## 2 Theoretical Basis  
While some of the finer details regarding the conversion of radiance to irradiance have evolved over time, the general approach remains the same and is outlined in this section. Irradiance \(F\) is related to radiance \(I\) by:  
\[F(\theta_{S})=\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}}I(\theta_{S},\ \theta_{V},\ \varphi)\cos\theta_{V}\sin\theta_{V}d\theta_{V}d_{\varphi} \tag{1}\]where \(\theta_{S}\) is the solar zenith angle, \(\theta_{V}\) is the viewing zenith angle, and \(\varphi\) is the relative azimuth angle between the sun and the satellite (Figure 1). For the special case of Lambertian or isotropic radiance \(I_{iso}\) where the radiance does not depend on viewing geometry (\(\theta_{V},\ \varphi\)), the calculation of irradiance \(F\) in Equation (1) simplifies to:  
\[F(\theta_{S})=\pi I_{iso}(\theta_{S}). \tag{2}\]  
It follows that, for the more general anisotropic case, Equation (1) can be written as:  
\[F(\theta_{S})= \frac{\pi I(\theta_{S},\ \theta_{V},\ \varphi)}{R(\theta_{S},\ \theta_{V},\ \varphi)}, \tag{3}\]  
where \(R\) is the anisotropic function that relates radiance to irradiance. The function \(R\) provides a correction to account for how much the radiance in direction (\(\theta_{V},\ \varphi\)) deviates from the equivalent isotropic case; a value greater than unity would be associated with a radiance \(I\) that is larger than the isotropic radiance \(I_{iso}\), and vice versa. Angular distribution models (ADMs) are, in essence, a discretized form of the anisotropic function \(R\) known as anisotropic factors. In order to derive anisotropic factors, and therefore generate ADMs, an empirical approach has traditionally been adopted. Discrete angular bins are defined, and a large number of observed radiances collected over an extended time period are assigned to each of those angular bins and averaged (e.g., Figure 2). Figure 1: Solar-viewing geometry represented by three angles: solar zenith angle (\(\theta_{S}\)), viewing zenith angle (\(\theta_{V}\)), and relative azimuth angle (\(\varphi\)). For solar zenith angle bin \(i\), viewing zenith angle bin \(j\), and relative azimuthal bin \(k\), the anisotropic factor \(R_{ijk}\) is calculated as:  
\[R_{ijk}=\frac{\pi\overline{I}_{ijk}}{F_{i}}. \tag{4}\]  
In Equation (4), \(\overline{I}_{ijk}\) is the mean of normalized instantaneous radiances \(I_{ijk}\) in angular bin \((i,\ j,\ k)\). A normalization is applied to each instantaneous radiance observation \(I^{\prime}_{ijk}\) to account for variations in Earth-sun distance and solar zenith angle as follows:  
\[I_{ijk}=I^{\prime}_{ijk}\frac{\mu_{i}}{\mu^{\prime}}\bigg{(}\frac{r^{\prime}} {r_{0}}\bigg{)}^{2}, \tag{5}\]  
where \(\mu^{\prime}\) and \(r^{\prime}\) are the cosine of the solar zenith angle and the Earth-sun distance at the time of observation, respectively, \(\mu_{i}\) is the cosine of the solar zenith angle at the mid-point of solar zenith angle bin \(i\), and \(r_{0}\) is the mean Earth-sun distance. Note that the instantaneous radiances \(I^{\prime}_{ijk}\) considered in Equation (5) are assumed to be pre-processed, including spectral unfitering and calibration [25; 26; 27], which is not discussed here. The corresponding irradiance \(F_{i}\) that appears in Equation (4) is the summation of \(\overline{I}_{ijk}\) over all viewing zenith angle bins \(j\) and all relative azimuthal angle bins \(k\) using a discrete form of Equation (1) with the functional form:  
\[F_{i}=\sum_{j=1}^{N_{j}}w_{j}\ \sum_{k=1}^{N_{k}}w_{k}\overline{I}_{ijk}, \tag{6}\]  
where \(N_{j}\) and \(N_{k}\) are the total number of angular bins in viewing zenith angle and relative azimuth angle, respectively, and \(w_{j}\) and \(w_{k}\) are weights associated with the hemispheric integral. The weights can be determined analytically for certain angular bin widths, or estimated numerically with fixed angular bin widths and an additional weighting by cosine of the viewing zenith angle (e.g., via Gaussian quadrature).