Monthly Sunspot Number Research Articles

An optimal method for determining the time of the minimum of the 23rd solar cycle, July 2008

The time of the minimum of the solar cycle is usually determined by the minimum of the average monthly sunspot number smoothed over 13 months, i.e., with a large time delay, not earlier than eight months after the event. A new optimal method which allows one to establish the time of the minimum of the cycle as early as four months after the event is proposed. In the new method, the indicator of the time of a cycle minimum is the time of reaching the minimum level of average monthly values of the solar constant after which four succeeding values of this constant are larger than the pre-ceding minimum level. It is shown that the minimum of the past 23rd solar cycle took place in July, and the new 24th cycle started in August 2008.

Predicting the Amplitude of a Solar Cycle Using the North – South Asymmetry in the Previous Cycle: II. An Improved Prediction for Solar Cycle 24

Recently, using Greenwich and Solar Optical Observing Network sunspot group data during the period 1874-2006, (Javaraiah, MNRAS, 377, L34, 2007: Paper I), has found that: (1) the sum of the areas of the sunspot groups in 0-10 deg latitude interval of the Sun's northern hemisphere and in the time-interval of -1.35 year to +2.15 year from the time of the preceding minimum of a solar cycle n correlates well (corr. coeff. r=0.947) with the amplitude (maximum of the smoothed monthly sunspot number) of the next cycle n+1. (2) The sum of the areas of the spot groups in 0-10 deg latitude interval of the southern hemisphere and in the time-interval of 1.0 year to 1.75 year just after the time of the maximum of the cycle n correlates very well (r=0.966) with the amplitude of cycle n+1. Using these relations, (1) and (2), the values 112 + or - 13 and 74 + or -10, respectively, were predicted in Paper I for the amplitude of the upcoming cycle 24. Here we found that in case of (1), the north-south asymmetry in the area sum of a cycle n also has a relationship, say (3), with the amplitude of cycle n+1, which is similar to (1) but more statistically significant (r=0.968) like (2). By using (3) it is possible to predict the amplitude of a cycle with a better accuracy by about 13 years in advance, and we get 103 + or -10 for the amplitude of the upcoming cycle 24. However, we found a similar but a more statistically significant (r=0.983) relationship, say (4), by using the sum of the area sum used in (2) and the north-south difference used in (3). By using (4) it is possible to predict the amplitude of a cycle by about 9 years in advance with a high accuracy and we get 87 + or - 7 for the amplitude of cycle 24.

Modeling of Sunspot Numbers by a Modified Binary Mixture of Laplace Distribution Functions

This paper presents a new approach for describing the shape of 11-year sunspot cycles by considering the monthly averaged values. This paper also brings out a prediction model based on the analysis of 22 sunspot cycles from the year 1749 onward. It is found that the shape of the sunspot cycles with monthly averaged values can be described by a functional form of modified binary mixture of Laplace density functions, modified suitably by introducing two additional parameters in the standard functional form. The six parameters, namely two locations, two scales, and two area parameters, characterize this model. The nature of the estimated parameters for the sunspot cycles from 1749 onward has been analyzed and finally we arrived at a sufficient set of the parameters for the proposed model. It is seen that this model picks up the sunspot peaks more closely than any other model without losing the match at other places at the same time. The goodness of fit for the proposed model is also computed with the Hathaway – Wilson – Reichmann \(\overline{\chi}\) measure, which shows, on average, that the fitted model passes within 0.47 standard deviations of the actual averaged monthly sunspot numbers.

Improved Subset Autoregression: WithRPackage

The FitAR R (R Development Core Team 2008) package that is available on the Comprehensive R Archive Network is described. This package provides a comprehensive approach to fitting autoregressive and subset autoregressive time series. For long time series with complicated autocorrelation behavior, such as the monthly sunspot numbers, subset autoregression may prove more feasible and/or parsimonious than using AR or ARMA models. The two principal functions in this package are SelectModel and FitAR for automatic model selection and model fitting respectively. In addition to the regular autoregressive model and the usual subset autoregressive models (Tong 1977), these functions implement a new family of models. This new family of subset autoregressive models is obtained by using the partial autocorrelations as parameters and then selecting a subset of these parameters. Further properties and results for these models are discussed in McLeod and Zhang (2006). The advantages of this approach are that not only is an efficient algorithm for exact maximum likelihood implemented but that efficient methods are derived for selecting high-order subset models that may occur in massive datasets containing long time series. A new improved extended {BIC} criterion, {UBIC}, developed by Chen and Chen (2008) is implemented for subset model selection. A complete suite of model building functions for each of the three types of autoregressive models described above are included in the package. The package includes functions for time series plots, diagnostic testing and plotting, bootstrapping, simulation, forecasting, Box-Cox analysis, spectral density estimation and other useful time series procedures. As well as methods for standard generic functions including print, plot, predict and others, some new generic functions and methods are supplied that make it easier to work with the output from FitAR for bootstrapping, simulation, spectral density estimation and Box-Cox analysis.

Computational Forecasting of Wavelet-converted Monthly Sunspot Numbers

Monthly average sunspot numbers follow irregular cycles with complex nonlinear dynamics. Statistical linear models constructed to forecast them are therefore inappropriate, while nonlinear models produce solutions sensitive to initial conditions. Two computational techniques – neural networks and genetic programming – that have their advantages are applied instead to the monthly numbers and their wavelet-transformed and wavelet-denoised series. The objective is to determine if modeling wavelet-conversions produces better forecasts than those from modeling series' observed values. Because sunspot numbers are indicators of geomagnetic activity their forecast is important. Geomagnetic storms endanger satellites and disrupt communications and power systems on Earth.

태양주기별 흑점수의 프랙탈 차원과 최대흑점수의 상관관계

프랙탈 차원은 연속적으로 관측된 불규칙한 자료의 성질을 정량적으로 표현하는 파라메터이다. 본 연구에서는 프랙탈 차원을 이용하여 태양 흑점수의 불규칙한 변동을 분석하고 다가오는 태양주기의 최대흑점수를 예측하였다. 이를 위하여 우리는 SIDC(Solar Influences Data analysis Center)에서 제공하는 1850년부터 2004년까지의 일일흑점수를 이용하여 Higuchi의 방법으로 각 태양주기의 프랙탈 차원을 결정하고 같은 태양주기 동안의 최대 월평균 흑점수와 비교하였다. 그 결과 프랙탈 차원과 최대 월평균 흑점수는 강한 역비례 관계를 보였다. 이러한 관계를 이용하여 태양활동 극소기부터 활동이 증가되는 기간인 4년동안의 흑점수만을 이용하여 프랙탈 차원을 구하고 다가올 극대기의 최대 월평균 흑점수를 예측한 결과 관측된 최대흑점수와 0.89의 좋은 상관관계를 보였다. The fractal dimension is a quantitative parameter describing the characteristics of irregular time series. In this study, we use this parameter to analyze the irregular aspects of solar activity and to predict the maximum sunspot number in the following solar cycle by examining time series of the sunspot number. For this, we considered the daily sunspot number since 1850 from SIDC (Solar Influences Data analysis Center) and then estimated cycle variation of the fractal dimension by using Higuchi's method. We examined the relationship between this fractal dimension and the maximum monthly sunspot number in each solar cycle. As a result, we found that there is a strong inverse relationship between the fractal dimension and the maximum monthly sunspot number. By using this relation we predicted the maximum sunspot number in the solar cycle from the fractal dimension of the sunspot numbers during the solar activity increasing phase. The successful prediction is proven by a good correlation (r=0.89) between the observed and predicted maximum sunspot numbers in the solar cycles.

Relationship Between Solar Maximum Amplitude and Max-Max Cycle Length

The maximum amplitude of a solar activity cycle is found to be inversely correlated (r = -0.769) with the newly defined max-max cycle length two cycles earlier in a 13 month mean of monthly sunspot numbers. Meanwhile, a 14 cycle periodicity is found in the fitted residuals. The max-max cycle length can be used as one of the indicators to predict amplitudes. As a result, the amplitudes of cycles 24 and 25 are estimated to be 150.3 ± 22.4 and 102.6 ± 22.4, respectively, where the indicated error is the standard error.

High- And Low-Latitude Solar Activities: Who Leads Whom In Time Phase?

Using the data from observations of polar faculae by the National Astronomical Observatory of Japan from July 1951 to December 1998, we investigate whether there is a time lag between high-latitude solar activity and low-latitude solar activity. The cross-correlation analysis of the smoothed monthly numbers of the polar faculae with the smoothed monthly sunspot numbers shows that, high-latitude solar activity should lead low-latitude solar activity in time phase. The periodic characteristics of both of them also indicate that high-latitude activity evidently leads low-latitude activity.

High Flare Activity in the Late Declining Phase of Cycle 23

In the declining phase of the current solar cycle (23), a large number of major flares were produced. In this cycle, the monthly sunspot number continuously remained below 100 since October 2002. However, during four epochs since then, flare activity became very high. Compared to this, each of cycles 21 and 22 produced only one epoch of high activity in the declining phase. In the declining phase of cycle 20, similarly to this cycle, there were four epochs of high flare activity. During 2003 and 2004, the distribution of flare sizes measured in GOES classes was much harder (i.e., proportionately more energetic flares) than during the maximum years. Such pronounced hardening of the size distribution was not observed in the previous cycles. It is of theoretical interest to understand why some cycles are very active in the declining phase, and the high level of activity in the declining phase has practical implications for planning solar observations and forecasting space weather.

A review of the solar cycle length estimates

New estimates of the solar cycle length are calculated from an up‐to‐date monthly sunspot record using a novel but mathematically rigorous method involving multiple regression, Fourier approximation, and analytical expressions for the first derivative based on calculus techniques. The sensitivity of the estimates to smoothing are examined and the analysis is used to identify possible systematic changes in the sun. The solar cycle length analysis indicates a pronounced change in the sun around 1900, before which the estimates fluctuate strongly and after which the estimates show little variability. There have been speculations about an association between the solar cycle length and Earth's climate, however, the solar cycle length analysis does not follow Earth's global mean surface temperature. A further comparison with the monthly sunspot number, cosmic galactic rays and 10.7 cm absolute radio flux since 1950 gives no indication of a systematic trend in the level of solar activity that can explain the most recent global warming.

The influence of the 11 yr solar cycle on the interannual–centennial climate variability

The monthly sunspot number (SSN) for January 1749–August 2004, the global sea-surface temperature (gmSST) and the regional SSTs in the northern N. Pacific (npSST) and the Nino3.4 (ninoSST) areas for the winters of 1870–2004 are analyzed by a wavelet transform to show their multi-scale nature. On the interdecadal timescales, both gmSST and npSST have similar variation tendencies with that of the intensity and cycle-length of the 11 yr SSN, with slight phase differences. The npSST and ninoSST are often out of phase on the decadal–interdecadal timescales. The ninoSST is predominated by the interannual timescales peaking around 3.8 yr. Moreover, the ninoSST exhibits an apparent 80–90 yr signal that is almost out of phase with that observed in SSN. Numerical experiments using a simple nonlinear system illustrate that the intensity of the seasonal forcing, modulated by the 11 yr solar activity, is likely an important factor causing different dominant timescales in regional SSTs. Even a small change in the “solar constant” by 0.04% on the 11 yr timescale may result in a regime change in the response (e.g. SST) with various dominant timescales, including the 77 and 88 yr signals that are similar to those of the “Gleissberg cycle” in observed SSN. The results show that part of the energy of the internal variability of the system is transferred to the forced variability that may have richer timescales than those in the forcing itself due to nonlinear resonance. This suggests that observed interannual–centennial climate signals are not purely internal, but also external because of the existence of the 11 yr solar activity cycle, which has changed the “solar constant” in the past and will continue doing so in the future. It also suggests that if the solar “Gleissberg cycle” is included in the forcing term, the 77 and 88 yr interdecadal signals and their subharmonics on centennial timescales may be more significant than what is shown here, which might have some implication to “global warming” research.

WAVELETS IN MULTI-STEP-AHEAD FORECASTING

This paper investigates the possibility of obtaining long-into-the-future reliable forecasts of observed nonlinear cyclical phenomena. Unsmoothed monthly sunspot numbers that are characteristically cyclical with nonlinear dynamics as well as their wavelet-transformed and wavelet-denoised series are forecasted through October 2008. The objective is to determine whether modelling wavelet-conversions of a series provides reasonable forecasts. Two computational techniques –neural networks and genetic programming –are used to model the dynamics of the series. Statistical comparison of their ex post forecasts is then used to identify the data set and computational technique to use under the circumstances.

Reconstruction of Monthly and Yearly Group Sunspot Numbers From Sparse Daily Observations

Some periods before 1820 are poorly covered by sunspot observations. In addition to apparent, long observational gaps, there are also periods when there are only few sparse daily sunspot observations during a long time. It is important to estimate the reliability of the monthly and yearly mean sunspot values obtained from such sparse daily data. Here we suggest a new method to estimate the reliability of individual monthly means. The method is based on comparing the actual sparse data (sample population) to the well-measured sunspot data in 1850–1996 (reference population), and assumes that the statistical properties of sunspot activity remain similar throughout the entire period. For each sample population we first found those months in the reference population that contain the same data set, and constructed the statistical distribution of the corresponding monthly means. The mean and standard error of this distribution represent the mean and uncertainty of a monthly mean sunspot number reconstructed from sparse daily observations. The simple arithmetic mean of daily values can be adequately applied for months which contain more than 4–5 evenly distributed daily observations. However, the reliability of monthly means for less covered months has to be estimated more carefully. Using the estimated, new monthly values, we have also calculated the weighted annual sunspot numbers.

Normalization of sunspot cycles and eigen mode analysis

The smoothed monthly sunspot numbers of the previous 22 complete sunspot cycles are normalized in time domain, and then an eigen mode analysis is carried out to draw the principle factors (or components) in the cycles. The results show that the main characteristics of the solar cycles can be described fairly well by the first 5 eigen modes. The obtained eigen modes are used to predict the declining phase of cycle 23 on the basis of the data prior to its maximum. The prediction indicates that cycle 23 will last for 127 months to December 2006, with the minimum of 6.2.

Evolutions of various solar indices around sunspot maximum and sunspot minimum years

Abstract. The smoothed monthly sunspot numbers showed that in many solar cycles, (a) during years around sunspot maxima, there was only one prominent maximum, but in some cycles there was a broad plateau. If the beginning and end of these are termed as first and second maxima (separated by several months), the first maximum was generally the higher one, and the valley in between was very shallow. Solar indices at or near the photosphere generally showed similar structures with maxima matching with sunspot maxima within a month or two. Indices originating in the chromosphere and above showed two peaks in roughly the same months as sunspots (with some exceptions, notably the Coronal green line, and the Total Solar Irradiance). Yet often, the second maximum was larger than the first maximum, and the valley between the two maxima was deeper, as compared to sunspot maxima, and (b) during years around sunspot minima, the smoothed sunspot minimum could be sharp and distinct, lasting for a month or two, or could spread over several months. Among the indices originating at or near the photosphere, the Ca K line intensity showed good matching with sunspots, but the Ca Plage area, the Sunspot Group Area, and the solar magnetic fields seemed to show minima earlier than the sunspots, indicating that these activities died out first. These also showed recoveries from the minima later than sunspots. Most of the other indices originating in the chromosphere and corona attained minima coincident with sunspot minima, but in some cases, minima earlier than sunspots were seen, while in some other cases minima occurred after the sunspot minima. Thus, the energy dissipation in the upper part of the solar atmosphere sometimes lagged or led the evolution of sunspots near sunspot minimum. In a few cases, after the minimum, the indices recovered faster than the sunspots. In general, the chromospheric indices seemed to evolve similar to sunspots, but the evolution of coronal indices was not always similar to sunspots, and may differ considerably between themselves.Key words. Solar physics, astrophysics and astronomy (Corona and transition region; Magnetic fields; Photosphere and chromosphere)

Energy Balance in the Flaring Solar Corona

Assuming only that flares derive their energy from a coronal source and that flaring is the dominant mechanism for depleting that source, the global coronal response time (time for flares to remove available coronal energy) is about 9 months. A detailed model for dynamic energy balance in the solar corona over the solar cycle is presented to describe how the magnetic free energy in the solar corona varies in response to changes in the supply of energy to the system and to changes in the flaring rate. The model predicts that both the flaring rate and the free energy of the system should lag behind the driving of the system because of the coronal response time (the detailed model gives a lag of ~11 months). This effect may account for hysteresis phenomena between certain solar activity indices. A specific example is presented in support of the model. Analysis of time series of monthly sunspot numbers and monthly numbers of soft X-ray flares over the years 1976-1999 indicates a tendency for flare numbers to lag behind sunspot numbers by ~6 months.

A method to predict amplitude and date of maximum sunspot number

A method to predict the amplitude and date of the maximum sunspot number is introduced. The regression analysis of the relationship between the variation rate of monthly sunspot numbers in the initial stage of solar cycles and both of the maximum and the time-length of ascending period of the cycle showed that they are closely correlative. In general, the maximum will be larger and the ascending period will be shorter when the rate is larger. The rate of sunspot numbers in the initial 2 years of the 23rd cycle is thus analyzed based on these grounds and the maximum of the cycle is predicted. For the smoothed monthly sunspot numbers, the maximum will be about 139.2 ±18.8 and the time-length of ascending period will be about 3.31±0.42 years, that is to say, the maximum will appear around the spring of the year 2000. For the mean monthly ones, the maximum will be near 170.1±22.9 and the time-length of ascending period will be about 3.42 ±0.46 years, that is to say, the appearing date of the maximum will be later.

On the Solar‐Cycle Modulation of the Homestake Solar Neutrino Capture Rate and the Shuffle Test

There exists no significant correlation between the Homestake neutrino data up to run 133 and the monthly sunspot number, according to a test that is based on certain optimality properties for this type of problem. It is argued that priorly reported highly significant results for segments of the data are due to a statistical fallacy: the usual methods for evaluating the significance of common tests for correlation are not applicable in the sunspot-neutrino context. Moreover, an appropriate evaluation of these tests gives results that are compatible with the hypothesis of no correlation. Some new methods are introduced for assessing the significance of common measures of correlation in a time series setting, with a special emphasis on the Spearman rank correlation coefficient.

Is there memory in solar activity?

The Hurst effect is a presumed and unexpected behavior of geoastrophysical time series by which these time series have persistence or ``memory.'' The application of Hurst analysis to monthly sunspot numbers [B. B. Mandelbrot and J. R. Wallis, Water Resour. Res. 5, 321 (1969)] yielded a Hurst exponent $H=0.86\ifmmode\pm\else\textpm\fi{}0.05,$ suggesting that solar activity shows persistence and that the underlying responsible mechanism can guarantee a positive correlation of solar activity during long time lapses, raising, at the same time, the possibility of the existence of long-term memory in solar activity. More recently, radiocarbon data have been used for a similar study [A. Ruzmaikin, J. Feynmann, and P. Robinson, Sol. Phys. 149, 395 (1994)] resulting in a constant value $H=0.84$ between 100 and 3000 years, which indicates persistence of solar activity in such time scales. Furthermore, Mount Wilson rotation measurements have also been analyzed in the same way [R. W. Komm, Sol. Phys. 156, 17 (1996)] and the results indicate that temporal variations of solar rotation on time scales shorter than the 11-year cycle are caused by a stochastic process which is characterized by persistence. Here, we have followed the scale of fluctuation approach to show that there is no incontrovertible evidence for the presence of the Hurst effect in sunspot areas and, therefore, that there is no proof of the existence of long-term memory in solar activity.

Response of the intensity of subtropical high in the northern hemisphere to solar activity

Using the intensity data of each northern subtropical high measured by monthly 500 hPa height charts for the recent 38 years (1954–1991), we calculate their correlations with the monthly sunspot number and monthly solar radio flux at 10.7 cm wave length, respectively. Through strict test, we further confirm a series of high correlations. Next, using a method called the non-integer (year) wave, the significant response of each subtropical high’s intensity to solar activity at its main period of 10.9-year length is found. Special attention is paid to that of the eastern Pacific high, the possible mechanism of such sensible response is also analysed.