Variation of the Earth’s Irradiance over Some Selected Towns in Nigeria

The extraterrestrial radiation is the solar radiation received at the top of the earth’s atmosphere on horizontal surface. This quantity over selected stations in the tropics was investigated. Daily data of the extraterrestrial radiation on the earth horizontal surface for the year 2018 for stations: Iwo, Abuja, Enugu, Port-Harcourt, Sokoto and Maiduguri obtained from the archive of HelioClim website were analyzed using MATLAB and Statistical Packages for Social Science (SPSS Version 20.0) to estimate the extraterrestrial radiation of the station considered. The results of the MATLAB revealed that the value of the coincidence is − 2 × 10 Wm across all stations. In January, the values between 15 20 peaks were observed in the year with the Irradiation (− 4 × 10 Wm) and the maximum (2 × 10 Wm). The results revealed the Root Mean Square Error RMSE for Sokoto (139.99), Abuja (162.72), Iwo (177.07), Maiduguri (171.34), Enugu (191.07), Port-Harcourt (212.27). The results also revealed that quadratic trend equation which accounted in the range 95.9% 41.9%. The results then concluded that Sokoto and Maiduguri have the highest solar Irradiance as revealed by the result. doi: 10.5829/ijee.2020.11.04.08


INTRODUCTION 1
Surface energy balance is the combination of the net radiation to or from the surface, the sensible (i.e direct) and the latent (i,e indirect) heat flux to or from the atmosphere, and the heat fluxes into or out of the submedium. Anthropogenic activities taking place on the surface of the earth has contributed significantly to the energy balance of the atmosphere. This solar radiation leads to the transportation and exchange of matters at the interface between the earth surface and the atmosphere which contribute to the climate change [1]. The more the solar radiation, the more the evaporation of water molecules [2]. However, this climate change brings about morbidity to the living and non-living things on the surface of the earth. The extraterrestrial radiation, which is the solar radiation received at the top of the earth's atmosphere on a horizontal surface can be singled out to see it variation latitudinally and temporally [3]. Extraterrestrial radiation is known to be affected by the change in the sun-earth distances and as such, the values of this parameter changing with time, latitude and season [4]. The general equation that models extraterrestrial radiation is employed in this research to investigate how well it simulates the actual extraterrestrial radiation in the *Corresponding Author E-mail: francisaweda4@gmail.com (F. O. Aweda) tropics. Several works have been done on the estimation of the earth's albedo globally but special attention is given to extraterrestrial radiation component of global radiation in this work. Mohandes, et al. [5]; Trajekovic, et al. [6]; Reddy and Ranjan [7] have used Artificial Neural Network (IA) Technique to estimate the solar radiation. As reported by Aweda et al. [8], seasonal effect on evaporation rate occurs as a result of the dry or wet temperature of the environment. The technical modelling approaches along with up performance of the fabricated solar systems for photovoltaic solar power generation and visible solar irradiation were extensively discussed in literature [9][10][11][12]. Babar and Bostrom [13] state that, owing to constraints all over the world, solar radiation equipment is not installed at any weather stations, but for the betterment of research, an internet data or satellite data can perform the same function with the installed equipment data.
The stations considered in this research are Iwo, Ilorin, Abuja, Enugu, Port-Harcourt, Sokoto and Maiduguri (all in Nigeria). The objectives of this research are: (1) comparative analysis of the modeled extraterrestrial and the actual values in prediction of the extraterrestrial radiation (2) significance of the coincidence of the estimated and the observed extraterrestrial radiation.

Data collection
The daily data of the extraterrestrial radiation ( ) of the earth horizontal surface for the year 2018 for six stations (Iwo, Abuja, Enugu, Port-Harcourt, Sokoto and Maiduguri), located in Nigeria were collected from the archive of HelioClim website of soda (http://www.sodapro.com) of MERRA-2 meteorological re-analysis. The data of twelve months (12) spanning from January to December 2018 were obtained as monthly average in Comma Separated Value (CSV) data format. Environmental running of MATLAB and statistical packages was done on the data for data plotting, curve fittings and other statistical analysis. These stations are spread over Nigeria. The assessment of the data was done on the 27th May, 2019 at about 8:51pm local time of Nigeria [9].

Extraterrestrial radiation modelling
The solar energy conservation requires radiation incident, short wave balancing at the edge of the atmosphere as reported by Audu and Isikwue [3].
where is the global solar radiation and is called clearness index, which is also the ratio of the global to the extraterrestrial radiation transmitted through the atmosphere to the ground surface. Barka et al. [14] is the absorbed solar radiation and is the fraction absorbed radiation, called the absorption coefficient or absorbance. is the extraterrestrial radiation towards the space.
is the extraterrestrial radiation incident on the surface of the earth at the edge of the Earth's atmosphere. According to Babatunde et al. [15], has found very small compared to others ratios. Therefore, the equation becomes However, the reflectivity or albedo can be estimated using But for this research, the extraterrestrial radiation employed was gotten from the Iqbal [16] which is: However, this equation estimates with a maximum error of 0.0006 rad (< 3) or, if the final two terms are omitted, with a maximum error of 0.0035 rad Iqbal [15]. This expression gives the approximate values of solar declination with varying degrees of accuracy which was developed by Spencer [17] 0 = ( 0 ) 2 = 1.000110 + 0.034221 cos Γ + 0.00128 sin Γ + 0.00719 cos 2Γ + 0.000077 sin 2Γ Γ is called the day angle (radians), where r is the sunearth distance and 0 is called one astronomical unit: 1AU = 1.496 × 10 8 km.
is the day number of the year, ranging from 1 on January to 365 on December. February is always assumed to have 28 days; because of the leap year cycles. However, it is more desirable to have the distance expressed in a simple mathematical form as shown in those equation above. For this reason, a number of mathematical expressions of varying complexities are available for the solution of Earth's albedo Iqbal [16]. Traditionally, the distance r is expressed in terms of a Fourier Series Expansion with a number of coefficients. This have a maximum error of 0.0001, but Spencer [17] developed the reciprocal of the square of the radius vector of the earth as shown in Equation (7) which is called the Eccentricity Correction Factor of the Earth's Orbit Iqbal [16].

Statistical analysis model
This study adopted the quadratic trend to estimate the irradiance in six locations (Sokoto, Abuja, Enugu, Iwo, Maiduguri and Port-Harcourt) in Nigeria. The quadratic trend equation is given as: where, is the irradiance at the time t and t is the time, t =1, 2, …., 365, is the error term. The normal equation for the quadratic trend in Equation (9) is given as: In matrix form, the systems of equations in Equation (10) can be written as: The parameters of the quadratic trend equation for irradiance in each of the six locations considered were estimated using the Statistical Package for Social Sciences (SPSS Version 20.0). The F-statistic was used to determine the significance of the quadratic trend equation while the p-values of the t-statistic was used to the significance of each of the term in the trend equation.
The extent to which time accounted for the variation in irradiance was determined using the coefficient of determination (R 2 ) while the forecasting performance of the trend equation was determined using the Root Mean Square Error (RMSE) given by:

The radiation of the solar attenuation across the earth surface
The Irradiation for both the estimated and the observed values agreed at the middle of the month i.e. January for all the stations considered except for station Enugu that shift a little to around 20 th of January. The results revealed that the value of the coincidence is −2 × 10 4 −2 across all the stations, except that of Enugu which occurred on the 20th day of the month (see Figure  1). This is in line with the fact that extraterrestrial radiation is determined not only by latitude and time of the day but also on the date of the month [18]. Figure 1 revealed that Maiduguri had its peak around the following days (1, 2, 3, 29, 30 and 31) in January 2018. With the minimum between (15)(16)(17)(18)(19)(20) in the year considered, with the irradiation (−4 × 10 4 −2 ) and the maximum is above the 2 × 10 4 −2 . For the observed value, it was noticed that the low ebb at the beginning of the month (January) rose to its maximum on the 31 st of January 2018, with the value 300   The value of irradiance across all the stations as shown in Figures 2a to 2f revealed that, all the stations follow the same pattern and there was a constant increase in Irradiance as compared to that of January and December until after the month of October when the irradiance decrease even more than that of November. However, months such as June, and July have their values less than 400 −2 . Other months such as February, March, April, May, August and September have their values of Irradiance greater than 400 −2 . While December have it value of irradiance around 380 −2 . The variation of irradiance across all the stations observed showed that the maximum irradiance occurred in the months February (420 −2 ) and March (419 −2 ), while the minimum irradiance occurred in the month January (372 −2 ) December (370 −2 ). Therefore, the solar radiation in the months February and March observed tend to be high due to the high increase of the Earth's Irradiation. While the solar radiation in the months January and December observed to be the minimum value due to winter period in Nigeria.

CONCLUSION
For this study, we used quadratic trend, error bar and extraterrestrial radiation equation on irradiance of the earth surface parameter. From the satellite data obtained from achieve of HelioClim website of soda, we then deduced the correlation between observed value of irradiance and the estimated value. This shows that there is a drastic difference between the estimated and the observed values as revealed by other authors. For this research, January has a sharp increase while October has sharp decrease. The two months signify high irradiance of the atmosphere across the entire country. Therefore, further studies are encouraged to reveal more details using some climatological parameters (rainfall, temperature, wind speed, pressure etc.) and the rate of sun incident angle and cloud to improve irradiance observation using other months of the year. Analytically, the modelling of solar irradiance using statistical package sometimes is difficult and requires some basic information for the completion of the Solar Irradiance Model Aweda and Samson [19]. However, this research concluded that solar irradiance across Nigeria got to its peak in the months February and March.