Deviation from reciprocity in bidirectional reflectance

The subject of the paper is to discuss and quantify deviations from recipro city of the bidirectional reflectance distribution function (BRDF), i.e., t he difference of BRDF obtained when inverting illumination and viewing dire ctions. Directional reciprocity is not valid in general, because when the i llumination beam has a spatial extension larger than the viewed area las is most often the case for BRDF measurements), some of the scatterers buildin g up the observed radiance are located at different places in reciprocal me asurements. The physical systems under consideration in the two experiments are different, hence the breakdown of reciprocity. The paper develops a th eory aiming at a quantitative estimation of deviations from directional rec iprocity due to this factor. The theory is based on integral forms of the r adiative transfer equation in a horizontal slab of heterogeneous absorbing and scattering media. The observed scene radiance is expanded in a series o f scattering orders. Integral expressions of the single- and multiple-scatt ering radiance are derived and put in a form suitable for the analysis of t he reciprocity problem. The first-order expression leads to an estimate of the order of magnitude E of the relative deviations from reciprocity, epsil on approximate to h/D deltaQ/Q root tan(2)theta (i) + tan(2) theta (v) - 2 tan theta (i) tan theta (v) cos phi, where D is the size of the viewed area (pixel size for imaging sensors), h is the vertical photon mean free path, deltaQ/Q is a measure of the scene heterogeneity, and theta (i), theta (v) , and phi are the illumination and view zenith angles and the relative azim uth between illumination and view directions. It is argued that this order of magnitude should remain approximately valid if all orders of scattering are taken into account. A discussion of practical applications in various f ields, laboratory optical reflectometry, Earth radiation budget monitoring, and terrestrial surfaces remote sensing is given.