Sunspot Number Series Research Articles

Long-term Evolution of Solar Activity and Prediction of the Following Solar Cycles

Solar activities have a great impact on modern high-tech systems, such as human aerospace activities, satellite communication and navigation, deep space exploration, and related scientific research. Therefore, studying the long-term evolution trend of solar activity and accurately predicting the future solar cycles are highly anticipated. Based on the wavelet transform and empirical function fitting of the longest recorded data of the annual average relative sunspot number (ASN) series of 323 yr to date, this work decisively verifies the existence of the solar century cycles and confirms that its length is about 104.0 yr, and the magnitude has a slightly increasing trend on the timescale of several hundred years. Based on this long-term evolutionary trend, we predict solar cycles 25 and 26 by using phase similar prediction methods. As for solar cycle 25, its maximum ASN will be about 146.7 ± 33.40, obviously stronger than solar cycle 24. The peak year will occur approximately in 2024, and the period will be about 11 ± 1 yr. As for solar cycle 26, it will start around 2030, and reach its maximum between 2035 and 2036, with maximum ASN of about 133.0 ± 3.200, and period of about 10 yr.

Is there a link between the length of the solar cycle and Earth’s temperature?

The Sun provides most of external energy to Earth’s system and thus has the potential of influencing it. Various studies reported a correlation between the solar cycle length and the northern hemisphere temperatures on Earth. Here, we reassess the cycle length record by incorporating the newly revised and updated sunspot number series as well as plage area composite, before comparing it to Earth temperature records. We find that cycle length series constructed from sunspot and plage data exhibit the same behaviour, both showing a downward trend after 1940. Our results suggest that the agreement between solar cycle lengths and temperatures found earlier is an artefact of (1) some arbitrary choices made by those studies when constructing the cycle length series as well as (2) a rather short time interval, to which the analyses were restricted. When considering the entire period of reliable sunspot and temperature data, these records diverge before about 1870 and after 1960. We also find a poor agreement between Earth temperatures and cycle length when using plage areas instead of sunspot data to derive cycle lengths. Our result of the divergence between cycle length series and Earth’s temperature after 1960 implies that the cycle length cannot be used to support a solar origin for the warming on Earth over the last 5 decades.

Sunspot observations at Kawaguchi Science Museum: 1972 – 2013

AbstractIndividual sunspot observations have formed a ground basis of international sunspot number, a unique reference for long‐term solar variability in the centennial timescale. The original datasets were subjected to exploitations and analyses upon the recalibrations of the sunspot number series. In this context, this study reviewed and analysed original sunspot records and their databases in the Kawaguchi Science Museum (KSM) in Japan. KSM hosts sunspot drawings and logbooks from 1972 to 2013. This dataset has a longer chronological coverage than what was known to the scientific community (1981–2010). These records have been digitized and publicized in a museum database, which allows users to access individual sunspot drawings and numerical data in KSM logbooks. These records are highly homogeneous as a single observer's dataset (Hitoshi Takuma), who used a 15‐cm refractor at the Kawaguchi Juvenile Museum in 1972–2003 and a 20‐cm refractor at KSM in 2003–2013. We also reviewed the Takuma data series, his monthly observation days (21.3 days/month), sunspot number in the whole disk and each hemisphere, and sunspot positions in a butterfly diagram. We also assessed Takuma's data stability in comparison with the international sunspot number and reference datasets of the SILSO. Takuma's data appear stable until 2003, when he changed the observation site and instrument. His data stability was quantitatively compared with the SILSO reference datasets, confirming the substantial long‐term stability of the data and establishing its reliability as an alternative reference for sunspot number recalibration.

Iterative Construction of the Optimal Sunspot-Number Series

The relative number of sunspots represents the longest evidence describing the level of solar activity. As such, its use goes beyond solar physics, e.g. towards climate research. The construction of a single representative series is a delicate task which involves a combination of observation of many observers. We propose a new iterative algorithm that allows to construct a target series of relative sunspot number of a hypothetical stable observer by optimally combining series obtained by many observers. We show that our methodology provides us with results that are comparable with recent reconstructions of both sunspot number and group number. Furthermore, the methodology accounts for the possible non-solar changes of observers' time series such as gradually changing observing conditions or slow change in the observers vision. It also provides us with reconstruction uncertainties. We apply the methodology to a limited sample of observations by \v{C}ESLOPOL network and discuss its properties and limitations.

Estimation methods for stationary Gegenbauer processes

This paper reviews alternative methods for estimation for stationary Gegenbauer processes specifically, as distinct from the more general long memory models. A short set of Monte Carlo simulations is used to compare the accuracy of these methods. The conclusion found is that a Bayesian technique results in the highest accuracy. The paper is completed with an examination of the SILSO Sunspot Number series as collated by the Royal Observatory of Belgium.

Chinese Sunspot Drawings and Their Digitizations-(VI) Extreme Value Theory Applied to the Sunspot Number Series from the Purple Mountain Observatory

Based on the daily sunspot number (SN) data (1954–2011) from the Purple Mountain Observatory, the extreme value theory (EVT) is employed for the research of the long-term solar activity. It is the first time that the EVT is applied on the Chinese SN. Two methods are used for the research of the extreme events with EVT. One method is the block maxima (BM) approach, which picks the maximum SN value of each block. Another one is the peaks-over-threshold (POT) approach. After a declustering process, a threshold value (here it is 300) is set to pick the extreme values. The negative shape parameters are obtained by the two methods, respectively, indicating that there is an upper bound for the extreme SN value. Only one value of the N-year return level (RL) is estimated: N = 19 years. For N = 19 years, the RL values of SN obtained by two methods are similar with each other. The RL values are found to be 420 for the POT method and the BM method. Here, the trend of 25th solar cycle is predicted to be stronger, indicating that the length of meridional forms of atmospheric circulation will be increased.

El Ni\xf1o Modoki can be mostly predicted more than 10 years ahead of time

The 2014–2015 “Monster”/“Super” El Niño failed to be predicted one year earlier due to the growing importance of a new type of El Niño, El Niño Modoki, which reportedly has much lower forecast skill with the classical models. In this study, we show that, so far as of today, this new El Niño actually can be mostly predicted at a lead time of more than 10 years. This is achieved through tracing the predictability source with an information flow-based causality analysis, which has been rigorously established from first principles during the past 16 years (e.g., Liang in Phys Rev E 94:052201, 2016). We show that the information flowing from the solar activity 45 years ago to the sea surface temperature results in a causal structure resembling the El Niño Modoki mode. Based on this, a multidimensional system is constructed out of the sunspot number series with time delays of 22–50 years. The first 25 principal components are then taken as the predictors to fulfill the prediction, which through causal AI based on the Liang–Kleeman information flow reproduces rather accurately the events thus far 12 years in advance.

Reconstruction of the Sunspot Number Source Database and the 1947 Zurich Discontinuity

The recalibration of the sunspot number series, the primary long-term record of the solar cycle, requires the recovery of the entire collection of raw sunspot counts collected by the Zurich Observatory for the production of this index between 1849 and 1980.Here, we report about the major progresses accomplished recently in the construction of this global digital sunspot number database, and we derive global statistics of all the individual observers and professional observatories who provided sunspot data over more than 130 years.First, we can announce the full recovery of long-lost source-data tables covering the last 34 years between 1945 and 1979, and we describe the unique information available in those tables. We then also retrace the evolution of the core observing team in Zurich and of the auxiliary stations. In 1947, we find a major disruption in the composition of both the Zurich team and the international network of auxiliary stations.This sharp transition is unique in the history of the Zurich Observatory and coincides with the main scale-jump found in the original Zurich sunspot number series, the so-called “Waldmeier” jump. This adds key historical evidence explaining why methodological changes introduced progressively in the early 20th century could play a role precisely at that time. We conclude on the remaining steps needed to fully complete this new sunspot data resource.

A Modern Reconstruction of Richard Carrington’s Observations (1853–1861)

The focus of this article is a re-count of Richard Carrington’s original sunspot observations from his book drawings (Carrington in Observations of the Spots on the Sun from November 9, 1853, to March 24, 1861 Made at Redhill, Williams and Norgate, London, 1863) by an observer from the World Data Center-SILSO (WDC-SILSO, http://www.sidc.be/silso/home ) network, Thomas H. Teague (UK). This modern re-count will enable the recomputation of the entire Sunspot Number series in a way Carrington’s original counts (Casas and Vaquero in Solar Phys. 289(1), 79, 2014) did not. Here we present comparison studies of the new re-counted series with contemporary observations, new data extracted from the Journals of the Zürich Observatory and other sources of Carrington’s own observations and conclude that Carrington’s group counting is very close to the modern way of counting while his method for counting individual spots lags significantly behind modern counts. We also test the quality and robustness of the new recount with methods developed in Mathieu et al. (Astrophys. J. 886(1), 7, 2019).

Time-Reversibility, Causality and Compression-Complexity.

Detection of the temporal reversibility of a given process is an interesting time series analysis scheme that enables the useful characterisation of processes and offers an insight into the underlying processes generating the time series. Reversibility detection measures have been widely employed in the study of ecological, epidemiological and physiological time series. Further, the time reversal of given data provides a promising tool for analysis of causality measures as well as studying the causal properties of processes. In this work, the recently proposed Compression-Complexity Causality (CCC) measure (by the authors) is shown to be free of the assumption that the "cause precedes the effect", making it a promising tool for causal analysis of reversible processes. CCC is a data-driven interventional measure of causality (second rung on the Ladder of Causation) that is based on Effort-to-Compress (ETC), a well-established robust method to characterize the complexity of time series for analysis and classification. For the detection of the temporal reversibility of processes, we propose a novel measure called the Compressive Potential based Asymmetry Measure. This asymmetry measure compares the probability of the occurrence of patterns at different scales between the forward-time and time-reversed process using ETC. We test the performance of the measure on a number of simulated processes and demonstrate its effectiveness in determining the asymmetry of real-world time series of sunspot numbers, digits of the transcedental number and heart interbeat interval variability.

On the Prediction of Solar Cycles

This article deals with the prediction of the upcoming solar activity cycle, Solar Cycle 25. We propose that astronomical ephemeris, specifically taken from the catalogs of aphelia of the four Jovian planets, could be drivers of variations in solar activity, represented by the series of sunspot numbers (SSN) from 1749 to 2020. We use singular spectrum analysis (SSA) to associate components with similar periods in the ephemeris and SSN. We determine the transfer function between the two data sets. We improve the match in successive steps: first with Jupiter only, then with the four Jovian planets and finally including commensurable periods of pairs and pairs of pairs of the Jovian planets (following Mörth and Schlamminger in Planetary Motion, Sunspots and Climate, Solar-Terrestrial Influences on Weather and Climate, 193, 1979). The transfer function can be applied to the ephemeris to predict future cycles. We test this with success using the “hindcast prediction” of Solar Cycles 21 to 24, using only data preceding these cycles, and by analyzing separately two 130 and 140 year-long halves of the original series. We conclude with a prediction of Solar Cycle 25 that can be compared to a dozen predictions by other authors: the maximum would occur in 2026.2 (± 1 yr) and reach an amplitude of 97.6 (± 7.8), similar to that of Solar Cycle 24, therefore sketching a new “Modern minimum”, following the Dalton and Gleissberg minima.

Robustness of Solar-Cycle Empirical Rules Across Different Series Including an Updated Active-Day Fraction (ADF) Sunspot Group Series

Empirical rules of solar-cycle evolution form important observational constraints for the solar-dynamo theory. This includes the Waldmeier rule relating the magnitude of a solar cycle to the length of its ascending phase, and the Gnevyshev–Ohl rule clustering cycles to pairs of an even-numbered cycle followed by a stronger odd-numbered cycle. These rules were established as based on the “classical” Wolf sunspot number series, which has been essentially revisited recently, with several revised sets released by the research community. Here we test the robustness of these empirical rules for different sunspot (group) series for the period 1749 – 1996, using four classical and revised international sunspot-number and group sunspot-number series. We also provide an update of the sunspot-group series based on the active-day fraction (ADF) method, using the new database of solar observations. We show that the Waldmeier rule is robust and independent of the exact sunspot (group) series: its classical and $n+1$ (relating the length of $n$ th cycle to the magnitude of ( $n+1$ )th cycle) formulations are significant or highly significant for all series, while its simplified formulation (relating the magnitude of a cycle to its full length) is insignificant for all series. The Gnevyshev–Ohl rule was found robust for all analyzed series for Solar Cycles 8 – 21, but unstable across the Dalton minimum and before it.

Anomalies of Shape of 11-year Solar Cycle in Sunspot Number Series

The shape of the 11-year cycle of solar activity can be described by a set of parameters that include its amplitude, the lengths of the ascending and descending branches and the relative position of the latter. Using the revised sunspot number series SN we demonstrate that large (more than 1.5 standard deviations) variations of those parameters are mainly sited in the first part of the series (from mid 18 to mid 19 centuries). Such an unbalanced distribution of the anomalies can be interpreted in two alternative ways. First, characteristics of the solar activity mechanism could really change over time. Second, and more probable, the first part of the series could contain errors that corrupted the shape of cycles.

Long–Term Variations in Solar Differential Rotation and Sunspot Activity, II: Differential Rotation Around the Maxima and Minima of Solar Cycles 12 – 24

Studies of variations in the solar differential rotation are important for understanding the underlying mechanism of solar cycle and other variations of solar activity. We analyzed the sunspot-group daily data that were reported by Greenwich Photoheliographic Results (GPR) during the period 1874 – 1976 and Debrecen Photoheliographic Data (DPD) during the period 1977 – 2017. We determined the equatorial rotation rate [ $A$ ] and the latitude gradient [ $B$ ] components of the solar differential rotation by fitting the data in each of the 3-year moving time intervals (3-year MTIs) successively shifted by one year during the period 1874 – 2017 to the standard law of differential rotation. The values of $A$ and $B$ around the years of maxima and minima of Solar Cycles 12 – 24 are obtained from the 3-year MTIs series of $A$ and $B$ and studied the long-term cycle-to-cycle modulations in these coefficients. Here we have used the epochs of the maxima and minima of the Solar Cycles 12 – 24 that were recently determined from the revised Version-2 international sunspot-number series. We find that there exits a considerably significant secular decreasing-trend in $A$ around the maxima of solar cycles. There exist no secular trends in both $A$ and $B$ around the minima of solar cycles. The secular trend in $B$ around the maxima of solar cycles is also found to be statistically insignificant. We fitted a cosine function to the values of $A$ , and also to those of $B$ , after removing the corresponding linear trends. The cosine-fits suggest that there exist ≈54-year (≈94-year) and ≈82-year (≈79-year) periodicities in $A$ ( $B$ ) around the maxima and minima of solar cycles, respectively. The amplitude of the cosine-profile of $A$ ( $B$ ) around the minima of solar cycles is about 41% (65%) larger than that of $A$ ( $B$ ) around the maxima. In addition, the cosine profiles of $A$ and $B$ suggest a large (up to $180^{\circ }$ ) phase difference between the long-term variations of $A$ , and also between those of $B$ , around maxima and minima of solar cycles. Implications of all these results are discussed briefly.

Predicting the Shape of Solar Cycle 25 Using a Similar-Cycle Method

A similar-cycle method is applied in predicting the shape of Solar Cycle 25, through a more scientific definition to select similar cycles. Using the current solar minimum $R_{\mathrm{min}}(25)$ as a reference, the six most similar cycles to Solar Cycle 25 are found to be Cycles 24, 15, 12, 14, 17, and 10 (in that order). The monthly values of sunspot-number series for the whole of Cycle 25 are predicted by weighted averaging the corresponding ones in the six similar cycles. As a result, Solar Cycle 25 is predicted to peak around October 2024 with an amplitude of about $R_{\mathrm{m}}=137.8\pm 31.3$ and to end around September 2030. As a by-product, there might be a secondary peak eight months earlier. The similar-cycle method considers only the solar cycles with similar parameters rather than all ones as for regression methods. It has an advantage that it does not depend so much on the accuracy of the observation.

Randomness in Sunspot Number: A Clue to Predict Solar Cycle 25

Forecasting of solar activity is extremely important, due to its impact on our space-based technology. In recent years, we have observed a continuous decline in the peak sunspot number for Solar Cycles (SCs) 21 to 24. We are more curious about peak activity of SC 25 because if the solar activity weakens further, then it may be an indication of new extended minima. So far, the daily sunspot-number data make up the longest observational series of the solar activity available, and recently this has been replaced with the corrected new Version 2.0 sunspot-number series. In general, different prediction models are available to forecast the peak smoothed sunspot number (SSN) of the upcoming SC but they are based on the older Version 1.0 sunspot data. Therefore, it is necessary to check the applicability of earlier proposed models to predict the peak SSN using the Version 2.0 sunspot-number series. Here, we re-evaluate one such earlier proposed prediction model, which is based on the estimates of the Shannon entropy, a measure of randomness, for Version 2.0 sunspot data. We find that this prediction model is applicable to the Version 2.0 sunspot-number series. To verify the robustness of the prediction model, we used the histogram and additionally the kernel density estimator (KDE) method to calculate the probability distribution function (PDF). The estimate of the PDF is a prerequisite to compute the Shannon entropy. We found that the prediction model is robust and the correlation coefficients between model parameters are 0.93 and 0.92, respectively, for these two approaches. This exercise provides a new prediction model for the peak SSN based on the Version 2.0 sunspot-number series. The model forecasts the peak smoothed sunspot number of $136.9\pm 24$ using a histogram-derived PDF and $150.7\pm 25$ using a KDE-derived PDF for the upcoming SC 25. These predictions for SC 25 are more reliable as up to date (December 2020) sunspot-number data have been utilized to get the entropy in the end phase of SC 24. It suggests that SC 25 would be similar or slightly stronger than SC 24.

EMD and LSTM Hybrid Deep Learning Model for Predicting Sunspot Number Time Series with a Cyclic Pattern

The prediction of a time series such as climate indices and the sunspot number (SSN) with long-term oscillatory behaviors has been a challenging task due to the complex combination of oscillations. Frequency extraction algorithms have been developed to separate a time series into different oscillation components according to frequency, such as empirical model decomposition (EMD) and wavelet analysis. In the current study, the deep learning long short-term memory (LSTM) model was employed to predict the oscillation components extracted using EMD. The SSN series was modeled with the hybrid EMD-LSTM model. The simulation study results indicate that the LSTM model reproduces the smooth cyclic pattern of the sine function, and only a few hidden units are needed to model it. The EMD-LSTM model achieves better performance than does the LSTM model for mid-range SSN predictions while the LSTM achieves better performance within the first few time lags. However, the cyclic prediction of the SSN requires mid-range lags; thus, the superior performance of the EMD-LSTM model for these lags cannot be ignored. Furthermore, the remaining components from the significant EMD signals can be modeled to reveal the variability (or uncertainty) in the prediction. The summed residual is fitted by k-nearest neighbor resampling. The final SSN prediction results show that the EMD-LSTM model predicts a later and larger SSN for Solar Cycle 25 than does the LSTM model. Overall, the results lead to the conclusion that the EMD-LSTM model might be a suitable alternative for modeling complex sunspot time series with cyclic patterns.

Relationship between Sunspot Numbers and Mean Annual Precipitation: Application of Cross-Wavelet Transform—A Case Study

Observations show that the Sun, which is the primary source of energy for the Earth’s climate system, is a variable star. In order to understand the influence of solar variability on the Earth’s climate, knowledge of solar variability and solar–terrestrial interactions is required. Knowledge of the Sun’s cyclic behavior can be used for future prediction purposes on Earth. In this study, the possible connection between sunspot numbers (SSN) as a proxy for the 11-year solar cycle and mean annual precipitation (MAP) in Iran were investigated, with the motivation of contributing to the controversial issue of the relationship between SSN and MAP. Nine locations throughout Iran were selected, representing different climatic conditions in the country. Cross-wavelet transform (XWT) analysis was employed to investigate the temporal relationship between cyclicities in SSN and MAP. Results indicated that a distinct 8–12-year correlation exists between the two time series of SSN and MAP, and peaks in precipitation mostly occur one to three years after the SSN maxima. The findings of this study can be beneficial for policymakers, to consider future potential droughts and wet years based on sunspot activities, so that water resources can be more properly managed.

Duffing Oscillator Model of Solar Cycles

Abstract We propose that the solar cycle variability could be described in the framework of an external quasi-sinusoidal influence on an oscillator with cubic nonlinearity and linear damping (Duffing oscillator). To demonstrate this, we compare the empirical amplitude–frequency dependence with the theoretical one obtained by the Krylov–Bogolyubov averaging method. The empirical data are a composite time series of 2.0 version of sunspot number series, which starts in 1700, and the sunspot group number series by Svalgaard & Schatten, scaled to sunspot number, for 1610–1699 interval. We find that while this interpretation of solar cycle is a mathematical approximation, it explains several properties of solar cycle variability.

Sunspot observations by Hisako Koyama: 1945–1996

ABSTRACT Sunspot records are the only observational tracer of solar activity that provides a fundamental, multicentury reference. Its homogeneity has been largely maintained with a succession of long-duration visual observers. In this article, we examine observations of one of the primary reference sunspot observers, Hisako Koyama. By consulting original archives of the National Museum of Nature and Science of Japan (hereafter, NMNS), we retrace the main steps of her solar-observing career, from 1945 to 1996. We also present the reconstruction of a full digital data base of her sunspot observations at the NMNS, with her original drawings and logbooks. Here, we extend the availability of her observational data from 1947–1984 to 1945–1996. Comparisons with the international sunspot number (Version 2) and with the group sunspot number series show a good global stability of Koyama's observations, with only temporary fluctuations over the main interval 1947–1982. Identifying drawings made by alternate observers throughout the series, we find that a single downward baseline shift in the record coincides with the partial contribution of replacement observers mostly after 1983. We determine the correction factor to bring the second part (1983–1996) to the same scale with Koyama's main interval (1947–1982). We find a downward jump by 9 per cent after 1983, which then remains stable until 1995. Overall, the high quality of Koyama's observations with her life-long dedication leaves a lasting legacy of this exceptional personal achievement. With this comprehensive recovery, we now make the totality of this legacy directly accessible and exploitable for future research.