Determining Real Permittivity From Fresnel Coefficients in GNSS-R

Abstract

Global Navigation Satellite System Reflectometry (GNSS-R) can be used to derive information about the composition or the properties of ground surfaces, by analyzing signals emitted by GNSS satellites and reflected from the ground. If the received power is measured with linearly polarized antennas, under the condition of smooth surface, the reflected signal is proportional to the modulus of the perpendicular and parallel polarization Fresnel coefficients, which depend on the incidence angle θ, and on the dielectric constant ε of the soil. In general, ε is a complex number; for non-dispersive soils, the imaginary part of ε can be neglected, and a real value of ε is sought. We solve the realvalued problem explicitly giving formulas that can be used to determine the dielectric constant ε and we compare the analytical solution with experimental data in the case of sand soil.

Explicit Complex Solutions to the Fresnel Coefficients

“…Thus e is a real number, e .1, and e coincides with the real solution e n found in [19]. (In the other limit case u ¼ b n cos h of (12), < e ð Þ ¼ 1, as expected.…”

“…1 if and only if C n , 0. In this case, (18) coincides with the real solution ( 17) of (3) found in [19], because c n ¼ C n j j. Similarly, the analogue of (4) has two real algebraic solutions e p1 and e p2 , with e p1 , 1 , e p2 if C p , 0.…”

“…In this case, e can still be found from either of (3) or (4), as long as the angle h and the coefficients c n and c p satisfy a mutual compatibility condition, which can be checked explicitly and ensures that the corresponding two solutions coincide. In [19], explicit equivalent formulas for < e ð Þ were given, assuming that = e ð Þ ¼ 0. This solution corresponds to the limit case u ¼ a n cos h; indeed, in this case, y ¼ 0 and…”

Soil Moisture Content from GNSS Reflectometry Using Dielectric Permittivity from Fresnel Reflection Coefficients

“…It is important to extend the SMC estimation to greater angles of incidence, not limited to 35 • , and thus to increase the number of SPs. This implies reviewing methods such as those proposed in [72], in which an inversion method based on the two linear Fresnel coefficients for GNSS-R data is undertaken. The authors based their work on the coherent component of the signal without considering the incoherent BSR and VOD effects.…”