The basis clear sky LUT consists of spectrally resolved RTM results for aerosols with different values of aerosol optical depth (aod), single scattering albedo (ssa) and asymmetry parameter (gg). The basis LUT is dedicated to consider the effect of aerosol scattering and absorption. The absorption by water vapour and ozone as well as the surface reflection are predominantly independent from the aerosol state and fixed values have been used for the calculation of the basis LUT,

i.e., a water vapor column of 15

mm, an ozone content of 345 DU, and a surface albedo of 0.2. The atmospheric values conform to the US standard atmosphere. The effect of the solar zenith angle on the transmission, hence the solar irradiance, is considered by the use of the Modified Lambert–Beer (MLB) function developed by Mueller

et al.[

22], which is given in

Equation (2).

where

τ_{0Λ} is the optical depth of the vertical column,

I_{Λ} is the solar radiation at ground for a solar zenith angle (SZA) of

θ_{z} and

I_{0}_{,enh,}_{Λ} is based on the extraterrestrial irradiance according to

Equation (3) at wavelength (wavelengths band) Λ. The cosine of the solar zenith angle accounts for the decrease of the surface flux density due to the increase of the solar zenith angle (the same amounts of photons are distributed over a larger area for increasing solar zenith angles). This relation is identical to the Lambert–Beer function (or MLB function) with exception of an additional “empirical” correction exponent

a, hence a correction of the parameter

$\frac{\tau}{\mathit{cos}({\theta}_{z})}$ [

22]. The correction parameters

a_{Λ} are calculated based on two RTM runs, one at

θ_{z} =0 and the other at

θ_{z} =60°, hence the correction parameters

a_{Λ} can be calculated without the need for a numerical fit.

I_{0}_{,enh} is based on the extraterrestrial irradiance at the top of atmosphere and estimated using

Equation (3)[

22].

Here

D_{Λ} and

B_{Λ} are the diffuse and direct (beam) component of the solar surface irradiance

I_{Λ} =

B_{Λ} +

D_{Λ} at a SZA of zero. The application of this equation is needed in order to preserve a good match of the MLB relation with RTM results for high optical depths, where the use of the extraterrestrial irradiance

I_{0Λ} fails. It is important to notice that the Modified Lambert–Beer relation (

Equation (2)) is also used for the calculation of the direct (beam) irradiance

B_{Λ}, with the exception that instead of using

I_{0}_{,enh,}_{Λ} the original extraterrestrial irradiance

I_{0Λ} must be used. The fitting parameters

a_{Λ} have different values for direct irradiance and (total) solar surface irradiance. Using the MLB function, the calculated direct irradiance, diffuse irradiance as well as the solar surface irradiance can be reproduced very well (see

Figure 2 as an example). The MLB function is discussed in detail by Mueller

et al.[

22], including proof and verification. Further validations are given by Ineichen

et al.[

23].

The MLB parameters

a_{Λ},

I_{0}_{,enh,}_{Λ} and

τ_{0Λ} are saved in the basis LUT for each wavelength band and different values of aerosol optical depth (aod), single scattering albedo (ssa) and asymmetry parameter (gg). The resulting basis LUT is three dimensional and contains the MLB-parameters to calculate

I_{basis} and

B_{basis} with

Equation (2). The effect of

aod is significantly larger than the effect of both

ssa and

gg, so the basis LUT contains MLB parameters for 23

aod values times three

ssa values times two

gg values (

ssa ∈ {0.7

, 0.85

, 1.0} and

gg ∈ {0.6

, 0.78}). The spectral effect of aerosols is considered by application of a standard aerosol model [

24,

25].

The calculation of the solar irradiance is done as follows: for any given aerosol state the nearest neighbors are selected from the LUT and the solar irradiance is calculated for these nearest neighbors at any given solar zenith angle by application of the MLB relation. Subsequent interpolation between the nearest neighbors provides the solar irradiance for a given aerosol state, but only for fixed water vapor, surface albedo and ozone values. In order to correct for variations from the fixed values occurring in nature, the parameterizations described in Sections 3.2 and 3.3 are applied as consecutive steps.

The sensitivity of surface irradiance on atmospheric variables has been evaluated for all steps in order to reduce the mesh points of the needed RTM calculations as much as possible. Decision criteria for the mesh width has been the effect of the parameters on the atmospheric transmission in combination with its degree of non-linearity, e.g., for a parameter with large effect on the atmospheric transmission but a predominantly linear behavior the interpolation grid can be wide-meshed, while with increasing non-linearity the distance between the grid-points has to be decreased.