In black, sketch demonstrating the construction of the {x_k,s_k} pairs starting from {x₀,s₀} as the photon is traced back from the detector through the medium. Vectors are pointed in the direction of photon propagation, which is the reverse of the direction of integration in the BMC algorithm. In principle, a {x_k,s_k} pair can lead to two new {x_ka,s_ka} and {x_kb,s_kb} pairs. In the MC implementation of the algorithm, a scheme based on the coefficients of the ℒ_ℬ+ℬ and + operators, Eq. (11), determines whether the photon’s next move occurs within the atmospheric medium or whether the photon moves onto the planet’s surface. In blue, one specific photon trajectory within the family of possible trajectories. For this specific trajectory, the red arrows denote the direction of the unscattered stellar photons.