A transport theory of pulse propagation in a strongly forward scattering random medium

The scalar time-dependent equation of radiative transfer is used to develop a theory of pulse propagation in a discrete random medium whose scatter function (phase function) consists of a strong, narrow forward lobe superimposed over an isotropic background. The situation analyzed is that of a periodic sequence of plane-wave pulses, incident from an air half-space, that impinges normally upon the planar boundary surface of a random medium half-space; the medium consists of a random distribution of particles that scatter (and absorb) radiation in accordance with the aforementioned phase function. After splitting the specific intensity into the reduced incident and diffuse intensities, the solution of the transport equation in the random medium half-space is obtained by expanding the angular dependence of both the scatter function and the diffuse intensity in terms of Legendre polynomials, and by using a point matching procedure to satisfy the boundary condition that the forward travelling diffuse intensity be zero at the interface. Curves of received power show that, at small penetration depths, the coherent (reduced incident) intensity dominates, whereas at large depths, the incoherent (diffuse) intensity is the strongest and causes the pulses to broaden and distort. The motivation for this study was to complement a test series, on mm-wave pulse propagation in vegetation, by a theory that provides understanding of overall trends and assistance in the interpretation of measured results. In the mm-wave region, all scatter objects in a forest have large dimensions compared to a wavelength and, therefore, produce strong forward scattering and a phase function of the type assumed in this paper.