  We propose a new approach to model ground penetrating radar signals that
propagate through a homogeneous and isotropic medium, and are scattered at thin
planar fractures of arbitrary dip, azimuth, thickness and material filling. We
use analytical expressions for the Maxwell equations in a homogeneous space to
describe the propagation of the signal in the rock matrix, and account for
frequency-dependent dispersion and attenuation through the empirical Jonscher
formulation. We discretize fractures into elements that are linearly polarized
by the incoming electric field that arrives from the source to each element,
locally, as a plane wave. To model the effective source wavelet we use a
generalized Gamma distribution to define the antenna dipole moment. We combine
microscopic and macroscopic Maxwell's equations to derive an analytic
expression for the response of each element, which describes the full electric
dipole radiation patterns along with effective reflection coefficients of thin
layers. Our results compare favorably with finite-difference time-domain
modeling in the case of constant electrical parameters of the rock-matrix and
fracture filling. Compared with traditional finite-difference time-domain
modeling, the proposed approach is faster and more flexible in terms of
fracture orientations. A comparison with published laboratory results suggests
that the modeling approach can reproduce the main characteristics of the
reflected wavelet.
