The most widely used equation relating radiation to sunshine duration is the Angstrom-Prescott relationship, which can be expressed as [5,116]:
where N is the total number of daylight hours in the month. Suehrcke equates this approximately to
nb, clean where Hb is the monthly average of daily horizontal surface beam (direct) radiation and HHb clean is the monthly average of daily clear sky horizontal surface beam radiation. In order to relate Hb to monthly mean daily horizontal surface radiation H Suehrcke uses Page  diffuse fraction relationship as
H -12 CK where Hd is the monthly mean daily horizontal surface diffuse radiation, C is a constant, and K is the monthly mean daily clearness index, defined as
K - H (26)
with H0 the monthly mean daily horizontal extraterrestrial radiation. Given that by definition
H - Hb + Hid (27)
Eqs. (23) and (26) lead to
Hb - CH0k2
The same relationship holds for Hbjcleim
Hb, clear — CH0Kclear (29)
with Kclear as the monthly average clear sky clearness index defined as
where H is the monthly average daily radiation on a horizontal surface, H0 is the monthly mean daily horizontal extraterrestrial radiation, n is the number of bright sunshine hours per month, N is the total number of daylight hours in the month, and finally, a and b are model constants. These constants are determined empirically from a given data set and they assume a wide range of values depending on the location considered [5,6,52,78, 121,142]. If it is not possible to estimate these parameters from measured data for a specific location, they can be inferred from correlations established at neighboring locations [97,131].
The empirical determination of a and b is undoubtedly the greatest shortcoming of the Angstrom-Prescott relationship and it limits the usefulness of the formula. The Suehrcke [116,118] derivation is presented here briefly. For a given month with a number n of hours of bright sunshine, the sunshine fraction fclear is defined as
Fclear - N (23)
where Hclear is the monthly mean daily horizontal surface clear sky radiation. Combination of Eqs. (28) and (29) leads to elimination of the constant C and hence Suehrcke’s relationship becomes
In this relation Kclear is the only semi-empirical constant, which is a measurable quantity, and it depends on the local atmospheric conditions and according to Suehrcke , it has values typically between 0.65 and 0.75.
On the other hand, by definition bright sunshine duration s is the number of hours per day that the sunshine intensity exceeds some predetermined threshold of brightness. Angstrom [5,6] proposed a linear relationship between the ratio of monthly-averaged global radiation H to cloudless global irradiation Hcg and monthly-averaged sunshine duration, s. It is given as
where cj = 0.25 and S is the monthly-averaged astronomical day duration (day length). Angstrom  determined the value of cj from Stockholm data, but it was not until more than 30 years later that he (Angstrom ) stated that Eq. (32) was obtained from mean monthly data and should not be used with daily data.
In order to eliminate H from sunshine records, Angstrom’s model required measurements of global radiation on completely clear days, Hcg. The limitation prompted Prescott  to develop a model that was a fraction of the extraterrestrial radiation on a horizontal surface Hо rather than Hcg, because HH0 can be calculated easily. Hence, the modified Angstrom model referred to as the Angstrom-Prescott formula is [52,83]
T = c2 + c3 "s (33)
where the over-bars denote monthly average values, and c2 = 0.22 and c3 = 0.54, are determined empirically by Prescott . Since then many empirical models have been developed that estimate global, direct and diffuse radiation amounts from the number of bright sunshine hours [3,53,54, 63,79,104,122]. All these models utilize coefficients that are site specific and/or dependent on the averaging period considered. This limits their application to stations where the values of the coefficients were actually determined or at best to localities of similar climate, and for the same average period.
Hay  lessened the spatial and temporal dependence coefficients by incorporating the effects of multiple-reflection but his technique requires surface and cloud albedo data. More recently, Suehrcke  has argued that the relationship between global radiation and sunshine duration is approximately quadratic and thus the linear models as given in Eqs. (32) and (33) have wrong functional forms. Few authors have considered the relationships between the sunshine duration, observed irradiation, and potential daily clear-sky beam radiation. Suehrcke and McCormick  first proposed the following relationship
HbL = s
where Hs bn is the monthly-averaged daily beam irradiation at normal incidence, Hbnc is the monthly-averaged potential daily clear-sky beam irradiation at normal incidence, and Sc is the monthly-averaged day-length modified to account for when the sun is above a critical solar elevation angle. The ratio s/Sc is similar to s/S in Eqs. (32)-(34), except Sc corrects for the irradiation threshold of sunshine recorders. The basis of Eqs. (34) and (35) implies that for a given day, the beam radiation incident at the surface (Hb or Hbn) is a fraction, s/S, of what would have been incident, if the sky had been clear all day . In the absence of clouds, Hbc and Hbnc are functions of atmospheric scattering and absorption processes. The appeal of Eqs. (34) and (35) is twofold. On the one hand they provide a means of estimating the potential beam irradiation, and on the other they do not contain empirically derived coefficients. However, a minimum averaging period is recommended when using these equations to estimate potential beam irradiation, which has been suggested as month . The time averaging is necessary since s is simply the total number of sunshine hours per day and provides no information about when the sky was cloudless during any given day. There are several other assumptions in Eqs. (34) and (35). Turbidity and precipitable water are the same during cloudless and partly cloudy days, measurements of s are accurate, and the sunshine recorder threshold irradiance is constant and known.
Different global terrestrial solar irradiation estimation models on the earth’s surface are proposed which use the sunshine duration data as the major predictor at a location. Some others include additional meteorological factors as the temperature and humidity, but all the model parameter estimations are based on the least squares technique and mostly linear regression equation is employed for the relevant relationship between the terrestrial solar irradiation and the predictor factors.
Angstrom  provided the first global solar irradiation estimation model from the sunshine duration data. This model expresses the ratio of the average global terrestrial irradiation, Hs to extraterrestrial irradiation which is the cloudless irradiation, H0, to the ratio of average sunshine duration, Ss to the cloudless sunshine duration S0 as
the superposition principle of two extreme cloud states, which are reflected in a + b summation. However, in actual situations superposition is not possible with respect to all possible combinations of atmospheric variables other than the cloud cover. This is the first indication why the summation, a + b, did not equal to 1.0 as suggested by Prescott .
In practical applications various nonlinear estimation models are also proposed in order to relieve the assumption of superposition. Another physical fact that the solar irradiation models should include nonlinear effects is that atmospheric turbidity and turbulence in the planetary boundary do not necessarily vary linearly with total cloud cover. There are numerous studies and proposals as alternatives of the linear model in the solar energy literature and with the expectations of more studies in the future, but Gueymard et al.  state that the studies related to solar irradiation should now be more fully scrutinized. In particular, it is understood that the mere use of Angstrom’s equation to estimate global irradiation from local sunshine data would generally be judged as not publishable unless a new vision in the model structure is documented. All these explanations indicate that linear models are very restrictive, and therefore, many researchers have tried to propose nonlinear models for better refinements in the solar terrestrial irradiation estimations.