Left: the coordinate systems of (r, θ) and (p, z). r_photo is the photosphere, r₁ and r₁ + Δr are the inner and outer radii of the molecular shell, respectively, and r_j is some arbitrary radius. These locations coincide with the equivalent velocities in the right panel. Right: a cartoon graphic of how MOFAT works. The molecular clumps have semiaxes of r_c and r_a with a density enhancement factor of f_c. When we integrate along p_j, the ray will have a certain probability (P, yellow star) of hitting a clump at v_j. The effects of the clump are then included, and are based on the chord length s. See Appendix A for descriptions of variables and implementation.