A New Calibrated Sunspot Group Series Since 1749: Statistics of Active Day Fractions
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Abstract
Although sunspotnumber series have existed since the midnineteenth century, they are still the subject of intense debate, with the largest uncertainty being related to the “calibration” of the visual acuity of individual observers in the past. A daisychain regression method is usually applied to intercalibrate the observers, which may lead to significant bias and error accumulation. Here we present a novel method for calibrating the visual acuity of the key observers to the reference data set of Royal Greenwich Observatory sunspot groups for the period 1900 – 1976, using the statistics of the activeday fraction. For each observer we independently evaluate their observational thresholds [\(S_{\mathrm{S}}\)] defined such that the observer is assumed to miss all of the groups with an area smaller than \(S_{\mathrm{S}}\) and report all the groups larger than \(S_{\mathrm{S}}\). Next, using a MonteCarlo method, we construct a correction matrix for each observer from the reference data set. The correction matrices are significantly nonlinear and cannot be approximated by a linear regression or proportionality. We emphasize that corrections based on a linear proportionality between annually averaged data lead to serious biases and distortions of the data. The correction matrices are applied to the original sunspotgroup records reported by the observers for each day, and finally the composite corrected series is produced for the period since 1748. The corrected series is provided as supplementary material in electronic form and displays secular minima around 1800 (Dalton Minimum) and 1900 (Gleissberg Minimum), as well as the Modern Grand Maximum of activity in the second half of the twentieth century. The uniqueness of the grand maximum is confirmed for the last 250 years. We show that the adoption of a linear relationship between the data of Wolf and Wolfer results in grossly inflated group numbers in the eighteenth and nineteenth centuries in some reconstructions.
Keywords
Solar activity Sunspots Solar observations Solar cycle1 Introduction
Solar activity regularly changes in the course of the 11year Schwabe cycle, in addition to which it also shows slower secular variability (see, e.g., the review by Hathaway, 2015). This is often quantified using the sunspotnumber series, which covers, with different levels of data quality, the period since 1610, starting with the first telescopic observations. It is generally accepted (Usoskin, 2013; Hathaway, 2015) that solar activity varies between very low activity, called grand minima, such as the Maunder Minimum during 1645 – 1715 (Eddy, 1976; Usoskin et al., 2015), and grand maxima such as the recent period of high activity in the second half of the twentieth century, called the Modern Grand Maximum (Solanki et al., 2004).
Several potential errors in the assessment of observers’ quality have been suggested recently, leading to discontinuities such as the Waldmeier discontinuity in the Wolf and International Sunspot Number (Clette et al., 2014; Lockwood, Owens, and Barnard, 2014) in the 1940s, which are related to a change in the sunspotcounting algorithm at the Zurich observatory; a jump between observations by Wolf and by Wolfer in the 1880s (Clette et al., 2014); and a discontinuity between Schwabe and Wolf data in 1848 (Leussu et al., 2013). Thus, a need for a revision of the sunspot series by recalibrating individual observers has become clear.

First, it is “spineless”, viz. while backbones may be solid (but see below), the connection between them is boneless, via stretchy multistore regressions that accumulate errors and cannot guarantee robust normalization. We note that although the backbone method is said by its authors to avoid daisychaining, this is not the case, since it includes calibration of data (between the backbones) derived by comparison with data from an adjacent interval using intercalibration over a period of overlap between the two. Even though they use the observer covering intermediate years to calibrate observers in earlier and later years, the errors at either step propagate between the beginning and the end of the series calibrated in this way. For example, the most surprising feature of the SS15 series relates to the amplitudes of solar cycles in the eighteenth and nineteenth centuries compared to modern cycles: these comparisons rely on a series of daisychained multistore regressions between the older and the modern data, regardless of the order in which they are carried out. Daisychaining is a major concern because errors in each intercalibration are compounded over the duration of the composite data series. Avoiding daisychaining requires a calibration method that can be applied for all data segments to the same reference conditions, independently of the calibration of temporally adjacent data series.

Second, the use of a linear proportionality between annually averaged data points is inappropriate (see Section 4) and may lead to serious biases. In addition, there are, in general, a great many problems and pitfalls associated with the regressions used by daisychaining (Lockwood et al., 2006, 2015). The errors in the data can violate assumptions involved in the technique, leading to grossly misleading fits, even when correlation coefficients are high. The relationship may not be linear and a regression derived for a period of low activity would inherently involve gross extrapolation if applied to largeramplitude cycles (see a discussion in Section 4). In addition, the use of the proportionality (regressions are forced through the origin) will, in general, cause amplification of the solarcycle amplitudes in data from lower visualacuity observers (Lockwood et al., 2015).
2 Calibration Method
The calibration method is based on a comparison of the statistics of the activeday fraction (ADF) of the data from the observer in question with that of the reference data set. The ADF (or the related fraction of the spotless days) is a very sensitive indicator of the level of solar activity around solar minima, which is more robust than the number of sunspots or groups (Harvey and White, 1999; Kovaltsov, Usoskin, and Mursula, 2004; Vaquero, Trigo, and Gallego, 2012; Vaquero et al., 2015). The method includes several stages: assessing the observational quality of individual observers, quantified as the area of sunspot groups that they would not have noticed; recalibrating individual observers to the reference data set; and compiling a composite time series. These stages are described below.
2.1 Reference Data Set
We normalized all observations to the reference data set, which was selected to be the RGO (Royal Greenwich Observatory) data^{1} of sunspot groups, since it provides all the necessary information (the observed sunspotgroup areas) on a regular basis. The RGO data set, often also called the Greenwich Photoheliographic Results (GPR: Baumann and Solanki, 2005; Willis et al., 2013b), was compiled using whitelight photographs (photoheliograms) of the Sun from a small network of observatories, giving a data set of daily observations between 17 April 1874 and the end of 1976, thereby covering nine solar cycles. The observatories used were the Royal Observatory, Greenwich (until 2 May 1949); the Royal Greenwich Observatory, Herstmonceux (3 May 1949 – 21 December 1976); the Royal Observatory at the Cape of Good Hope, South Africa; the Dehra Dun Observatory, in the NorthWest Provinces (Uttar Pradesh) of India; the Kodaikanal Observatory, in southern India (Tamil Nadu); and the Royal Alfred Observatory in Mauritius. Any remaining data gaps were filled using photographs from many other solar observatories, including the Mount Wilson Observatory, the Harvard College Observatory, Melbourne Observatory, and the US Naval Observatory.
The sunspot areas were measured from the photographs with the aid of a large position micrometer (see Willis et al., 2013b and references therein). The original RGO photographic plates from 1918 onwards have survived and have been digitized by the Mullard Space Science Laboratory in the UK. Automated scaling algorithms can derive sunspot areas (Çakmak, 2014), and it has been shown that the RGO data reproduce the manually scaled daily sunspotgroup numbers very well with a correlation of over 0.93 (A. Tlatov and V. Ershov, Private communication, 2015). However, the RGO data may be subject to an unstable dataquality problem before 1900 (Clette et al., 2014; Cliver and Ling, 2016; Willis, Wild, and Warburton, 2016). After 1977 the RGO data have been replaced by USAF–NOAA data with a different definition of sunspotgroup areas (Balmaceda et al., 2009). Accordingly, we limited the RGO reference data set to the period 1900 – 1976, which includes 28,644 daily records (924 months) with full coverage. This period includes moderatetohigh solar activity cycles and thus leads to a conservative upper bound of the observer calibration, as discussed below. We assume that the RGO dataset corresponds to one observed by a “perfect” observer, who reports all of the sunspot groups, including the smallest one. Although we cannot a priori be sure that this assumption is correct, it does not affect the calibration method. As we show below, we did not find observers in the eighteenth and nineteenth century the quality of whose data would be better than that of the RGO series.
2.2 Assessing the Quality of Observers
We assumed that the “quality” of observers reporting sunspot groups is related to the size (area) of sunspot groups that they can see or report. It is quantified as the threshold area [\(S_{\mathrm{S}}\)] (in millionths of the solar disk: msd) of the sunspot group, so that the observer would miss all of the groups with an area smaller than \(S_{\mathrm{S}}\) and observe all of the groups with an area greater than \(S_{\mathrm{S}}\). We used the apparent area as seen by the observer, not corrected for foreshortening. Here we estimate this threshold area [\(S_{\mathrm{S}}\)].
2.2.1 Calibration Curves
 i)
For each day during the reference period, we counted in the reference data set the number of sunspot groups with the observed whole area exceeding a given value.
 ii)
For each month of the reference data set, we calculated the activeday fraction (ADF) index defined as \(A={{n_{\mathrm{a}}}/n}\), where \(n_{\mathrm{a}}\) is the number of days with activity (at least one group is observed), and \(n\) is the number of observational days in the month. The ADF index [\(A\)] takes values from zero (no spots observed during the month) to unity (some sunspot groups observed for every day with observations during the month).
 iii)For the whole reference period (924 months), we constructed a cumulative probability function of the ADF [\(P(A*)= N(A\leq A*)/N\)], where \(A*\) is the given ADF value ranging between 0 and 0.9, \(N(A\leq A*)\) is the number of months with values of \(A\) lower than or equal to \(A*\), and \(N\) is the total number of months analyzed. For example, \(P(0.1)=0.029\) implies that 27 months (2.9 %) out of all the months in the reference data set have the ADF index \(A\leq0.1\) This distribution is shown in Figure 1 as the solid curve for \(S_{\mathrm{S}}=0\). Statistical uncertainties of the \(P\)values are defined as \(\sigma _{0}(A*,S_{\mathrm{S}})=\sqrt{N(A\leq A*,S_{\mathrm{S}})/N(S_{\mathrm{S}})}\).
 iv)
We repeated steps 2 and 3 above, but applying an artificial observational threshold for the group area, viz. \(S_{\mathrm{S}}>1\), 5, 10, 15, 20, etc. (msd) using the reference data set. This emulates observations of an “imperfect” observer who cannot see or does not report groups with an area smaller than the threshold value. Distributions similar to that in item iii) were constructed for different values of \(S_{\mathrm{S}}\) to form a set of calibration curves \(P(A,S_{\mathrm{S}})\), as shown in Figure 1, for the range of \(S_{\mathrm{S}}\) from 5 to 300 msd.
 v)Since real observers usually make observations on only a fraction \(f\leq1\) of days, we also emulated the effect of this and estimated the related uncertainties by a MonteCarlo method. For each observer we defined the fraction [\(f\)] for the entire period of their observation used for calibration. We repeated steps ii) – iv) above, but randomly removed the fraction (1\(f\)) of daily values from the reference data set. We did this 1000 times for each combination of \(f\) and \(S_{\mathrm{S}}\) and thus defined an ensemble of calibration curves \(P(A,S_{\mathrm{S}},f)\). Next we defined the mean values of \(P(A,S_{\mathrm{S}},f)\) over the ensemble that are equal within the uncertainties to \(P(A,S_{\mathrm{S}},f=1)\), and their 68 % uncertainties \(\sigma_{1}(A,S_{\mathrm{S}},f)\) defined as the upper and lower 16 % quantiles. The final uncertainty of the calibration curves \(P(A,S_{\mathrm{S}},f)\) is defined as \(\sigma(A,S_{\mathrm{S}},f)=\sqrt{\sigma_{0}^{2}+\sigma_{1}^{2}}\), where \(\sigma_{0}(A,S_{\mathrm{S}},f)\) is defined similarly to step iii) above, but for the random subset of the whole reference data set. An example of the cumulative probability function \(P\) is shown, along with uncertainties, in Figure 2 for \(S_{\mathrm{S}}=45~\mbox{msd}\) and \(f=0.66\) as corresponding to the observations of R. Wolf from Zurich. The value of \(f\) is known from the observer’s (in this case R. Wolf) records, and the value of \(S_{\mathrm{S}}=45~\mbox{msd}\) gives the best fit to the observed pdf for this \(f\).
Thus, a set of calibration curves [\(P(A,S_{\mathrm{S}},f)\)] was constructed that was used to evaluate the quality of an individual observer’s sunspotgroup detection as quantified in terms of “missing” spots with the area below the threshold \(S_{\mathrm{S}}\).
2.2.2 Assessing the Observational Quality of Individual Observers
To assess the quality of individual observers, we compared the functions [\(P\)] constructed for the data from that observer with the calibration curves as follows.
For a given observer with the daily observational coverage fraction [\(f\)], we defined, similarly to step ii) in Section 2.2.1, the monthly ADF index [\(A\)] and its distribution [\(P(A,f)\)]. Uncertainties of the \(P\)values were considered random, similar to those in step iii) of Section 2.2.1. Then, for the given value of \(f\) we fit the observer’s distribution to the calibration curves [\(P(A,S_{\mathrm{S}},f)\)] described above to find the value of \(S_{\mathrm{S}}\), corresponding to the observer, and its uncertainty. The fit was made using the \(\chi^{2}\)method for \(P(A,S_{\mathrm{S}},f)\) in the range of \(A\) between 0.1 and 0.8 (eight degrees of freedom). Low values of \(A<0.1\) and higher values \(A>0.8\) were not used because of low statistics and required high daily coverage within a month. Moreover, definition of very low and very high values requires a large number of observational days per month, which may distort the corresponding statistics for observers with poor coverage. The bestfit value of \(S_{\mathrm{S}}\) is defined by minimizing the \(\chi ^{2}\)values, while uncertainties are defined as the range of \(S_{\mathrm{S}}\) where the value of \(\chi ^{2}(S_{\mathrm{S}})\) lies below \(\chi^{2}_{0}+1\) (\(\chi^{2}_{0}\) being the minimum value) corresponding to the 68.3 % confidence interval.
Results of calibration of the key observers used here. Columns list the name of the observer; the period of observation [\(T_{\mathrm{obs}}\)]; the period used for calibration [\(T_{\mathrm{cal}}\)]; the number of observational days [\(N\)] used for calibration; the data coverage in % [\(f\)]; the fraction of spotless days [SDF] in the record; the threshold area [\(S_{\mathrm{S}}\)] in uncorrected msd; values in parentheses denote the upper and lower \(1\sigma\) bound. For details see text.
Observer  \(T_{\mathrm{obs}}\)  \(T_{\mathrm{cal}}\)  N  f  SDF  \(S_{\mathrm{S}}\) 

RGO  1874 – 1976  1900 – 1976  28,644  \({\approx}\,100~\%\)  16 %  0 
Quimby  1889 – 1921  1900 – 1921^{c}  10,830  92 %  23 %  \(22 \bigl({}^{28}_{16} \bigr)\) 
Wolfer  1876 – 1928  1900 – 1928^{c}  7165  68 %  21 %  \(6 \bigl({}^{12}_{ 0} \bigr)\) 
Winkler^{a}  1882 – 1910  1889 – 1910^{d}  4812  60 %  24 %  \(53 \bigl({}^{66}_{45} \bigr)\) 
Tacchini  1871 – 1900  1879 – 1900^{d}  6235  78 %  19 %  \(10 \bigl({}^{14}_{ 7} \bigr)\) 
Leppig  1867 – 1881  1867 – 1880^{d}  2463  52 %  26 %  \(45 \bigl({}^{33}_{55} \bigr)\) 
Spoerer  1861 – 1893  1865 – 1893^{d}  5386  53 %  15 %  \(3 \bigl({}^{5}_{0} \bigr)\) 
Weber  1859 – 1883  1859 – 1883  6981  79 %  19 %  \(22 \bigl({}^{28}_{16} \bigr)\) 
Wolf  1848 – 1893  1860 – 1893^{d}  8102  66 %  21 %  \(45 \bigl({}^{53}_{36} \bigr)\) 
Shea  1847 – 1866  1847 – 1866  5538  79 %  20 %  \(25 \bigl({}^{33}_{18} \bigr)\) 
Schmidt  1841 – 1883  1841 – 1883  6887  49 %  21 %  \(10 \bigl({}^{15}_{6} \bigr)\) 
Schwabe  1825 – 1867  1832 – 1867  8570  65 %  18 %  \(13 \bigl({}^{18}_{8} \bigr)\) 
Pastorff  1819 – 1833  1824 – 1833^{d}  1451  41 %  14 %  \(5 \bigl({}^{10}_{0} \bigr)\) 
Stark^{a}  1813 – 1836  1813 – 1836^{e}  2406  30 %  41 %  \(60 \bigl({}^{70}_{50} \bigr)\) 
Derfflinger^{a}  1802 – 1824  1816 – 1824^{e}  346  11 %  38 %  \(50 \bigl({}^{80}_{40} \bigr)\) 
Herschel^{a}  1794 – 1818  1795 – 1815  344  4 %  16 %  \(23 \bigl({}^{35}_{10} \bigr)\) 
Horrebow  1761 – 1776  1766 – 1776^{e}  1365  34 %  27 %  \(75 \bigl({}^{95}_{60} \bigr)\) 
Schubert  1754 – 1758  1754 – 1757^{e}  404  36 %  37 %  \(10 \bigl({}^{16}_{5} \bigr)\) 
Staudacher^{b}  1749 – 1799  1761 – 1776  1035  14 %  5.5 %  – 
To check the dependence of the \(S_{\mathrm{S}}\) on the activity level, we composed synthetic pseudoobservers as subsets of the RGO data for the periods of low activity 1902 – 1923 (called \(\mathrm{RGO}_{\mathrm{low}}\)) and high activity 1944 – 1964 (\(\mathrm{RGO}_{\mathrm{high}}\)). By construction, these pseudoobservers have the true value of \(S_{\mathrm{S}}=0\) since they are subsets of the reference RGO data set. Next we calibrated these pseudoobservers using the ADF method. We found that the formal threshold for the \(\mathrm{RGO}_{\mathrm{low}}\) observer is \(S_{\mathrm{S}}=27\pm10~\mbox{msd}\), i.e. the observer making observations during periods of low activity is likely to be overcalibrated (the threshold appears too high and, as a result, their observations, normalized to the reference data set, are overestimated). For the \(\mathrm{RGO}_{\mathrm{high}}\) pseudoobserver we found a value of \(S_{\mathrm{S}}=16\pm5~\mbox{msd}\). The negative \(S_{\mathrm{S}}\) means that we had to apply the observational threshold of 16 msd to the \(\mathrm{RGO}_{\mathrm{high}}\) data set to reproduce the ADF statistics for the reference data set with \(S_{\mathrm{S}}=0\). This would potentially lead to a slight underestimate of the corrected data for this observer. However, we note that the \(\mathrm{RGO}_{\mathrm{high}}\) pseudoobserver is an extreme case since the period around Solar Cycle 19 was characterized by the highest activity in the entire sunspotnumber series in all of the existing sunspot series, including SS15. We did not obtain a “negative” threshold for any real observer considered here. This implies that the method tends to provide an upper estimate of solar activity that lies on the high side, particularly during periods of lowtomoderate activity. However, the test for \(\mathrm{RGO}_{\mathrm{low}}\) shows that during the grand minima the method cannot be applied. Future work will aim to establish calibrations for observers working during the Maunder Minimum that are consistent with those derived here so that the data series can be extended back into the seventeenth century and the Maunder Minimum.
We have identified 18 observers whose records can be calibrated by this method and form the core of the sunspot group historical series since the mideighteenth century. They are listed in Table 1, along with the full ranges of observational dates, the periods used for calibration, the spotlessday fraction and the obtained observational threshold [\(S_{\mathrm{S}}\)] with its \(1\sigma\) uncertainties.
For the period of the late nineteenth and the twentieth century we considered only a few key observers with long stable records since the quality and density of data during the past hundred years were high, and thorough studies of their intercalibration have been performed (Clette et al., 2014). For the observers Quimby and Wolfer, who have a significant overlap with the reference data set, we used the overlap period for a direct calibration.
For the period before the midnineteenth century, all of the observers were considered and calibrated whenever possible following the method described here (i.e. whenever they had sufficient observations to apply our technique). For each observer we used data of sunspotgroup counts from the HS98 database (Hoyt and Schatten, 1998) except for Schwabe and Staudacher (see comments below). Whenever possible, we used data for complete solar cycles. Unless indicated otherwise, we considered for each observer only months with three or more daily observations.
We also show in Table 1 the spotlessday fraction [SDF], which is the number of days with the reported absence of spots to the total number of observational days during the calibration period [\(T_{\mathrm{cal}}\)], for each observer. The reference RGO data set contains \({\approx}\,16~\%\) of spotless days. SDF is in the range of 15 % to 26 % for most of the observers in the nineteenth century except for Stark and Derfflinger, whose observations were likely reflecting the low activity around the Dalton Minimum. Apparently different from all others was Staudacher, with only 57 spotless days (5.5 %) reported. He was probably more interested in drawing spots than in reporting their absence. Regardless of the reason, his statistic of spotless and active days is distorted and cannot be used to calibrate his quality directly to the reference period, as discussed in Sect. 2.3.1.
Some specific comments are given below.
S.H. Schwabe observed the Sun during 1825 – 1867; but we considered only the period of 1832 – 1867 for calibration. This period covers three full cycles, since the earlier part of his record is thought to be of less stable quality (Leussu et al., 2013). Sunspotgroup numbers for Schwabe’s observations were taken not from the HS98 database, but from a new revised collection by Arlt et al. (2013) (available at www.aip.de/Members/rarlt/sunspots/schwabe as version 1.3 from 12 August 2015).
For J.W. Pastorff from Drossen we used data for observer 263 in the HS98 database. His SDF is low (14 % – see Table 1), indicating that he might have skipped reporting some spotless days. If this is true, it may lead to an overestimate of his observational quality and consequently to an underestimate of the sunspotgroup number based on his record. On the other hand, his corrected data are consistent with those of Schwabe and Stark (Figure 5).
C. Horrebow from Copenhagen (observer 180 in the HS98 database) observed the Sun during 1761 – 1776. Here we used for the calibration the period of 1766 – 1776 (one solar cycle) because the data are very sparse before 1766.
T. Derfflinger from Kremsmünster observed the Sun during 1802 – 1824, but we used for the calibration data covering 1816 – 1824 to exclude the Dalton Minimum so as to avoid the potential problems discussed above associated with a mismatch in the range of the data compared to that for the reference data set.
J.M. Stark from Augsburg observed the Sun during 1813 – 1836, and we used all data for observer 255 of the HS98 database, while not considering the generic nosunspot day records (observer #254 called “STARK, AUGSBURG, ZERO DAYS” in the HS98 database).
J.C. Schubert from Danzig observed the Sun during 1754 – 1758. We used for the calibration the period of 1754 – 1757, which is \({\pm}\,2\) years around the formal cycle minimum in 1755.2. This includes 404 daily observations with 36 % coverage. To assess the quality of Schubert’s observations, we used the RGO statistics, as described in Section 2.2.1, but using RGO data only within \({\pm}\,\mbox{two years}\) around the solarcycle minima to be consistent with Schubert’s cycle coverage.
J.C. Staudacher from Nürnberg, while providing about 1035 daily drawings for the period 1749 – 1795 (\({\approx}\,6~\%\) coverage), as published by Arlt (2008), cannot be directly calibrated in the way proposed here because he did not properly report days without sunspots. He reported no spots for only 5.5 % days, which is less than all other observers (15 – 40 %, see Table 1). Moreover, there are no zerospot months (with the number of daily observations more than two) in his record, in contrast to all other observers. This distorts the ADF statistics and prevents a direct calibration as described above. The case of Staudacher is considered separately in Section 2.3.1.
The following observers produced sufficiently long observational records but cannot be calibrated in the manner described above because of sparse or unevenly distributed observations or because they did not report spotless days: Lindener, Tevel, Arago, Heinrich, Flaugergues, and Hussey. We also did not consider observers during and around the Maunder Minimum because of the very low level of activity (Usoskin et al., 2015) when the method cannot be applied. The period between the end of the Maunder Minimum and the mideighteenth century cannot be studied because of a lack of sufficient observations (Vaquero and Vázquez, 2009).
In addition, we also checked the record by H. Koyama from Tokyo, who observed the Sun over the period 1947 – 1984 with about 56 % daily coverage; these data formed one backbone for the method by SS15. We used for the calibration the period of 1953 – 1976 to cover full cycles, and to be more consistent with the RGO time interval; a total of 4778 daily observations were processed. The calibration was performed using the reference RGO dataset for the same period of time. The threshold [\(S_{\mathrm{S}}\)] value was found to be \(8\pm5~\mbox{msd}\), yielding a result fully consistent with the RGO data (see Figure 5). We stress that this record was not used in the compilation of the final series, but only to test the method.
The calibration method works after 1754 when Schubert started observing. If Staudacher’s data are included (see Section 2.3.1), the calibration starts in 1749. Before Staudacher there is a paucity of sufficiently long timeseries of observations by single observers, so that the method cannot work due to too poor statistics, and before 1715 the method is not applicable because of the Maunder Minimum, where the statistics of the reference data set cannot be applied. We stopped the calibration in 1900 since the reference data set of RGO data was used after 1900.
2.3 Corrections of Individual Observers
After we defined the observational threshold [\(S_{\mathrm{S}}\)] and its uncertainty for each observer (see above), observations (the number of sunspot groups) by this observer were calibrated to the reference data set. All corrections were made on the daily scale because of the nonlinearity of correction that may otherwise distort the relation, as discussed below.
 i)A test value \(S_{\mathrm{S}}^{*}\) is randomly selected, using the normally distributed random numbers, from the distribution of \(S_{\mathrm{S}}\) values for the observer (see Table 1). A degraded subset of the reference daily data set is constructed by considering only sunspot groups with the (uncorrected) area \(\geq S_{\mathrm{S}}^{*}\), i.e. what an observer with the observational limit of \(S_{\mathrm{S}}^{*}\) would have recorded. For each daily value \(G_{S_{\mathrm{S}}^{*}}\) from the degraded data set we construct a distribution of the \(G_{\mathrm{ref}}\)values from the reference data set (\(S_{\mathrm{S}}=0\)), similar to that shown in Figure 4b.
 ii)
Step i) above is repeated 1000 times, each time randomly selecting an \(S_{\mathrm{S}}^{*}\) value for a given observer, and summing up all of the distributions of \(G_{\mathrm{ref}}\) for a given \(G_{S_{\mathrm{S}}^{*}}\) The probability density function (pdf) of the \(G_{\mathrm{ref}}\) values for each \(G_{S_{\mathrm{S}}^{*}}\) value (i.e. reported by the observer) is constructed. Finally, the correction matrix for the particular observer is constructed as illustrated in Figure 4a.
 iii)
For each daily recorded value [\(G\)] of the observer, the corresponding mean and the 68 % upper and lower quantiles of the \(G_{\mathrm{ref}}\) were calculated, giving the mean corrected daily group number and its uncertainties.
The method is illustrated in Figure 4 by calibration of the observational quality of R. Wolf from Zurich. The correction of an “imperfect” observer (R. Wolf in this example) is based on an assessment of how many sunspot groups the “perfect” observer (the reference RGO in our case) would see for a day when the imperfect observer reported \(G_{\mathrm{Wolf}}\) groups. Thus, for a given \(G_{\mathrm{Wolf}}\) value (\(x\)axis) we obtain a pdf of the reference values [\(G_{\mathrm{ref}}\)] to yield the mean and the uncertainties of the corrected number of sunspot groups. We note that the relation is well approximated by a power law with the spectral index 0.83 (the dashedred curve in Figure 4), but this functional form is shown only for illustration and was not used in the construction of correction matrices. An important feature observed is that the mean \(G_{\mathrm{ref}}\)value is nonzero (0.38) for spotless days reported by an imperfect observer, R. Wolf in our example. Accordingly, we cannot say whether zero spots by Wolf implies a true spotless day or whether groups were small and went undetected. It is important that the correction implied by the matrix cannot be approximated by a linear (or worse still, a proportional) regression. Panel c of Figure 4 depicts the daily correction factor defined as the ratio of the \(G_{\mathrm{ref}}\) (the number of groups the real observer would see if they were a perfect observer) to \(G_{\mathrm{Wolf}}\) (the number of groups the observer R. Wolf actually reported). The ratio gradually drops from \({\approx}\,1.8\) for one group reported by Wolf to 1.19 for 15 groups reported by Wolf. If Wolf saw 25 groups, the correction would have been only 1.02. This implies that the larger the number of sunspot groups (the higher the activity is), the smaller the relative error of the imperfect observer. It is clear that a simple linear regression cannot be used to correct an imperfect observer (see Section 4). This was also emphasized by Lockwood et al. (2015) from a study of the effects of imposing different observational thresholds on the RGO data.
2.3.1 Calibration of Staudacher via Horrebow
J.C. Staudacher is a key observer to evaluate solar activity in the second half of the eighteenth century and a backbone observer for SS15. It is crucially important to evaluate the quality of the data that he produced. However, since he apparently did not properly (see Table 1) report spotless days, being primarily interested in drawing sunspots, the ADF method used here cannot be directly applied to his data, and it would yield an unrealistically high quality of observations. Accordingly, we calibrated the Staudacher data in two steps to the reference data set. This is the only exception to the method described above.
2.3.2 Test of the Correction: Wolfvs.Wolfer
A good and important example to test our (and other) calibration methods on is the relation between data reported by J.R. (Rudolf) Wolf and H.A. (Alfred) Wolfer, both from the Zurich observatory. First, a proper comparison of the two observers is crucially important as Wolf was the reference observer for the Wolf sunspotnumber series, while Wolfer is the reference observer for the ISN (v.2) and a backbone observer for the SS15 reconstruction. Without an adequate comparison between them, it is difficult to compare the various sunspot series now available. Second, a long period of their overlap (4385 days during 1876 – 1993) exists when both observers independently reported sunspots. HS98, using a linear regression applied to the daily values of the overlapping periods, proposed that the two observers are quite close to each other in the quality of their observations. In contrast, SS15 proposed that the relation between them is a linear and proportional scaling so that \(G_{\mathrm{Wolfer}}=1.66 G_{\mathrm{Wolf}}\), based on a linearregression analysis of the annually averaged number of sunspot groups.
Figure 7d shows the relation of the simultaneousday group numbers by the two observers, after the correction performed here (reduction to the reference RGO data set). The data are nearly perfectly corrected – the corrected data lie around the diagonal implying that both series report the same quantity. The best fit to the orange balls (the last point excluded) is \(G^{*}_{\mathrm{Wolfer}}=(0.98\pm0.03) G^{*}_{\mathrm{Wolf}}\), where the asterisks denote the values calibrated to the reference data set. We recall that Wolf and Wolfer were calibrated to the reference data set independently of each other, and thus this comparison provides a direct test of the validity of the method. The ratio \(G^{*}_{\mathrm{Wolfer}}/G^{*}_{\mathrm{Wolf}}\) should, of course, be unity if the calibration is done correctly: that this is the derived value within the uncertainty shows that both have been properly estimated. Thus, since the ratio between the daily group numbers (for the days when both observers made observations) of Wolf and Wolfer is consistent with unity (i.e. onetoone relation) after each of them was independently corrected, we conclude that the method works well. We note that this example is shown only for illustration and was not used in the actual calibration.
3 Corrected Series of SunspotGroup Numbers
In this section we construct a composite series of sunspotgroup numbers for the period since 1749 using the selected observers.
3.1 Compilation of the Corrected Series
For each day [\(t\)] we considered all those observers whose reports are available for that day. For each such observer \(i\) we took the reported \(G_{\mathrm{obs},i}\) value for that day and corrected it to the reference data set using the correction matrix for that observer (as described in Section 2.3) to define the pdf of the corresponding values [\(G_{\mathrm{ref}}\)]. If several observers were available for the day, the corresponding pdfs were multiplied and renormalized to unity again. To avoid possible voiding of the observational days with an outlying datum, we set the minimum pdf values to \(10^{4}\). Then, from this composite (over all of the observers for the day) pdfs we calculated the mean daily value of \(G_{\mathrm{ref}}\) and its 68 % error as the standard deviation from the gathered pdf divided by \(\sqrt{N}\), where \(N\) is the number of observers for the day.
From these composite daily series we constructed a monthly composite series as a standard weighted average (see details given in, e.g. Usoskin, Mursula, and Kovaltsov, 2003) of the composite daily values. This series is available in the electronic supplement to the article.
 i)
For each day with existing data within the given year, we randomly took a value of daily \(G\) from the composite pdfs of corrected daily \(G\)values obtained above.
 ii)
From these randomly taken daily \(G\)values we computed the monthly values as the simple arithmetic mean.
 iii)
The annual value \(G{'}\) was computed as the arithmetic mean of the monthly values.
 iv)
Steps i) – iii) were repeated 1000 times so that a distribution of the annual \(G{'}\)values was obtained. Finally, the mean and the 68 % (divided by the \(\sqrt{n}\), where \(n\) is the number of the months with data in the year) were computed from the distribution. Errors propagate naturally and without biases in this approach.
The activity remains at a moderate level in the nineteenth century and is higher in the eighteenth century. However, activity (sunspotgroup numbers) in both the eighteenth and nineteenth centuries remained significantly lower than the Modern Grand Maximum in the second half of the twentieth century.
We also note that there is another source of nonlinearity not accounted for here (neither was it considered in previous series, such as HS98, ISN, or SS15). It is related to the calculation of monthly and annual values from a small number of sparse daily observations. Under these conditions, the simple arithmetic average tends to overestimate the number of sunspots (groups) for active periods if the number of daily observations per month is smaller than three (Usoskin, Mursula, and Kovaltsov, 2003). The overestimate can be as much as 20 – 25 %. This may affect the values for the eighteenth century, where data coverage was low. Since this effect leads to a possible overestimate of the monthly (and thus annual) values, it keeps the averaged series provided here as a conservative upper limit. However, this effect does not influence the calibration and correction procedure that works with the original daily data. Neither is the corrected daily series affected. This effect will be taken into account in forthcoming studies.
3.2 Comparison to Previous Reconstructions
The new series is close to the GSN series by HS98, being consistent with it within uncertainties after \({\approx}\,1830\) and yielding Solar Cycles 10 and 11 to be slightly higher than in the HS98 series. The newly reconstructed cycles are significantly higher than the HS98 values before 1830. On the other hand, the new series is consistently lower than the values proposed by SS15, except for the years of the sunspotcycle minima. This difference is mostly due to the linear regression used by SS15 to correct the Wolfvs.Wolfer records, which leads to a bias, as discussed in Section 4. Before 1830 the new series lies between the HS98 and SS15 series, on one hand being consistent with the conclusion by SS15 that the HS98 GSN is most likely too low before the Dalton Minimum, but on the other hand implying that the revision by SS15 is too high. We recall that because of the moderatetohigh activity underlying the reference data set, our reconstruction tends to overestimate reconstructed activity at earlier times with lower activity. We note that the RGO data and the SS15 series also diverge around 1945, and tests by Lockwood, Owens, and Barnard (2016) against independent data indicate that this is another error in the backbone reconstruction. The two errors (around 1945 and the Wolfvs.Wolfer relationship) both act in the same direction, to make early group sunspot numbers too large in relation to those in the Modern Grand Maximum.
The new reconstruction suggests that the sunspot activity (quantified as the number of groups) was somewhat higher in the mideighteenth century than it was in the midnineteenth century, but significantly lower than the Modern Grand Maximum (Usoskin et al., 2003; Solanki et al., 2004) in the second half of the twentieth century. The Dalton (ca. 1800) and Gleissberg (ca. 1900) Minima of solar activity are clearly seen as the reduced magnitude of solar cycles, but they are not considered as grand minima of activity in contrast to the Maunder Minimum (Usoskin, 2013).
4 A Note on Nonlinearity and the Lack of Proportionality
The main reason for the difference between the sunspotactivity levels resulting from this work and previous recent reconstructions (e.g. Clette et al., 2014; Svalgaard and Schatten, 2016) is that the latter were based on linear regressions between annually averaged data of individual observers, which can seriously distort the results. Furthermore, SS15 assumed proportionality between the data (by forcing linear regressions through the origin and using only scaling correction factors without offsets), which is also incorrect and leads to possible errors that accumulate because the backbone calibrations are daisychained (Lockwood et al., 2015). For example, the linear regression suggests a ratio of \(1/0.6=1.66\) between the numbers of sunspot groups reported by Wolf and Wolfer. This assumes that Wolf was missing 40 % of all groups that would have been observed by Wolfer, regardless of the activity level, viz. a nospot record by Wolf would correspond to no spots observed by Wolfer, but 15 groups reported by Wolf would correspond to 25 groups by Wolfer. As we show here, a linear regression (proportionality) based on the annually averaged data may lead to significant biases leading to a heavy overestimation of the number of sunspot groups recorded by Wolf during the periods of high activity.
The relation between records of different observers is nonlinear (see, e.g., Figure 4a for Wolf). First, it does not necessarily go through the origin (0 – 0 point). Thus \(G_{\mathrm{Wolf}}=0\) yields a nonzero \(G_{\mathrm{ref}}\) with a mean of 0.49 and the 68 % range from 0 to 2 groups. This indicates that when Wolf reported no spots, there might have been 0 – 2 groups on the Sun. This feature is totally missed when assuming a linear proportionality. The relation between Wolf and Wolfer is quite steep (the slope of a regression forced through the origin is about 1.7) for low activity days (1 – 2 groups), but then the slope of the relation drops (see Figure 4c) to almost unity for days with more than 20 groups. It is obvious that a single correction factor is not applicable for such a relation as it assumes that an imperfect observer (Wolf in this case) misses the same fraction of spots regardless of the activity level. However, the nonlinearity of the relation arises because an observer with a poorer instrument or eyesight, or observing in poorer conditions, does not see and report the smallest spots as defined by their observational threshold [\(S_{\mathrm{S}}\)].
Another problem with linear regressions is that because of the nonlinearity, the averaging procedure is not transmissive for corrections, i.e. the correction of the averaged values is different from the average of corrected values. Corrections must be applied to daily values using a matrix as shown in Figure 4, and only after that can the corrected values be averaged to monthly or yearly resolution, as described in Section 2.3. Corrections applied to (annually) averaged values miss the nonlinearity and, since the annual values are dominated by low and moderate numbers of groups, lead to an overestimate of the relation and, as a consequence, to too high solar cycles. Moreover, the use of simple arithmetic means from sparse daily values may lead to an additional overestimate of the monthly (or annual) values, thus distorting the relation (Usoskin, Mursula, and Kovaltsov, 2003, see discussion in Section 3.1). A weighted average should be used instead.
Accordingly, the use of annual (or even monthly) averaged values for a linear scaling correction is not appropriate and is grossly misleading.
A different average size of sunspot groups as a function of activity level producing such a nonlinearity is known in solar physics (e.g. Solanki and Unruh, 2004). E.g., the ratio of faculae to spots changes with activity level (i.e. the ratio of number of small to large magnetic flux tubes). Of course, neither faculaetospot area ratio nor sunspot areas directly enter into the number of sunspot groups or the size of groups, but they are examples of other (somewhat related) quantities showing a nonlinear behaviour.
5 Conclusions
 i)
constructing calibration curves for the reference data set;
 ii)
assessing the quality of individual observers in terms of the observational threshold of the sunspotgroup area [\(S_{\mathrm{S}}\)];
 iii)
normalizing raw data by the individual observers to the reference data set;
 iv)
compiling the composite series.

The reference data set is of the “perfect” quality, and the ADF statistic for this set is representative for the entire period under investigation. The selection of the reference data set ensures that violations of this assumption may only lead to a possible overestimate of the activity.

The quality of an observer is quantified as the observational threshold [\(S_{\mathrm{S}}\)], so that the observer misses all groups with an area smaller than \(S_{\mathrm{S}}\) and reports all groups with an area greater than \(S_{\mathrm{S}}\), and the quality remains constant throughout their entire period of observations, but this can be revisited in the future by a piecewise calibration.

The correction of an “imperfect” observer is based on an estimate of the number of groups that they would see if they were a “perfect” observer (i.e. with the data quality corresponding to the reference data set).
The series presented here is a basic skeleton, or core, of the reconstruction of the number of sunspot groups, to which other observers with shorter sunspot records can and will be added later by means of direct normalization to this core series. The raw series includes daily numbers of sunspot groups, the mean values and their 68 % confidence intervals, reduced to the reference data set, for each individual observer listed in Table 1. The composite series exists, in addition to raw daily resolution, as monthly and annual averages provided in the electronic supplementary material.
The new series is consistent with the group sunspot number (Hoyt and Schatten, 1998) after about 1830, but is systematically higher than that in the eighteenth century, implying that the sunspot activity was higher than proposed by HS98 before the Dalton Minimum. Conversely, the new series is significantly and systematically lower than the backbone sunspotgroup number SS15 in the nineteenth and eighteenth century, implying that the backbone reconstruction grossly overestimates solar activity during that period. We demonstrated that the overestimate of the backbone method was caused by the incorrect use of linear proportionality to normalize the observers to each other, which led to propagation and accumulation of errors. Furthermore, the use of annual means of sparse data led to additional errors, enhancing those for observers active at earlier times.
The new series depicts lows of solar activity around 1800, known as the Dalton Minimum, and around 1900, known as the Gleissberg Minimum, although they are not considered as grand minima of activity. We note that the new series provides an upper limit for the group numbers during most of the times, in particular around the Dalton Minimum, when both the activity level and the observation density were low. The Modern Grand Maximum of activity in the twentieth century is confirmed as a unique event over the last 250 years. This confirms and enhances the established pattern of the secular variability of solar activity (e.g. Hathaway, 2015; Usoskin, 2013).
Footnotes
 1.
We used the version of the RGO data available at the Marshall Space Flight Center (MSFC) solarscience.msfc.nasa.gov/greenwch.shtml , as compiled, maintained, and corrected by D. Hathaway. This data set is slightly different from other versions of the RGO data stored elsewhere, e.g. at the National Geophysical Data Center in Boulder CO (Willis et al., 2013a.).
Notes
Acknowledgements
We are thankful to Rainer Arlt for the revised data of Staudacher. Contributions from I. Usoskin, K. Musrsula, and G. Kovaltsov were done in the framework of the ReSoLVE Centre of Excellence (Academy of Finland, project no. 272157). Work at the University of Reading is supported by the UK Science and Technology Facilities Council under consolidated grant number ST/M000885/1. G. Kovaltsov acknowledges partial support from Programme No. 7 of the Presidium RAS. This work was partly supported by the BK21 plus program through the National Research Foundation (NRF) funded by the Ministry of Education of Korea.
Disclosure of Potential Conflicts of Interest
The authors declare that they have no conflicts of interests.
Supplementary material
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