The sketch demonstrates how a differential element of solid angle dΩ fans out when traced sunwards from the observer’s site. The dashed line represents the trajectory passing through the middle-point of the solid angle element. In a transparent atmosphere, the ratio of radiance and the squared refraction index is conserved on the ray trajectory, and the corresponding irradiance for the differential solid angle element is simply L(x_O,s_O)s_O·n_xOdΩ(s_O) ~ L(x_⊙,s_⊙)dΩ(s_O). For the calculation of Eq. (3), a mathematical transformation is applied to rewrite the integral in the auxiliary variables ξ and η.