Effective dielectric constant for a random medium with different scattering species after renormalization of the Helmholtz equation

The main objective of the present study is to investigate the problem of coherent scattering from a half-space of discrete scatterers by importing a simple version of the renormalization formalism, which has been of great use in other fields such as solid state physics or particle physics. The problem of scattering from vegetation involves a great number of parameters including shape, size and spatial distributions of the scattering elements. Regarding the spatial distribution, not much is known about the correlation functions [7] between particles. Although an ultimate aim should be a more complicated and detailed renormalization model, the theoretical paradigm proposed here starts with the simplest geometry. Namely, renormalization is applied to solve the Dyson equation for the scattering problem of a half-space of uncorrelated and isotropic, uniformly distributed scatterers of two species: spherical and cylindrical shape. Thus, the assumption of uncorrelated scatterers allows us to obtain a very simple dispersion relation, which is valid only for sparse distributions and low frequencies.