In the near future a number of global, multiyear, satellite-borne ultraviolet remote sensing missions will scan or image the limb of the Earth to measure the O+ concentration within the ionospheric F layer. On the day side we will use discrete inverse theory (DIT) to retrieve vertical profiles of the O+ number density from measurements of the 834-¿ airglow. Here we describe the theoretical foundations and the components of the retrieval code, which computes both a maximum likelihood solution and the associated covariance matrix. New results include enhancement of the DIT formalism to distinguish between random and nonrandom (systematic) errors and comparisons of Millstone Hill incoherent scatter radar (ISR) data with generalized Chapman-type representations of the O+ density profile. Of the candidates, the Chapman-type profile with a constant-gradient scale height provides the best fits to the ISR data.¿ 1997 American Geophysical Union