J. Nig. Soc. Phys. Sci. 4 (2022) 911 Journal of the Nigerian Society of Physical Sciences Performance Evaluation and Statistical Analysis of Solar Energy Modeling: A Review and Case Study Samy A. Khalil∗ Solar and Space Department, National Research Institute of Astronomy and Geophysics (NRIAG), 11421, Helwan, Cairo, Egypt Abstract The main target of this research is a quantitative review of literature on global solar radiation (GSR) models available for different stations around the world. The statistical analysis of 400 existing sunshine-based GSR models on a horizontal surface is compared using 40-year meteorological data in the selected locations in Egypt. The measured data is divided into two sets. The first sub-data set from 1980 to 2019 was used to develop empirical correlation models between the monthly average daily global solar radiation fraction (H/H0) and the monthly average of desired meteorological parameters. The second sub-data set from 2015–2019 was used to validate and evaluate the derived models and correlations. The developed models were compared with each other and with the experimental data of the second subset based on the statistical error indicators such as RMSE, MBE, MABE, MPE, and correlation coefficient (R). The statistical test of the correlation, coefficient (R), for all models gives very good results (above 0.92). The smallest values of t-Test occur around the models (M 272, M 261, M 251, and M 238). The accuracy of each model is tested using ten different statistical indicator tests. The Global Performance Indicator (GPI) is used to rank the selected GSR models. According to the results, the Rietveld model (Model 272) has shown the best capability to predict the GSR on horizontal surfaces, followed by the Katiyar et al. model (Model 251) and the Aras et al. model (Model 261). DOI:10.46481/jnsps.2022.911 Keywords: Solar energy, Models, Statistical Indicators, Performance and Sunshine duration. Article History : Received: 02 July 2022 Received in revised form: 26 August 2022 Accepted for publication: 30 August 2022 Published: 30 September 2022 c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: B. J. Falaye 1. Introduction Solar energy has remained the most dependable source of energy capable of supporting and maintaining all of the activ- ities and cycles that support life on the planet for plants, crea- tures, and other material substances. Solar energy is generally acknowledged as a key energy hotspot for the future all over the ∗Corresponding author tel. no: +20225560645, Fax: +20225548020 Email address: samykhalil2014@gmail.com; samynki@yahoo.com (Samy A. Khalil ) planet due to the natural issues related to fossil fuels as well as their restricted stores. Different wellsprings of non-sustainable power cause perceptible ecological risks. Sustainable power sources, for example, geothermal, wind, sun-based and flowing, are harmless to the ecosystem since they have a lot lower nat- ural effect than regular sources like petroleum derivatives. As a result, solar energy is widely regarded as the most promising and reasonable type of energy capable of resolving the natural issues that humanity will face in the future [1-3]. Sustainable power sources, for example, solar energy, can possibly moderate a few negative ecological issues, in- 1 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 2 cluding environmental change brought about by concentrated petroleum derivative abuse. With quick mechanical advance- ment and decreasing costs, solar energy will play an important role in future energy frameworks. In the forecast, study, and planning of solar energy frameworks, information on solar ra- diation and its parts in a specific area is extremely fundamental. Global solar radiation on a level surface is the most fundamen- tal piece of information used in the planning and forecasting of the display of a solar energy device at a specific location. Sun- based light can be assessed by handling pictures from satellites or by on-ground estimations with pyrometers in meteorologi- cal stations. Satellite radiation estimations are not exact in as- sessing ground solar power radiation since they require envi- ronmental models to gauge the solar radiation at ground level [4-8]. Solar radiation showing up on earth is the most basic and sustainable power source in nature. Solar energy, brilliant light, and people since antiquated times, utilizing a scope of truly ad- vancing advancements, have bridled hotness from the sun. The energy from the sun could play a key part in de-carbonizing the worldwide economy through enhancements in energy ef- ficiency and forcing costs on ozone-harming substance pro- ducers. In solar energy research, information on sun-powered radiation and its components in a given area is a significant contribution for sun-based energy applications such as photo- voltaic, sun-based warm frameworks, sun-based heaters, and aloof sun-based plans. The information ought to be dependable and promptly accessible for plan, advancement, and execution assessment of nearby planet groups for a specific area. The most effective way to decide how much worldwide sunlight-based radiation at any site is to introduce estimating instruments, for example, pyranometer and pyrheliometer, at that specific spot and to screen and record its everyday record- ing, which is actually an extremely drawn-out and expensive activity [5]. Regardless of the significance of sunlight-based radiation estimations, in many developing countries this data is not immediately accessible due to a lack of the ability to bear the cost of the estimating equipment and procedures involved. Hence, various relationships and techniques have been created to assess every day or monthly worldwide solar power radiation in view of the more promptly accessible meteorological infor- mation at a larger part of weather conditions stations. Experimental models that have been utilize to work out sun- oriented radiation are normally find on the accompanying el- ements; Astronomical elements (sun based steady, earth-sun distance, sun based declination and hour point), Geographical variables (scope, longitude and height of the site), Geometri- cal elements (azimuth point of the surface, slant point of the surface, sun rise point, sun azimuth point), Physical elements (dissipating of air particles, water fume content, dispersing of residue and other climatic constituents like O2, N2, CO2, O and so forth) and meteorological elements (extraterrestrial sunlight based radiation, sun-sparkle term, temperature, precipitation, relative stickiness, impacts of shadiness, soil temperature, van- ishing, reflection of the environs, and so on) [5]. The number of connections that have been distributed and attempted to assess the monthly average global solar radia- tion is generally high, making it difficult to select the most appropriate technique for a specific reason and site. Choos- ing a suitable technique from different existing models depends on their information prerequisites and model exactness. Fore- casts of monthly daily worldwide sunlight-based radiation for a large number of areas are being introduced in various research works. It is essential that every one of the proposed models con- tain exact constants, which depend upon the season and the to- pographical area of a specific spot. The models examined in this review article are sequentially introduced and thus valuable for choosing the suitable model to assess worldwide sun-based radiation for a specific spot of interest [5-7]. The first model to predict the worldwide solar powered ra- diation was developed by Angstrom, which relates the nor- mal monthly worldwide solar powered radiation to crisp morn- ing radiation and to the normal part of conceivable daylight hours [9]. Prescott [10] modified the Angstrom model. From that point on, numerous scientists have modified the Angstrom model by utilizing different boundaries, for example, daylight hours, minimum temperature, mean temperature, maximum temperature, relative moistness, elevation, scope, longitude, overcast cover, and so on, or blends of the boundaries. Nev- ertheless, the most broadly utilized boundary to date is the day- light hour. Iziomon and Mayer [11] have presumed that the presenta- tion of the daylight-based model is far superior to the meteo- rological boundary-based models. The most commonly used boundary up to this point is daylight length because it is ac- cessible in many areas and can be effectively and consistently estimated [12, 13]. When the first request Angstrom, type re- lationship is examined, the second and third requests do not improve the precision to a significant level [14, 15]. In addi- tion, Almorox and Hontoria [16] proposed the direct, quadratic, third degree, and the logarithmic relapse models for Spain and inferred that the third request gave the general best outcome, however, re-complimented to utilize the straight relapse model since the blunder between the straight and third request relapse models is extremely small. Angstrom’s worldwide sunlight- based radiation model is one of the pioneering works in the field of solar radiation. In the long term, it tends to be seen that numerous scientists have generally utilized this relation- ship by adjusting coefficients for diversified areas (like [17-42] and some more). Numerous scientists have modified the Angstrom model by utilizing different boundaries like daylight hours, scope, longi- tude, and height. Glover and McCullum [17] have proposed a relapse model that considers the effect of scope and reasoned that the worth of “a” will be an element of scope, though the worth of “b” is effectively consistent. Correspondingly, [42] was proposed for the relapse condition as far as scope and day- light hours, utilizing the metrological information of 40 areas (37 areas from Pakistan and 3 areas from India). Hussain et al. [19] have proposed a relapse model considering the scope effect, utilizing the meteorological information of 40 stations all over the planet, and presumed that alongside the elevation effect, the half-yearly apportionment of the year ought to be utilized. [20] used meteorological data from 40 different loca- 2 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 3 tions around the world as a component of scope, height, and the proportion of daylight hours, and concluded that the proposed model can be used in any region of the world with a 10% error. For the Kingdom of Saudi Arabia [43], the relapse model is a component of daylight length, scope, longitude, and elevation. Jin it et al. [21] proposed higher request relapse models based on meteorological data from 69 locations in China. A compar- ative review was finished by [22] for 86 different areas across China and proposed the higher request relapse model regarding scope, longitude and elevation. As of late, Gadiwala et al. [23], Onyango, and Ongoma [24] have proposed the relapse model for Pakistan and Kenya separately. Bakirci [25] proposed the direct, quadratic, and third re- quest relapse models for the Eastern Anatolia Region (EAR) of Turkey and inferred that the third request polynomial con- dition gives the best outcome. The temperature is based on Quansah et al. [26] for the Ashanti Region of Ghana, where the deliberate and determined values show excellent concurrence with one another. Likewise, the constants “a” and “b” are site- specific. Yaniktepe and Genc [27] have proposed a straight, quadratic, and cubic relapse model for Osmaniye. The exhi- bition of the proposed model was contrasted with nine differ- ent models left in writing utilizing measurable tests (MAPE, MABE, and RMSE). It was inferred that the cubic relapse model performs better when contrasted with direct and quadratic models. Namrata et al. [28] proposed a straight relapse model for three different ar- eas in Jharkhand, India. The exhibition of the proposed model has contrasted with the deliberate information and the relapse model proposed by Rietveld [29], Ogleman [30], Akinoglu [31], Glover [17] and Gopinathan [20]. It was inferred that the relapse consistent “a” and “b” does not shift with height and scope. It was seen that the amount of relapse consistent is prac- tically the same for the three different urban areas (for example, Jamshedpur 0.717, Ranchi 0.700, and Bokaro 0.714). Das et al. [32] chose eleven nonlinear and six straight models acces- sible in writing to assess worldwide sunlight-based radiation over South Korea. It was presumed that the nonlinear models perform far superior to the straight models, likewise utilizing the normal Kriging strategy solar maps have plotted for South Korea. Using nine years’ worth of meteorological data He- jase and Assi [33] fostered a direct, quadratic, cubic, dramatic, log-straight, and logarithmic model for the United Arab Emi- rates and presumed that the cubic model gives minimal blun- ders among the whole proposed model. The main objective of this paper is to evaluate and perform extensive global solar radi- ation (GSR) models available in the present research by using the statistical indicators and the most accurate models chosen in this study, and apply these models for case study using the available meteorological parameters 2. Methodology The global solar radiation models are classified according to the basis of the input parameters they employ in correlating with the clearness index. The clearness index (kt) shows the proportional depletion via the sky of the incoming photovoltaic radiation and consequently offers the stage of availability of photovoltaic radiation and adjustments in the atmospheric sit- uation in a given environment [34]. Mathematically, the clear- ness index is the ratio of horizontal global solar radiation to the extraterrestrial solar radiation (Ho) on a monthly basis, princi- pally calculated theoretically as given by [35]. H0 = (24/π)I sc[1 + 0.033 cos(360n/365)] ×[cosϕ cos δ sin ωs + (2πωs/360) sin ϕ sin δ (1) IS C is the solar constant, (ϕ) is the latitude of the location, (δ) is the solar declination, (ωs) is the mean sunrise hour angle for the given month and n is the number of days of the year start- ing from firstly January. The solar declination (deltaup) and mean sunrise hour angle (omegaups) for a given month can be calculated using equations 2 and 3, respectively. δ = 23.45 sin[360(n + 284)/365] (2) ωs = cos−1(− tan ϕ tan δ) (3) It has been establish that world photovoltaic radiation is excep- tionally affect by means of meteorological parameters, astro- nomical factors, geographical factors, and geometrical factors. This may want to be attribute to the strong point of nearby lo- cal weather in figuring out the meteorological and atmospheric parameters that exceptional match a given locality. This addi- tionally relies upon on the availability of enter meteorological/ atmospheric parameter(s) that a given radiometric station or an person is successful of measuring robotically which in the end grew to become out to be the great enter parameter at the dis- posal of the researcher for predicting world photo voltaic radi- ation in that place [36-39]. 3. Statistical indicators The accuracy and performance of the derived correlations in predicting global solar radiation were evaluated because of the following statistical error tests; mean bias error (MBE), root mean square error (RMSE), mean percentage error (MPE), maximum absolute relative error (MARE), mean absolute error (MAE), root mean square relative error (RMSRE), coefficient of determination (R2) and t-Test statistic. These error indices are defined as [36]: MBE = 1 n n∑ i =1(Hi,m−Hi,c) (4) RMSR = √√√ 1 n n∑ i=1 (Hi,m−Hi,c) 2 (5) MPE = 1 n n∑ i=1 ( Hi,m−Hi,c Hi,m )x100 (6) MARE = max (∣∣∣∣∣∣ (Hi,m−Hi,cHi,m ∣∣∣∣∣∣ ) (7) MAE = 1 n n∑ i=1 |Hi,m−Hi,c| (8) 3 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 4 Table 1: Models of GSR used in the present research for calculation of monthly average daily of GSR on horizontal surface. No. of Model Modeling of GSR Author/References 1 H/H0 = 0.365 + 0.351 (S/S0) El-Sebaii et al. [14] 2 H/H0 = 0.332 + 0.407 (S/S0) Raja et al. [40] 3 H/H0 = 0.280 + 0.493 (S/S0) Hay [41] 4 H/H0 = 0.412 + 0.211 (S/S0) Elagih et al. [42] 5 H/H0 = 0.272 + 0.384 (S/S0) Yao et al. [43] 6 H/H0 = 0.230 + 0.620 (S/S0) Glover et al. [17] 7 H/H0 = 0.361 + 0.366 (S/S0) Khogali [44] 8 H/H0 = 0.315+ 0.402 (S/S0) Khogali [44] 9 H/H0 = 0.278 + 0.467 (S/S0) Khogali [44] 10 H/H0 = 0.357 + 0.374 (S/S0) Khogali [44] 11 H/H0 = 0.433 + 0.271 (S/S0) Khogali [44] 12 H/H0 = 0.350 + 0.353 (S/S0) Khogali [44] 13 H/H0 = 0.339 + 0.359 (S/S0) Khogali [44] 14 H/H0 = 0.402 + 0.234 (S/S0) Khogali [45] 15 H/H0 = 0.211 + 0.572 (S/S0) Khogali [45] 16 H/H0 = 0.208 + 0.544 (S/S0) Khogali [45] 17 H/H0 = 0.460 + 0.208 (S/S0) Khogali [45] 18 H/H0 = 0.325 + 0.423 (S/S0) Khogali [45] 19 H/H0 = 0.325 + 0.356 (S/S0) Khogali [45] 20 H/H0 = 0.287 + 0.463 (S/S0) Khogali [45] 21 H/H0 = 0.341 + 0.446 (S/S0) Garg [46] 22 H/H0 = 0.309 + 0.482 (S/S0) Garg [46] 23 H/H0 = 0.302 + 0.464 (S/S0) Garg [46] 24 H/H0 = 0.327 + 0.399 (S/S0) Garg [46] 25 H/H0 = 0.293 + 0.459 (S/S0) Garg [46] 26 H/H0 = 0.291 + 0.464 (S/S0) Garg [46] 27 H/H0 = 0.330+ 0.453 (S/S0) Garg [46] 28 H/H0 = 0.286+ 0.467 (S/S0) Garg [46] 29 H/H0 = 0.279+ 0.514 (S/S0) Garg [46] 30 H/H0 = 0.340+ 0.399 (S/S0) Garg [46] 31 H/H0 = 0.393+ 0.357 (S/S0) Garg [46] 32 H/H0 = 0.334+ 0.444 (S/S0) Hussain et al. [19] 33 H/H0 = 0.217+ 0.593 (S/S0) Hussain et al. [19] 34 H/H0 = 0.308+ 0.407 (S/S0) Hussain et al. [19] 35 H/H0 = 0.292+ 0.473 (S/S0) Hussain et al. [19] 36 H/H0 = 0.300+ 0.359 (S/S0) Hussain et al. [19] 37 H/H0 = 0.291+ 0.454 (S/S0) Hussain et al. [19] 38 H/H0 = 0.283+ 0.469 (S/S0) Hussain et al. [19] 39 H/H0 = 0.314+ 0.421 (S/S0) Hussain et al. [19] 40 H/H0 = 0.292+ 0.433 (S/S0) Hussain et al. [19] 41 H/H0 = 0.274+ 0.470 (S/S0) Hussain et al. [19] 42 H/H0 = 0.310+ 0.400 (S/S0) Hussain et al. [19] 43 H/H0 = 0.312+ 0.418 (S/S0) Hussain et al. [19] 44 H/H0 = 0.296+ 0.554 (S/S0) Hussain et al. [19] 45 H/H0 = 0.375+ 0.350 (S/S0) Hussain et al. [19] 46 H/H0 = 0.170+ 0.590 (S/S0) Ibrahim [47] 47 H/H0 = 0.290+ 0.460 (S/S0) Ibrahim [47] 48 H/H0 = 0.220+ 0.550 (S/S0) Ibrahim [47] 49 H/H0 = 0.140+ 0.610 (S/S0) Ibrahim [47] 50 H/H0 = 0.230+ 0.540 (S/S0) Ibrahim [47] 51 H/H0 = 0.520+ 0.230 (S/S0) Ibrahim [47] 52 H/H0 = 0.700+ 0.030 (S/S0) Ibrahim [47] 53 H/H0 = 0.130+ 0.604 (S/S0) Srivastava et al. [48] 54 H/H0 = 0.126+ 0.600 (S/S0) Srivastava et al. [48] 55 H/H0 = 0.336+ 0.339 (S/S0) Srivastava et al. [48] Continue on the next page 4 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 5 Table 1: Models of GSR used in the present research for calculation of monthly average daily of GSR on horizontal surface. No. of Model Modeling of GSR Author/References 56 H/H0 = 0.121+ 0.582 (S/S0) Srivastava et al. [48] 57 H/H0 = 0.105+ 0.569 (S/S0) Srivastava et al. [48] 58 H/H0 = 0.052+ 0.644 (S/S0) Srivastava et al. [49] 59 H/H0 = 0.093+ 0.632 (S/S0) Srivastava et al. [49] 60 H/H0 = 0.138+ 0.556 (S/S0) Srivastava et al. [49] 61 H/H0 = 0.491+ 0.263 (S/S0) Jemaa et al. [50] 62 H/H0 = 0.305+ 0.429 (S/S0) Khorasantzadeh et al. [51] 63 H/H0 = 0.133+ 0.647 (S/S0) Jin et al. [21] 64 H/H0 = 0.570+ 0.011 (S/S0) Suthar et al. [52] 65 H/H0 = 0.269+ 0.489 (S/S0) Marwal [53] 66 H/H0 = 0.228+ 0.311 (S/S0) Katiyar et al. [15] 67 H/H0 = 0.262+ 0.395 (S/S0) Katiyar et al. [54] 68 H/H0 = 0.223+ 0.512 (S/S0) Katiyar et al. [54] 69 H/H0 = 0.229+ 0.531 (S/S0) Katiyar et al. [15] 70 H/H0 = 0.228+ 0.509 (S/S0) Katiyar et al. [15] 71 H/H0 = 0.190+ 0.620 (S/S0) Soler [55] 72 H/H0 = 0.072+ 0.694 (S/S0) Hejase et al. [33] 73 H/H0 = 0.200+ 0.510 (S/S0) Flocas [56] 74 H/H0 = 0.657+ 0.023 (S/S0) Ahmad et al. [7] 75 H/H0 = 0.458+ 0.175 (S/S0) Ahmad et al. [7] 76 H/H0 = 0.177+ 0.692 (S/S0) Jain [57] 77 H/H0 = 0.175+ 0.552 (S/S0) Bahel et al. [58] 78 H/H0 = 0.360+ 0.340 (S/S0) Maduekwe et al. [59] 79 H/H0 = 0.347+ 0.352 (S/S0) Rehman [18] 80 H/H0 = 0.324+ 0.405 (S/S0) Ahmad et al. [7] 81 H/H0 = 0.222+ 0.652 (S/S0) Li et al. [60] 82 H/H0 = 0.170+ 0.680 (S/S0) Yohanna et al. [61] 83 H/H0 = 0.250+ 0.500 (S/S0) Gadiwala et al. [23] 84 H/H0 = 0.309+ 0.368 (S/S0) Chegaar et al. [62] 85 H/H0 = 0.367+ 0.366 (S/S0) Chegaar et al. [62] 86 H/H0 = 0.233+ 0.591 (S/S0) Chegaar et al. [62] 87 H/H0 = 0.175+ 0.712 (S/S0) Onyango et al. [24] 88 H/H0 = 0.180+ 0.620 (S/S0) Onyango et al. [24] 89 H/H0 = 0.278+ 0.648 (S/S0) Ezekwe and Ezeilo [63] 90 H/H0 = 0.260+ 0.431 (S/S0) Sambo [64] 91 H/H0 = 0.250+ 0.450(S/S0) Jackson et a [65] 92 H/H0 = 0.291+ 0.306 (S/S0) Kuye et al. [66] 93 H/H0 = 0.242+ 0.641 (S/S0) Safari et al. [67] 94 H/H0 = 0.320+ 0.309 (S/S0) Tijjani [68] 95 H/H0 = 0.239+ 0.585 (S/S0) Ituen et al. [69] 96 H/H0 = 0.138+ 0.488 (S/S0) Isikwue et al. [70] 97 H/H0 = 0.220+ 0.430 (S/S0) Quansah et al. [26] 98 H/H0 = 0.110+ 0.790 (S/S0) Nwokoye et al. [71] 99 H/H0 = 0.249+ 0.566 (S/S0) Adaramola [72] 100 H/H0 = 0.422+ 0.128 (S/S0) Ampratwum et al. [73] 101 H/H0 = 0.267+ 0.475 (S/S0) Ulgen et al. [74] 102 H/H0 = 0.318+ 0.449 (S/S0) Togrul et al. [75] 103 H/H0 = 0.352+ 0.361 (S/S0) Aziz et al. [76] 104 H/H0 = 0.343+ 0.322 (S/S0) Bakirci [25] 105 H/H0 = 0.288+ 0.547 (S/S0) Sherif [77] 106 H/H0 = 0.244+ 0.415 (S/S0) Okonkwo et al. [78] 107 H/H0 = 0.183+ 0.530 (S/S0) Assi et al. [79] 108 H/H0 = 0.183+ 0.647 (S/S0) Assi et al. [79] 109 H/H0 = 0.495+ 0.593 (S/S0) Katiyar [15] 110 H/H0 = 0.215+ 0.527 (S/S0) Said et al. [80] Continue on the next page 5 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 6 Table 1: Models of GSR used in the present research for calculation of monthly average daily of GSR on horizontal surface. No. of Model Modeling of GSR Author/References 111 H/H0 = 0.226+ 0.418 (S/S0) Tiris et al. [81] 112 H/H0 = 0.340+ 0.320 (S/S0) Veeran et al. [82] 113 H/H0 = 0.270+ 0.650 (S/S0) Veeran et al. [82] 114 H/H0 = 0.140+ 0.570 (S/S0) Lewis [83] 115 H/H0 = 0.313+ 0.474 (S/S0) Jain [57] 116 H/H0 = 0.307+ 0.488 (S/S0) Jain [57] 117 H/H0 = 0.309+ 0.599 (S/S0) Jain [84] 118 H/H0 = 0.241+ 0.488 (S/S0) Luhanga et al. [85] 119 H/H0 = 0.240+ 0.513 (S/S0) Jain and Jain [84] 120 H/H0 = 0.191+ 0.571 (S/S0) Khogali et al. [45] 121 H/H0 = 0.262+ 0.454 (S/S0) Khogali et al. [45] 122 H/H0 = 0.297+ 0.432 (S/S0) Khogali et al. [45] 123 H/H0 = 0.174+ 0.615 (S/S0) Alsaad [86] 124 H/H0 = 0.230+ 0.380 (S/S0) Hinrichsen [87] 125 H/H0 = 0.220+ 0.420 (S/S0) Hinrichsen [87] 126 H/H0 = 0.230+ 0.480 (S/S0) Page [88] 127 H/H0 = 0.307+ 0.312 (S/S0) Abdalla et al. [89] 128 H/H0 = 0.441+ 0.292 (S/S0) Ogolo [90] 129 H/H0 = 0.248+ 0.509 (S/S0) Ogolo [90] 130 H/H0 = 0.278+ 0.483 (S/S0) Ogolo [90] 131 H/H0 = 0.222+ 0.580 (S/S0) Ogolo [90] 132 H/H0 = 0.176+ 0.563 (S/S0) Rensheng et al. [22] 133 H/H0 = 0.206+ 0.546 (S/S0) Louche et al. [91] 134 H/H0 = 0.180+ 0.615 (S/S0) Newland [92] 135 H/H0 = 0.154+ 0.787 (S/S0) Gopinathan et al. [93] 136 H/H0 = 0.196+ 0.721 (S/S0) Gopinathan et al. [93] 137 H/H0 = 0.308+ 0.417 (S/S0) Aras et al. [94] 138 H/H0 = 0.275+ 0.039 (S/S0) Sivamadhavi et al. [95] 139 H/H0 = 0.300+ 0.415 (S/S0) Amoussa [96] 140 H/H0 = 0.238+ 0.522 (S/S0) Amoussa [96] 141 H/H0 = 0.279+ 0.416 (S/S0) Bakirci [25] 142 H/H0 = 0.228+ 0.527 (S/S0) El-Metwally [97] 143 H/H0 = 0.174+ 0.615 (S/S0) Alsaad [86] 144 H/H0 = 0.309+ 0.368 (S/S0) Chegaar and Chibani [62] 145 H/H0 = 0.367+ 0.367 (S/S0) Chegaar and Chibani [62] 146 H/H0 = 0.233+ 0.591 (S/S0) Chegaar and Chibani [62] 147 H/H0 = 0.340+ 0.320 (S/S0) Veeran and Kumar [82] 148 H/H0 = 0.270+ 0.650 (S/S0) Veeran and Kumar [82] 149 H/H0 = 0.267+ 0.475 (S/S0) Ulgen and Hepbasli [74] 150 H/H0 = 0.140+ 0.570 (S/S0) Lewis [83] 151 H/H0 = 0.180+ 0.620 (S/S0) Tiris et al. [81] 152 H/H0 = 0.217+ 0.545 (S/S0) Almorox and Hontoria [16] 153 H/H0 = 0.335+ 0.367 (S/S0) Raja and Twidell [40] 154 H/H0 = 0.215+ 0.527 (S/S0) Said et al. [80] 155 H/H0 = 0.242+ 0.501 (S/S0) Ulgen and Hepbasli [74] 156 H/H0 = -2.81 + 3.78 (S/S0) El-Sebaii et al. [14] 157 H/H0 = 0.324+ 0.405 (S/S0) Ahmed and Ulfat [98] 158 H/H0 = 0.133+ 0.647 (S/S0) Jin et al. [21] 159 H/H0 = 0.230+ 0.380 (S/S0) Akpabio and Etuk [99] 160 H/H0 = 0.332+ 0.311 (S/S0) Ampratwum and Dorvlo [73] 161 H/H0 = 0.242+ 0.356 (S/S0) Ampratwum and Dorvlo [73] 162 H/H0 = 0.180+ 0.600 (S/S0) Benson et al. [100] 163 H/H0 = 0.240+ 0.530 (S/S0) Benson et al. [100] 164 H/H0 = 0.180+ 0.660 (S/S0) Rietveld [29] 165 H/H0 = 0.200+ 0.600 (S/S0) Rietveld [29] Continue on the next page 6 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 7 Table 1: Models of GSR used in the present research for calculation of monthly average daily of GSR on horizontal surface. No. of Model Modeling of GSR Author/References 166 H/H0 = 0.220+ 0.580 (S/S0) Rietveld [29] 167 H/H0 = 0.200+ 0.620 (S/S0) Rietveld [29] 168 H/H0 = 0.240+ 0.520 (S/S0) Rietveld [29] 169 H/H0 = 0.240+ 0.530 (S/S0) Rietveld [29] 170 H/H0 = 0.230+ 0.530 (S/S0) Rietveld [29] 171 H/H0 = 0.220+ 0.550 (S/S0) Rietveld [29] 172 H/H0 = 0.200+ 0.590 (S/S0) Rietveld [29] 173 H/H0 = 0.290+ 0.600 (S/S0) Rietveld [29] 174 H/H0 = 0.170+ 0.660 (S/S0) Rietveld [29] 175 H/H0 = 0.180+ 0.650 (S/S0) Rietveld [29] 176 H/H0 = 0.276+ 0.648 (S/S0) Ezekwe and Ezeilo [63] 177 H/H0 = 0.260+ 0.620 (S/S0) Ezekwe and Ezeilo [63] 178 H/H0 = 0.210+ 0.490 (S/S0) Ezekwe and Ezeilo [63] 179 H/H0 = 0.200+ 0.740 (S/S0) Arinze and Obi [101] 180 H/H0 = 0.413+ 0.241 (S/S0) Sambo [64] 181 H/H0 = 0.160+ 0.530 (S/S0) Folayan and Ogunbiyi [102] 182 H/H0 = 0.263+ 0.374 (S/S0) Banna and Gnininri [103] 183 H/H0 = 0.253+ 0.357 (S/S0) Banna and Gnininri [103] 184 H/H0 = 0.271+ 0.300 (S/S0) Banna and Gnininri [103] 185 H/H0 = 0.208+ 0.748 (S/S0) Falayi et al. [104] 186 H/H0 = 0.018+ 1.139 (S/S0) Augustine and Nnabuchi [105] 187 H/H0 = 0.295+ 0.306 (S/S0) Augustine and Nnabuchi [105] 188 H/H0 = 0.191+ 0.433 (S/S0) Augustine and Nnabuchi [105] 189 H/H0 = 0.278+ 0.331 (S/S0) Augustine and Nnabuchi [105] 190 H/H0 = 0.290+ 0.420 (S/S0) Augustine and Nnabuchi [105] 191 H/H0 = 0.290+ 0.253 (S/S0) Augustine and Nnabuchi [105] 192 H/H0 = 0.336+ 0.247 (S/S0) Augustine and Nnabuchi [105] 193 H/H0 = 0.219+ 0.638 (S/S0) Olayinka [106] 194 H/H0 = 0.211+ 0.629 (S/S0) Olayinka [106] 195 H/H0 = 0.163+ 0.806 (S/S0) Olayinka [106] 196 H/H0 = 0.450+ 0.274 (S/S0) Olayinka [106] 197 H/H0 = 0.320+ 0.308 (S/S0) Tijjani [68] 198 H/H0 = 0.170+ 0.680 (S/S0) Yohanna and Itodo [107] 199 H/H0 = 0.239+ 0.585 (S/S0) Ituen [69] 200 H/H0 = 0.249+ 0.566 (S/S0) Adaramola [72] 201 H/H0 = 0.300+ 0.530 (S/S0) Yakubu and Medugu [108] 202 H/H0 = 0.287+ 0.547 (S/S0) Musa et al. [109] 203 H/H0 = 0.138+ 0.488 (S/S0) Isikwue et al. [70] 204 H/H0 = 0.239+ 0.717 (S/S0) Kolebaje and Mustapha [110] 205 H/H0 = 0.258+ 0.612 (S/S0) Kolebaje and Mustapha [110] 206 H/H0 = 0.250+ 0.643 (S/S0) Kolebaje and Mustapha [110] 207 H/H0 = 0.286+ 0.537 (S/S0) Kolebaje and Mustapha [110] 208 H/H0 = 0.318+ 0.513 (S/S0) Kolebaje and Mustapha [110] 209 H/H0 = 0.310+ 0.540 (S/S0) Kolebaje and Mustapha [110] 210 H/H0 = 0.194+ 0.398 (S/S0) Ohunakin et al. [111] 211 H/H0 = 0.115+ 0.567 (S/S0) Solomon [112] 212 H/H0 = 0.389+ 0.358 (S/S0) Gana and Akpootu [113] 213 H/H0 = 0.417+ 0.316 (S/S0) Gana and Akpootu [113] 214 H/H0 = 0.334+ 0.449 (S/S0) Gana and Akpootu [113] 215 H/H0 = 0.416+ 0.317 (S/S0) Gana and Akpootu [113] 216 H/H0 = 0.453+ 0.268 (S/S0) Gana and Akpootu [113] 217 H/H0 = 0.386+ 0.360 (S/S0) Gana and Akpootu [113] 218 H/H0 = 0.083+ 0.684 (S/S0) Afungchui and Neba [114] 219 H/H0 = 0.286+ 0.579 (S/S0) Afungchui and Neba [114] 220 H/H0 = 0.253+ 0.427 (S/S0) Afungchui and Neba [114] Continue on the next page 7 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 8 Table 1: Models of GSR used in the present research for calculation of monthly average daily of GSR on horizontal surface. No. of Model Modeling of GSR Author/References 221 H/H0 = 0.303+ 0.484 (S/S0) Afungchui and Neba [114] 222 H/H0 = 0.314+ 0.479 (S/S0) Afungchui and Neba [114] 223 H/H0 = 0.005+ 1.313 (S/S0) Sarsah and Uba [115] 224 H/H0 = 0.244+ 0.415 (S/S0) Okonkwo and Nwokoye [78] 225 H/H0 = 0.220+ 0.430 (S/S0) Quansah et al. [26] 226 H/H0 = 0.288+ 0.547 (S/S0) Sheriff [77] 227 H/H0 = 0.192+ 0.442 (S/S0) Ike [116] 228 H/H0 = 0.045+ 0.051 (S/S0) Sani et al. [117] 229 H/H0 = 0.010+ 0.750 (S/S0) Adesina et al. [118] 230 H/H0 = 0.240+ 0.310 (S/S0) Olatona and Adeleke [119] 231 H/H0 = 0.110+ 0.790 (S/S0) Okonkwo et al. [78] 232 H/H0 = 0.295+ 0.532 (S/S0) Innocent et al. [120] 233 H/H0 = 0.250+ 0.522 (S/S0) Boluwaji and Onyedi [121] 234 H/H0 = 0.314+ 0.491 (S/S0) Boluwaji and Onyedi [121] 235 H/H0 = 0.315+ 0.433 (S/S0) Boluwaji and Onyedi [121] 236 H/H0 = 0.130+ 0.620 (S/S0) Coulibaly and Ouedoraogo [122] 237 H/H0 = 0.210+ 0.450 (S/S0) Coulibaly and Ouedoraogo [122] 238 H/H0 = 0.170+ 0.470 (S/S0) Coulibaly and Ouedoraogo [122] 239 H/H0 = 0.210+ 0.460 (S/S0) Coulibaly and Ouedoraogo [122] 240 H/H0 = 0.150+ 0.460 (S/S0) Coulibaly and Ouedoraogo [122] 241 H/H0 = 0.180+ 0.530 (S/S0) Coulibaly and Ouedoraogo [122] 242 H/H0 = 0.210+ 0.430 (S/S0) Coulibaly and Ouedoraogo [122] 243 H/H0 = 0.230+ 0.400 (S/S0) Coulibaly and Ouedoraogo [122] 244 H/H0 = 0.180+ 0.490 (S/S0) Coulibaly and Ouedoraogo [122] 245 H/H0 = 0.219+ 0.614 (S/S0) Okundamiya et al. [123] 246 H/H0 = 0.236+ 0.621 (S/S0) Okundamiya et al. [123] 247 H/H0 = 0.479+ 0.246 (S/S0) Okundamiya et al. [123] 248 H/H0 = 0.270+ 0.240 (S/S0) Ayodele and Ogunjuyigbe [124] 249 H/H0 = 0.292+ 0.284 (S/S0) +0.211 (S/S0)2 Katiyar et al. [15] 250 H/H0 = -0.197+ 0.177 (S/S0) -0.912 (S/S0)2 Katiyar et al. [15] 251 H/H0 = 0.129+ 0.933 (S/S0) -0.503 (S/S0)2 Katiyar et al. [15] 252 H/H0 = 0.234+ 0.465 (S/S0) -0.043 (S/S0)2 Katiyar et al. [15] 253 H/H0 = 0.184+ 0.845 (S/S0) -0.280 (S/S0)2 Almorox et al. [16] 254 H/H0 = 0.181+ 0.895 (S/S0) -0.361 (S/S0)2 Akinoglu et al. [125] 255 H/H0 = 0.222+ 0.705 (S/S0) -0.217 (S/S0)2 Ogelman [30] 256 H/H0 = 1.111- 1.828 (S/S0) +1.660 (S/S0)2 Ogelman [30] 257 H/H0 = 0.168+ 0.835 (S/S0) +0.320 (S/S0)2 Ogolo [90] 258 H/H0 = 0.154+ 0.933 (S/S0) +0.370 (S/S0)2 Ogolo [90] 259 H/H0 = 0.719- 1.067 (S/S0) +1.145 (S/S0)2 Ogolo [90] 260 H/H0 = 0.339+ 0.287 (S/S0) +0.119 (S/S0)2 Singh et al. [5] 261 H/H0 = 0.154+ 1.172 (S/S0) -0.705 (S/S0)2 Aras et al. [94] 262 H/H0 = 0.100+ 0.874 (S/S0) -0.255 (S/S0)2 Togrul et al. [126] 263 H/H0 = 0.180+ 1.160 (S/S0) -0.910 (S/S0)2 Said et al. [80] 264 H/H0 = 0.348+ 0.320 (S/S0) +0.070 (S/S0)2 Maduekwe et al. [59] 265 H/H0 = 0.222+ 0.538 (S/S0) +0.152 (S/S0)2 Ahmad et al. [7] 266 H/H0 = 0.069+1.326 (S/S0) - 0.667 (S/S0)2 Onyango et al. [24] 267 H/H0 = -2.767+ 9.207 (S/S0) -6.350 (S/S0)2 Safari et al. [67] 268 H/H0 = -2.767+9.207 (S/S0) -0.674 (S/S0)2 Tijjani [68] 269 H/H0 = 0.025+ 1.125 (S/S0) -0.308 (S/S0)2 Nwokoye et al. [71] 270 H/H0 = 0.282+ 0.572 (S/S0) -0.224 (S/S0)2 Bakirci [25] 271 H/H0 = 0.100+ 1.020 (S/S0) -0.440 (S/S0)2 Rietveld [29] 272 H/H0 = 0.326+ 0.344 (S/S0) +0.102 (S/S0)2 Rietveld [29] 273 H/H0 = 0.692- 0.068 (S/S0) +0.118 (S/S0)2 Sekhar et al. [40] 274 H/H0 = 0.264+ 0.516 (S/S0) -0.005 (S/S0)2 Sekhar et al. [40] 275 H/H0 = 0.494+ 0.064 (S/S0) +0.173 (S/S0)2 Sekhar et al. [40] Continue on the next page 8 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 9 Table 1: Models of GSR used in the present research for calculation of monthly average daily of GSR on horizontal surface. No. of Model Modeling of GSR Author/References 276 H/H0 = 0.273+ 0.468 (S/S0) +0.018 (S/S0)2 Sekhar et al. [40] 277 H/H0 = 0.263+ 0.514 (S/S0) -0.008 (S/S0)2 Sekhar et al. [40] 278 H/H0 = 0.403+ 0.788 (S/S0) -0.224 (S/S0)2 Sekhar et al. [40] 279 H/H0 = 0.279+ 0.037 (S/S0) +0.009 (S/S0)2 Sekhar et al. [40] 280 H/H0 = 0.285+ 0.036 (S/S0) +0.002 (S/S0)2 Sivamadhavi et al. [95] 281 H/H0 = -0.067+ 1.772 (S/S0) -1.249 (S/S0)2 Sivamadhavi et al. [95] 282 H/H0 = 0.635+ 0.097 (S/S0) -0.026 (S/S0)2 Elagib et al. [42] 283 H/H0 = 0.140+ 0.613 (S/S0) +0.035 (S/S0)2 Assi et al. [79] 284 H/H0 = 0.487+ 0.612 (S/S0) -0.018 (S/S0)2 Jin et al. [21] 285 H/H0 = 0.434+ 0.233 (S/S0) +0.166 (S/S0)2 Katiyar et al. [15] 286 H/H0 = -0.359+ 1.763 (S/S0) -0.659 (S/S0)2 Jemaa et al. [50] 287 H/H0 = 0.215+ 0.625 (S/S0) -0.221 (S/S0)2 Hejase et al. [33] 288 H/H0 = 0.748- 0.949 (S/S0) +1.042 (S/S0)2 Togrul et al. [126] 289 H/H0 = 0.231+ 0.463 (S/S0) – 0.044 (S/S0)2 Ravichandran et al. [127] 290 H/H0 = 0.189+ 0.845 (S/S0) – 0.390 (S/S0)2 Okonkwo et al. [78] 291 H/H0 = 0.388+ 0.580 (S/S0) -0.407 (S/S0)2 Assi et al. [79] 292 H/H0 = 0.086+ 1.158 (S/S0) -0.556 (S/S0)2 Suthar et al. [52] 293 H/H0 =-0.140+2.52 (S/S0) -3.71 (S/S0)2+2.24(S/S0)3 Samuel [128] 294 H/H0 =-2.43+11.95 (S/S0) -16.75 (S/S0)2+7.96(S/S0)3 Ertekin and Yaldiz [129] 295 H/H0 =0.230+0.380 (S/S0)+0.469 (S/S0)2-0.366(S/S0)3 Almorox and Hontoria [16] 296 H/H0 =0.81-3.34(S/S0)+7.38 (S/S0)2-4.51(S/S0)3 Lewis [83] 297 H/H0 =0.241+0.363(S/S0)+0.459 (S/S0)2-0.371(S/S0)3 Ulgen and Hepbasli [130] 298 H/H0 =0.152+1.133(S/S0)-1.113 (S/S0)2+0.452(S/S0)3 Tahran and Sari [131] 299 H/H0 =0.128+0.725(S/S0)-0.229 (S/S0)2+0.184(S/S0)3 Jin et al. [21] 300 H/H0 =0.483-0.616(S/S0)+1.893 (S/S0)2-1.098(S/S0)3 Aras et al. [94] 301 H/H0 =0.150+1.145(S/S0)-1.474 (S/S0)2+0.963(S/S0)3 Rensheng et al. [22] 302 H/H0 =0.631-0.725(S/S0)+1.208 (S/S0)2-0.463(S/S0)3 Bakirci [25] 303 H/H0 =0.171+0.026(S/S0)+2.100 (S/S0)2-1.640(S/S0)3 Burari et al. [132] 304 H/H0 =0.197+0.981(S/S0)+0.053 (S/S0)2-1.317(S/S0)3 Kolebaje and Mustapha [110] 305 H/H0 =0.231+0.548(S/S0)+1.884 (S/S0)2-4.215(S/S0)3 Kolebaje and Mustapha [110] 306 H/H0 =0.400+0.137(S/S0)+0.001 (S/S0)2+1.139(S/S0)3 Kolebaje and Mustapha [110] 307 H/H0 =0.306+0.464(S/S0)+0.001 (S/S0)2-0.189(S/S0)3 Kolebaje and Mustapha [110] 308 H/H0 =0.091+1.401(S/S0)-0.928 (S/S0)2-0.001(S/S0)3 Kolebaje and Mustapha [110] 309 H/H0 =0.851+0.869(S/S0)+0.001 (S/S0)2-1.364(S/S0)3 Kolebaje and Mustapha [110] 310 H/H0 =0.050+0.971(S/S0)-0.200 (S/S0)2+0.001(S/S0)3 Nwokoye and Okonkwo [71] 311 H/H0 =1.561-0.365(S/S0)+0.037 (S/S0)2-0.001(S/S0)3 Sani et al. [117] 312 H/H0 =0.25+0.38(S/S0)-0.210 (S/S0)2+0.074(S/S0)3 Ayodele and Ogunjuyigbe [124] 313 H/H0 =2.72-11.01(S/S0)+ 17.43 (S/S0)2-8.654(S/S0)3 Katiyar et al. [15] 314 H/H0 =1.38-6.33(S/S0)+ 12.75 (S/S0)2-7.66(S/S0)3 Katiyar et al. [15] 315 H/H0 =0.294+0.086(S/S0)+ 0.77 (S/S0)2-0.436(S/S0)3 Katiyar et al. [15] 316 H/H0 =0.512-0.993(S/S0)+ 2.53 (S/S0)2-1.33(S/S0)3 Katiyar et al. [15] 317 H/H0 =0.16+0.87(S/S0)- 0.61(S/S0)2+0.34(S/S0)3 Bahel et al. [58] 318 H/H0 =0.230+ 0.381(S/S0)+ 0.469(S/S0)2- 0.366(S/S0)3 Almorox et al. [15] 319 H/H0 =0.285+0.259(S/S0)+0.617 (S/S0)2-0.483(S/S0)3 Ulgen et al. [130] 320 H/H0 =0.179+0.981(S/S0)-0.296 (S/S0)2-0.266(S/S0)3 Togrul et al. [126] 321 H/H0 =0.475-0.898(S/S0)+2.773 (S/S0)2-1.54(S/S0)3 Onyango et al. [24] 322 H/H0 =0.400+0.214(S/S0)+0.153 (S/S0)2+0.119(S/S0)3 Jemaa et al. [50] 323 H/H0 =1.324-4.942(S/S0)+8.714 (S/S0)2-4.549(S/S0)3 Khorasanizadeh et al. [51] 324 H/H0 =-0.898+5.936(S/S0)-7.493 (S/S0)2+3.326(S/S0)3 Khorasanizadeh et al. [51] 325 H/H0 =0.147+0.099(S/S0)-0.009 (S/S0)2+0.0005(S/S0)3 Sivamadhavi et al [95] 326 H/H0 =0.160+0.093(S/S0)-0.008 (S/S0)2+0.0004(S/S0)3 Sivamadhavi et al [95] 327 H/H0 =-1.88+12.65(S/S0)-21.87 (S/S0)2+12.37(S/S0)3 Yaniktepe et al. [27] 328 H/H0 =0.420+0.529(S/S0)-0.777 (S/S0)2+0.521(S/S0)3 Assi et al. [79] 329 H/H0 =0.441+0.829(S/S0)-0.977 (S/S0)2+0.421(S/S0)3 Assi et al. [79] 330 H/H0 =0.164-683(S/S0)+1.073 (S/S0)2-0.0009(S/S0)3 Soler [55] Continue on the next page 9 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 10 Table 1: Models of GSR used in the present research for calculation of monthly average daily of GSR on horizontal surface. No. of Model Modeling of GSR Author/References 331 H/H0 =10.09-38.46(S/S0)+50.75 (S/S0)2-21.83(S/S0)3 Hejase et al. [33] 332 H/H0 =0.346+0.422(S/S0)+2.33 (S/S0)2-1.634(S/S0)3 Marwal [53] 333 H/H0 =0.11-0.01(S/S0)+5.3 (S/S0)2-10.4(S/S0)3+5.6(S/S0)4 Marwal [53] 334 H/H0 =0.40-0.85(S/S0)+4.1 (S/S0)2-4.9(S/S0)3+2.1(S/S0)4 Zabara [132] 335 H/H0 =0.550 + 0.656(S/S0) + 0.721 log(S/S0) Sarsah and Uba [115] 336 H/H0 = -0.337 + 0.656(S/S0) + 0.406 exp(S/S0) Sarsah and Uba [115] 337 H/H0 = 0.809 + 0.386 log(S/S0) Onyango et al. [24] 338 H/H0 = 0.8336 + 1.0573 log(S/S0) Ampratwum et al. [73] 339 H/H0 = 0.6050 + 0.1917 ln(S/S0) Yao et al. [43] 340 H/H0 = 0.34 + 0.40 (S/S0)+ 0.17 ln(S/S0) Newland [92] 341 H/H0 = 0.753 + 0.561 ln(S/S0) Hejase et al. [33] 342 H/H0 = 0.7166 + 0.2705 ln(S/S0) Marwal [53] 343 H/H0 = 0.6290 + 0.1053 log(S/S0) Assi et al. [79] 344 H/H0 = 0.6858 + 0.01531 log(S/S0) Assi et al. [79] 345 H/H0 = 0.880 − 0.1295 (S/S0) + 0.666 ln(S/S0) Hejase et al. [33] 346 H/H0 = 0.7621− 0.1627(S/S0) + 0.08576 log(S/S0) Assi et al. [79] 347 H/H0 = 0.7081− 0.0127(S/S0) + 0.08576 log(S/S0) Assi et al. [79] 348 H/H0 = 0.3396 e0.8985(S/So) Togrul et al. [126] 349 H/H0 = 0.3593 e 0.6093(S/So) Khorasanizadeh et al. [51] 350 H/H0 = 0.3593 e0.6093(S/So) Yao et al. [43] 351 H/H0 = 0.2674 e1.0391(S/So) Elagib et al. [42] 352 H/H0 = 0.4248 e0.3947(S/So) Hejase et al. [33] 353 H/H0 = 0.263 e1.088(S/So) Assi et al. [79] 354 H/H0 = 0.5399 e0.1630(S/So) Marwal [53] 355 H/H0 = 0.3158 e0.9251(S/So) Assi et al. [79] 356 H/H0 = 0.6416 e0.099(S/So) Almorox et al. [15] 357 H/H0 = −0.0271 + 0.3096 × e(S/So) Khorasanizadeh et al. [51] 358 H/H0 = 0.3661 + 0.1133 e(S/So) Togrul et al. [126] 359 H/H0 = 0.7316 (S/So)0.4146 Nwokoye et al. [71] 360 H/H0 = 0.6673 (S/So)0.5343 Yao et al. [43 ] 361 H/H0 = 0.6128 (S/So)0.2499 Elagib et al. [42] 362 H/H0 = 0.7399 (S/So)0.5201 Marwal [53] 363 H/H0 = e−0.4698 (S/So)0.034 Assi et al. [79] 364 H/H0 = e−0.4058 (S/So)0.00014 Assi et al. [79] 365 H/H0 = e−0.1361 (S/So)1.7156 Ampratwum et al. [73] 366 H/H0 = -0.864 +1.862 (S/So)2.344 El-Sebaii et al. [14] 367 H/H0 = -3.386 +0.220(Tmax)- 0.003 (Tmax)2 Okonkwo and Nwokoye [78] 368 H/H0 = -3.388.434 +0.638(Tmean)-0.012 (Tmean)2 Ohunakin [111] 369 H/H0 = -0.987 +5.256(TR)- 4.536 (TR)2 Okonkwo and Nwokoye [78] 370 H/H0 = 0.24 + e0.064( T/S o) Ayodele and Ogunjuyigbe [124] 371 H/H0 = 0.45 +0.39 log(∆T/So) Ayodele and Ogunjuyigbe [124] 372 H/H0 = 0.219 + 0.526 (S/So) + 0.004 w El-Metwally [97] 373 H/H0 = -0.107 + 0.70 (S/S0) – 0.0025T + 0.004 RH Maghrabi [12] 374 H/H0 = -0.139+0.229 (S/So)+0.01T+0.004V+0.002RH+0.002PS Trabea and Shaltout [134] 375 H/H0 = -1.92 + 2.60 (S/So) + 0.006T El-Sebaii et al. [14] 376 H/H0 = -1.62 + 2.24 (S/So) + 0.332RH El-Sebaii et al. [14] 377 H/H0 = 0.388 cosφ + 0.367 (S/So) Raja et al. [40] 378 H/H0 = 0.388 cosφ + 0.407 (S/So) Raja et al. [40] 379 H/H0 = 0.3092 cosφ + 0.4931 (S/So) Ulgen et al. [74] 380 H/H0 = 0.12 + 0.58 (S/So) + 7.56 x 10−5h Lewis [83] 381 H/H0 = 0.28 – 0.141 cosφ + 0.026h + 0.542(S/So) Rensheng et al. [22] 382 H/H0 = 0.122 + 0.001 cosφ + 2.57 x 10−2h+ 0.543(S/So) Rensheng et al. [22] 383 H/H0 =0.275+4.27x10−5ψ+0.141cosφ + 2.63x10−2h+0.542(S/So) Rensheng et al. [22] 384 H/H0 =0.117+4.11x10−5ψ+0.001cosφ + 2.59x10−2h+0.543(S/So) Rensheng et al. [22] 385 H/H0 =-0.117+4.11x10−5ψ+0.001cosφ + 2.59x10−2h+0.543(S/So) Rensheng et al. [22] Continue on the next page 10 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 11 Table 1: Models of GSR used in the present research for calculation of monthly average daily of GSR on horizontal surface. No. of Model Modeling of GSR Author/References 386 H/H0 =0.001φ+2.41x10−2h+0.109+1.03(S/So)-1.22(S/So)2+0.79(S/So)3 Rensheng et al. [22] 387 H/H0 = 3.614 – 0.364 (RH)2 Kolebaje et al. [110] 388 H/H0 = 0.507 – 0.863 (C/Co)2 Augustine and Nnabuchi [105] 389 H/H0 = 1.309 + 0.601 (S/So) – 0.999(TR) – 0.0129 (Tmean) Falayi et al. [104] 390 H/H0 = -0.839 + 0.247 (S/So) + 0.0439 (Tmean) Olayinka [106] 391 H/H0 = 1.395 + 1.591 (S/So) - 0.046 (Tmean) Ituen et al. [69] 392 H/H0 = -6.874 + 7.31 (S/So)0.0237 + 0.117 (ln∆T) Okundamiya et al. [123] 393 H/H0 = 5.981 – 1.991[(∆T+RH)/N]0.5 Kolebaje et al. [110] 394 H/H0 = 2.931 – 0.570[(∆T+RH)/N]0.5 – 1.214 (TR) Kolebaje et al. [110] 395 H/H0 = 0.00689 + 0.0221 (S/So) + 0.968 (C/Co)) Olayinka [106] 396 H/H0 = -0.423 + 0.301 (S/So) + 0.0256 (Tmax) + 0.0725 (RH/100) Olayinka [106] 397 H/H0 = -0.183 + 0.279 (S/So) + 0.170 (TR)) + 0.0179 (Tmax) – 0.0128 (C) Okundamiya et al. [123] 398 H/H0 = -0.012+0.001(S/So)+0.999(C/Co)+0.00053(Tmax) -0.001 (RH/100) Olayinka [106] 399 H/H0 = -0.518 cosφ +19.219 cos n +5.513(Tmax)+125.78(S/So) +21.683 (Tmax/RH)+5.634(Tmax/RH)2-2.693(cos φ cos n) – 33.15 Ajayi et al. [135] 400 H/H0 = 4.251-4.188 cosφ +0.0437h +[(-10.577+11.451 cosφ-0.0832h) (S/So)] +[(12.725-13.099 cosφ+0.1h) (S/So)2] Jin et al. [21] Table 2: The variable statistical indicator results and GPI of models in the present research. Model MBE RMSE MPE MARE MAE RMSRE R2 t-test U95 GPI 1 -2.314 2.985 -8.365 0.225 2.356 0.214 0.4689 1.355 3.256 0.2546 2 -2.561 3.214 -7.325 0.365 1.325 0.145 0.5221 1.214 4.235 0.3165 3 -3.214 3.654 -11.325 0.335 1.965 0.125 0.4462 1.569 3.245 0.1245 4 -1.685 4.321 -9.356 0.229 2.365 0.0985 0.4391 1.689 5.325 -0.3269 5 -2.345 3.698 -8.365 0.256 3.245 0.168 0.4506 1.474 6.325 0.2148 6 -2.865 2.654 -15.365 0.654 2.654 0.265 0.4406 1.677 7.235 -0.3215 7 -1.324 1.356 -3.985 0.452 3.126 0.354 0.4425 1.654 3.256 -0.2352 8 -2.365 3.245 -11.325 0.362 2.547 0.247 0.5397 0.792 4.325 0.3256 9 -1.954 4.325 -9.365 0.289 2.658 0.298 0.5674 0.608 4.329 0.4243 10 -2.451 5.326 -14.325 0.654 3.658 0.345 0.6025 0.476 3.456 0.5263 11 1.365 3.265 -11.365 0.345 1.654 0.145 0.5698 0.589 6.354 0.4512 12 -3.214 2.965 -8.325 0.542 1.857 0.189 0.5385 0.795 4.358 0.3252 13 -2.324 3.321 15.362 0.654 1.365 0.247 0.3256 2.397 3.658 -1.2358 14 1.895 3.478 11.325 0.356 5.326 0.268 -0.5485 2.952 2.359 -2.3145 15 -1.987 2.689 -6.325 0.412 1.325 0.214 0.3251 2.399 5.326 -1.2452 16 -2.654 1.356 -11.524 0.325 4.325 0.236 -0.4529 2.482 4.365 -1.5641 17 -2.345 1.325 -14.356 0.421 3.256 0.314 0.5374 0.796 5.326 0.3248 18 -2.852 3.654 -3.568 0.229 1.325 0.198 0.5621 0.658 6.314 0.3849 19 -1.358 4.245 -13.654 0.325 4.325 0.145 0.4658 1.361 5.325 0.2508 20 -3.256 4.321 -9.365 0.361 2.698 0.168 -0.4789 2.689 4.325 -2.1465 21 -2.378 3.265 -8.325 0.358 2.754 0.325 0.4215 1.785 3.658 -0.359 22 -1.896 2.658 -7.256 0.568 2.568 0.254 0.3551 2.363 6.325 -0.6548 23 1.325 2.965 11.356 0.689 2.314 0.315 -0.5491 3.214 4.356 -2.3149 24 1.658 3.354 -13.654 0.452 1.658 0.125 0.2398 2.416 5.326 -1.3267 25 -2.356 3.658 9.324 0.365 2.359 0.168 -0.4582 2.511 4.325 -1.6549 26 -2.089 1.965 5.326 0.654 1.325 0.215 0.4428 1.595 7.321 0.0214 27 1.325 2.314 -3.865 0.235 2.365 0.314 0.4431 1.592 5.326 0.0658 28 -1.958 3.254 -9.541 0.214 1.658 0.326 0.4457 1.572 6.325 0.1241 29 -1.689 1.658 -12.654 0.425 2.365 0.289 0.4388 1.691 5.659 -0.3275 30 -1.365 2.365 -15.325 0.325 3.256 0.135 0.3631 2.325 6.458 -0.5425 31 -2.123 3.654 -9.325 0.658 2.358 0.147 0.5369 0.798 7.321 0.3246 32 2.912 2.365 -14.235 0.552 1.358 0.168 0.4504 1.478 6.325 0.2144 Continue on the next page 11 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 12 Table 2: The variable statistical indicator results and GPI of models in the present research. Model MBE RMSE MPE MARE MAE RMSRE R2 t-test U95 GPI 33 1.325 1.698 -8.325 0.356 4.325 0.128 0.6098 0.362 5.326 0.6523 34 -2.365 1.965 6.325 0.654 5.326 0.068 0.4469 1.556 4.325 0.1259 35 -2.345 2.356 7.325 0.658 3.245 0.087 0.4501 1.479 7.231 0.2141 36 5.632 3.654 -21.324 0.456 2.359 0.256 0.5405 0.785 6.325 0.3264 37 -1.356 4.325 -23.654 0.425 2.145 0.326 0.6061 0.382 5.326 0.6247 38 -2.356 4.658 -9.325 0.268 1.365 0.411 0.6023 0.479 4.365 0.5245 39 -2.314 3.265 -11.456 0.689 1.987 0.358 0.2356 2.418 5.326 -1.3269 40 -1.325 2.324 -13.698 0.754 2.365 0.265 -0.4585 2.524 6.541 -1.6587 41 -1.325 1.658 -23.654 0.452 2.895 0.124 0.5481 0.703 7.325 0.3548 42 -1.365 3.256 -22.35 0.556 3.245 0.325 0.5233 0.988 5.325 0.3179 43 -2.895 4.325 -4.325 0.542 1.658 0.258 0.4355 1.693 4.369 -0.3277 44 -3.256 3.658 5.365 0.562 5.325 0.168 0.4404 1.679 3.256 -0.3246 45 -2.365 1.689 9.356 0.345 4.235 0.178 0.4451 1.575 4.325 0.1237 46 -1.365 1.885 6.325 0.658 2.356 0.165 0.4478 1.551 4.658 0.1582 47 -1.985 2.658 -9.325 0.425 1.547 0.245 0.4597 1.367 3.256 0.2479 48 -1.568 2.365 -3.567 0.635 1.325 0.187 0.3634 2.322 3.256 -0.5247 49 -3.245 3.258 -5.632 0.411 2.356 0.314 0.4216 1.784 3.652 -0.3456 50 -3.325 3.326 -11.125 0.635 1.854 0.265 0.4509 1.472 4.214 0.2245 51 -1.235 4.457 -6.325 0.458 1.658 0.124 0.3691 2.242 5.635 -0.4154 52 1.325 3.334 -8.658 0.362 2.314 0.197 0.5036 1.239 6.785 0.2874 53 2.356 2.654 -15.452 0.578 2.224 0.311 0.4218 1.762 7.224 -0.3364 54 -1.635 1.865 -3.235 0.632 1.589 0.368 0.3685 2.254 3.568 -0.4159 55 -2.658 2.356 -3.589 0.478 2.125 0.254 0.5148 1.235 4.457 0.2958 56 -1.563 3.258 -9.785 0.365 2.256 0.234 0.5363 0.802 4.892 0.3242 57 -2.654 5.689 -11.325 0.574 3.324 0.214 0.6055 0.405 3.635 0.6217 58 1.758 3.754 -11.235 0.392 1.254 0.214 0.6235 0.263 5.325 0.7457 59 -3.635 2.635 12.356 0.457 1.541 0.136 0.6124 0.359 3.258 0.6548 60 -2.568 3.547 9.325 0.458 1.254 0.225 0.6065 0.379 4.325 0.6256 61 1.235 3.245 -3.256 0.398 5.124 0.475 -0.6165 4.214 2.865 -2.4647 62 2.356 2.365 -3.658 0.689 1.254 0.194 -0.4588 2.529 4.256 -1.6589 63 -2.235 1.568 -11.425 0.478 4.145 0.241 0.5984 0.493 4.321 0.4586 64 -2.698 1.856 -11.325 0.547 4.325 0.378 0.5511 0.689 5.785 0.3574 65 -2.445 3.224 -5.326 0.368 2.314 0.236 0.5408 0.782 5.325 0.3275 66 -1.658 4.785 -11.326 0.457 4.325 0.168 0.4892 1.245 4.925 0.2691 67 -3.568 4.658 -8.325 0.658 2.214 0.192 -0.6152 3.342 4.658 -2.4593 68 1.325 4.325 -6.324 0.457 2.325 0.378 0.3456 2.375 5.647 -0.6587 69 -1.632 2.478 -7.658 0.365 2.412 0.178 0.3325 2.386 6.478 -0.7456 70 1.785 1.547 11.785 0.745 1.389 0.265 -0.6189 4.465 5.324 -2.6582 71 1.542 1.352 -12.658 0.653 1.658 0.197 -0.4251 2.475 3.958 -1.4576 72 -2.457 3.325 4.259 0.447 2.658 0.127 -0.4325 2.476 5.236 -1.4579 73 -2.245 1.568 -4.658 0.556 1.785 0.247 0.4432 1.589 6.526 0.0854 74 1.587 2.658 -3.325 0.785 2.325 0.368 0.4467 1.558 5.458 0.1252 75 -1.458 3.564 -9.659 0.347 1.478 0.295 0.4408 1.675 5.325 -0.2569 76 -1.254 1.455 4.325 0.658 2.365 0.223 0.3614 2.337 5.952 -0.6245 77 -1.147 2.365 11.325 0.457 3.447 0.189 0.5416 0.772 6.258 0.3459 78 -2.658 3.235 6.354 0.857 2.365 0.245 0.5869 0.558 5.635 0.4563 79 2.365 2.125 -10.325 0.478 1.857 0.195 0.5098 1.238 6.854 0.2952 80 1.568 1.258 -11.325 0.457 4.447 0.168 0.6028 0.474 5.445 0.5268 81 -2.689 1.635 6.985 0.458 3.256 0.092 0.4495 1.524 4.658 0.2136 82 -2.785 2.689 7.658 0.687 3.365 0.124 0.5361 0.805 3.352 0.3238 83 5.258 3.411 -15.324 0.578 2.411 0.369 0.5686 0.596 5.935 0.4257 84 -1.658 4.857 -25.324 0.258 2.245 0.458 0.6034 0.465 5.589 0.5281 85 -2.689 4.325 -9.689 0.345 3.214 0.658 0.5924 0.495 4.578 0.4584 86 -2.457 3.635 -14.895 0.785 1.325 0.411 0.3275 2.394 5.658 -1.1954 87 -1.658 2.854 -16.321 0.658 2.658 0.365 0.3059 2.405 4.698 -1.2547 Continue on the next page 12 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 13 Table 2: The variable statistical indicator results and GPI of models in the present research. Model MBE RMSE MPE MARE MAE RMSRE R2 t-test U95 GPI 88 -1.547 1.441 -25.647 0.365 2.365 0.245 0.6325 0.235 6.325 0.7548 89 -1.896 3.458 -29.356 0.348 3.478 0.265 0.3547 2.365 4.358 -0.6552 90 -2.425 4.875 -4.684 0.447 1.326 0.194 0.4671 1.356 3.256 0.2542 91 1.325 3.658 5.895 0.457 4.325 0.192 0.4325 1.695 2.958 -0.3279 92 -2.125 1.356 9.457 0.689 3.214 0.132 0.4488 1.542 4.478 0.1895 93 -1.658 1.457 6.698 0.532 2.658 0.256 0.4588 1.374 4.856 0.2458 94 -1.632 2.254 -9.784 0.547 1.892 0.124 0.5413 0.775 3.478 0.3453 95 -2.568 1.457 6.589 0.632 3.256 0.135 0.4483 1.547 5.326 0.1863 96 -1.458 1.365 9.325 0.547 2.114 0.245 0.4492 1.528 4.784 0.2132 97 -1.567 2.235 -7.365 0.365 1.214 0.158 0.5419 0.769 3.247 0.3468 98 -3.256 2.456 -7.325 0.423 2.425 0.145 0.5348 0.807 2.356 0.3235 99 -2.125 3.658 -6.245 0.449 1.658 0.245 0.5863 0.561 3.254 0.4558 100 -3.658 2.356 -12.356 0.658 1.547 0.189 0.5326 0.809 3.457 0.3231 101 -1.247 3.254 -9.556 0.523 2.457 0.089 0.3452 2.377 4.325 -0.6589 102 -2.568 4.325 -8.658 0.452 3.356 0.168 0.5852 0.563 5.326 0.4552 103 -2.145 2.325 -12.365 0.457 2.547 0.158 0.4211 1.985 6.325 -0.3566 104 -1.658 2.365 -3.658 0.589 3.589 0.257 0.3617 2.336 3.254 -0.5892 105 -2.457 3.245 -10.658 0.658 2.235 0.289 0.6162 0.281 4.568 0.7434 106 -1.356 4.568 -9.785 0.457 2.125 0.195 0.5896 0.524 4.785 0.4581 107 -2.925 4.235 -11.325 0.526 3.456 0.248 0.5631 0.641 3.658 0.4117 108 1.558 3.457 -9.325 0.457 1.358 0.235 0.5593 0.669 6.785 0.3694 109 -3.657 2.235 -8.658 0.589 1.457 0.245 0.5894 0.541 4.698 0.4576 110 -2.457 3.658 15.125 0.358 1.658 0.312 -0.3657 2.466 3.122 -1.3654 111 1.263 3.256 11.658 0.457 5.425 0.247 -0.6124 3.335 2.852 -2.456 112 -1.145 2.325 -6.365 0.658 1.578 0.158 -0.4483 2.478 5.125 -1.4582 113 -2.789 1.547 -8.325 0.658 4.452 0.125 0.2985 2.408 4.857 -1.2563 114 -2.457 1.657 -11.367 0.567 3.358 0.245 0.5592 0.672 5.245 0.3686 115 -2.536 3.856 -5.364 0.365 1.658 0.325 0.5842 0.566 5.236 0.4547 116 -1.438 4.547 -11.258 0.457 4.458 0.245 0.6021 0.482 4.325 0.5241 117 -3.625 4.658 -9.635 0.547 2.425 0.245 -0.5236 2.788 3.256 -2.1691 118 -2.547 3.635 -8.857 0.534 2.624 0.236 0.3311 2.391 3.258 -0.8571 119 -1.587 2.245 -7.568 0.349 2.425 0.189 0.4036 2.128 5.236 -0.3655 120 1.559 2.658 10.325 0.524 2.415 0.235 -0.6163 3.458 4.258 -2.4632 121 2.354 3.547 -13.124 0.658 1.524 0.158 -0.4591 2.534 5.124 -1.6592 122 1.256 3.254 9.568 0.471 2.658 0.198 0.2547 2.411 4.325 -1.2582 123 -3.256 1.568 6.345 0.511 1.578 0.195 0.5476 0.705 6.325 0.3543 124 1.635 2.547 -3.258 0.325 2.458 0.247 0.4447 1.577 4.325 0.1235 125 -1.456 3.625 -9.785 0.452 1.258 0.254 0.5422 0.767 5.236 0.3508 126 -1.325 1.245 -11.356 0.652 2.125 0.314 0.3669 2.258 4.325 -0.4163 127 -1.758 2.145 -12.689 0.725 3.415 0.245 0.3365 2.379 5.362 -0.6592 128 -2.658 3.365 -7.689 0.457 2.458 0.325 0.5671 0.609 6.325 0.4155 129 2.456 2.452 -12.458 0.657 1.625 0.245 0.5324 0.845 4.352 0.3227 130 1.558 1.457 -8.785 0.258 4.356 0.235 0.6032 0.468 2.356 0.5274 131 -2.457 1.256 6.658 0.658 5.659 0.145 0.5548 0.674 1.356 0.3682 132 -2.698 2.556 7.895 0.457 3.415 0.125 0.5403 0.787 6.325 0.3261 133 5.245 3.457 -15.356 0.756 2.425 0.236 0.4582 1.375 4.235 0.2455 134 -1.924 4.658 -21.245 0.857 2.356 0.248 0.5892 0.544 3.256 0.4574 135 -2.657 4.452 -9.478 0.365 1.758 0.356 0.6017 0.485 4.254 0.5237 136 -2.458 3.547 -11.689 0.245 1.659 0.256 -0.3685 2.468 5.124 -1.3663 137 -1.745 2.568 -13.75 0.356 2.415 0.198 0.2937 2.409 4.356 -1.2574 138 -1.865 1.245 -22.356 0.754 2.356 0.258 0.5668 0.612 6.325 0.4151 139 -1.365 2.356 -21.356 0.758 3.451 0.235 0.6089 0.365 4.236 0.6357 140 -2.652 3.254 -4.785 0.635 1.259 0.129 0.3638 2.319 2.365 -0.4583 141 -3.568 3.256 5.689 0.758 5.652 0.154 0.3625 2.329 2.356 -0.5482 142 -2.758 1.785 9.125 0.457 4.425 0.354 0.4577 1.377 3.256 0.2451 Continue on the next page 13 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 14 Table 2: The variable statistical indicator results and GPI of models in the present research. Model MBE RMSE MPE MARE MAE RMSRE R2 t-test U95 GPI 143 -1.554 1.653 5.326 0.457 2.452 0.256 0.4481 1.549 4.254 0.1647 144 -1.645 2.457 -8.654 0.547 1.411 0.547 0.4896 1.242 3.547 0.2856 145 -2.235 1.245 -9.325 0.326 2.356 0.425 0.5401 0.789 2.356 0.3259 146 -1.224 1.354 4.235 0.456 3.256 0.325 0.4465 1.568 1.598 0.1248 147 -1.324 2.235 -11.325 0.625 4.325 0.094 0.4281 1.696 4.325 -0.3281 148 -2.356 2.658 -2.547 0.425 2.456 0.521 0.3648 2.312 2.365 -0.4259 149 -1.235 3.425 -4.356 0.658 2.358 0.425 0.3286 2.392 1.569 -0.8579 150 -2.347 3.578 -10.325 0.775 1.325 0.325 0.6082 0.371 5.214 0.6328 151 -1.896 4.245 -8.356 0.556 1.452 0.452 0.3645 2.314 6.321 -0.4267 152 2.356 3.658 -6.325 0.632 2.365 0.652 0.5521 0.684 5.263 0.3662 153 3.456 2.356 -11.345 0.758 2.159 0.325 0.3622 2.331 6.256 -0.5549 154 -1.235 1.457 -4.326 0.458 1.423 0.425 0.3597 2.342 3.452 -0.6327 155 -2.457 2.658 -3.452 0.758 3.256 0.456 0.5789 0.568 4.214 0.4543 156 -1.145 3.452 -8.658 0.654 2.657 0.365 0.6049 0.425 4.254 0.5646 157 -2.356 5.455 -9.658 0.758 3.754 0.452 0.4572 1.379 3.985 0.2448 158 1.654 3.524 -10.457 0.658 1.658 0.098 0.6078 0.374 5.365 0.6325 159 -3.478 2.256 6.325 0.758 1.412 0.189 0.4571 1.381 3.158 0.2445 160 -2.254 3.415 9.458 0.659 1.523 0.326 0.4512 1.471 4.785 0.2254 161 1.547 3.659 -3.789 0.458 5.632 0.214 -0.6136 3.39 2.365 -2.4584 162 2.589 2.356 -3.365 0.658 1.456 0.354 -0.3588 2.459 4.658 -1.3372 163 -2.789 1.345 -14.214 0.658 4.854 0.569 0.4879 1.254 4.854 0.2594 164 -2.325 1.425 -13.269 0.457 4.432 0.412 0.5508 0.692 5.254 0.3565 165 -2.145 3.658 -5.547 0.569 2.623 0.425 0.5536 0.676 5.658 0.3675 166 0.765 4.356 -10.547 0.325 4.568 0.356 0.8549 0.152 4.457 0.7859 167 -3.425 4.754 -8.658 0.457 2.459 0.278 -0.5245 2.789 4.365 -2.2452 168 1.587 4.547 -6.458 0.658 2.752 0.322 0.4418 1.668 5.452 -0.2455 169 -1.245 2.785 -7.258 0.658 2.623 0.269 0.4023 2.132 6.124 -0.3667 170 1.356 1.326 11.325 0.658 1.456 0.145 -0.5597 3.226 5.356 -2.3542 171 1.145 1.587 -8.325 0.554 1.256 0.356 -0.2584 2.419 4.325 -1.3275 172 -2.658 3.325 4.569 0.457 2.985 0.265 -0.4592 2.547 3.259 -1.6595 173 -2.457 1.425 -4.758 0.325 1.452 0.159 0.4865 1.256 5.244 0.2591 174 1.258 2.256 -3.658 0.658 2.785 0.456 0.5267 0.857 4.122 0.3223 175 -1.325 3.411 -9.785 0.745 1.328 0.325 0.4236 1.758 3.259 -0.3295 176 -1.578 1.658 4.455 0.325 2.159 0.452 0.3658 2.259 4.325 -0.4238 177 -1.547 3.256 8.658 0.689 3.358 0.259 0.5264 0.865 5.326 0.3221 178 -2.547 2.356 5.324 0.754 2.425 0.324 0.5263 0.895 4.325 0.3218 179 2.457 1.356 -13.255 0.658 1.541 0.425 0.5765 0.586 5.325 0.4516 180 1.248 2.356 -10.355 0.453 4.953 0.354 0.5666 0.615 4.326 0.4142 181 -2.325 1.425 6.456 0.589 3.458 0.245 0.5214 1.234 3.256 0.3142 182 -2.248 2.325 7.258 0.745 3.635 0.245 0.5657 0.622 1.235 0.4135 183 4.658 3.245 -12.356 0.259 2.895 0.159 0.5683 0.598 3.256 0.4253 184 -1.258 4.456 -21.458 0.458 2.874 0.432 0.5472 0.707 4.235 0.3541 185 -2.457 3.254 -8.326 0.754 3.562 0.425 0.5263 0.912 6.321 0.3216 186 -2.254 3.145 -11.256 0.365 1.654 0.524 -0.5694 3.263 2.356 -2.3585 187 -1.356 2.658 -13.325 0.852 2.524 0.411 -0.5697 3.266 3.256 -2.3593 188 -1.145 1.245 -22.659 0.756 2.854 0.239 0.5469 0.742 5.326 0.3539 189 -1.658 3.225 -25.325 0.458 3.745 0.324 0.4423 1.659 1.236 -0.2364 190 -2.458 4.145 -4.425 0.321 1.852 0.245 0.5695 0.591 3.256 0.4326 191 1.625 3.325 5.325 0.658 4.423 0.148 0.3558 2.357 4.225 -0.6528 192 -2.478 1.785 8.325 0.785 3.657 0.425 0.4523 1.463 3.269 0.2358 193 2.356 1.547 5.269 0.689 2.456 0.257 0.4563 1.383 1.359 0.2442 194 -3.256 3.456 -7.659 0.365 1.652 0.457 0.5628 0.644 4.235 0.4113 195 -1.547 2.356 6.245 0.457 3.587 0.325 0.4751 1.344 3.259 0.2559 196 -2.658 1.658 6.325 0.547 2.923 0.154 0.5411 0.779 4.352 0.3451 197 -3.145 1.658 -3.245 0.421 1.547 0.325 0.6045 0.435 2.596 0.5642 Continue on the next page 14 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 15 Table 2: The variable statistical indicator results and GPI of models in the present research. Model MBE RMSE MPE MARE MAE RMSRE R2 t-test U95 GPI 198 -2.658 2.785 5.326 0.754 1.325 0.238 0.6324 0.247 3.124 0.7547 199 -1.356 3.325 -8.658 0.652 2.356 0.214 0.6054 0.412 2.356 0.5685 200 -2.687 2.754 -11.596 0.458 2.658 0.169 0.6144 0.329 4.235 0.6588 201 -1.546 3.689 -7.365 0.658 1.985 0.157 0.3321 2.388 5.326 -0.7459 202 -2.865 4.456 -8.452 0.358 3.245 0.235 0.6158 0.295 4.325 0.6895 203 -2.324 2.785 -12.958 0.754 2.458 0.168 0.3694 2.235 5.326 -0.4115 204 -1.987 2.788 -3.254 0.658 3.325 0.278 0.3593 2.345 3.258 -0.6329 205 -2.689 3.965 -10.325 0.457 2.745 0.325 0.8625 0.145 4.458 0.8292 206 -1.758 2.356 -7.654 0.457 2.658 0.457 0.6183 0.269 4.365 0.7454 207 -2.325 3.256 -16.324 0.325 3.145 0.325 0.6145 0.325 3.145 0.6853 208 1.456 3.875 -11.562 0.158 1.584 0.168 0.5885 0.546 5.326 0.4571 209 -3.325 2.685 -8.256 0.856 2.356 0.452 0.6141 0.335 2.356 0.6586 210 -2.214 3.425 11.569 0.289 3.256 0.245 -0.4526 2.481 3.547 -1.4594 211 1.568 3.635 -7.325 0.536 4.325 0.158 -0.6183 4.457 2.658 -2.5874 212 -1.654 2.457 -6.458 0.495 2.325 0.214 -0.3947 2.471 4.325 -1.3674 213 -2.452 1.257 -8.369 0.258 4.654 0.189 -0.4595 2.568 4.356 -1.6599 214 -2.158 1.453 -10.589 0.557 3.785 0.324 0.5785 0.571 3.256 0.4539 215 0.857 3.365 -6.578 0.754 1.458 0.258 0.9154 0.129 5.365 0.8575 216 -1.258 4.235 -8.256 0.659 3.658 0.325 0.6087 0.369 4.659 0.6353 217 -3.269 4.457 -7.356 0.458 2.856 0.156 0.2456 2.415 3.547 -1.3254 218 -1.654 3.356 -8.256 0.365 2.452 0.324 0.3329 2.385 3.458 -0.6635 219 -1.248 2.869 -7.785 0.547 2.369 0.324 0.3352 2.381 5.658 -0.6595 220 2.356 2.458 12.359 0.985 2.756 0.254 -0.5479 2.874 4.445 -2.2659 221 1.895 3.452 -2.356 0.458 1.269 0.189 -0.4125 2.473 5.658 -1.4524 222 1.895 3.857 9.245 0.547 2.562 0.145 -0.5829 3.268 4.456 -2.3624 223 -3.452 1.369 6.785 0.623 1.627 0.241 0.5881 0.549 5.326 0.4568 224 1.354 2.125 -3.589 0.457 2.568 0.324 0.4665 1.358 4.962 0.2512 225 2.356 3.365 -9.356 0.658 2.142 0.159 0.6287 0.254 5.854 0.7545 226 -1.654 1.785 -8.325 0.458 2.125 0.425 0.3589 2.346 4.632 -0.6333 227 -1.364 2.578 6.324 0.554 3.658 0.268 0.3628 2.327 5.547 -0.5473 228 -2.425 3.856 -4.325 0.657 2.985 0.245 0.6135 0.345 6.458 0.6582 229 2.689 2.753 -11.569 0.365 1.452 0.165 0.6165 0.274 4.856 0.745 230 -2.356 1.685 -8.652 0.458 4.753 0.189 0.6134 0.351 2.958 0.6575 231 0.825 1.689 6.441 0.458 5.425 0.145 0.8674 0.137 1.857 0.8468 232 -2.156 2.785 7.365 0.689 3.665 0.264 0.6269 0.258 6.857 0.7542 233 4.325 3.125 -11.325 0.458 2.785 0.159 0.6129 0.356 4.547 0.6572 234 0.755 4.259 -18.329 0.653 2.411 0.325 0.8137 0.169 3.568 0.7856 235 1.659 4.785 -9.125 0.455 1.625 0.369 0.6156 0.324 4.458 0.6891 236 -2.687 3.985 -11.574 0.562 1.258 0.425 -0.4578 2.489 4.326 -1.5742 237 -1.254 2.457 -12.658 0.754 2.658 0.362 0.6051 0.415 4.856 0.5681 238 0.758 1.857 -21.587 0.658 2.459 0.457 0.6351 0.199 6.325 0.7586 239 -1.965 2.698 -24.269 0.547 3.632 0.159 0.5467 0.751 5.326 0.336 240 -2.354 3.547 -8.236 0.356 1.547 0.246 0.3341 2.382 3.254 -0.6597 241 -3.875 3.785 4.326 0.456 5.785 0.256 0.3315 2.389 2.458 -0.7543 242 -2.459 1.689 7.658 0.857 4.632 0.457 0.6125 0.358 3.358 0.6568 243 -1.456 1.356 6.547 0.658 2.356 0.365 -0.6458 4.469 5.326 -3.2561 244 -1.547 2.758 -7.236 0.745 3.245 0.457 0.5517 0.686 3.258 0.3657 245 -2.589 1.587 -6.258 0.458 2.658 0.632 0.5872 0.555 2.657 0.4566 246 -1.935 2.658 3.256 0.345 3.325 0.524 0.4522 1.465 1.258 0.2351 247 1.356 1.658 -7.326 0.562 2.569 0.254 0.3268 2.396 3.256 -1.2326 248 -1.562 1.569 -1.569 0.742 3.256 0.652 0.4856 1.259 6.321 0.2586 249 -2.365 3.245 -3.256 0.523 2.657 0.325 0.5501 0.695 5.263 0.3558 250 -1.698 2.356 -12.365 0.689 3.754 0.425 0.5531 0.677 2.365 0.3673 251 0.785 3.245 -6.325 0.456 1.658 0.456 0.7536 0.175 1.569 0.7855 252 2.458 2.365 -5.326 0.854 1.412 0.365 -0.547 2.792 5.214 -2.2459 Continue on the next page 15 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 16 Table 2: The variable statistical indicator results and GPI of models in the present research. Model MBE RMSE MPE MARE MAE RMSRE R2 t-test U95 GPI 253 3.365 2.852 -11.689 0.369 1.523 0.452 0.4416 1.669 6.321 -0.2459 254 -1.589 1.658 -4.758 0.689 5.632 0.098 0.3972 2.145 5.263 -0.3675 255 -2.785 2.325 -3.698 0.754 1.456 0.425 -0.5624 3.251 6.256 -2.3549 256 -1.658 3.895 -8.325 0.856 2.657 0.356 -0.2698 2.422 3.452 -1.3283 257 -2.587 5.689 -9.654 0.689 1.523 0.278 -0.3592 2.461 4.214 -1.3379 258 1.359 3.457 -13.659 0.459 5.632 0.322 0.4854 1.324 4.325 0.2582 259 -3.658 2.658 6.758 0.365 1.456 0.269 0.5495 0.697 5.326 0.3554 260 -2.457 3.245 9.635 0.458 2.657 0.145 0.5527 0.679 4.325 0.3671 261 0.752 3.425 -3.258 0.556 3.754 0.356 0.7165 0.189 5.325 0.7853 262 2.256 2.689 -3.698 0.469 1.658 0.265 -0.3652 2.463 4.326 -1.3385 263 -2.556 1.785 -7.326 0.458 1.412 0.159 0.4852 1.325 3.256 0.2579 264 -2.852 1.358 -11.325 0.842 1.523 0.456 0.5483 0.701 1.235 0.3551 265 -2.624 3.589 -5.852 0.758 3.458 0.325 0.5524 0.682 5.214 0.3668 266 -1.458 4.365 -9.326 0.536 3.635 0.265 0.6056 0.385 6.321 0.6238 267 -3.852 4.658 -7.325 0.659 2.895 0.159 -0.5456 2.795 5.263 -2.2467 268 1.654 4.265 -6.689 0.745 2.874 0.456 0.4414 1.672 2.365 -0.2462 269 -1.785 2.256 -7.985 0.458 3.562 0.325 0.3956 2.148 1.569 -0.3683 270 1.258 1.477 14.325 0.925 1.654 0.452 -0.5632 3.254 5.214 -2.3562 271 1.568 1.632 -8.325 0.638 2.524 0.259 -0.2965 2.425 6.321 -1.3288 272 -2.452 3.458 4.852 0.852 2.854 0.324 0.9345 0.124 5.263 0.8652 273 -2.852 1.689 -4.458 0.754 3.745 0.148 -0.5476 2.857 6.256 -2.2477 274 1.365 2.589 -3.256 0.539 1.852 0.265 0.4411 1.673 3.452 -0.2468 275 -1.745 3.236 -9.689 0.657 4.423 0.159 0.3698 2.225 4.214 -0.3694 276 -1.325 1.458 4.689 0.658 3.657 0.456 -0.5689 3.259 4.325 -2.3577 277 -1.452 3.658 8.256 0.785 2.456 0.325 -0.3254 2.426 5.326 -1.3294 278 -2.658 2.589 6.356 0.358 1.652 0.452 -0.4597 2.569 1.569 -1.6624 279 2.125 1.458 -12.356 0.754 3.587 0.259 0.4785 1.329 4.214 0.2576 280 1.852 2.698 -10.562 0.658 2.923 0.324 0.5258 0.915 4.325 0.3213 281 -1.325 1.857 6.895 0.785 3.458 0.148 0.4235 1.759 5.326 -0.3359 282 -2.658 2.698 7.415 0.658 3.635 0.425 0.3652 2.261 1.569 -0.4245 283 4.258 3.758 -11.325 0.754 2.895 0.257 0.5254 0.924 5.214 0.3207 284 -1.458 4.689 -8.325 0.892 4.423 0.457 0.5251 0.935 6.321 0.3205 285 -2.632 3.458 -6.325 0.589 3.657 0.325 0.5247 0.952 5.263 0.3201 286 -2.552 3.645 -10.325 0.457 2.456 0.154 -0.6014 3.269 6.256 -2.3635 287 -1.639 2.589 -14.952 0.635 1.652 0.259 0.6839 0.193 3.452 0.7851 288 -1.451 1.658 -21.457 0.452 3.587 0.324 0.5624 0.652 5.326 0.4111 289 -1.574 3.753 -15.623 0.852 2.923 0.257 0.4721 1.347 1.569 0.2554 290 -2.853 4.478 -4.589 0.625 3.458 0.457 0.4517 1.466 5.263 0.2343 291 1.426 3.689 7.326 0.587 3.635 0.325 0.4554 1.385 6.256 0.2438 292 -2.753 1.658 6.325 0.632 3.458 0.154 0.5623 0.656 3.452 0.4108 293 2.573 1.785 4.325 0.458 3.635 0.325 0.4695 1.351 5.326 0.2551 294 -3.475 3.852 -7.265 0.458 3.587 0.148 0.5692 0.593 1.569 0.4322 295 -1.689 1.658 -5.325 0.354 2.923 0.411 0.3555 2.359 5.214 -0.6537 296 -3.569 3.256 4.589 0.658 3.458 0.239 0.4515 1.469 6.321 0.2337 297 -1.569 2.356 -6.325 0.542 3.635 0.324 0.4551 1.386 5.263 0.236 298 -1.954 1.356 11.356 0.452 2.365 0.145 -0.4562 2.484 5.326 -1.5649 299 -1.356 3.245 -13.654 0.362 1.658 0.168 0.5245 0.958 4.325 0.3198 300 -2.356 4.325 9.324 0.289 2.365 0.325 0.5597 0.659 7.321 0.3842 301 -2.314 5.326 5.326 0.654 3.256 0.254 0.4633 1.363 5.326 0.2507 302 -1.325 3.265 -3.865 0.345 2.358 0.315 -0.4793 2.698 6.325 -2.1491 303 -2.345 2.965 -9.541 0.542 1.358 0.125 0.4158 1.988 5.659 -0.3569 304 5.632 2.985 -12.654 0.654 4.325 0.168 0.3546 2.367 6.458 -0.6563 305 -1.356 3.214 -15.325 0.358 5.326 0.215 -0.5495 3.216 7.321 -2.3165 306 -2.345 2.658 -9.325 0.568 3.245 0.145 -0.3256 2.428 6.325 -1.3299 307 5.632 2.965 -14.235 0.689 2.359 0.168 -0.4599 2.582 5.326 -1.6635 Continue on the next page 16 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 17 Table 2: The variable statistical indicator results and GPI of models in the present research. Model MBE RMSE MPE MARE MAE RMSRE R2 t-test U95 GPI 308 -2.314 2.985 -8.325 0.452 2.145 0.325 0.5453 0.754 4.325 0.3532 309 -2.561 3.214 6.325 0.365 1.365 0.254 0.5231 0.991 3.658 0.3175 310 -3.214 2.658 7.325 0.654 1.987 0.315 0.4277 1.698 6.325 -0.3283 311 -1.685 2.965 -21.324 0.235 2.365 0.145 0.4402 1.681 4.356 -0.3249 312 -2.345 3.354 -6.325 0.214 2.895 0.168 0.4445 1.579 5.326 0.1231 313 -1.954 3.658 -11.524 0.425 3.245 0.325 0.4476 1.553 4.325 0.1574 314 -2.345 1.965 -14.356 0.654 1.658 0.254 0.4595 1.369 7.321 0.2475 315 5.632 2.314 -9.541 0.345 5.325 0.315 0.4258 1.702 5.326 -0.3286 316 -2.314 1.658 -12.654 0.542 4.235 0.125 0.44 1.683 6.325 -0.3254 317 -2.561 2.365 -15.325 0.654 2.356 0.168 0.4442 1.581 5.659 0.1228 318 -3.214 3.654 -9.325 0.358 1.547 0.215 -0.4569 2.485 6.458 -1.5654 319 -1.685 2.365 -14.235 0.568 1.658 0.314 0.5243 0.961 7.321 0.3195 320 -2.345 1.698 -8.325 0.689 5.326 0.326 0.5596 0.663 6.325 0.3835 321 -2.865 1.965 6.325 0.452 3.245 0.145 0.4625 1.364 5.326 0.2505 322 -1.324 2.356 7.325 0.365 2.359 0.168 -0.4875 2.754 4.325 -2.1556 323 -2.365 3.654 -21.324 0.654 2.145 0.325 0.4148 1.991 7.231 -0.3572 324 -1.954 4.325 -6.325 0.235 1.365 0.254 0.3544 2.369 5.326 -0.6574 325 -1.356 4.658 -11.524 0.214 1.987 0.315 -0.5578 3.219 6.325 -2.3181 326 -2.356 3.265 -14.356 0.425 2.365 0.125 -0.3259 2.431 5.659 -1.3314 327 -2.314 2.365 -3.568 0.325 2.895 0.168 0.3248 2.403 6.458 -1.2459 328 -1.325 3.654 -13.654 0.358 3.245 0.215 -0.4574 2.487 7.321 -1.5665 329 -2.345 2.365 -9.365 0.568 1.658 0.314 0.5241 0.968 6.325 0.3191 330 5.632 3.654 -8.325 0.689 5.325 0.326 0.5594 0.666 5.326 0.3831 331 -1.356 4.325 -7.256 0.452 4.235 0.289 0.4612 1.366 4.325 0.2501 332 -2.345 4.658 11.356 0.365 2.356 0.135 -0.5224 2.784 3.658 -2.1683 333 5.632 3.265 -13.654 0.654 1.547 0.147 0.4125 2.125 6.325 -0.3579 334 -2.314 2.365 9.324 0.235 1.658 0.168 0.3459 2.374 4.356 -0.6581 335 -2.561 3.654 5.326 0.214 2.314 0.125 -0.5593 3.225 5.326 -2.3199 336 5.632 2.365 -3.865 0.425 1.658 0.168 -0.3264 2.435 4.325 -1.3325 337 -1.356 1.698 -3.568 0.325 2.359 0.215 -0.4628 2.632 7.321 -1.6641 338 -2.356 1.965 -13.654 0.658 1.325 0.314 0.5448 0.756 5.326 0.3529 339 -2.314 2.356 -9.365 0.552 2.365 0.326 0.5227 0.993 6.325 0.3172 340 -1.325 3.654 -8.325 0.356 1.658 0.145 0.4256 1.705 5.659 -0.3289 341 -1.325 3.654 -7.256 0.654 2.365 0.087 0.4397 1.685 6.458 -0.3258 342 -1.365 4.325 11.356 0.658 3.256 0.256 0.4438 1.583 7.321 0.1226 343 -2.895 4.658 -13.654 0.456 2.358 0.326 0.4472 1.555 6.325 0.1571 344 -3.256 3.265 9.324 0.425 5.326 0.411 0.4591 1.371 5.326 0.2471 345 -2.365 2.365 5.326 0.268 3.245 0.358 0.4254 1.754 4.325 -0.3292 346 -1.365 3.654 -3.865 0.689 2.359 0.265 0.4394 1.688 7.231 -0.3263 347 -1.985 2.365 -9.541 0.358 2.145 0.124 0.4435 1.586 6.325 0.1224 348 2.356 1.326 -3.365 0.556 2.459 0.456 0.3569 2.348 5.263 -0.6339 349 3.456 1.587 -14.214 0.632 2.752 0.365 0.5782 0.574 6.256 0.4532 350 -1.235 3.325 -13.269 0.758 2.623 0.165 0.6041 0.457 3.452 0.5637 351 -2.457 1.425 -14.214 0.458 1.456 0.245 0.4547 1.387 4.214 0.2433 352 -1.145 2.256 -13.269 0.758 1.256 0.521 0.6073 0.377 4.254 0.6321 353 -2.356 3.411 -5.547 0.654 2.985 0.425 0.4541 1.415 3.985 0.2429 354 1.654 1.658 -10.547 0.758 1.452 0.325 0.3642 2.315 5.365 -0.4273 355 -3.478 3.256 -8.658 0.658 2.785 0.452 0.5515 0.687 3.158 0.3652 356 -2.254 2.356 -6.458 0.758 5.632 0.652 0.3621 2.333 4.785 -0.5564 357 1.547 1.356 -7.258 0.659 1.456 0.325 0.3568 2.351 3.158 -0.6347 358 2.589 2.356 -3.365 0.458 4.854 0.425 0.5776 0.579 4.785 0.4528 359 -2.789 1.425 -14.214 0.658 4.432 0.456 0.6036 0.459 5.214 0.5632 360 -2.325 2.325 -13.269 0.325 2.623 0.365 0.4538 1.435 5.263 0.2426 361 -2.145 3.245 -5.547 0.658 2.623 0.452 0.6014 0.488 6.256 0.4984 362 1.258 4.456 -10.547 0.745 4.568 0.098 0.4535 1.441 3.452 0.2422 Continue on the next page 17 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 18 Table 2: The variable statistical indicator results and GPI of models in the present research. Model MBE RMSE MPE MARE MAE RMSRE R2 t-test U95 GPI 363 -1.325 3.254 -14.214 0.325 2.459 0.189 -0.3269 2.452 4.214 -1.3345 364 -1.578 3.145 -13.269 0.556 2.752 0.326 -0.4781 2.658 4.254 -1.6657 365 2.589 2.658 -5.547 0.632 2.623 0.214 0.4784 1.333 3.985 0.2571 366 -2.789 1.245 -10.547 0.758 1.456 0.354 0.3565 2.353 5.365 -0.6356 367 -2.325 2.785 -8.658 0.458 1.256 0.569 0.5771 0.582 3.158 0.4525 368 -2.145 1.326 -6.458 0.325 2.985 0.325 0.6032 0.463 4.785 0.5284 369 1.258 1.587 -7.258 0.658 1.452 0.452 0.4783 1.337 3.985 0.2568 370 -1.325 3.245 -3.365 0.325 2.459 0.259 0.5642 0.632 5.365 0.4128 371 -1.578 4.456 -14.214 0.556 2.752 0.324 0.5681 0.603 3.158 0.4251 372 -2.457 3.254 -13.269 0.632 2.623 0.425 0.5439 0.758 4.785 0.3526 373 -1.145 3.145 -5.547 0.758 1.456 0.354 0.5238 0.985 3.158 0.3188 374 -2.356 2.658 -10.547 0.458 1.256 0.245 -0.6021 3.271 4.785 -2.3641 375 1.654 1.245 -14.214 0.758 2.985 0.245 -0.6023 3.274 5.214 -2.3652 376 -2.457 2.785 -13.269 0.654 1.452 0.159 0.5435 0.759 6.321 0.3524 377 -1.145 1.326 -5.547 0.758 2.785 0.365 0.4533 1.448 5.263 0.2418 378 -2.356 1.587 -6.458 0.658 1.328 0.452 0.6031 0.469 6.256 0.5272 379 1.654 3.325 -7.258 0.659 2.159 0.098 0.4531 1.457 3.452 0.2415 380 -3.478 1.425 11.325 0.458 3.358 0.189 -0.3574 2.457 4.214 -1.3367 381 -2.254 2.256 -8.325 0.658 2.623 0.326 -0.4785 2.664 5.452 -1.6674 382 1.547 3.411 4.569 0.325 4.568 0.214 0.4754 1.341 6.124 0.2563 383 2.589 3.325 -14.214 0.658 2.459 0.354 0.5634 0.635 5.356 0.4126 384 -2.789 1.425 -13.269 0.745 2.752 0.569 0.5677 0.606 4.325 0.4247 385 -2.325 2.256 -5.547 0.325 2.623 0.325 0.5431 0.761 3.259 0.3521 386 -2.145 3.411 -10.547 0.689 1.456 0.452 0.5236 0.986 5.452 0.3182 387 1.258 1.658 -8.658 0.754 1.256 0.259 -0.6025 3.275 6.124 -2.3669 388 -1.325 3.256 -6.458 0.658 2.985 0.324 -0.6032 3.277 5.356 -2.3672 389 -1.578 2.356 -7.258 0.325 1.452 0.425 0.528 0.763 4.325 0.3518 390 -1.547 1.356 11.325 0.658 2.785 0.354 0.4528 1.459 3.259 0.2412 391 -2.457 2.356 -3.365 0.325 1.328 0.245 0.6012 0.491 5.244 0.4654 392 -2.254 1.425 -14.214 0.556 2.159 0.245 0.4525 1.461 4.122 0.2407 393 -1.356 2.325 -13.269 0.632 3.358 0.159 -0.6035 3.329 3.259 -2.3685 394 -1.145 3.245 -5.547 0.758 2.425 0.432 -0.6051 3.332 4.325 -2.3697 395 -1.658 4.456 -10.547 0.458 1.541 0.452 0.5425 0.766 5.326 0.3514 396 -2.458 3.254 -8.658 0.365 4.953 0.259 0.4421 1.666 4.325 -0.2369 397 1.625 3.145 -6.458 0.457 3.458 0.324 0.5689 0.595 5.325 0.4316 398 2.356 3.325 -10.547 0.758 2.785 0.652 0.3562 2.355 5.214 -0.6367 399 3.456 1.425 -8.658 0.654 5.632 0.325 0.5768 0.585 5.263 0.4522 400 -3.245 2.314 -5.236 0.589 1.547 0.235 0.5632 0.639 4.356 0.4121 Table 3: The Rank of models in the present research Score of GPI Model Rank Score of GPI Model Rank Score of GPI Model Rank Score of GPI Model Rank 0.8652 272 1 0.7545 225 13 0.6575 230 25 0.6238 266 37 0.8575 215 2 0.7542 232 14 0.6572 233 26 0.6217 57 38 0.8468 231 3 0.7457 58 15 0.6568 242 27 0.5685 199 39 0.8292 205 4 0.7454 206 16 0.6548 59 28 0.5681 237 40 0.7859 166 5 0.7450 229 17 0.6523 33 29 0.5646 156 41 0.7856 234 6 0.7434 105 18 0.6357 139 30 0.5642 197 42 0.7855 251 7 0.6895 202 19 0.6353 216 31 0.5637 350 43 0.7853 261 8 0.6891 235 20 0.6328 150 32 0.5632 359 44 0.7851 287 9 0.6853 207 21 0.6325 158 33 0.5284 368 45 0.7586 238 10 0.6588 200 22 0.6321 352 34 0.5281 84 46 0.7548 88 11 0.6586 209 23 0.6256 60 35 0.5274 130 47 0.7547 198 12 0.6582 228 24 0.6247 37 36 0.5272 378 48 Continue on the next page 18 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 19 Table 3: The Rank of models in the present research Score of GPI Model Rank Score of GPI Model Rank Score of GPI Model Rank Score of GPI Model Rank 0.5268 80 49 0.3671 260 104 0.3172 339 159 0.2136 81 214 0.5263 10 50 0.3668 265 105 0.3165 2 160 0.2132 96 215 0.5245 38 51 0.3662 152 106 0.3142 181 161 0.1895 92 216 0.5241 116 52 0.3657 244 107 0.2958 55 162 0.1863 95 217 0.5237 135 53 0.3652 355 108 0.2952 79 163 0.1647 143 218 0.4984 361 54 0.3574 64 109 0.2874 52 164 0.1582 46 219 0.4654 391 55 0.3565 164 110 0.2856 144 165 0.1574 313 220 0.4586 63 56 0.3558 249 111 0.2691 66 166 0.1571 343 221 0.4584 85 57 0.3554 259 112 0.2594 163 167 0.1259 34 222 0.4581 106 58 0.3551 264 113 0.2591 173 168 0.1252 74 223 0.4576 109 59 0.3548 41 114 0.2586 248 169 0.1248 146 224 0.4574 134 60 0.3543 123 115 0.2582 258 170 0.1245 3 225 0.4571 208 61 0.3541 184 116 0.2579 263 171 0.1241 28 226 0.4568 223 62 0.3539 188 117 0.2576 279 172 0.1237 45 227 0.4566 245 63 0.336 239 118 0.2571 365 173 0.1235 124 228 0.4563 78 64 0.3532 308 119 0.2568 369 174 0.1231 312 229 0.4558 99 65 0.3529 338 120 0.2563 382 175 0.1228 317 230 0.4552 102 66 0.3526 372 121 0.2559 195 176 0.1226 342 231 0.4547 115 67 0.3524 376 122 0.2554 289 177 0.1224 347 232 0.4543 155 68 0.3521 385 123 0.2551 293 178 0.0854 73 233 0.4539 214 69 0.3518 389 124 0.2546 1 179 0.0658 27 234 0.4532 349 70 0.3514 395 125 0.2542 90 180 0.0214 26 235 0.4528 358 71 0.3508 125 126 0.2512 224 181 -0.2352 7 236 0.4525 367 72 0.3468 97 127 0.2508 19 182 -0.2364 189 237 0.4522 399 73 0.3459 77 128 0.2507 301 183 -0.2369 396 238 0.4516 179 74 0.3453 94 129 0.2505 321 184 -0.2455 168 239 0.4512 11 75 0.3451 196 130 0.2501 331 185 -0.2459 253 240 0.4326 190 76 0.3275 65 131 0.2479 47 186 -0.2462 268 241 0.4322 294 77 0.3264 36 132 0.2475 314 187 -0.2468 274 242 0.4316 397 78 0.3261 132 133 0.2471 344 188 -0.2569 75 243 0.4257 83 79 0.3259 145 134 0.2458 93 189 -0.3215 6 244 0.4253 183 80 0.3256 8 135 0.2455 133 190 -0.3246 44 245 0.4251 371 81 0.3252 12 136 0.2451 142 191 -0.3249 311 246 0.4247 384 82 0.3248 17 137 0.2448 157 192 -0.3254 316 247 0.4243 9 83 0.3246 31 138 0.2445 159 193 -0.3258 341 248 0.4155 128 84 0.3242 56 139 0.2442 193 194 -0.3263 346 249 0.4151 138 85 0.3238 82 140 0.2438 291 195 -0.3269 4 250 0.4142 180 86 0.3235 98 141 0.236 297 196 -0.3275 29 251 0.4135 182 87 0.3231 100 142 0.2433 351 197 -0.3277 43 252 0.4128 370 88 0.3227 129 143 0.2429 353 198 -0.3279 91 253 0.4126 383 89 0.3223 174 144 0.2426 360 199 -0.3281 147 254 0.4121 400 90 0.3221 177 145 0.2422 362 200 -0.3283 310 255 0.4117 107 91 0.3218 178 146 0.2418 377 201 -0.3286 315 256 0.4113 194 92 0.3216 185 147 0.2415 379 202 -0.3289 340 257 0.4111 288 93 0.3213 280 148 0.2412 390 203 -0.3292 345 258 0.4108 292 94 0.3207 283 149 0.2407 392 204 -0.3295 175 259 0.3849 18 95 0.3205 284 150 0.2358 192 205 -0.3359 281 260 0.3842 300 96 0.3201 285 151 0.2351 246 206 -0.3364 53 261 0.3835 320 97 0.3198 299 152 0.2343 290 207 -0.3456 49 262 0.3831 330 98 0.3195 319 153 0.2337 296 208 -0.359 21 263 0.3694 108 99 0.3191 329 154 0.2254 160 209 -0.3566 103 264 0.3686 114 100 0.3188 373 155 0.2245 50 210 -0.3569 303 265 0.3682 131 101 0.3182 386 156 0.2148 5 211 -0.3572 323 266 0.3675 165 102 0.3179 42 157 0.2144 32 212 -0.3579 333 267 0.3673 250 103 0.3175 309 158 0.2141 35 213 -0.3655 119 268 Continue on the next page 19 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 20 Table 3: The Rank of models in the present research Score of GPI Model Rank Score of GPI Model Rank Score of GPI Model Rank Score of GPI Model Rank -0.3667 169 269 -0.6563 304 302 -1.3345 363 335 -2.1691 117 368 -0.3675 254 270 -0.6574 324 303 -1.3367 380 336 -2.2452 167 369 -0.3683 269 271 -0.6581 334 304 -1.3372 162 337 -2.2459 252 370 -0.3694 275 272 -0.6587 68 305 -1.3379 257 338 -2.2467 267 371 -0.4115 203 273 -0.6589 101 306 -1.3385 262 339 -2.2477 273 372 -0.4154 51 274 -0.6592 127 307 -1.3654 110 340 -2.2659 220 373 -0.4159 54 275 -0.6595 219 308 -1.3663 136 341 -2.3145 14 374 -0.4163 126 276 -0.6597 240 309 -1.3674 212 342 -2.3149 23 375 -0.4238 176 277 -0.6635 218 310 -1.4524 221 343 -2.3165 305 376 -0.4245 282 278 -0.7456 69 311 -1.4576 71 344 -2.3181 325 377 -0.4259 148 279 -0.7459 201 312 -1.4579 72 345 -2.3199 335 378 -0.4267 151 280 -0.7543 241 313 -1.4582 112 346 -2.3542 170 379 -0.4273 354 281 -0.8571 118 314 -1.4594 210 347 -2.3549 255 380 -0.4583 140 282 -0.8579 149 315 -1.5641 16 348 -2.3562 270 381 -0.5247 48 283 -1.1954 86 316 -1.5649 298 349 -2.3577 276 382 -0.5425 30 284 -1.2326 247 317 -1.5654 318 350 -2.3585 186 383 -0.5473 227 285 -1.2358 13 318 -1.5665 328 351 -2.3593 187 384 -0.5482 141 286 -1.2452 15 319 -1.5742 236 352 -2.3624 222 385 -0.5549 153 287 -1.2459 327 320 -1.6549 25 353 -2.3635 286 386 -0.5564 356 288 -1.2547 87 321 -1.6587 40 354 -2.3641 374 387 -0.5892 104 289 -1.2563 113 322 -1.6589 62 355 -2.3652 375 388 -0.6245 76 290 -1.2574 137 323 -1.6592 121 356 -2.3669 387 389 -0.6327 154 291 -1.2582 122 324 -1.6595 172 357 -2.3672 388 390 -0.6329 204 292 -1.3254 217 325 -1.6599 213 358 -2.3685 393 391 -0.6333 226 293 -1.3267 24 326 -1.6624 278 359 -2.3697 394 392 -0.6339 348 294 -1.3269 39 327 -1.6635 307 360 -2.456 111 393 -0.6347 357 295 -1.3275 171 328 -1.6641 337 361 -2.4584 161 394 -0.6356 366 296 -1.3283 256 329 -1.6657 364 362 -2.4593 67 395 -0.6367 398 297 -1.3288 271 330 -1.6674 381 363 -2.4632 120 396 -0.6528 191 298 -1.3294 277 331 -2.1465 20 364 -2.4647 61 397 -0.6537 295 299 -1.3299 306 332 -2.1491 302 365 -2.5874 211 398 -0.6548 22 300 -1.3314 326 333 -2.1556 322 366 -2.6582 70 399 -0.6552 89 301 -1.3325 336 334 -2.1683 332 367 -3.2561 243 400 RMSRE = √√√ 1 n n∑ i=1 [(Hi,m−Hi,c)/Hi,m] 2 (9) R2 = 1 − n∑ i=1  (Hi,m−Hi,c)2∑n i=1 ( Hi,m−Hm. avg )  × 100 (10) t-Test = √ (n−1) (MBE)2√ (RMS E)2− √ (MBE)2 (11) 4. Uncertainty at 95% (U95) The expanded uncertainty in the 95% confidence interval is using to represent the data of the model deviation [36]. U 95 = 1.96(S D 2 + RMS E2)1/2 (12) Among them, SD is the percentage standard deviation (W/m2) of the difference between the predicted value and the measured value. In the above formula, 1.96 is the coverage factor corresponding to the 95% confidence interval. 20 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 21 5. Global Performance Index (GPI) The Global Performance Indicator (GPI) is a statistical in- dicator. The values of the statistical indicators have to be scaled down between zero and one, with the median value being sub- tracted from the scaled value of the individual statistical indica- tors. Finally, using the appropriate weight factor for individual statistical indicators, GPI is obtained. The mathematical ex- pression for GPIi of the ithmodel is defined as [36]: GPIi = 1∑ 0 j=1α j ( Ỹ j − Yi j ) (13) where αj equals to (-1) for the statistical indicator coefficient of determination, while for all other statistical indicators it is equal to 1, Ỹ j is the median of scaled values of indicator j, Yi j is the scaled value of indicator j for model i. A higher value of GPI represents better accuracy of the model. 6. Evaluation of GSR modeling According to a review of the literature, the sunshine hour is the most widely used parameter for calculating the monthly av- erage global photovoltaic radiation (MADGSR). The following variables (either single or in combination) are viewed whilst deciding on the world photovoltaic radiation (GSR) models; Sunshine Hour, Latitude and Sunshine Hour, Altitude and Sun- shine Hour, Latitude, altitude and sunshine hour, Latitude, lon- gitude, altitude and sunshine hour, Latitude, longitude and sun- shine hour. The GSR fashions chosen from the literature are in a range of varieties (linear, quadratic, cubic, power, exponen- tial, logarithmic, and greater order). Throughout the method of a GSR fashion resolution, care has been taken to avoid mod- els that characterize a widespread deviation from the measured value. Table 1 shows that the 400 models of GSR are used in the current lookup for calculation of monthly to daily world photo- voltaic radiation on a horizontal surface. The symptoms of the statistical evaluation of four hundred GSR fashions have been carried out and considered for unique chosen areas over Egypt to the use of the statistical error checks as noted in desk two Without the coefficient of dedication (R 2), the best price for all statistical error exams is zero. The ratio of world photovoltaic radiation to that of extraterrestrial photovoltaic radiation is rec- ognized as the clearness index (H/Ho). Table 2 shows that the results of the statistical indicators viewed in the current lookup were once shocking to see that character statistical check esti- mate in exceptional fashion. The negative sign values of (MBE) and (MPE) indicated overestimation, while positive signs in- dicated underestimation as compared to the measured values. From the table we notice that the values of MBE, RMSE, MPE, MARE, MAE, RMSRE, and t-Test vary between (5.632 to - 3.875, 5.689 to 1.245, 15.362 to-29.356, 0.985 to 0.158, 5.785 to 1.214, 0.658 to 0.068, and 4.469 to 0.124) respectively. Table 3 shows the rank of the GSR Model based on the Global Performance Indicator (GPI). According to table 3, the most accurate model for the selected locations in the current study is model no 272 (GPI = 0.8652), which ranked first Table 4. The information of selected sites in the present study Location WMO No. Lat. (0N) Lon. (oE) Elevation (m) Marsa- Matrouh 62.306 31.20 27.13 25 Asyut 62.392 27.12 31.30 52 Aswan 62.414 23.58 32.47 192 among all GSR models considered in the current study and was proposed by the Rietveld model. The GSR model 215 (0.8575), which was proposed by Gana and Akpotu, is ranked second, and model no. 231 (0.8468), which ranks third amongst the en- tire GSR models considered in the present studies. Also from table 3, we see that the GPI score varies between 0.8652 and 3.2561 for the entire models considered in the present study. The GSR model 243 shows the worst ranking, having a GPI score of −3.2561. 7. A case study 7.1. Observations and experimental Data Used Observations of the global solar radiation have been carried out by an Eppley normal incidence pyranometer. Its title im- plies that it is designed for the dimension of world photovoltaic radiation. It is already mentioned that the essential goal of this research is to comprehensively accumulate and evaluate the in- ternational photovoltaic radiation fashions reachable in the lit- erature and categorize them primarily based on the employed meteorological parameters. In order to consider the applica- bility and accuracy of the amassed fashions for computing the month-to-month common day-to-day international photovoltaic radiation on a horizontal surface. The Asyut site is called the urban site, and the Aswan location is called the western desert. The information of the selected locations in the present research is summarized in table 4 and figure 1. In the present research, the values of measured data of daily global solar radiation on a horizontal surface in the different se- lected locations during the period 1980–2019 are obtained from the Egyptian Meteorological Authority (EMA). The measured global solar radiation data was checked and controlled for er- rors and inconsistencies. The purpose of data quality control was to eliminate faulty data and inaccurate measurements. Af- ter the quality control, the measured data was averaged to ob- tain the monthly mean daily values by taking the data for the average day of each month. The measured data was then di- vided into two sets. The first sub-data set from 1980 to 2019 was employed to develop empirical correlation models between the monthly average daily global solar radiation fraction (H/H0) and the monthly average of desired meteorological parameters. The second sub-data set from 2015–2019 was then used to val- idate and evaluate the derived models and correlations. 8. Results and Discussion Figure 2 shows the spatial distribution of the long-term monthly mean of the GSR time series in the present research 21 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 22 during the climate study period 1980–2019 over Egypt. It is noticed that the GSR increases towards the south during all months. During winter months, the GSR fluctuates from 11.25 to 9.35 MJ/m2 in December, from 10.82 to 8.95 MJ/m2 in Jan- uary, and from 16.95 to 12.41 MJ/m2 in February in the selected locations in the present study. During spring months, the GSR values increase markedly from March (17.91-21.39 MJ/m2) through April (21.29-24.89 MJ/m2) up to May (24.92-28.75 MJ/m2). During summer months, GSR is maximum (27.52- 31.65 MJ/m2) in Jun followed by July (25.1-28.46 MJ/m2) and August (21.35-24.71 MJ/m2). During autumn months, the monthly GSR is similar to winter months, where September has the maximum values in (17.27-21.91 MJ/m2) followed by Octo- ber (14.39–19.25 MJ/m2) and November has the minimum val- ues (11.81–14.86 MJ/m2). Also from figure 1, we indicate that the values of the GSR are nearly zonal during most of the winter and autumn months, while it has a different spatial distribution (not zonal) in the spring and summer months. Moreover, the Egyptian Nile Delta location is frequently included in the low- est GSR values for the duration of all months, which might also be due to the excessive population and current of city and indus- trial areas that multiplied the attention of aerosol air pollution. On the other hand, the highest values of the GSR are detected over south Egypt due to low latitude. The monthly variation of the GSR values is due to changes in the atmospheric properties; the prevailing weather pattern in winter is the middle latitude, and the formed clouds reduce the direct sunlight with low at- mospheric turbidity. The relation between the average monthly clearness index versus sunshine duration ratio is clear in figure 3. It is clearly observed that a positive linear association exists between the clearness index and sunshine duration ratio. The evaluation of developed correlations in estimating the global so- lar radiation on the horizontal surface is assessed by comparing each model’s outputs and using the top ten models according to the rank order, which is found in table 5. The measured val- ues from the second subset of data from 2015 to 2019 based on the RMSE, MBE, MABE, R, MPE, and t-Test are summa- rized in table 5. The statistical indicators listed in this table generally provide reasonable criteria to compare models but do not objectively indicate whether the estimates from a model are statistically significant. The t-statistic (Eq. 11) allows mod- els to be compared and, at the same time, indicates whether a model’s estimate is statistically significant at a particular confi- dence level or not. The smaller the t-value, which gives the accuracy of the model performance, A summary of all the statistical parame- ters is present in table 5. For higher modeling accuracy, RMSE, MBE, ABE, and MPE indices should be closer to zero, but the correlation coefficient (R) should approach one as closely as possible. The performance of the most accurate models in each category for estimating the monthly average daily global solar radiation in selected locations in the present research is present and compared in table 5. From the statistical analysis, it can be seen that the estimated values of monthly mean daily global so- lar irradiation are in good agreement with the measured values for all models in the selected sites. It is found that the values of RMSE are in the range of 1.145 Figure 1. Map of the selected sites in the present study Figure 2. The measured values of monthly average daily global solar radiation for the selected sites during the period from 1980 to 2019 in the present research to 2.895 (in MJ/m2 day) in the selected sites in the present re- search. The MBE achieved in this study for all models is in the acceptable range. Negative values of MBE in temperature and cloud-based models indicate an underestimation of measured global solar radiation by these models. As an overestimation of an individual observation may cancel an underestimation in a separate observation, using the MABE index is more appropri- ate than using MBE. The mean percentage errors (MPE) of all models are in the range of acceptable values between 1.178% and - 1.129%. Also, according to the statistical test of the correlation co- efficient (R), for all models, very good results are given (above 0.92). The highest values of correlation coefficient according to models (M 272, M 261, M 238 and M 251) are 0.998, 0.996, 0.987, and 0.982 respectively for the Marsa-Matrouh site, while for the Asyut location it is (0.992, 0.987, 0.975, and 0.971), but (0.995, 0.992, 0.971, and 0.969) for the Aswan site. Also, from table 5, it is indicated that, the smallest values of t-Test occur around the models (M 272, M 261, M 251, and M 238) for all selected sites in the present research. This means that these models have good accuracy for predicting the monthly global solar radiation (GSR) in the selected sites during the pe- riod of 2015 to 2019 and compare with measured data in the 22 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 23 Table 5. The evaluation models by using statistical indicators results during the period time from 2015 to 2019 in the present research Marsa-Matrouh Model M272 (Rank 1) M215 (Rank 2) M231 (Rank 3) M205 (Rank 4) M166 (Rank 5) M234 (Rank 6) M251 (Rank 7) M261 (Rank 8) M287 (Rank 9) M238 (Rank 10) MBE -1.356 2.356 -2.854 -2.248 1.547 -2.857 -1.789 2.345 -1.658 1.985 RMSE 1.329 1.325 1.235 2.895 1.357 2.356 2.114 1.598 1.654 1.245 MPE% -2.547 1.958 2.687 -1.785 -3.456 1.985 2.356 -2.547 -1.874 -1.548 MABE 1.658 1.235 2.589 1.325 1.257 2.315 1.325 1.487 1.347 1.198 R2 0.998 0.948 0.947 0.938 0.924 0.918 0.982 0.996 0.968 0.987 t-Test 0.121 0.245 0.368 0.457 0.365 0.541 0.127 0.122 0.169 0.125 Asyut Month M272 (Rank 1) M215 (Rank 2) M231 (Rank 3) M205 (Rank 4) M166 (Rank 5) M234 (Rank 6) M251 (Rank 7) M261 (Rank 8) M287 (Rank 9) M238 (Rank 10) MBE 1.568 -2.378 -1.658 1.658 -1.875 1.586 -2.356 -1.856 2.356 -1.524 RMSE 1.215 2.356 1.245 1.456 2.387 1.145 2.456 1.654 2.345 1.115 MPE% -1.235 -2.654 1.658 1.324 -1.985 2.345 1.178 1.325 1.456 -1.325 MABE 1.325 1.198 1.654 2.312 1.658 1.325 1.235 1.115 1.245 1.102 R2 0.992 0.932 0.958 0.928 0.947 0.958 0.971 0.987 0.954 0.975 t-Test 0.114 0.542 0.329 0.412 0.524 0.328 0.124 0.118 0.165 0.138 Aswan Month M272 (Rank 1) M215 (Rank 2) M231 (Rank 3) M205 (Rank 4) M166 (Rank 5) M234 (Rank 6) M251 (Rank 7) M261 (Rank 8) M287 (Rank 9) M238 (Rank 10) MBE -1.179 1.325 -1.325 -2.314 -1.354 1.247 -1.289 1.327 2.158 -1.248 RMSE 1.234 2.128 1.189 1.624 2.214 1.278 1.301 1.425 1.356 1.369 MPE% -1.129 -2.189 1.412 1.547 -1.254 2.148 1.245 -1.658 -1.309 -1.415 MABE 1.214 1.235 1.325 2.127 1.324 1.191 1.168 1.325 1.114 1.208 R2 0.995 0.947 0.947 0.938 0.939 0.967 0.969 0.992 0.972 0.971 t-Test 0.132 0.358 0.411 0.314 0.419 0.289 0.147 0.132 0.185 0.141 Figure 3. Average monthly of clearness index versus with sunshine duration ratio in the present study during the period time 1980 to 2019 same period of the present research. The measured values of the monthly average daily global solar radiation and corresponding calculated values by employing the Coulibaly and Ouedoraogo model (M 238) [124], Katiyar et al. model (M 251) [127], Aras et al. model (Model 261) [94] and Rietveld model (M 272) [29] for the selected locations in the present study are illustrated in figure 4. As may be seen, agreement between the values ob- tained from the selected models and the measured data is very 23 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 24 good with the highest accuracy. Figure 4. Comparison between measured and calculated values of monthly average GSR models for selected sites in the present study from 2015 to 2019 9. Conclusion A global study of the global distribution of global solar radi- ation (GSR) requires knowledge of the radiation data in various countries. For this purpose, the designers and manufacturers of photovoltaic gear will want to recognize that in the ordinary world, photovoltaic radiation is on hand in exceptional regions. Solar energy technologies offer opportunities for use of a clean, renewable, and domestic energy resource and are essential com- ponents of a sustainable energy future. In the present research, a quantitative review of literature on global solar radiation (GSR) models available for different sta- tions around the world is discussed. The statistical analysis of 400 existing sunshine-based GSR models on a horizontal sur- face is compared using 40-year meteorological data in the se- lected locations in Egypt (Marsa-Matrouh, Asyut, and Aswan). Furthermore, to evaluate the accuracy and applicability of the various models for estimating the monthly average daily global radiation on a horizontal surface, the long-term measured mete- orological and solar data for the selected site in the preset work is employed. The developed models are then compared with each other and with the experimental data of the second subset based on the statistical error indicators such as RMSE, MBE, MABE, MPE, and correlation coefficient (R). In the present research, the measured data was then divided into two sets. The first sub-data set from 1980 to 2019 was used to develop empirical correlation models between the monthly average daily global solar radiation fraction (H/H0) and the monthly average of de- sired meteorological parameters. The second sub-data set from 2015–2019 was then used to validate and evaluate the derived models and correlations. The monthly variation of the GSR values is due to changes in the atmospheric properties; the pre- vailing weather pattern in winter is the middle latitudes, and the formed clouds reduce the direct sunlight with low atmospheric turbidity. It is established that there is a relationship between the av- erage monthly clearness index and the sunshine duration ratio. It is clearly observed that a positive linear association exists be- tween the clearness index and sunshine duration ratio. Accord- ing to the statistical analysis, the estimated values of monthly mean daily global solar irradiation for all models at the se- lected sites are in good agreement with the measured values. It is found that the values of RMSE are in the range of 1.145 to 2.895 (in MJ/m2 day) in the selected sites in the present re- search. The MBE achieved in this study for all models is within the acceptable range. Negative values of MBE in temperature and cloud-based models indicate an underestimation of measured global solar ra- diation by these models. As an overestimation of an individual observation may cancel an underestimation in a separate obser- vation, using the MABE index is more appropriate than using MBE. The mean percentage errors (MPE) of all models are in the range of acceptable values between 1.178% and - 1.129%. Also, according to the statistical test of the correlation, coef- ficient (R), for all models, very good results are given (above 0.92). The highest values of correlation coefficient according to models (M 272, M 261, M 238 and M 251) are (0.998, 0.996, 0.987, and 0.982) respectively for the Marsa-Matrouh site, while for the Asyut location are (0.992, 0.987, 0.975, and 0.971), and (0.995, 0.992, 0.971, and 0.969) for the Aswan site. The smallest values of t-Test occur around the models (M 272, M 261, M 251, and M 238) for all selected sites in the present research. This means that these models have good accuracy for predicting the monthly global solar radiation (GSR) in the selected sites during the period of 2015 to 2019 and can be compared with measured data in the same period of the present research. The accuracy of each model is tested using ten different statistical indicator tests. Furthermore, the Global Performance Indicator (GPI) is used to rank the selected GSR models. According to the results, the Rietveld model (Model 272) has shown the best capability to predict the GSR on horizontal surfaces, followed by the Katiyar et al. model (Model 251) and the Aras et al. model (Model 261). The 24 Khalil / J. Nig. Soc. Phys. Sci. 4 (2022) 911 25 findings of this study are particularly valuable for developing international locations and remote areas where there are few metrological stations available due to high technology costs. Nomenclature Isc = Solar Constant (= 1367 W/m2). H = Monthly average daily global solar radiation (MJ/m2). 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