Top: definition of the incident, s′, and emergent, s, photon directions at a scattering event within the atmosphere. The xyz axes form a rest reference frame fixed to the planet. The differential solid angle dΩ(s′) = sinθdθdφ is defined with s serving as polar axis. Angles θ ∈ [ 0,π] and φ ∈[0, 2π]. In the backtracing of photons of BMC algorithms, s is known at each collision and s′ must be sampled from the relevant scattering phase function. Angles i′ and i, both ∈[0, π], are needed for consistent referencing of the Stokes vector throughout the scattering process. Vectors {, and } and {e₁, e₂ and e₃} define right-handed coordinate systems at the meridional planes of the incident and emergent photon directions, respectively. Bottom: definition of the incident, s′, and emergent, s, photon directions at a reflection event at the local surface (plane ). Here, n is the inward-pointing normal vector at the surface, and ẑ is oriented along −n. The differential solid angle dΩ(s′) = sinθdθdφ is defined with ẑ serving as polar axis.