Exact equations are derived for calculating the brightness temperature of a medium that is bounded by a plane surface and whose properties (dielectric constant and thermometric temperature) vary only with depth. Although no approximations are made in the development of the principal results, the final equations are presented in a form most convenient for use in the area of microwave radiometry (that is, in the limit where the Rayleigh‐Jeans approximation to the Planck blackbody radiation law is valid). Some special cases in which the differential equations arising in the theory can be solved analytically are presented as examples. A practical numerical method for evaluating the exact equations by use of a digital computer is also discussed.
A. Stogryn (Tue,) studied this question.