Physical Modeling and Some Recent Models

1.1 Introduction

A physical model for the transmission of radiation through a semi-transparent mat­ter of finite thickness, like the atmosphere, should start with monochromatic, non­coherent electromagnetic wave of initial intensity І0(Я), where Я is the wavelength of the electromagnetic wave. If such a wave traveling in free space enters a non­dispersive, homogeneous and isotropic layer of matter then the energy conservation reads:

І0(Я)=Ір(Я)+Іт(Я)+Іа(Я) (5.6)

where Ір(Я), ІТ(Я) andІа(Я) are the reflected, transmitted and absorbed intensities, respectively at wavelength Я, by the layer, atmosphere for our case. In this expres­sion, even if the incident monochromatic energy is specular (directional) – which is the case for extraterrestrial radiation, impinging on the atmosphere – reflected and transmitted radiation will have also a diffuse component due to the scattering by dif­ferent constituents of the matter. Another physical fact is the irradiative properties of the heated matter, and in our case it is the atmosphere, the temperature of which is increased due to the absorption of electromagnetic wave propagating through it (Coulson 1975). Hence, propagation of the sun rays through the atmosphere is a sophisticated phenomenon possessing many different physical mechanisms of inter­action of matter with the electromagnetic waves. Nevertheless, dividing Eq. (5.6) by І0(Я) and calling р(Я), а(Я) and т(Я) as the spectral reflectance, absorbance and transmittance of the layer at a specific wavelength Я, one can write the expression:

1 = р(Я) + а(Я) + т(Я). (5.7)

For the solar spectrum, most of the emitted radiation is in the range of wavelengths from 0.20 to 4.0 pm, typical of the spectrum of a blackbody at around 6000 K. The values of р(Я), а (Я) and т(Я) are different for each wavelength (or in wavelength intervals) for the atmosphere and one needs to integrate to find the transmitted amount of instantaneous radiation over all wavelengths, knowing the value of the incident radiation І0 (Я).

Some models starts from spectral calculations, assigning different values (most of the time as a function air mass) to the transmittance of the atmosphere for dif­ferent wavelengths bands considering the interaction of electromagnetic radiation
with various constituents of our atmosphere, some examples are Leckner (1978); Atwater and Ball (1978) and Bird (1984). These interacting constituents are mainly water vapor, greenhouse gases, ozone, dust and aerosol, some of which are either changed or given by man-made interventions to the atmospheric and climatic cycles of our globe. The important issues of this approach rest on the definition, formula­tion and/or measurement of the spectral values for the transmittance due to different components of a dynamic atmosphere whichbring in quite cumbersome calculations to reach the instantaneous irradiation values on the earth surface. Scattering due to the atmospheric constituents and the reflection from the ground introduce an extra diffuse component to the irradiation values, the transmittance of which must be han­dled with care and on a different base. Spectral models define these parameters in different wavelength bands and reach instantaneous transmission of the atmosphere for the whole solar spectrum.

Some other types of models are based on a critical assumption that the trans­missions of the atmosphere for different components of irradiation (mainly beam and diffuse) can be obtained using some spectrally averaged values of the transmit – tances and some other optical properties of the atmospheric components. Actually, such properties should be written in an integral form for a spectral average value, for example for the transmittance of the beam component, as:

In this expression the angle of incidence for the light rays are not taken into ac­count which of course will introduce extra complexity in the determination of these spectrally averaged properties. Amin and Amax are the minimum and maximum wavelengths of the energy source under consideration which in our case is the solar spectrum. Such spectrally averaged properties can be directly measured and used to construct models without any concern of their spectral variations. Such approaches of course should be supported and validated by measurements of the total, diffuse and beam components of the global solar irradiation on the surface of the Earth. In any case, one needs assumptions such as that the spectral property is constant at least for some wavelength band to allow a numerical integration unless an analytical form for these properties can be given.

A two-band model for example was developed to estimate clear-sky solar irradi – ance which divided the solar spectrum into UV/VI (0.29-0.7 pm) and IR (0.7-2.7 pm) bands. In this model effect of atmospheric constituents are parameterized using pre­liminary integrations of spectral transmission functions (Gueymard 1989).

May be the most important dilemma of all these models is the need of a fur­ther integration of these properties for a specified time interval which are mostly an hour or a day for our case. Such integration procedures should contain air mass, and since the incidence angle and thus the column of atmosphere traversed is a
function of time, this introduces new complications in modeling. If however instan­taneous spectrally averaged properties can be defined, then by making analogy that the time integrations will not change the form of the analytically derived models, some inter-relationships between the regression coefficients of the empirical rela­tions and physically defined average properties can be obtained, a typical example is the coefficients of Angstrom equation. Hence, solar irradiation on the surface of the earth may be directly written in a similar manner as given by Davies and McKay (1982):

G = Go І^,/(ра, pg, Pa) (5.9)


where G and G0 are global and cloudless sky irradiance, is ith cloud layer trans­

mittance as defined by Davies and McKay (1982) which indeed is also a function of the reflectance, transmittance and absorbance of the atmospheric constituents. The function /(pc, pg, pa) depends on cloud-base, ground and clear sky atmospheric reflectance which stands to take into account the multiple reflections between the ground and the atmosphere. In this expression G0 may be the solar radiation above the atmosphere at the location of interest but of course this replacement may in­troduce modification in the definition of ¥. In this approach if one starts with the instantaneous values of the solar radiation and spectrally averaged physical proper­ties, then the only requirement will be the time integrations within a preferred time interval, predominantly hourly and daily.

Following sub-sections review two recent models together with their physical basis which also make use of the simple Angstrom-Prescott approach. The reasons of choosing these two models are as they represent two different approaches in the search for a physical model for the relationship between the solar irradiation and bright sunshine hours and both end up with a simple Angstrom-Prescott type cor­relation. I should note that there exist various models and correlations with good performances but it is rather hard to discuss them all within the content of one chap­ter. I hope also that the models we discuss herein would be enough to comprehend the subject matter of interest, namely exploring the recent advances between the so­lar irradiation and bright sunshine hours. Another point is that the following models are compared with some other models and showed good performances as appeared in the literature.

Updated: August 2, 2015 — 12:50 am