Explicit Complex Solutions to the Fresnel Coefficients

Global Navigation Satellite System Reflectometry (GNSS-R) is a microwave remote sensing technique which can be used to derive information about the composition or the properties of ground surfaces. The received power of the GPS signals reflected by the ground is proportional to the magnitude of the reflection Fresnel coefficients In particular, it depends on the incidence angle $\theta$ and on the ground's permittivity $\epsilon$. The knowledge of $\epsilon$ is important for determining various conditions and characteristics of the surface (e.g., soil moisture, salinity, freeze-thaw transitions). The value of $\epsilon$ can be found from the Fresnel reflection coefficients, for a given incidence angle $\theta$. For dispersive media, $\epsilon$ is a complex quantity; we present explicit formulas, which express both $\Re(\varepsilon)$ and $\Im(\varepsilon)$ as a function of the incident angle $\theta$ and of the magnitude of the linearly polarized Fresnel reflection coefficients.