Wavelet analysis of sunspot relative numbers

NOTESWavelet analysis of sunspotsmoothed SN maximum is around the spring of 2000, andpredicted that the date of the maximum of monthlyrelative numberssmoothed SN in 3 months is later than that of the former.The date is actually in July, 2000. Although the error ofHAN Yanben' & HAN Yonggang:2P| Athe prediction is not large; lhe advance time of the predic-tion is only about 2 yearslo. At present, great allention wasNational Astronomical Observatories, Chinese Academy of Sciences,paid to predicting SN by using the geomagnetic aa-index.Beijing 100012, China;Obridko reported that the relationship between aa indexDeparment of Applied Mathematics, Beiing Polytechnic University,and the variation of SN at the descending branch of theBejing 100022. Chinasolar cycle is close. He found that the maximum of 23rdCorrespondence should be adressed to Han Yanben (e-mail: hyb@bao.cycle is (203.2土10.7) while with the method based on theac.cn)ratio of the even-odd cycles. the value is (74.7土6.9) ac-Abstract The time series of the monthly smoothed sun.cording to his improved method'. For the contradictoryspot pumbers in 1749- 20000 is analyzed with the wavelet. resul, he thought that a cerain anomaly might be in theThe result shows that besides the known time-variation of23rd solar cycle. Therefore, we believe that the variationthe period about 11 years, other main periods of the sunspot regularity and the prediction method of the solar activitynumbers, such as the periods of about 100 years and so on, need a thorough study. The purpose of this work is tovary with time. We suggest that the time-variation of theanalyze the time series of monthly smoothed SN furthermais periods is the manifestation of the complex variation of and discuss the time- variation of some periodic compo-sunspot numbers. It is significant to make a thorough study nents.of the character and mecbanism of the time-variation of the1 Complicated variation of sunspot numbersperiods for proving prediction of sunspot numbers, espe-cally for understanding the variation process of sunspotThe data of monthy SN that started in 1749 are con-numbers.sidered as a systematic time series. The preliminary inves-Keywords: sunspot numbers, predlction, wavelet, time-variation.tigation of the series shows that the change exists in themaximum peaks and lhe length of the solar cycles. ForSunspots are considered as the main part of the solar example, the maximum of the monthly smoothed SN inactiviy. The variation of the sunspot number (SN) is al_ 19th cycle is 2013, but that of 6th cycle is only 48.7. Theways uaken as an impotant way to indicate the activit. range of the cycle length lies from about 8.6 l0 17.5 yearsSimultaneously, it has been found that the solar activityif the length is measured according to the interval betweenpossibly influences some changes and anomalous phe- the two maxima, or it ranges from about 99 to 12.5 yearsnomena of the slreresrial environment, climate, if it is measured acording t0 the minimas!. Fris- Chris-weather, seismicity, and so on!. So the studies on thetensen et al. investigated the variation of the cycle lengthsolar activity including SN and pediction are paid great in the study of the rlaionship berween the length and theatentin. In the studies, the 11-year-period was found and atnospheric temperature!.confirmed and some periodic components were obtained.There are some periodic components in the SN seriesHowever, it is dffcult to predict SN exaculy by the which are obtained through the spectrum analysis. such asperiodic parameters and some methods have sill been the periods ofabout 11 years, 100yearsand so on.frequently used at present. For example, many predictions Someone thought that there is a period of about 200aboul the maximum of monuhly smoothed SN for the 23rd yers!. However, some dferences exist in parameters ofsolar eycle are quite dfferent from the observed value periods that are obtained from data in dfferenet time iner-.later. 120.8. A panel of the NOAA Space Environment vals. This showed that lhe variation of SN is very complex.Center (SEC) with 12 scientists commented 28 predictionsand the varation is nonlinear'.derived from 6 kinds of techniques. The predications areOchadlich et al. analyzed the series with the nonlin-distributed in a large range, (115+ 40)-(200土35), and ear character using the wavelet and dscussed the variationthe ideal resut was thoughi i0 be abou (160士30). The of the cycle lnghl"!. Frick et al. studied the time seriesforecasted span of the appearance date of the maximum, of sunspot groups during 1610- 1994 by the wavelet andfrom January 1999 to June 2001, was also lager". The found that the weak solar activity is related to long solarobserved resut showed that the dale is in Apri of 2000 cycles!]. Feng et al. analyzed the yearly mean SN inWang gave a btter prediction, (119+ 30), but tbere isan 1700- 1993, and the resul showed that the range of theerror of one year on the dale prediction of the maximumi. cycle length is 9-15 years and the highest ocuensZhang et al. provided a prediction method about2 years are found to be asocialed with the 10 and 11 years.ahead of the maximum', their result is(115.4 14.9). Fligeg et al. analyzed several data sets by the MoreOne of the authors predicted that the date of the monhly wavelet, such as the Zurich relative SN, SN. the group1) Joselyn, J. A. Anderson. J. Coffeg. H. et al, Solar Cycle 23 prijet: summary of panal findings. 1996. 11. 8 al ht/:/./.wa.../nforCycle 23. html.中国煤化工Chinese Science Bulletin Vol. 47 No.7 April 2002609MHCNMHGNOTESsunspot number, the sunspot area, the facular area, Caseries of the monthly smoothed SN in 1749- 2000 isplage area, and "Be records. By combining the resultsshown in fig. 2. The upper part of the figure is the originalwith all indicators, a composite of the cycle length wassignal and the wavelet transformation map is sel inconstrucled duning the last 600 years. The result showedfig. 2(b) (see cover of this issue). Originally. the verticalthat the activity period around 1I years is strong duringaxis of the wavelet map denotes the frequency. Forthe last 50 years. They used the Zurich relative SN to ob-reader's convenience, the vertical axis has been displayedtain a mean period of 10.7 years, and the values derivedby the period values. The analyzed periodic range is fromfrom other proxies lie from about 10.5 10 10.7 years!4!.5 to 3000 months.Liao et al. considered that the I1-year-period for SN seriesin 1818- 1999 is constituted by two periods, 11.3 and10.3 years.8].GaussianCosineMany authors paid atention to the study of thevariation of the main period about 11 years. However, theactual variation of the amplitude and the length of thesolar cycles is also influenced by the century period(about00 years) and other periods. It seems that thevariation of other periods has not been fully studied al-though the variation of the 158 day-period has been in-vestigated by the wavelet's. In this nole. the variation ofthe periods, from about I to 200 years, will be analyzedwith he wavelet.2 Wavelet analysis of sunspot numbersHig. 1. Askech mnap of the wavclel.The parameters of periodic components of a timeseries are easily obtained by the spectrum analysis or3 Time-variation in periodic components of sunspotsimilar methods. Generally, the results can only show thenumberscharacter of the signal in the frequency domain, bul not inthe time-frequency domain. Especially for the time seriesIt can be seen from fig. 2 that the character of thewith shaky periodic components, the variation of peridictime and frequency (period) of the original signal com-components cannot be found by the spectrum analysis.bined with each other. The change of the gray shadow ofCompared with the spectrum analysis, the wavelet trans-the blocks with different shapes in the map denotes theformation is a mathematical method with some advan-time-variation of the energy density. In the cover, the al-tages. The method is able to display lots of information oftemation between the red and blue denotes that betweena time series because it can simultaneously show the localthe peak and valley of the energy density. respectively.character of the signal in the time domain and frequencyThe color shadow displays the relative variation of thedomainI6.7.energy density.In general, an unstable time series of the signal in-In order 10 display the fluctuation of the period morecludes some periodic components from the low to the highclearly, the contour line map of the wavelet transformationfrequency and noise possibly. In order to determine itsis also drawn in fig. 3. Both kinds of blocks encircled byinformation well, we need a relatively narTow time-win-the solid and dotted lines express the places and intensitydow so as t0 ensure he precision for the high frequency,of he positive and negative energy densities of the peri-and a relatively wide time-window for acquiring the com-ods, respectively. There are 30 contour lines from theplete information of the low frequency. The wavelet pro-minimum to the maximum energy (15 for the positive andvides such a time-frequency window that it will auto-I5 for the negative energy. respectively). The dashed linematically become narrow for the high frequency, or willis the boundary of both kinds of blocks. The contour linesbecome wide for the low frequency. In the waveletclose to the center express the much larger energy densilytransformation, it is important to select a ft mother wave-in every block. The oumber of the lines becomes lesslet y() in order to show more significant information ofwhen the energy (amplitude) gets smaller, and vice versa.the signal. For the 1-D wavelet transformation, the motherThey show the change of the dense energy distribution. i.e.wavelet should satisfy the "admissible condition'":the periods of the montly smoothed SN are changingwith time,厂V()=d1=0. The wavelet we selected is shown inFigs. 2 and 3 show that almost none of periods of themonthly smoothed SN are stable. The variation of the pe-fig. 1. it is the inner product of the Gaussian and cosineriodic length and the energy density is obvious. Some-function.times. the energy density becomes too small to fit the fig-The I-D wavelet transformation result for the timeures (or does not exist at all). However. such periods中国煤化工610MYHCNMHGNo.7 April 2002NOTES250r200eg100816.3268512-;102420481745 1765 1785 1805 1825 1845 1865 1885 1905 1925 1945 1965 1985 2005YearFig.2. 1-D wavelet rastormation of the montly smoothed ss in 1749- 2000.0 (0) Original signal; (0) waveler mup.250150|100-.16-256”17451768 1785 1805 1825 1845 1865 1885 1905 1925 1945 1968 1985 2005ng.3. Contour map of the 1-D wavelct trunsformation ol the monthly smothedSN in 1749- 200中国煤化工Chinese Science Bulletin Vol. 47 No.7 April 2002MHCNMHGNOTEScould be found through he spectrum analysis since their tha the solar activity is very complicated. And this is alsoenergy is relatively strong at that time; but sometimes,the main reason why the different periodic parameters aretheir amplitude must be smaller. Such periods are thusderived from the time series in distinct ime. At present,called semi-periods.however, the reason why the SN variation is so complexIn the wavelet maps, there are periods about 1 and 2has nol yet been fully understood. Especially, the time-years and more in the high frequency area. But these pe-variation of some periods leads 1o nonlinearity which af-riods are not notable and with only very small energy. Infects the precision of the prediction for the future solarthe area between the high and the middie frequency. there cycle. It is significant to make a thorough study on theis a period about 5 years, which is obvious when the solar properties and mechanisms of the time varaion of thecycle maximum is very large. There also exists a period periods for understanding the variation reguarly and theabout 8 years, but the length and the energy density areprediction of SN more precisely.different in the distinct time. For example, the period isThe result of this note shows that the wavelet analy-ambiguous around the years 1810 and 1900 when the peak sis is able to display the significant information for thevalue of the sunspot number is smal. The next location is kind of time series including periodic components withthe period abou 11 years, it will be explained in the fol-time-variation, such as the series of SN.lowing part separately. For another part that is near theAcknowldgements The authors would like to thank Drs. Luo Ali,low frequency, there are several areas where the energy isZhao Juan for their kindy discussions about this note. This work wascentalied, and they crrespond t0 the periods about 21 sppote by lhe Nainal Natural Science Foundation or China (Grantand 29 years, respectively. The 2l-year-period is stronger Nos. 19973011 and 19833030).around the years 1800, 1870 and 1960, but the weak ap-Referencespeared around the year 1920. The variation of the periodabout 29 years is almost similar to that of the 21-year-1. Herman, J. R. Goldberg. R. A.. Sun, Wieather and Clmate. Wash.ineton:NASA. 1978period. There are also the centralized areas of the energyFris-Chnistensen, E.FisChisense,n E. Lsse. K. Lengh of the soiar cycle. An in.corresponding l0 the period about 40 years, but the energydicalor of solar activiy closly ascited with chmale. Science,density is small. In the low frequency area, the period ofabout 100 years (ccntury period) is displayed. Only theButler, C. J.. JcC儿Jobmston, !D. J.. A provisional long mean air tem-peralure series for Amagh Observatory. J. Atmospheric and Ter-:two areas of the negative energy and three of the positiveenergy exist in the figure since the length of the SN seriesWang, J. L A prediction of solar activities in the later phase ofis only about 250 years in the note. Before the middle ofsolar cycle 22 and in solar cycle 23. Acta Astrophysica Sunica (m19th century, the periodic length is shorter lhan 100 years.Chinese). 1992, 1214): 369..Zhang, G Q.. Wang. H. N.. Prediction of maximum sunspot num-Then the length became longer and even longer than 100ber in solar cycle 23, Solar Physics. 199. 188: 397.years but the energy became smaller. The indication ofs. Han, Y. B.. A methoxd to predict ampliude and date of maximumabout 200-year-period is also displayed in the figure andof solar cycle, Chinese Science Bulletin. 2000. 45( I4): 1287.two centralized areas of the negative energy are around， Obridko V. N. Some comments on the prublem of solar cycleprediction. Solar Physics. 1995, 156(1): 179.the years 1780 and 1880. Its energy density is obviously8. Liao. D. C. Liao, X. H, New endence for pssible impact of so.smaller than that of the period about 100 years.lar activity om long-term fluctuation of the earth rotation, ChineseOf course, the period about 11 years in the middleScience Bullein, 2001. 46(| 1); 905.frequency area of the figure is most interesting. Its energyRomanov.Yu, S.,. 7gonyaiko. N. S.. The periodcity o[ solur eycles.is the largest in all periods. Although the period is consid-Solar Physis.1994, 152: 31.ered 10 include two peaks, 11.3 and 10.3 years', theyI0. Serre, T. Nesme -Ribes, E.. Nonlinear analyis of solar cycles.cannot be divided in the figure since the time-windows are11. Ochadick. A. R." Kitikos, H. N. Giegengack. R., Vanations invery close and the amplitude of one of the peaks is small.the period of the suspot cycle, Geophysical Research Lters.In the figure, the centralized energy area varies with time1993, 2014): 147.obviously. The area moved t0 the location where the12. Frick, P. Galyagin, D.. Hoyt, D. V. et al. Wavelet unalysis of solaractivity recorded by sunspot grvups, Astron. and Astruphy. 1997.shorter length of the period is involved when the solarcycle maximum is very large, and vice versa. For example,13. Feng, B.. Ke, X. Z, Ding, H. .... Waveler analysis ot sumspouin several decades around the year 1770 (about 1750一numbers, Chin, Astron. Asrophys. (in Chinese). 1998, 2211): 83.1780), the large energy density is located in the area14. Fligge. M. Solanki, s. K. Beer, J.. Determinaton of solar cyelelength variations using the continuous wavelet transform, Astron.where the period is smaller than 11 years but longer than11 years during 1800- -1830. The variation character is 15. Baleser, 1 L. Oliver R. Baudin F. Disovery ot the near 158similar to Fligge's resut!4.day periodicity in group sumspot numbers during the eigheenthcentury. Astrophysical J.. 1999. S22: LI53.4 Conclusion and discussion6. Meyer, Y. Wavelets and Operators. Cambridge. Cambridge Uni.versity Press, 1992.The result of the wavelet analysis of the monthly17. Xu. K.. Xu,J. W. Ban. X. J.. Forcasting of some non sationarysmoothed relative SN shows that the series includes sometime senes based on wavelet decomposition. Acta Electronicaperiodic components. The length and energy density ofSinica (in Chinese), 2001. 2914): 566.some main components change with time. This expresses(Received Dcember 29, 2001)中国煤化工612YHCNMHGNo.7 April 202