Global radiation

4.4.1. Theoretic background and model

The commonly used distinction between longwave radiation (thermal radiation) and shortwave radiation (global radiation or solar radiation) becomes obsolete if the source of the global radiation is considered. Global radiation is, of course, nothing but the thermal radiation of the sun reaching the earth’s surface.
Therefore, the irradiance of the global radiation reaching the top of the earth’s atmosphere (known as extraterrestrial radiation) can be calculated with surprising accuracy by only using Stefan-Boltzmann’s law and simple geometric considerations.
Using Stefan-Boltzmann’s law and applying the effective temperature of the sun 𝑇𝑠𝑜𝑙𝑠𝑢𝑟𝑓= 5778 K (NSSDC, 1998) the radiant exitance of the sun’s surface 𝑀𝑠𝑜𝑙𝑠𝑢𝑟𝑓 can be determined as:

Considering that the irradiance decreases with 1/𝑟 2 the radiant exitance at the sun’s surface can be scaled by multiplying it by the ratio of the radius of the sun 𝑟𝑠𝑜𝑙 (CNEOS NASA, 2022) to the average earth-sun distance, known as astronomical unit 𝐴𝑈 (CNEOS NASA, 2022), to determine the irradiance reaching the earth’s atmosphere:

In order to get the spectral radiant exitance 𝑀 of extraterrestrial radiation, Planck’s law for black-body radiation can be applied:

Relevant for solar calculations, however, is the radiation reaching the earth’s surface. In the earth’s atmosphere, solar radiation is attenuated, i.e. absorbed and scattered by atmospheric gases. Mostly water vapour, carbon dioxide, ozone and oxygen molecules absorb in this wavelength range (Finlayson-Pitts and Pitts, 2000), leading to the characteristic absorption bands of the global radiation spectrum (see Figure 29). The optical path length through the atmosphere also has to be considered. This is generally done by referring to the air mass coefficient AM, whereby AM1 is used for the path length of a strictly vertical incidence angle (zenith angle). The quantity AM0 is sometimes used to refer to extraterrestrial radiation. Finally, AM1.5 is the commonly used case to characterise average irradiance at a typical incidence angle (48° zenith angle).

The actual spectral distribution and total irradiance fluctuate strongly and depend on many parameters, such as weather, pollution, solar angle etc.
For reference purposes, standard spectra are used. The most commonly used spectrum is provided by NREL and the American Society for Testing and Materials and is called ASTM G-173 (ASTM, 2020). The spectrum reflects the irradiance on a 37° tilted surface, facing directly into the sun under clear sky conditions. Alternatively, the international standard EN 410 ( 2011) used to calculate the total solar transmittance of shaded and unshaded glazings provides the spectral distribution in normalised form. A comparison between the two spectral distributions can be seen in Figure 31. Using the developed modules (see section 4.4.3), some statistical key figures characterising the global radiation spectrum are derived and presented in Table 2. The analysis was performed using the ASTM G-173 reference spectrum (ASTM, 2020).

It can be seen that the relatively narrow visible wavelength range contributes more than half of the total energy of the entire spectrum and, therefore, also contains the wavelengths with the highest spectral irradiance values. Ultraviolet radiation plays a minor role, as wavelengths within this range are significantly scattered away or absorbed by atmospheric ozone and water molecules (FinlaysonPitts and Pitts, 2000).

4.4.2. Implementation

For further use, only the normalised spectral distribution of the global radiation is relevant. For later use in the Monte-Carlo raytracer, it is essential to be able to generate random wavelengths with a probability distribution function (PDF) matching the desired normalised spectrum.

In order to achieve this, the spectrum’s PDF function is generated by normalising the spectrum, maintaining the discrete interval steps as provided in the specific standard. Also, based on the defined interval steps, the cumulated distribution function CDF is found, simply by summation. Since this discrete function cannot be inverted to explicitly achieve CDF -1, the inversion is carried out during the sampling algorithm: firstly, a random number r between 0 and 1 is created. The random number reflecting the CDF-function value is used to determine the function’s argument, i.e. the associated wavelength interval. This is performed using a binary search algorithm. Finally, a random wavelength is selected within the interval, assuming an even distribution. For efficiency reasons, only one random number generation can be used to cover both steps. The implementation is contained in the function getSampleWL of module TGlobRad of the library rc_globalradiation.

The module can be initialised with different spectral distributions. Currently, the ASTM G173-03 global tilt, ASTMG173-03 direct+circum solar and the EN 410 spectra are available.

The desired spectrum (gsASTMtilt, gsASTMcircDir, gsEN 410) is specified in the call of the constructor method Create when an instance of the class is created.
The additionally included method getSampleWLat function allows specifying a value for the CDF parameter instead of using a random number. Hence, the function can be used to pull a set of energetically equidistant sampling wavelengths of the global radiation spectrum. The method is applied in the inversion algorithm to determine refractive index functions (see section 4.12).

4.4.3. Testing and validation

In order to validate the implementation, one billion random wavelength samples are generated using the getSampleWL function (the calculation time required for this is approx. one minute on a single CPU). The distribution of the sampled data is visualised as a histogram with 5 nm intervals and compared against the normalised reference spectrum ASTM G173-03. Figure 31 shows the retrieved results for the gsEN 410 and gsASTMtilt spectrum, as well as the normalised ASTM G173-03 spectrum for reference. In general, the spectra show similar distributions; however, the ASTM spectra offer a higher resolution, as the EN 410 spectral distribution is provided in intervals ranging from 20 nm to 100 nm only. For most practical cases, the higher resolution will not be relevant, however.

As can be seen in Figure 31, the implemented code is able to generate wavelength samples following the distribution of the global radiation spectrum. This is essential for its later application and proves the validity of the algorithm.