The average number of photon scattering events in vertically inhomogeneous atmospheres

The paper is devoted to the description of the novel approach to calculate the vertical distribution of the average number of photon scattering events N¯ in a plane-parallel turbid layer having arbitrary local optical scattering characteristics and optical thickness. The derived equation for the vertical distribution of the average number of photon scattering events is a generalization of the well-known Ambartsumian's formula (valid for homogeneous turbid media only) to the more general case of media with the vertically varying single scattering albedo. Numerical calculations are performed using the discrete ordinates technique in combination with the adjoint formulation of the radiative transfer equation. The performance of the numerical techniques introduced is studied using well-known asymptotic analytical equations valid for semi-infinite weakly absorbing layers. We also compare results obtained with published calculations of N¯ derived from a superposition technique.