, solid] decreases from 102.3 [with an error of $\Delta S_{\mathrm{p}}=|S_{\mathrm{p}}-S_{\mathrm{m}}(24)|=20.4$, dashed] at $\Delta m=1$ to 82.8 ($\Delta S_{\mathrm{p}}= 0.9$) at $\Delta m=60$. There is a local peak, $S_{\mathrm{p}}=96.0$ ($\Delta S_{\mathrm{p}}=14.1$) at $\Delta m=32$, two (three) months earlier than the peak of Cycle 22 (3) and six months earlier than the secondary peak of Cycle 24 [66.9, February 2012]. If $\Delta m>50$, $r(\Delta m)$ may decrease. The mean absolute (relative) prediction error is $\overline{\Delta S}_{\mathrm{p}}= 7.2$ ($\overline{E}_{\mathrm{r}}=8.8\%$) over $\Delta m=[1, 60]$.

Figure 6

Predicting the maximum amplitude of Cycle 24 [$S_{\mathrm{p}}(24)$, solid] by (a) the decreasing ($\Delta m<0$) and rising ($\Delta m>0$) rate, and (b) the sunspot number [$S_{\mathrm{N}}$] of Version V1.0 at the declining ($\Delta m<0$) and rising ($\Delta m>0$) phase as a function of $\Delta m$ (months) from the solar minimum [$\Delta m=0$]. The horizontal dash-dotted line represents the observed $S_{\mathrm{m}}(24)=81.9$. The dashed line indicates the prediction error, $\Delta S_{\mathrm{p}}=|S_{\mathrm{p}}(24)-S_{\mathrm{m}}(24)|$. The dotted line represents the correlation coefficient [$r(\Delta m)$]. The left vertical dash-dotted line indicates $\Delta m=-39$ months. The right vertical dash-dotted line in (a) indicates $\Delta m=27$ months at which Du and Wang (2012) obtained the result: $r(27)=0.88$ and $S_{\mathrm{p}}=84.0$.

Du (2020b) analyzed the correlation between $S_{\mathrm{m}}$ and the decrease rate [$\beta _{\mathrm{d}}$], calculated from the timing of solar minimum [$t_{\mathrm{min}}$] to $\Delta m$ months earlier. They found that the correlation coefficient [$r(\Delta m)$] between $S_{\mathrm{m}}$ and $\beta _{\mathrm{d}}$ for Cycles 1 – 24 is the highest at $\Delta m=-39$ months, using the sunspot numbers of Version 2.0. Now, we use the sunspot numbers of Version 1.0 to predict $S_{\mathrm{m}}(24)$ by $\beta _{\mathrm{d}}$, as shown in the left part of Figure 6a (for $\Delta m<0$). As $\Delta m$ varies from −1 to −60, $r(\Delta m)$ for Cycles 1 – 23 tends to increase (if $\Delta m\geqslant -50$) and the predicted value [$S_{\mathrm{p}}$] decreases from 109.3 ($\Delta S_{\mathrm{p}}= 27.4$) at $\Delta m=-1$ to 58.0 ($\Delta S_{\mathrm{p}}=23.9$) at $\Delta m=-60$. The correlation coefficient decreases if $\Delta m<-50$ and may be negative if $\Delta m<-58$. At $\Delta m=-39$ months, $r(-39)=0.78$, and $S_{\mathrm{p}}=87.1$ ($\Delta S_{\mathrm{p}}= 5.2$). The mean prediction error is $\overline{\Delta S}_{\mathrm{p}}= 10.3$ ($\overline{E}_{\mathrm{r}}=12.6 \%$) over $\Delta m=[-1,-60]$.

Figure 6b shows the predicted $S_{\mathrm{p}}(24)$ by the sunspot number [$S_{\mathrm{N}}$] at the preceding declining ($\Delta m<0$) and rising ($\Delta m>0$) phase $\Delta m$ months from the solar minimum. As $\Delta m$ increases from 1 to 60, the correlation coefficient [$r( \Delta m)$] between $S_{\mathrm{m}}$ and $S_{\mathrm{N}}(\Delta m)$ for Cycles 1 – 23 increases (if $\Delta m\leqslant 50$), and $S_{\mathrm{p}}$ varies in the range [73.9, 93.0] with the largest prediction error [$\Delta S_{\mathrm{p}}=11.1$] at $\Delta m=32$. As $\Delta m$ varies from −1 to −60, $r(\Delta m)$ tends to increase (if $\Delta m\geqslant -50$), and $S_{\mathrm{p}}$ decreases from 87.0 ($\Delta S_{\mathrm{p}}= 5.1$) at $\Delta m=-1$ to 58.3 ($\Delta S_{\mathrm{p}}=23.6$) at $\Delta m=-60$. The correlation coefficient decreases if $\Delta m<-50$ and may be negative if $\Delta m<-58$. At $\Delta m=-39$ months, $r(-39)=0.79$ and $S_{\mathrm{p}}=83.2$ ($\Delta S_{\mathrm{p}}= 1.3$). Over $\Delta m=[1,60]$, the mean prediction error is $\overline{\Delta S}_{\mathrm{p}}= 4.4$ ($\overline{E}_{\mathrm{r}}=5.3\%$), smaller than that, 7.2 (8.8%), using $\beta _{\mathrm{a}}$ in Figure 6a. Over $\Delta m=[-1,-60]$, the mean prediction error is $\overline{\Delta S}_{\mathrm{p}}= 5.9$ ($\overline{E}_{\mathrm{r}}=7.3\%$), smaller than that, 10.3 (12.6%), using $\beta _{\mathrm{d}}$ in Figure 6a.

In summary, the prediction error by the sunspot number [$S_{\mathrm{N}}$] is smaller than the one computed by the rate. The lowest correlation coefficient is around the solar minimum, $r=0.16$ at $\Delta m=-1$ for the rate and $r=0.56$ at $\Delta m=0$ for $S_{\mathrm{N}}$. At the rising phase, $r(\Delta m)$ increases with the increase of $\Delta m$ and $r(\Delta m)>0.75$ if $\Delta m\geqslant 16$. At the declining phase, $r(\Delta m)$ also tends to increase with the increase of $|\Delta m|$ and $r(\Delta m)>0.75$ if $-52\leqslant \Delta m \leqslant -29$. The prediction error near the solar minimum tends to be larger than those computed at other $\Delta m$ in the range [−50, 50].

Appendix B: Predicting $S_{\mathrm{m}}$ of Cycle 24 Using the Sunspot Numbers of Version 2.0

Figure 7 shows the results obtained by using the sunspot numbers of Version 2.0 in a similar manner as was done in Figure 6.

• i) Using the decrease ($\Delta m<0$) and rising ($\Delta m>0$) rate (Figure 7a), the lowest correlation coefficient for Cycles 1 – 23 is $r(\Delta m)=0.19$ at $\Delta m=3$, also close to the solar minimum in time. With the increase of $|\Delta m|$, $r(\Delta m)$ tends to increase and the prediction error [$\Delta S_{\mathrm{p}}=|S_{\mathrm{p}}-S_{\mathrm{m}}(24)|$] tends to decrease. $r(\Delta m)>0.75$ if $\Delta m\leqslant -30$ or $\Delta m\geqslant 20$. The maximum prediction error is $\Delta S_{\mathrm{p}}=61.1$ ($E_{\mathrm{r}}=52.5\%$) at $\Delta m=1$. The mean prediction error is $\overline{\Delta S}_{\mathrm{p}}= 24.7$ ($\overline{E}_{\mathrm{r}}=21.2 \%$) for $\Delta m>0$ and $\overline{\Delta S}_{\mathrm{p}}= 23.4$ ($\overline{E}_{\mathrm{r}}=20.1\%$) for $\Delta m<0$. At $\Delta m=-39$, $r(-39)=0.80$ and $\Delta S_{\mathrm{p}}=18.8$ (16.2%). At $\Delta m=20$ (similar to the time for predicting Cycle 25), $r(20)=0.77$, $S_{\mathrm{p}}=132.2$ (asterisk), and $\Delta S_{\mathrm{p}}=15.8$ (13.6%).

  Figure 7

  

  Similar to Figure 6 but using the sunspot numbers of Version 2.0. The horizontal dash-dotted line indicates the maximum amplitude [$S_{\mathrm{m}}(24)=116.4$] of Cycle 24.

• ii) Using the sunspot number at the declining ($\Delta m<0$) and rising ($\Delta m>0$) phase (Figure 7b), the lowest correlation coefficient for Cycles 1 – 23 is $r(\Delta m)=0.51$ at $\Delta m=9$. With the increase of $|\Delta m|$, $r(\Delta m)$ tends to increase and $r(\Delta m)>0.75$ if $\Delta m\leqslant -16$ or $\Delta m\geqslant 18$. The mean prediction error is $\overline{\Delta S}_{\mathrm{p}}= 14.4$ ($\overline{E}_{\mathrm{r}}=12.3 \%$) for $\Delta m>0$ and $\overline{\Delta S}_{\mathrm{p}}= 14.5$ ($\overline{E}_{\mathrm{r}}=12.4\%$) for $\Delta m<0$. At $\Delta m=-39$, $r(-39)=0.80$ and $\Delta S_{\mathrm{p}}=13.2$ ($E_{\mathrm{r}}=11.3\%$). For $\Delta m>0$, the prediction error is minimum (maximum), $\Delta S_{\mathrm{p}}=0.4$ (30.5), at $\Delta m=51$ (31) and has a local minimum [6.4] at $\Delta m=21$. At $\Delta m=20$ (similar to the time for predicting Cycle 25), $r(20)=0.81$, $S_{\mathrm{p}}=123.2$ (asterisk), and $\Delta S_{\mathrm{p}}=6.8$ ($E_{\mathrm{r}}=5.8\%$).

Using the sunspot number (Figure 7b), the correlation coefficient is slightly higher, and the prediction error tends to be smaller than that using either a decrease or rising rate (Figure 7a). Near the solar minimum, the correlation coefficient tends to be weaker, and the prediction error tends to be larger than those at other $\Delta m$. The mean relative prediction error for Cycle 24 using the data of the new version is larger than that using the data from Version 2.0.