In this paper we derive the constraints imposed on an arbitrary 4x4 real matrix such that it correspond to a physically realisable Mueller matrix M. These constraints are important, in practice, when M is derived from experimental measurements. Under such circumstances the measured matrix may be filtered to yield two components, one of which is physically realisable plus a nonphysical remainder term. By comparing the relative magnitudes of these two, we obtain a quantitative measure of system fidelity. In the course of this development, we outline two new matrix descriptors of polarised scattering, the system covariance and coherency matrices. These 4x4 Hermitian operators are linearly related to X but yield better physical insight into the scattering phenomena causing depolarisation. We illustrate by reference to rough surface scattering under physical optics.© (1990) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.