Theoretically, the global solar radiation that reaches at the earth’s surface is a result of complex interactions between the solar radiations at the top of the earth’s atmosphere evaluated by the solar constant G_sc = 1369 ± 6 W/m² and the climatic conditions of the location. This evaluation does take also into account the rotation of the earth about its own axis and by its elliptical orbit about the sun and the solar declination (δ). Excluding the meteorological and climatic conditions, the daily extraterrestrial solar radiation H₀ on a horizontal surface for a day in a month is well approximated by the given expression [14] :

H 0 = G s c × ( R 0 R ) 2 × 24 × 3600 π × [ cos ( L ) cos ( δ ) cos ( ω ) + sin ( L ) sin ( δ ) ] (1)

L is the latitude and ω is the solar hour angle. The solar declination ( δ ) is expressed by:

δ = 23.45 sin [ 0.986 × ( J + 284 ) ] (2)

Equation (1) overestimates the global irradiation as it does not take into account the interaction with atmosphere’s components. Indeed, the intensity of solar radiation through the atmosphere is attenuated by various atmospheric constituents, namely gases (air, molecules, ozone, CO₂, OH, etc.), liquid and solid particles (aerosols PM_2.5, PM₁₀ and clouds). This attenuation is mainly due to clouds, and the path length through atmosphere is also critical.

Facing with the insufficiency and the quality of the in-situ measurement global solar radiation data, and the overestimation of the global irradiation by Equation (1), let us used in this paper the general empirical approach formula pioneered by angstrom in 1942 [15] , and later completed by Prescott 1940 [16] . This formula relates average daily global radiation with average daily sunshine hours:

H ¯ = H ¯ 0 ( a + b d ¯ d ¯ 0 ) (3)

where a, b are the constants to be determined. d ¯ is the monthly mean daily bright sunshine duration obtained by the Campbell-Stokes recorder, and d ¯ 0 is the highest monthly mean daily sunshine duration that can be calculated using the followed expression:

d ¯ 0 = 2 15 cos − 1 ( tan δ × tan L ) (4)

Despite the lowest availability of the radiation in-situ instruments at meteorological stations in Cameroon, we used in this paper, the sunshine duration of nine (9) stations and five (5) global solar radiations of them. These data were furnished by the Cameroon National Meteorological Office (DNM).

Based on the Trewarth’s climatic classification [17] , the 9 stations were grouped by region. Owing to its geographical position stretching from latitude 4˚N to 14˚N, Cameroon belongs to three following types of climate. Douala, Yaoundé, Manfé and Dschang cities belonging to latitude 3˚50'N and 5˚45'N make up the first region with a humid subtropical climate. The second region stretching from latitude 5˚50'N to 7˚30'N with a tropical wet climates contains the cities of Yoko, Koundja and N’Gaoundéré, while a warm and dry climate make up the third region with Garoua and Maroua. The 9 stations shared out in the 3 regions are presented in Table 1.

Among the 9 meteorological stations where these data were recorded, only 5 had the both types of data: the hour’s solar length in a day (d) and the global solar radiation (H) (Table 2).

For the 9 stations, the monthly average daily values over 10 to 20 years of measured data for the sunshine length hours were computed. Figure 1(a); Figure 2(a); Figure 3(a) are the curve of the measured values of the monthly average daily sunshine duration of the 3 climatic regions. And Figure 1(b), Figure 2(b), and Figure 3(b) are the mean value for each of them.

All the obtained plots brought out the similar trend of the monthly average daily sunshine. However, when we draw the 3 region curves on the same graph (Figure 4), it appears curiously that they have the same horizontal trend while their vertical positions is related to the latitudinal coordinate and the type of climate from the wet to the dry one.

Taking into account the above climatic classification, the only 5 stations that had in-situ recorded global solar radiation were grouped by region according to Table 2.

After the sunshine duration, we draw as one can see on Figure 5 the trend of the monthly average daily global solar radiation for the 5 stations that have these data. Once again, the 5 form curves have the same annual trend, while the graph

position depends to the region. Based on these curves trend and the way they gathered together in Figure 5, we draw in Figure 6 the blue curve to express the dry climate global solar radiation H_d and in red the humid subtropical one H_h.

To estimate the unknown global solar radiation values of the 4 other stations, we computed the regression coefficients a and b using the following equations:

a = [ ( ∑ H ¯ H ¯ 0 ) ( ∑ ( d ¯ d ¯ 0 ) 2 ) ] − [ ( ∑ d ¯ d ¯ 0 ) ( ∑ ( d ¯ d ¯ 0 ) ( H ¯ H ¯ 0 ) ) ] [ M ∑ ( d ¯ d ¯ 0 ) 2 − ( ∑ d ¯ d ¯ 0 ) 2 ] (5)

b = M ( ∑ ( d ¯ d ¯ 0 ) ( H ¯ H ¯ 0 ) ) − [ ( ∑ d ¯ d ¯ 0 ) ∑ ( H ¯ H ¯ 0 ) ] [ M ∑ ( d ¯ d ¯ 0 ) 2 − ( ∑ d ¯ d ¯ 0 ) 2 ] (6)

The obtained values were then used in Equation (3) to estimate the values of the global solar radiation H ¯ e s t which were compared to its corresponding measured values H ¯ m e s at each of the 5 stations of Table 2. The deviation between the estimated and the measured values were summarized by calculating the following statistical parameters: Mean Bias Error (MBE), Root Mean Square Error (RMSE) and correlation coefficient r.

M B E ( % ) = 100 ( 1 H ¯ M ) ( ∑ i E i M ) R M S E ( % ) = 100 ( 1 H ¯ M ) ( ∑ i E i 2 M ) (7)

where E i = H ¯ e s t − H ¯ m e s with i = 1, M was the total number of the observations points.

r = ∑ ( H ¯ e s t − H ¯ E ) ( H ¯ m e s − H ¯ M ) ( ∑ ( H ¯ e s t − H ¯ E ) 2 ∑ ( H ¯ m e s − H ¯ M ) 2 ) (8)

Angstrom constants and statistical parameters are compiled in Table 3.

Noticing that the both solar parameters were almost the same at a close latitudinal coordinate we determined the mean values of each region (Table 4).

To validate our model, we tested it by using Rietveld and Turton models [3] [18] that could be considered most likely to be appropriate.

Let’s recall that the Turton’s model analyzes on a long term monthly average sunshine duration and global solar radiation data from 25 stations especially in humid tropical countries around the world in the both hemispheres. The formulation of his model is:

H ¯ H ¯ 0 = 0.30 + 0.40 ( d ¯ d ¯ 0 ) (9)

While Rietveld’s model applied to 42 stations in different countries around the world that were selected to collect the same data, expecting to derive a unified correlation applicable anywhere in the world. The follow relation was established:

H ¯ H ¯ 0 = 0.18 + 0.62 ( d ¯ d ¯ 0 ) (10)

Based on this two Equations (9) and (10) for the test, we used the length day d ¯ 0 and the sunshine duration averages d ¯ to estimate the value of the global solar radiation for the 3 regions.

Afterwards we compared the outcome estimated values of global solar radiation with the local values of the regions (Table 5).

Table 7 contains the must values of a, b constants calculated around the world. These values let the constants of the region 1 to be the best to estimate the global solar radiation of the four remain cities namely Manfé, Koundja, Yoko and Dschang.

Amongst all these a, b constants in Table 7, it stand out that the best one to estimate the unknown global solar radiations of the Manfé, Koundja, Yoko and Dschang cities are those of the region 1 of our country model with the closeness of their geographical coordinates.

The outcome computed values are compiled in Table 8 and Figure 7. We have the graph of the estimated global unknown global solar radiation and the figure shows once again the trend coherence of the vertical position that it is linked to the geographic position.