4.2 Smoothing

The monthly averages of the daily International Sunspot Number are noisy and must be smoothed in some manner in order to determine appropriate values for parameters such as minima, maxima, and their dates of occurrence. The daily values themselves are relatively uncertain. They depend upon the number and the quality of observations as well as the time of day when they are taken (the sunspot number changes over the course of the day as spots form and fade away). The monthly averages of these daily values are also problematic. The Sun rotates once in about 27-days but the months vary in length from 28 to 31 days. If the Sun is particularly active at one set of longitudes then some monthly averages will include one appearance of these active longitudes while other months will include two. This aspect is particularly important for investigations of short-term (months) variability (see Section 7). For long-term (years) variability this can be treated as noise and filtered out.

The traditional 13-month running mean (centered on a given month with equal weights for months –5 to +5 and half weight for months –6 and +6) is both simple and widely used but does a poor job of filtering out high frequency variations (although it is better than the simple 12-month average). Gaussian shaped filters are preferable because they have Gaussian shapes in the frequency domain and effectively remove high frequency variations (Hathaway et al., 1999Jump To The Next Citation Point). A tapered (to make the filter weights and their first derivatives vanish at the end points) Gaussian filter is given by

−t2∕2a2 −2 ( 2 2) W (t) = e − e 3 − t ∕2a (4 )
with
− 2a + 1 ≤ t ≤ +2a − 1 (5 )
where t is the time in months and 2a is the FWHM of the filter (note that this formula is slightly different than that given in Hathaway et al. (1999Jump To The Next Citation Point). There are significant variations in solar activity on time scales of one to three years (see Section 7). These variations can produce double peaked maxima which are filtered out by a 24-month Gaussian filter. The frequency responses of these filters are shown in Figure 21View Image.
View Image

Figure 21: Signal transmission for filters used to smooth monthly sunspot numbers. The 13-month running mean and the 12-month average pass significant fractions (as much as 20%) of signals with frequencies higher than 1/year. The 24-month FWHM Gaussian passes less than 0.3% of those frequencies and passes less than about 1% of the signal with frequencies of 1/2-years or higher.

Using the 24-month FWHM Gaussian filter on the data used to create Table 3 gives far more consistent results for both maxima and minima. The results for maxima are shown in Table 4. The ranges of dates for the last five maxima become: 1, 10, 13, 4, and 11 months – roughly half the ranges found using the 13-month running mean.


Table 4: Dates and values of maxima using the 24-month FWHM Gaussian with sunspot number data, sunspot area data, and 10.7 cm radio flux data as in Table 3.
Cycle
24-month Gaussian Maximum
24-Month Gaussian Sunspot Area
24-Month Gaussian 10.7 cm Flux
  Date Value Date R-Value Date R-Value
1 1761/05 72.9        
2 1770/01 100.5        
3 1778/09 137.4        
4 1788/03 130.6        
5 1804/06 45.7        
6 1816/08 43.8        
7 1829/10 67.1        
8 1837/04 146.9        
9 1848/06 115.7        
10 1860/03 92.1        
11 1870/11 138.5        
12 1883/11 64.7 1883/10 70.8    
13 1893/09 81.4 1893/09 84.7    
14 1906/05 59.6 1906/04 62.4    
15 1917/12 88.6 1918/01 79.6    
16 1927/12 71.6 1926/12 75.9    
17 1937/11 108.2 1938/02 118.1    
18 1948/03 141.7 1947/09 140.0    
19 1958/02 188.0 1958/03 192.0 1958/03 188.1
20 1969/03 106.6 1968/09 95.5 1969/07 104.6
21 1980/05 151.8 1981/06 140.2 1980/11 153.1
22 1990/02 149.2 1990/06 141.7 1990/06 156.1
23 2000/12 112.7 2001/11 106.2 2001/06 136.4


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