Conventionally, air mass factors are used to relate so-called slant trace-gas column densities measured by differential optical absorption spectroscopy (DOAS) using extraterrestrial light sources to desired vertical trace-gas column densities. We show that two fundamental problems arise when the traditional air mass factor A concept, which was originally developed for direct radiation measurements, is applied to spectroscopy of scattered radiation: (1) the inaccuracy of the traditional formula to determine vertical column densities V from measured slant column densities S, i.e., the equation V=S/A and (2) the inaccuracy of the traditional air mass factor computation method itself. We show that the reason for these inaccuracies is the substitution of absolute by differential slant column densities during a DOAS evaluation. It may be carried out and thus the traditional air mass factor concept is valid if and only if the optical density of a certain absorber is proportional to the product of its absorption cross section and its vertical column density. Since for spectroscopy of scattered radiation we have a nonlinear relationship between these quantities, the traditional concept cannot be, in general, extended to this application. The nonlinearity of this relationship is mathematically shown by a path integral formulation of the radiative transfer, which is developed. Its practical relevance is shown by Monte Carlo model calculations of the radiative transfer. The systematic error in both the air mass factor and the vertical column density associated with applying the traditional concept to spectroscopy of scattered radiation is also determined by Monte Carlo model calculations. Improved air mass factor concepts are developed in this work, which are also valid for absorption spectroscopy of scattered radiation (and not only for spectroscopy of direct radiation as the traditional concept) and include the traditional concept as a linear limit case. By applying one of these improved concepts the two fundamental problems mentioned above as well as the fundamental inconsistency of the method, i.e., the a priori knowledge of the vertical column density, which is required when applying the traditional concept, can be resolved. ¿ 2000 American Geophysical Union