The theory of heterochromatic irradiance fields is developed from the fundamental principles of radiative transfer. The irradiance field in natural waters that have fluorescing stratified layers of chlorophyll and other organic material is determined. This is the direct solution of the irradiance field and assumes knowledge of the depth distribution of the inherent optical properties, in particular the spectral absorption and scattering functions. Conversely, these optical properties are determined from in situ irradiance measurements by inverting the direct solution for the irradiance field. This is the inverse solution of the irradiance field. These direct and inverse solutions provide the mathematical framework for optical assays of fluorescing matter in lakes and seas, using measurements of the irradiance field within such media. In particular, the concentrations of the various fluorescing substances within the water are determined from in situ spectral irradiance measurements. ¿ American Geophysical Union 1988