Modeling Solar Radiation at the Earth’s Surface

Correlation Between Clear Day Global Index Kc and Kj

It is of interest to utilize the clear day solar global irradiation to determine a ‘daily clear day index’, Kc, which we define as

Kc = H/Hc, (4.9)

viz., the ratio of the daily solar global irradiation, H, to the daily clear day solar global irradiation, Hc. The daily clear sky solar global irradiation was calculated using the Berlynd model, since it gave the values that most closely agreed to those determined using the Iqbal filter, cf., previous section. Poldmaa (1975) has applied a similar analysis to investigate the effect that the cloud cupola exerts on the global irradiation in Estonia, viz., he normalized the measured solar global irradiation by the virtual solar global irradiation under a cloudless sky. He calculated the latter by applying the Berlynd model.

The monthly average daily Kc values together with the corresponding median and coefficient of variation (Cv) were determined. The coefficient of variation is defined as the ratio of the standard deviation to the average value and is a measure of the degree of scatter of the data around the mean value. The results of such an analysis are reported in Table 4.5.

Table 4.5 Monthly average Kc, median and coefficient of variation (%) for Beer Sheva

Month

Average

Median

Cv

January

0.728

0.792

29.6

February

0.748

0.795

27.8

March

0.783

0.839

23.2

April

0.842

0.889

18.6

May

0.899

0.945

13.7

June

0.958

0.972

5.0

July

0.945

0.956

4.9

August

0.935

0.943

4.2

September

0.925

0.939

4.3

October

0.852

0.876

11.0

November

0.774

0.837

20.4

December

0.731

0.808

27.8

It is observed from Table 4.5 that the monthly average Kc values range from a maximum of 0.958 for June to a minimum of 0.728 for January and that the magnitude of the median exceeds that of the average for all months, i. e., more than half of the data values exceed the average value. In addition, the magnitude of the coefficient of variation, Cv, is lowest for the months June through Septem­ber (values range from 4.2 to 4.3%) and highest for the months January through March, November and December (values range from 20.4 to 29.6%). This, once again, testifies to the prevalence of clear sky conditions during the summer months as reported for Beer Sheva and as expected for the Negev region, cf., Ianetz et al. 2000.

The daily clearness index has been utilized, in the past, by many researchers as an indication of the degree of cloudiness at a particular site, i. e., 1 - KT. Intu­itively, it appears that the daily clear day index, Kc, is better suited for this task. The magnitude of KT is a measure of both atmospheric transparency and cloudi­ness. Consequently, the degree of cloudiness as determined on the basis of KT also includes the other parameters contributing to the sky transparency. The daily clear sky index, Kc, is defined such that the effect of cloudiness is not a parameter. Viz., the daily clear day solar global irradiation, Hc, is determined under clear sky con­ditions and only those parameters contributing to sky transparency are involved. Consequently, the term 1 - Kc is a much better indication of the degree of cloudi­ness at a particular site, viz., (1 - Kc) ^ 0 for a clear day and (1 - Kc) ^ 1 for a cloudy day. We believe that this is sufficient justification for determining such an index.

A linear regression analysis was performed on the individual monthly databases to determine the correlation between the Kc and KT, viz.,

K = aKj. (4.10)

A priori, it is assumed that the intercept of the linear regression curve should be at the origin of the axes. The results of this analysis, i. e., the slope ‘a’ of the monthly linear regression curve and the corresponding correlation coefficient (R2) are reported in Table 4.4. In Figs. 4.2 (a) and (b) the data and linear regression curves for January and July, which are representative of those obtained for all months, are shown.

It is observed from Table 4.6 and Fig. 4.2 that Kc and KT are highly correlated. A Fisher’s statistic (F) analysis performed on the monthly databases showed that the regression equations explain almost 100% of the data variance. Consequently, it can be assumed that these correlations will be applicable to future measurements, since the data are representative (each individual monthly database consists of a minimum of 250 measurements).

It is apparent from Table 4.6 that there is a very small variation in the slopes of the individual monthly linear regression equations, i. e., they vary between 1.3666 and 1.4461. Thus an average annual correlation between Kc and KT has been determined based upon a database consisting of all 12 months of data, viz., 4087 data pairs. The data and linear regression curve are shown in Fig. 4.3. The coefficient of correlation

KT

Fig. 4.2 Linear regression analysis of Kc as a function of KT for (a) January, (b) July

of the annual linear regression curve is only minimally smaller than those obtained for the individual monthly databases. A statistical analysis was performed on the clear day global index as calculated utilizing the annual linear regression curve, Kccalc, i. e., inter-comparison with that determined using the Berlynd model Kc. The Mean Bias Error (MBE) and Root Mean Square Error (RMSE) were determined to be -0.023 and 0.019, respectively, whereas the value of the annual average Kc is 0.835. It can be concluded that from this analysis that an annual correlation may be sufficient in the case of Beer Sheva.

Table 4.6 Monthly slope and correlation coefficient for Kc = aKT

Month (data)

a

R2

January (415)

1.4213

0.9997

February (373)

1.3996

0.9995

March (403)

1.3937

0.9991

April (363)

1.3981

0.9997

May (354)

1.4067

0.9999

June(263)

1.4013

0.99995

July (373)

1.3666

0.9824

August (250)

1.3915

0.99998

September (318)

1.4192

0.99998

October (265)

1.3833

0.9996

November (326)

1.4267

0.99998

December (384)

1.4461

0.9995

Fig. 4.3 Linear regression analysis of Kc as a function of KT for all months

Modeling Solar Radiation at the Earth’s Surface

Quality Assessment Based Upon Comparison with Models

Many models based on the physics of radiation transfer through the clear atmo­sphere have been developed (Lacis and Hansen 1974; Atwater and Ball 1978; Hoyt 1978; Bird and Hulstrom 1981a, …

Solar Horizontal Diffuse and Beam Irradiation on Clear Days

There exist a number of models to determine the solar horizontal diffuse irradia­tion on a clear day (Kondratyev 1969) but they are complex and have very stringent conditions. Similarly, there …

Foreword

Reading the twenty chapters of this book caused me mixed reactions, though all were positive. My responses were shaped by several factors. Although I have main­tained a “watching brief’ on …

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