Predicting the maximum aa/Ap index through its relationship with the preceding minimum

Abstract. Predicting the strength and peak time of geomagnetic activity for the ensuing cycle 25 is important in space weather service for planning future space missions. The minimum aa geomagnetic index around the solar minimum has been often used to predict the maximum amplitude of sunspot cycle, but seldom used to directly predict the maximum aa index. This study analyzed the relationships between the maxima and minima of both the geomagnetic aa and Ap indices for the 11-year cycle. The maximum aa index is found to be well correlated to the preceding minimum with a correlation coefficient of r = 0.860. As a result, the maximum aa index for the ensuing cycle 25 is predicted to be aamax(25) = 26.9 ± 2.6. This value is equivalent to Apmax(25) = 17.3 ± 1.8 ± 1.2 if employing the high correlation between aa and Ap (r = 0.939). The maximum Ap index is also found to be well correlated to the preceding minimum with a correlation coefficient of r = 0.862. Based on this correlation, the maximum Ap index is predicted to be a slightly higher value of Apmax(25) = 19.0 ± 1.6. The rise time of the aa (Ap) index for the 11-year cycle is found to be nearly uncorrelated to the following maximum, r = −0.16 (−0.17). If the data point for cycle 24 (which is far from others) were not considered, the rise time of the Ap index for the 11-year cycle would be weakly correlated to the following maximum, r = −0.404 at a confidence level of 62 %. The rise time for cycle 25 would be roughly estimated to be 89.9 ± 31.6 (months), implying that the geomagnetic activity for the ensuing cycle 25 would peak around April 2025 ± 32 months.



Introduction
Studying and predicting geomagnetic activities are important in both geophysics and space weather. Severe geomagnetic activities may cause intense geomagnetic storms (Gonzalez et al., 1989(Gonzalez et al., , 1994Chen et al., 2019), leading to disruptions in communica- 25 tion and deviations of spacecrafts. With the current solar cycle 24 approaching its end, 2 https://doi.org /10.5194/angeo-2020-15 Preprint. Discussion started: 3 April 2020 c Author(s) 2020. CC BY 4.0 License. satellite and spacecraft-related departments want to know the strengths of both solar and geomagnetic activities in the ensuing cycle 25 for planning future space missions.
Among the various indices to quantitatively describe the geomagnetic activity, the aa index (Mayaud, 1972), derived from the 3-hourly K indices at two near-antipodal midlatitude stations in England and Australia, is the longest time series (since 1868) 5 and has been widely used for analyzing long-term trends in the global geomagnetic activity (Russell and Mulligan, 1995;Marat et al., 2017;Du, 2011a;El-Borie et al., 2019) and for analyzing its correlation with both climate change (Cliver et al., 1998;Dobrica et al., 2009;Gavrilyeva et al., 2017) and solar activity (Echer et al., 2004;Prestes et al., 2006;Lukianova et al., 2009;Du, 2011b,c;Singh and et al., 10 2019). The minimum aa index (aa min ), at or near the solar minimum of the solar cycle, has been widely used in predicting the maximum amplitude of the sunspot cycle (R m ), the so-called Ohl's precursor method (Ohl, 1979;Brown and Williams, 1969;Du et al., 2009). But it is seldom used to directly predict the maximum aa index (aa max ) of an ensuing cycle. 15 The planetary geomagnetic index Ap (Bartels, 1963) available since 1932, derived from the average of the measurements at 13 observatories around the globe, is a daily measure of the response of geomagnetic field to variations in the interplanetary magnetic field (IMF) and the solar wind (Li, 1997;McPherron, 1999;Tsurutani et al., 2006). It is the main global magnetic index forecasted by government agencies (McPherron, 20 1999). Most works on forecasting geomagnetic activity have been over short intervals, on the order of hours or days (McPherron, 1999;Abunina et al., 2013). At earlier years, Kane (1988) pointed out that it is impossible to forecast the long-term geomagnetic activity through analyzing the daily, monthly and annual values of the Ap and aa indices. Gordon (2015) demonstrated that long-term geomagnetic activity can only be predicted 25 to within a limited threshold of accuracy due to the irregular trends and cycles in the annual data and nonlinear variability in the monthly series, through analyzing the aa index.
In this study, we analyze the relationships between the maxima and minima of smoothed monthly values for the 11-year cycle of both the aa and Ap indices. It is found that the minima of both the aa and Ap indices are well correlated to the following maxima, and thus, the latter can be predicted by the former. This study is arranged as follows. The data used in the current work are shown in Sect. 2. Section 3 is devoted for the results. First, in Sect. 3.1, we simply analyze 5 the relationship between the smoothed monthly mean Ap and aa indices. Then, in Sect. 3.2, we predict the maximum aa index (aa max ) for cycle 25, through analyzing the relationship between aa max and the preceding minimum aa index (aa min ). Similar analysis is performed in Sect. 3.3 for predicting the maximum Ap index (Ap max ) for cycle 25, through analyzing the relationship between Ap max and the preceding minimum Ap 10 index (Ap min ). In Sect. 3.4, we analyze the relationships between the rise times of the aa and Ap indices for the 11-year cycle and the following maxima, followed by estimating the peak time of geomagnetic activity for the ensuing cycle. Some conclusions are discussed and summarized in Sect. 4.

15
In this study, we use the (13-month) smoothed monthly mean aa index since 1868 from the International Service of Geomagnetic Indices (ISGI) 1 . We also employ the "equivalent" aa index, based on the declination data from Helsinki 2 (Nevanlinna and Kataja, 1993;Lukianova et al., 2009), during the period from 1844 to 1867 for expanding the data back to 1844. This index is assimilated by multiplying the mean scale factor (1.14) 20 of this index to that by ISGI for the overlapping period from July 1868 through December 1879. The smoothed monthly mean Ap index 3 is also used in this study for comparison.  The upper dashed and lower dash-dotted lines represent the maxima (aa max ) and minima (aa min ) of the aa index, respectively, for the 11-year solar cycle. 5 The parameters used in the current work are listed in Table 1, in which, T a is the rise time of the aa index for the 11-year cycle; Ap max and Ap min are the maxima and minima of the Ap index for the 11-year cycle, respectively; T r is the rise time of the Ap index for

Relationship between Ap and aa
First, we simply analyze the relationship between the Ap and aa indices.
where ± represents the 1σ deviation. The standard deviation of fitting is σ = 1.2, and the correlation coefficient between the fitted and observed values is r = 0.939 at a confidence level greater than 99%. It is obvious that Ap is highly correlated with aa.

Relationship between aa max and aa min
Now, we analyze the relationship between aa max and aa min . Figure. 2(a) depicts the scatter plot of aa max against aa min for cycles 9-24 (triangles). The solid line represents the linear fit of aa max to aa min with the least-squares-fit regression equation given by 10 The standard deviation of fitting is σ = 2.6, and the correlation coefficient between aa max and aa min is r = 0.860 at a confidence level greater than 99%. Therefore, aa max is well correlated with the preceding aa min , and one can use the latter to predict the former. Substituting the value of aa min (12.78) for cycle n = 25 into this equation, one obtains aa max (25) = 26.9 ± 2.6 (labelled by asterisk). It implies that 15 the maximum aa index for the ensuing cycle 25 would be similar to or slightly higher than the average (25.34) over the past cycles, but would be much higher than that (18.81) of cycle 24 by about 35%.
If using the relationship between the smoothed Ap and aa in Eq. (1), the above prediction is equivalent to Ap max (25) = 17.3 ± 1.8 ± 1.2, here ±1.8 is the uncertainty 20 derived by the uncertainty (±2.6) of aa min (25) and ±1.2 is the standard deviation of the regression.

Relationship between Ap max and Ap min
Then, we analyze the relationship between Ap max against Ap min . Figure 2  figure that Ap max is also well correlated to Ap min , with a correlation coefficient of r = 0.862 at a confidence level greater than 99%. The linear fit of Ap max to Ap min (solid) is with a standard deviation of σ = 1.6. From this relationship, one can roughly estimate the maximum Ap index for cycle 25, Ap max (25) = 19.0 ± 1.6, by substituting the value of 5 Ap min (8.77) for cycle n = 25 into this equation. This value is slightly higher than the average (18.55) over the past cycles, but much higher than that (11.72) of cycle 24, similar to the case for aa max (25).

Estimating roughly the peak time of geomagnetic activity for the ensuing cycle 25
At last, in this section, we try to analyze whether or not the rise time of the geomagnetic index for the 11-year cycle could be estimated through its relationship with the following maximum, as the case often used in sunspot cycle (Waldmeier, 1939). 5 Figure 3(a) shows the scatter plot of the rise time of aa index (T a ) for the 11-year cycle against the maximum (aa max ). It is seen in this figure that the data points are much scattered, and so T a is nearly uncorrelated to the following aa max , r = −0.158. One can not use this correlation to estimate the rise and peak times of aa max .
Similarly, the rise time of the Ap index for the 11-year cycle (T r ) is also nearly un- scatter plot of T r against Ap max . Thus, this correlation is also unable to be used to estimate the peak time of Ap max . One may note in Figs. 2(b) and 3(b) that the data point of cycle 24 is far from others. The related relationships depend largely on the data point of cycle 24. If this point were not considered, there would be only seven data points left ( Fig. 4(a)), and the correla- with a standard deviation of σ = 1.5. The maximum Ap index would be predicted to be Ap max (25) = 19.4 ± 1.5, slightly higher than the original one (19.0 ± 1.6). However, if the data point of cycle 24 in Fig. 3(b) were not considered as shown in Fig. 4(b), the correlation coefficient between T r and Ap max would increase to r = −0.404 at a confidence level of about 62%. The linear regression equation of T r to Ap max would become T r = 212.2 ± 140.4 − (7.07 ± 7.16)Ap max , 5 with a standard deviation of σ = 27.7. From this relationship, one could roughly estimate the rise time, T r (25) = 89.9 ± 12.7 ± 8.5 ± 27.7 ∼ 89.9 ± 31.6 (months), by substituting Ap max (25) = 17.3 ± 1.8 ± 1.2 into this relationship, in which √ 12.7 2 + 8.5 2 + 27.7 2 = 31.6.
According the time of Ap min (25) (October 2017), the peak time of Ap max (25) would be predicted to be around October 2017 +T r (25) = April 2025±32 (months). 10

Discussions and Conclusions
There are many methods that can be used to predict the maximum amplitude of sunspot cycle (R m ), such as 1) statistical methods, employing the relationship between the inter-cycle parameters (Thompson, 1988;Hathaway et al., 1994) or the early rising rate (Thompson, 1988;Cameron and Schüssler, 2008;; 2) the 15 functional methods, using mathematical functions containing a few parameters (Hathaway et al., 1994;Du, 2011d) for extrapolating the following monthly values; 3) the geomagnetic precursor methods (Ohl, 1979;Brown and Williams, 1969;Du et al., 2009), using the geomagnetic activity near the solar minimum; and 4) the solar precursor ones (Schatten et al., 1978;Pesnell and Schatten, 2018), using the previous cycle's polar 20 field.
In contrast, there are less methods found to predict the maximum amplitude of geomagnetic index. Geomagnetic activity forecast has been over the order of hours or days (McPherron, 1999;Abunina et al., 2013). The annual or monthly prediction on the geomagnetic activity is within a limited accuracy (over 20%) due to the irregular 25 variation in the time series (McPherron, 1999;Gordon, 2015). At earlier years, Kane 11 (1988) even pointed out that it is impossible to forecast the long-term geomagnetic activity through analyzing the time series of the Ap and aa indices. The geomagnetic activity near the solar minimum or at the decreasing phase of the solar cycle has been widely used to predict the maximum amplitude of sunspot cycle, but were seldom used to predict the maximum amplitude of the geomagnetic activity itself. 5 In the current work, we analyzed the relationships between the maxima (aa max ) and minima (aa min ) of the aa index for the 11-year solar cycle, and between the maxima (Ap max ) and minima (Ap min ) of the Ap index. It is found that aa max (Ap max ) is well correlated to the preceding minimum, aa min (Ap min ), with a correlation coefficient of about r = 0.86. So, these relationships can be used to predict the strength 10 of geomagnetic activity for the ensuing cycle, aa max (25) = 26.9 ± 2.6, or equivalently Ap max (25) = 17.3 ± 0.5 ± 1.2. It implies that the strength of the ensuing cycle 25 would be similar to the average over the past cycles, but higher than that of cycle 24.
The well known 'Waldmeier effect' (Waldmeier, 1939) that the rise time of a solar cycle is well anti-correlated to the amplitude has been widely used to estimate the rise 15 and peak times of a solar cycle if the amplitude has been predicted. However, such a correlation disappears in the aa geomagnetic index. If not considering the data of cycle 24, the rise time of Ap index for the 11-year cycle would be found to be weakly (r = −0.404) correlated to the following maximum (Ap max ). Using this correlation, one could roughly estimate the rise time, T r (25) ∼ 89.9 ± 31.6 (months), and the peak time, 20 April 2025 ±32 months, of geomagnetic activity for the ensuing cycle 25. Certainly, this estimate is much less reliable than the predictions on the peak size.
According the analysis above, the main conclusions of this study may be summarized as follows, 1. The maximum aa index for the 11-year cycle (aa max ) is found to be well correlated 25 to the preceding minimum (aa min ), with a correlation coefficient of r = 0.860. As a result, the maximum for the ensuing cycle 25 is predicted to be aa max (25) = 26.9 ± 2.6. This value is equivalent to Ap max (25) = 17.3 ± 1.8 ± 1.2 if employing the high correlation between aa and Ap (r = 0.939).