Ground-reflected light: The invariance principle and the effect of luminaire height, emission pattern, and non-uniform albedo

The weighted contribution of ground-reflected component of light emissions from artificial sources gradually increases as the transitioning from bad-shielded to modernized light sources with low or zero direct uplight takes place in cities or towns. In this work, we demonstrate that the modeling of reflected light on a large domain for Lambertian flat surfaces does not require information about the height of the light sources and directional distribution of photons their produce. This kind of "invariance principle" becomes invalid when the homogeneity condition for the surface albedo is violated. However, we have shown that an analytical solution exists also for position-dependent albedo and even for angle-dependent reflectance which is the effect we now include to the light pollution models for the first time. This effect is known from the observation of sunbeams entering uneven surfaces at different zenith angles. Here in analogy with that daylight model we derive formulae for ground surfaces illuminated by artificial lights located at different heights above the surrounding terrain.