## 2. Basis function and cycle shape parametersFor our analysis, we use quarterly averages of the sunspot areas. They extend from the year 1875 to 1996 and consist of two parts having different reliability. The first part comes from the Greenwich Photoheliographic Results, extending from 1875 to 1971, but the graph given by White & Trotter (1976) was used as we have no primary digital data. Five points divided one year of the abscissa into four equidistance parts in the enlarged graph, then we regarded the average of coordinate values of adjacent two points as quarterly average of the corresponding time. The rest comes from sunspot observations of Yunnan Observatory (Hong & Wang, 1988). The Greenwich Photoheliographic Results are available until the year 1976, a comparison of the first part with the second was done for the overlapping period (1971 to 1976). Deviation between the two parts is less than , and for most values of sunspot areas it is less than . These quarterly averages are plotted in Fig. 1 for the entire interval (1875-1996).
Inspection of Fig. 1 reveals that individual cycles show a wide range of temporal behavior. For example, some cycles (like 12-16, and 20) are small in amplitude (about half the size of cycle 19), while others are considerably larger. Most cycles show substantial asymmetry with the rise to maximum being faster than the fall to minimum. Although the average cycle length (minimum to minimum) is about 11 years, individual cycles vary in length from about 9 to 14 years. All these properties are similar to the characteristics of the solar cycles described by the international sunspot relative numbers. Initiated by the work of Hathaway, et al. (1994), a function of the form is proposed to describe the shape of the sunspot area cycles, where
parameter
The correlation coefficient . This
effectively reduces again the number of parameters by one, leaving
only two. Then the two-parameter fit is attempted to be done for all
11 cycles. Table 1 also lists the best-fit parametric values for
the two-parameter function. Fig. 1 shows a comparison of the sunspot
area data and the two-parameter functional fit for the years 1875 to
1996 (cycles 12-22). A measure of the goodness-of-fit is given in the
fifth and eighth columns respectively for the three-parameter
functional fit and the two-parameter functional fit, which is defined
by , where
is the quarterly-averaged value of
the sunspot area, the functional fit
value, and © European Southern Observatory (ESO) 1999 Online publication: April 28, 1999 |