Doppler transform between two frequency grids: the inertial-frame grid q (bottom) and the comoving-frame grid q^⋆ (top). Both grids are identical, but the comoving one is displaced in increasing direction of frequency by Doppler shift q_μ: q^⋆ = q−q_μ. The forward transform (orange) consists of forward interpolation for the specific intensity I → I^⋆ and an increment of the angle-averaged intensity J^⋆. The backward transform (green) is backward interpolation for the profile ratio ρ^⋆ → ρ.