Propagation and scattering of collimated beam wave in vegetation using scalar transport theory

The scalar time-dependent equation of radiative transfer is used to develop a theory of pulse beamwave propagation and scattering in a medium character ized by many random discrete scatterers which scatter energy strongly in the forward scattering direction. Applications include the scattering of highly collimated millimeter waves in vegetation and optical beams in the atmosphere. The specific problem analyzed is that of a periodic sequence of Gaussian shaped pulses normally incident from free space onto the planar boundary surface of a random medium half-space, such as a forest, that possesses a power scatter (phase) function consisting of a strong, narrow forward lobe superimposed over an isotropic background. After splitting the specific intensity into the reduced incident and diffuse intensities, the solution of the transport equation expressed in cylindrical coordinates 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 Associate Legendre polynomials, by using a Fourier series/Hankel transform to obtain the equation of transfer for each spatial frequency, and by employing the weighted residual method to satisfy the boundary condition that the forward traveling diffuse intensity be zero at the interface. Data generated from the solution will be compared to results obtained from a computationally intensive second method of solution, which follows the procedure used by Chang and Ishimaru to study the propagation and scattering of monochromatic beam waves in random media. In this second method, the timedependent scalar transport equation is solved using a Fourier Series/Hankel transform along with the two-dimensional Gauss quadrature formula and an eigenvalue eigenvector technique. Numerical results are given for received power at different penetration depths, different beam sizes and different scatter directions