**3. Composing of annual cloudless-sky cycles**

The interpolated annual cycles of clear-sky daily *Gclear* and *I'clear* for the study site (Eerme et al, 2006) are presented in Fig. 1.

Fig. 1. Graph of the daily global and direct broad-band radiation in MJm-2 on cloudless day vs day in the year for Tartu-Tõravere Meteorological Station site

Selection of cloudless days was based on the daily sums of direct irradiance and sunshine duration with inclusion of the hourly values, if necessary, and on the cloud visual inspection data. The used data enabled us to confirm that the solar disk was not obscured by clouds but did not exclude a possibility of appearance non-obscuring clouds during the day. Such small cloud amounts have minor or practically no influence on the recorded daily values of solar radiation. The effect of variation of the distance between the Earth and the Sun is considered in the data. The smoothed annual cycles have been composed, using a moving average of 5 to 10 days with balanced positive and negative deviations of the AOD from its seasonal climatological value. For the period before 2002 the AOD values for broad-band solar radiation prepared by Russak (Russak, 2006; Russak et al., 2005, 2007) have been used. The data for years 1983-1985 and 1992-1995 were excluded for reason of containing significantly higher values from El Chichon and Pinatubo major eruptions than the usual contribution of large values. Major volcanic eruptions can be considered as stochastic natural fluctuations of atmospheric conditions in time scales of the performed study. The variations of global volcanic activity appear at much lower frequencies than the studied variations of available solar radiation.

Fig. 2. Probability density distributions of broadband aerosol optical depth in 1955-2003 in normal conditions and in volcanically disturbed atmosphere in 1983-1985 and 1992-1995

Fig 2. illustrates the distribution of broad-band AOD in summer half-year in normal and volcanically disturbed conditions. Since February 2002 the AERONET Cimel-318a sunphotometer (http://aeronet.gsfc.nasa.gov) operates at Tartu-Tõravere Meteorological Station. The cloud corrected AOD data at level 2.0 are used. Reliable relationship was established between the broadband AOD and AERONET AOD at 500 nm (Teral et al., 2004). The summer half-year distribution of AOD at 500 nm is presented in Fig. 3. The major part of AOD data are recorded in April to September. In February and November as well as often in October and March the amount of data is too small for statistical conclusions. In December and January almost no data have been recorded due to very low noon solar elevation. Thus, in November to February the cycles of *Gclear* and *I'clear* are less reliable than in March to October. It should be mentioned that at the study site about 80 % of global solar radiation and 87 % of direct irradiance are received during the bright period of the year from spring equinox to autumnal equinox. The average contribution, from the period November to February, to the annual amount of gobal irradiance is around 5.9 % and that of direct irradiance around 3.3 %.

solar radiation prepared by Russak (Russak, 2006; Russak et al., 2005, 2007) have been used. The data for years 1983-1985 and 1992-1995 were excluded for reason of containing significantly higher values from El Chichon and Pinatubo major eruptions than the usual contribution of large values. Major volcanic eruptions can be considered as stochastic natural fluctuations of atmospheric conditions in time scales of the performed study. The variations of global volcanic activity appear at much lower frequencies than the studied

> 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 *AOD*

Fig. 2. Probability density distributions of broadband aerosol optical depth in 1955-2003 in normal conditions and in volcanically disturbed atmosphere in 1983-1985 and 1992-1995

Fig 2. illustrates the distribution of broad-band AOD in summer half-year in normal and volcanically disturbed conditions. Since February 2002 the AERONET Cimel-318a sunphotometer (http://aeronet.gsfc.nasa.gov) operates at Tartu-Tõravere Meteorological Station. The cloud corrected AOD data at level 2.0 are used. Reliable relationship was established between the broadband AOD and AERONET AOD at 500 nm (Teral et al., 2004). The summer half-year distribution of AOD at 500 nm is presented in Fig. 3. The major part of AOD data are recorded in April to September. In February and November as well as often in October and March the amount of data is too small for statistical conclusions. In December and January almost no data have been recorded due to very low noon solar elevation. Thus, in November to February the cycles of *Gclear* and *I'clear* are less reliable than in March to October. It should be mentioned that at the study site about 80 % of global solar radiation and 87 % of direct irradiance are received during the bright period of the year from spring equinox to autumnal equinox. The average contribution, from the period November to February, to the annual amount of gobal irradiance is around 5.9 % and that of

Average 1955-2003

1992-95

Average in 1982-85 and

variations of available solar radiation.

0.0

direct irradiance around 3.3 %.

0.1

0.2

Frequency

0.3

0.4

The cloudless days exhibiting large deviations of *Gclear* and *I'clear* from the current normal value were excluded. The seasonal probability density distributions (see Fig. 2 and Fig. 3) of the AOD are skewed with a sharp maximum at small values and long tail of large values (Eerme et al., 2006). In the case of such distribution the conventional mean is not an appropriate measure of central tendency because it is shifted toward larger values than the major body of distribution. Median coincides with the peak of distribution much better and its value is about 20-30 % less than the conventional mean. However, median does not consider the differences in both wings of distribution. In the cases of equal median the distribution of values in wings may be substantially different.

Fig. 3. Probability density distribution of AERONET measured AOD in summer half-year at 500 nm in 2002-2010

Proposed by British statistican Bowley and popularised in the classic book by Tukey (Tukey, 1977), trimean takes into consideration the distribution in wings through inclusion of the 0.25 and 0.75 quartilles. In calculating trimean these values are considered with single and median with double weight. For AOD as well as for later *G/Gclear,* three central tendency measures – conventional mean, median and trimean, have been calculated and compared. We have a reason to consider trimean as the most appropriate measure for central tendency of skewed distributions.
