Modeling Solar Radiation at the Earth’s Surface

Angstrom-Prescott Relation and its Physical Significance

Prescott at 1940 used the data recorded in Mount Stramlo Observatory in Canberra- Australia. Instead of perfect clear sky value of Angstrom, he used the published data of Angot’s (as supplied by Brunt (1934)) values which is the solar radiation that would be received if the atmosphere were transparent. He obtained the regression equation:

H = 0.25 + 0.54N . (5.2)

In this expression, H is the monthly averages of daily global solar irradiation on hor­izontal surface, Ho is the monthly average daily solar radiation on horizontal surface if there were a transparent atmosphere, n is the monthly average of daily bright sun­shine hours and N is the maximum possible sunshine on cloudless day. Instead of measured perfectly clear sky value in Eq. (5.1), use of H0 makes it possible to utilize such correlations to estimate the irradiation values in locations where no radiation data exists. This is because it is possible to calculate H0 in any time interval using the solar radiation values reaching outside the atmosphere.

The main reason for the large difference between the coefficient of Prescott, 0.54 and that of Angstrom’s, 0.75 is the use of new normalizing value H0 instead of the a perfectly clear day value of the site of interest. Now we know that even similar calculated values of H0 of the site are used for normalizing the solar irradiation, such coefficients span a wide range of values varying with the location (mainly latitude), climatic, atmospheric and seasonal meteorological variations in the site under consideration.

Later, Angstrom at 1956 has written the expression:

Angstrom stated that for the fractional bright sunshine period n/N = 0, H = aHc while for n/N = 1, H = Hc; clarifying also his a value of 0.25 for Stockholm. This
value 0.25 means that H for an overcast sky has a value 25% of that of a perfectly clear sky.

In his 1956 article, Angstrom also gave a detailed physical understanding of the coefficient a, stating that Eq. (5.4) is an idealized form and the coefficient depends on many parameters. Some of these parameters are frequency of atmospheric distur­bances, cloud amounts and types, month of the year, altitude and ground reflectance. In the work by Martinez-Lozano et al. (1984) a values for a number of locations are presented; they span a range of 0.22 to 0.68. Now it is very clear that all such parameters mentioned above and others are effective on the regression coefficients of the linear sunshine based models, namely the Angstrom-Prescott relation:

Calculations of H0 and N values are explained in details in Duffie and Beckmann (1991), which start with the solar constant, 1367W/m2. Solar constant is defined as the extraterrestrial solar intensity outside of the atmosphere incident on a perpendic­ular surface at the mean sun-earth distance. It is reduced to an instantaneous value outside the atmosphere on a horizontal surface for any location, by taking into ac­count the varying sun-earth distance and multiplying by cosine of the zenith angle of the location of interest. For the total values in any time interval of course one should integrate the instantaneous values in the required time interval, mainly hourly and daily. Monthly averages of daily values, H0 and N, can be obtained simply by taking the averages of daily values or to simplify by using the values at a specific day num­ber of the year which gives directly the monthly average daily value of that specific month, as explained in Duffie and Beckman (1991).

Angstrom also derived simply the value of a1 in Eq. (5.3) in terms of Angstrom coefficients a and b as a1 = a/(a + b) (Angstrom 1956) with a definition that his value Hc in Eq. (5.3) is equal to H in Eq. (5.5) for n = N. He calculated the values of a1 using regression constants a and b of different stations and obtained values in quite a large range of from 0.218 to 0.583. He thus concluded that the values of a or the other coefficients depends on different climatic and geographic parameters stated above. Later as stated by Gueymard et al. (1995), for the mean value of irradiation after monthly averaging, it is a question of “superposability” of two extreme cloud states for two limiting idealized days (fully overcast and perfectly cloudless) which is only a simplifying assumption. Essentially, for n/N = 0 and for n/N = 1 the values of H/H0 would certainly be different than a and a + b, even for different days of the same month.

It is rather easy to attribute rough physical meanings to the coefficients a and b in Eq. (5.5) using the extreme values of n/N. If there is no cloud obscuring the sun within a day, then n/N = 1 and H/H0 = a + b can be interpreted as the monthly average daily value for the transmittance of a clear day. Note that clear day (n/N = 1 in this case) does not always mean a perfectly clear day without appearance of any cloud all the day. Even sometimes the presence of clouds that do not obscure the sun may increase the irradiation reaching the site due to high reflections. Another fact is that the days without any cloud may have different solar irradiation reaching the

Earth due to differences in the air mass and also due to some atmospheric conditions such as dense turbidity. For a completely overcast day, n/N = 0 and H/H0 = a, which essentially accounts for the diffuse component. It may represent the average daily transmission of an overcast sky of the site under consideration. The range of values obtained for a and b given below however show that they are affected by many geographical and atmospheric parameters.

Angstrom coefficients a and b in Eq. (5.5) have quite a wide range of different values, a ranging from 0.089 to 0.460 and b from 0.208 to 0.851 as tabulated for 100 locations in the review article by Akinoglu (1991), or from 0.06 to 0.44 for a and from 0.19 to 0.87 for b as given for 101 locations in the paper of Martinez-Lozano et al. (1984). This variation may be considered to recognize the importance of the above mentioned parameters affecting the regression constants of the empirical cor­relation Eq. (5.5). In addition, one should think that the variation of a with b might be hindering another conceptual information about the relation between global solar irradiation and bright sunshine hours, to be used in developing an estimation model with higher accuracy and better universal applicability. Another important fact that must not be overlooked is the measurement errors both for the irradiation values and bright sunshine hours.

Nevertheless, many researchers expressed Angstrom coefficients in terms of dif­ferent geographical and climatic parameters such as the latitude, altitude, sunshine fraction (see for example Akinoglu and Ecevit (1989); Gopinhathan (1988); Rietvel (1978); Abdalla and Baghdady (1985)). An overall conclusion that can be derived from all these works might be summarized as: these coefficients depend on all phys­ical, spatial and the dynamic properties of the atmosphere at the region of interest. One may even state that, for a region the coefficients derived from a long term data of some number of years can be different than those obtained by using the data of same length for the same region but for another set of years. This is of course another research of interest which necessitates long-term reliable data with high accuracy from different regions.

One of the most important facts about the wide range of values of the re­gression constants is hidden in the diffuse component of the energy income, as Angstrbm (1924) concluded. Diffuse component has mainly three different parts as explained in section 4.3, each of which depends on different physical properties of the elements of our environment and the atmosphere. Use of Hc instead of H0 may reduce the wide range of the values of the regression coefficients since Hc includes the diffuse irradiation characteristics of the atmosphere of the location of interest. Then, of course, Hc must be calculated by some other means for the locations where the long term measured data is missing.

Gueymard (1993b) developed a model to calculate Hc using reasonable estimates of precipitable water w and Angstrom turbidity coefficient в which makes it possi­ble to use Angstrom Equation in its original form. As mentioned above Hc intrinsi­cally contains information on the atmospheric characteristics of the site of interest. Hence, he recommended that the researches should be directed toward the determi­nation of a of Angstrom expression, namely the Eq. (5.4).

Two self-explanatory excel worksheet are included in the CD-ROM supplied with the book, which calculate the daily and monthly mean daily values of H0 and N, namely ‘daily-calculations-Ho. xls’ and ‘monthly-mean-daily-calculations - Ho. xls’. Calculations only need to input the latitude of the location of interest to the cell B3. Solar constant is taken to be 1367W/m2.