Page 69 - ITU Journal: Volume 2, No. 1 - Special issue - Propagation modelling for advanced future radio systems - Challenges for a congested radio spectrum

ITU Journal: ICT Discoveries, Vol. 2(1), December 2019

ANALYTIC MODELS FOR BI-STATIC SCATTERING FROM A RANDOMLY ROUGH SURFACE WITH
COMPLEX RELATIVE PERMITTIVITY

Mostafa A. Karam , Ryan S. McDonough
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1 ASRC Federal Technical Services, McLean, Virginia, 22101, USA. NASA Glenn Research Center, Cleveland, Ohio, 44135, USA.

Abstract – This study provides explicit mathematical formulations for the bi-static scattering coefficient
from a randomly rough surface with a complex relative permittivity based on the following analytic models:
Small perturbation model (SPM), Physical optics model (PO), and Kirchhoff approximation model (KA). Then
it addresses the two shortcomings associated with each of the three models: i) limited applicability domain,
and ii) null predicted values for the cross-polarized bi-static scattering coefficients within plane of incidence.
The plane of incidence contains both backscattering direction and forward (specular reflection) direction
which are of interest to the spectrum community.

Keywords – Bi-static scattering coefficient, height correlation length, height variance, mean square slope,
reflection coefficient
capabilities of those models in predicting
1. INTRODUCTION cross-polarized bi-static scattering in the plane of
incidence. The basis for the mathematical models in
Several natural and manmade surfaces affecting
radio wave propagation can be treated as randomly this study are:
rough surfaces with complex relative permittivity. – Small perturbation model (SPM),
Among those surfaces are: – Physical optics model (PO), and
– Earth surface including land surface and sea – Kirchhoff approximation model (KA).
surface, The plane of incidence contains the incident
– Earth surface covers such as snow and sea ice, direction and the normal direction to the surface.
– town and city buildings and structures at the This plane contains the backscattering and the
HF frequencies and lower frequencies, forward (specular reflection) directions.

– and building walls at millimeter wave and The study is organized as follows. In Section 2, the
higher frequencies, etc. randomly rough interface characteristics are

The bi-static scattering coefficient from the above introduced along with the polarizations of the
surfaces is required for assessing several incident and scattered fields, as well as the two
parameters which are of interest to the spectrum components of the bi-static scattering coefficient:
community such as [1-3]; these are: the coherent component and diffuse component.
The analytical expression for the coherent
– fading depth due to reflection from the component is also given in Section 2. Explicit
Earth’s surface [2], expressions for the diffuse bi-static scattering
– interference power due to reflection from the coefficient component are developed in section 3,
Earth’s surface [1], section 4 and section 5 based on SPM, PO and KA
– radio noise due to microwave thermal models respectively. Finally, a summary and
emission from sea surface [3], etc. conclusion for the study is provided in section 6.
Deriving analytical expressions for the bi-static 2. PROBLEM FORMULATION
scattering coefficients could enhance as well as
extend the applicability domains of the existing Consider a randomly rough surface separating two
prediction methods of the above parameters [1-3]. non-magnetic media: an upper medium with
Based on the above, this study aims at: i) providing complex relative permittivity of unity (free space),
explicit expressions for the bi-static scattering and a lower medium with complex relative
coefficient based on three mathematical models permittivity . The surface height ( , ), which is

widely used in assessing bi-static scattering from aligned along the direction, is randomly
randomly rough surfaces, and ii) evaluating the fluctuating around an average value of zero and
with a constant variance .
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© International Telecommunication Union, 2019 53