**4. Probability density distributions of daily G/Gclear and measures of their central tendency and spread**

The daily ratios of *G/Gclear* were studied statistically within four conventional seasons of the year. The winter season extends from Dec 22 to March 20, the spring season from March 21 to June 20, the summer season from June 21 to Sept 22 and the autumnal season from Sept 23 to Dec 21. The summer, half-year which includes both spring and summer seasons was also studied separately. Similar study was performed on monthly level. The maximum, mean and minimum values of calculated central tendency measures for relative global irradiance in all seasons are presented in Table 1. Also the spread characteristics, StDevTri and StDevMed, were calculated relative to the trimean and median like the conventional standard deviation (StDev) is calculated relative to the mean (Wilks, 2006). For almost all seasons and months the StDevTri is the smallest and StDevMed tends to be the largest.


Table 1. Seasonal central tendency measures of the relative global irradiance *G/Gclear* and relative direct irradiance *I'/I'clear* in 1955–2010

The seasonal and monthly probability density distributions of daily G/Gclear for every year were studied for their symmetry and deviation from the Gaussian distribution. The seasonal as well as monthly probability density distributions in most cases were flatter than the Gaussian distribution and skewed. Sharper distributions appeared less frequently. Both the negatively and positively skewed ones were found. In dark half-year the monthly and seasonal probability density distributions tend to be positively and in bright half-year negatively skewed. Median and trimean of distributions in separate years tend to be larger than conventional mean in the prevailing sunny conditions. In cloudy conditions, on the contrary, conventional mean tends to be larger than median and trimean. The difference between median and mean is usually larger than that between trimean and mean. Year-to year variation of the ratios, Mean/Trimean and Mean/Median within seasons of the dark half-year was significantly larger than in the summer half-year.

In cloudy dark half-year, when small values of G/Gclear dominate, a few sunny days appearing in a month may cause positive skewness of distribution and enlarge the value of the conventional mean. The biases between mean and trimean, and between mean and median may exceed 20 %, while the majority of G/Gclear are very small values. In the months of bright half-year, a few very small values due to heavily cloudy days shift the conventional mean toward lower value. The annual monthly values of mean, median and trimean of G/Gclear and their year-to-year variation were studied. The seasonal and halfyearly differences between trimean, median and conventional mean are small because during the long periods different situations are encountered, and the distribution becomes more symmetric than it is for shorter intervals. Using the conventional mean as a reference

23 to Dec 21. The summer, half-year which includes both spring and summer seasons was also studied separately. Similar study was performed on monthly level. The maximum, mean and minimum values of calculated central tendency measures for relative global irradiance in all seasons are presented in Table 1. Also the spread characteristics, StDevTri and StDevMed, were calculated relative to the trimean and median like the conventional standard deviation (StDev) is calculated relative to the mean (Wilks, 2006). For almost all seasons and months the StDevTri is the smallest and StDevMed tends to be the largest.

Quantity Winter Spring Summer Autumn Summer half-

Mean 0.568 0.658 0.652 0.467 0.650 Median 0.530 0.653 0.646 0.412 0.675 Trimean 0.552 0.656 0.648 0.429 0.668 Min 0.406 0.563 0.567 0.371 0.586 Max 0.767 0.771 0.768 0.605 0.748

Mean 0.306 0.425 0.400 0.230 0.412 Min 0.163 0.249 0.283 0.090 0.305 Max 0.521 0.601 0.557 0.378 0.542 Table 1. Seasonal central tendency measures of the relative global irradiance *G/Gclear* and

The seasonal and monthly probability density distributions of daily G/Gclear for every year were studied for their symmetry and deviation from the Gaussian distribution. The seasonal as well as monthly probability density distributions in most cases were flatter than the Gaussian distribution and skewed. Sharper distributions appeared less frequently. Both the negatively and positively skewed ones were found. In dark half-year the monthly and seasonal probability density distributions tend to be positively and in bright half-year negatively skewed. Median and trimean of distributions in separate years tend to be larger than conventional mean in the prevailing sunny conditions. In cloudy conditions, on the contrary, conventional mean tends to be larger than median and trimean. The difference between median and mean is usually larger than that between trimean and mean. Year-to year variation of the ratios, Mean/Trimean and Mean/Median within seasons of the dark

In cloudy dark half-year, when small values of G/Gclear dominate, a few sunny days appearing in a month may cause positive skewness of distribution and enlarge the value of the conventional mean. The biases between mean and trimean, and between mean and median may exceed 20 %, while the majority of G/Gclear are very small values. In the months of bright half-year, a few very small values due to heavily cloudy days shift the conventional mean toward lower value. The annual monthly values of mean, median and trimean of G/Gclear and their year-to-year variation were studied. The seasonal and halfyearly differences between trimean, median and conventional mean are small because during the long periods different situations are encountered, and the distribution becomes more symmetric than it is for shorter intervals. Using the conventional mean as a reference

relative direct irradiance *I'/I'clear* in 1955–2010

half-year was significantly larger than in the summer half-year.

*G/Gclear*

*I'/I'clear*

year

leads to underestimation of the contrast between the winter and summer months availability of solar irradiance.

The ratio of Mean/Trimean of G/Gclear is positively correlated to the skewness of distribution during a whole year. The monthly coefficients of linear correlation vary between 0.55 and 0.88. In September to February the ratio, Mean/Trimean is positively correlated with the kurtosis of distribution, and in March to August the correlation was negative. In March to August the main tendency and the spread measures were negatively correlated. The monthly coefficients varied between –0.20 and -0.65. It means that instead of large values of G/Gclear dominating, the contribution of large deviations is small. In October to December positive correlation, with coefficients 0.60 and 0.65 respectively were obtained. It means that the probability of large deviations increases with increasing contribution of relatively small values.

### **5. Year-to-year variations of half-yearly, seasonal and monthly availability of global irradiance**

The annual availability of solar radiation at the latitude of study was determined by the contribution of summer half-year since the contribution of winter half-year was only about 20 %. At the same time the interannual variations tend to be larger in winter months. The variation in every summer half-yearly total, during the 56 years considered remains within ±10 % about the average, except for the two extremely sunny years 1963 and 2002, when the totals exceeded the average by 15 % and reached 75 % of the climatological cloudless-sky value.

Fig. 4. Time series of summer half-year average *G/Gclear* (above) and *I'/I'clear* (below) in 1955- 2010

In most years, the deviations from the mean are significantly less than 10 %. For the summer half-yearly totals, the difference between conventional mean and trimean is about 0.3 % and that between conventional mean and median about 0.6 %. The conventional mean could be considered here as acceptable as the measure of the general trend. The year-to-year variations are most strongly correlated with cloudiness. The coefficients of linear correlation of the average *G/Gclear* with the average total cloud amount, low cloud amount and the number of overcast days in all seasons have been around –0.80. The linear correlation between the summer half-yearly sums of relative global and relative direct irradiance is equal to 0.90, which is much higher than that for the winter half-year, when it is only 0.60, indicating larger contribution from the direct radiation to the variation of global irradiance at higher solar elevations and snow-free conditions. At the seasonal level the correlation was the highest, 0.96, in summer, and somewhat lower, 0.92, in spring when the ground albedo was not stable.

In the time series of yearly averages of *G/Gclear* and *I'/I'clear* in the summer half-year (Fig.4), a remarkable feature is observed: an interval of reduced values in the years 1976–1993, when 15 out of 18 values were lower than the 1955–2010 average. The period of low values was more contrasting in terms of direct than global irradiance. Searchers of linear trends could easily find a dimming trend between the 1960s and the mid-1980s and a brightening trend from the middle of the 1980s. Similar behaviour of annual totals from late 1950s and early 1960s has been obtained also in other sites of Northern Europe and European Russia (Wild et al., 2005; Chubarova, 2007). The finding of approximately 35-year periodicity of alternation of cloudy and bright conditions in Western Europe was attributed to sir Francis Bacon who declared it in first decade of 17th century. Later it was forgotten and found again in several cases. The dimming and following brightening presented in Fig. 4 is rather a manifestation of this periodicity. A little more than one full period is presented.

Fig. 5. Long-term variation of *G/Gclear* with year in spring, 1955–2010: Annual values and 7 year moving average

In most years, the deviations from the mean are significantly less than 10 %. For the summer half-yearly totals, the difference between conventional mean and trimean is about 0.3 % and that between conventional mean and median about 0.6 %. The conventional mean could be considered here as acceptable as the measure of the general trend. The year-to-year variations are most strongly correlated with cloudiness. The coefficients of linear correlation of the average *G/Gclear* with the average total cloud amount, low cloud amount and the number of overcast days in all seasons have been around –0.80. The linear correlation between the summer half-yearly sums of relative global and relative direct irradiance is equal to 0.90, which is much higher than that for the winter half-year, when it is only 0.60, indicating larger contribution from the direct radiation to the variation of global irradiance at higher solar elevations and snow-free conditions. At the seasonal level the correlation was the highest, 0.96, in summer, and somewhat lower, 0.92, in spring when the ground albedo

In the time series of yearly averages of *G/Gclear* and *I'/I'clear* in the summer half-year (Fig.4), a remarkable feature is observed: an interval of reduced values in the years 1976–1993, when 15 out of 18 values were lower than the 1955–2010 average. The period of low values was more contrasting in terms of direct than global irradiance. Searchers of linear trends could easily find a dimming trend between the 1960s and the mid-1980s and a brightening trend from the middle of the 1980s. Similar behaviour of annual totals from late 1950s and early 1960s has been obtained also in other sites of Northern Europe and European Russia (Wild et al., 2005; Chubarova, 2007). The finding of approximately 35-year periodicity of alternation of cloudy and bright conditions in Western Europe was attributed to sir Francis Bacon who declared it in first decade of 17th century. Later it was forgotten and found again in several cases. The dimming and following brightening presented in Fig. 4 is rather a

> 1950 1960 1970 1980 1990 2000 2010 Year

Fig. 5. Long-term variation of *G/Gclear* with year in spring, 1955–2010: Annual values and 7-

manifestation of this periodicity. A little more than one full period is presented.

was not stable.

0.50

year moving average

0.55

0.60

0.65

*G/G clear* 0.70

0.75

0.80

Considering the spring and summer seasons separately, some significant differences in the long-term behaviour of the available solar irradiance become evident. The smooth line representing 7-year moving average of interannual variation of *G/Gclear* in the spring season exhibits maxima around 1965 and 2000 and minima between 1980 and 1985 as well as around 1995 (Fig. 5).

Fig. 6. Probability density histogram of *G/Gclear* in spring season

Fig. 7. Long-term variation of *G/Gclear* with year in summer, 1955–2010:Annual values and 7 year moving average

A 7-year moving average turned out to be effective for smoothing the short-term variations and emphasizing the expected trends. This has also been approved in the case of other Estonian meteorological and hydrological data (Järvet and Jaagus, 1996).

Fig. 8. Probability density histogram of *G/Gclear* in summer season

The probability density distribution of G/Gclear for the spring of years 1955–2010 (Fig. 6) is symmetric and the values were close to the average with the highest frequency. The major part , 65 %, of the annual values occurred between 0.60 and 0.70; 20% were above 0.70 and 15% were below 0.60. The observations by several authors of the long-term dimming trend up to the middle of the 1980s and those of the following brightening trend (Che et al., 2005; Wild et al., 2005; Sanchez-Lorenzo et al., 2007) were generally in agreement with the G/Gclear observed in the spring.

In the summer, the distribution was less symmetric and the amounts of very large and small values were about 27% for both. In summer periods of large and small values of *G/Gclear* (Fig. 7) were revealed. Large values dominated in 1966–1975 and more so from 1994. No obvious long-term trend of dimming or brightening was found. Up to the late 1960s each fourth summer was sunny. During the mentioned bright periods, approximately each second summer was sunny. The probability density distribution of summer *G/Gclear* shown in Fig. 8 does not fit the normal distribution and exhibits rather bimodal nature. This suggests that the existence of two different regimes associated with mean wet or dry summers (D'Andrea et al., 2006), being partly driven by soil moisture, may be the reason of such distribution observed in Estonia, and presumably in other parts of Northern Europe. Such behaviour is expected to result from multiple equilibria in the continental water balance containing the contributions of large scale weather patterns as well as of the local soil water contents. The spring and summer mean *G/Gclear* are poorly correlated with each other.

A 7-year moving average turned out to be effective for smoothing the short-term variations and emphasizing the expected trends. This has also been approved in the case of other

Measured data

distribution

Corresponding normal

40 45 50 55 60 65 70 75 80 85 90 95 *G/Gclear* (%)

The probability density distribution of G/Gclear for the spring of years 1955–2010 (Fig. 6) is symmetric and the values were close to the average with the highest frequency. The major part , 65 %, of the annual values occurred between 0.60 and 0.70; 20% were above 0.70 and 15% were below 0.60. The observations by several authors of the long-term dimming trend up to the middle of the 1980s and those of the following brightening trend (Che et al., 2005; Wild et al., 2005; Sanchez-Lorenzo et al., 2007) were generally in agreement with the G/Gclear

In the summer, the distribution was less symmetric and the amounts of very large and small values were about 27% for both. In summer periods of large and small values of *G/Gclear* (Fig. 7) were revealed. Large values dominated in 1966–1975 and more so from 1994. No obvious long-term trend of dimming or brightening was found. Up to the late 1960s each fourth summer was sunny. During the mentioned bright periods, approximately each second summer was sunny. The probability density distribution of summer *G/Gclear* shown in Fig. 8 does not fit the normal distribution and exhibits rather bimodal nature. This suggests that the existence of two different regimes associated with mean wet or dry summers (D'Andrea et al., 2006), being partly driven by soil moisture, may be the reason of such distribution observed in Estonia, and presumably in other parts of Northern Europe. Such behaviour is expected to result from multiple equilibria in the continental water balance containing the contributions of large scale weather patterns as well as of the local soil water contents. The spring and summer mean *G/Gclear* are poorly correlated with each

Estonian meteorological and hydrological data (Järvet and Jaagus, 1996).

Fig. 8. Probability density histogram of *G/Gclear* in summer season

0

observed in the spring.

other.

5

10

15

20

*p* (%)

25

30

35

40

Fig. 9. Typical histogram of monthly distribution of relative daily sum of global irradiance G/Gclear in dark half-year

At the site, the daily mean of global irradiance in the summer half-year is approximately 0.65 of its climatological cloudless-sky value. It is almost equal to that of spring and summer. In two extremely fine weather years of 1963 and 2002, its value reached 0.75 of the clear-sky value. In three most cloudy summer half-years the value remained slightly below 0.60 of the climatological clear- sky value.

The monthly trimean values of G/Gclear presented in Table 2, varied over wide range. The ratio of maximal to minimal value varied from about 3.4 in December to 1.6 in June. The lowest values in November and December were about 0.20, while the highest was recorded in March and was about to 0.99.

Typical histogram of monthly distribution in dark half-year is presented in Fig 9. Sharp high value was recorded during the prevailing thick cloudiness. In November, about 50 % and in December and January about 25 % of the distributions were sharper than the Gaussian. In bright half-year the distribution was flatter than the Gaussian. In summer months sharp distributions may occur in the extremely fine weather conditions. Typical histogram of monthly distribution in bright half-year is shown in Fig 10.

In Table 3, the monthly average together with maximum and minimum values of the ratio Mean/Trimean are presented as well as the coefficients of linear correlation of this ratio with the kurtosis and skewness of distribution.

Variations in spread characteristics have been largest in StDevMed, but only moderately larger than those in two other spread characteristics. For all the months except November, the StDevTri in most cases happened to be the smallest. The ratio StDev/StDevTri exhibits sharper and more symmetric distributions than the ratio of StDevMed/StDevTri. Its values remained in the range between 0.98 and 1.02 with a very few exceptions lower than 0.98 and


only one exception above 1.02 out of 648. The distribution of the ratio StDevMed/StDevTri exhibits only one case of value below 0.98, but the tail often reaches the range between 1.06 and 1.10.

Table 2. The monthly values of assumed normal *Gclear* in physical units and recorded minimal, mean and maximal values of the trimean of monthly relative global *G/Gclear*. Monthly assumed normal *I'clear* and recorded minimal, mean and maximal values of *I'/I'clear* in 1955-2010
