The reflection coefficients at normal incidence are found for a large class of one-dimensionally inhomogeneous or stratified half-spaces, which contain a homogeneous half-space. The formulation of the problem involves a combination of the classical boundary value technique, and the nonclassical principle of invariant imbedding. Solutions are in closed form and expressible in terms of Bessel functions. All results are given in terms of the ratio of the distance between free space and the homogeneous half-space to the wavelength in vacuo. One special case is that of an arbitrary number of layers lying on a homogeneous half-space where the dielectric constant of each layer has a constant gradient. A number of other special cases, limiting cases, and generalizations are developed including one in which the thickness of the top layer obeys a probability distribution, and another formulation that is computationally efficient in which there is an extremely small change in the dielectric constant such as with atmospheric inversion layers.