A Novel Analytical Formulation of the Correlation of GNSS-R Signals Scattered by a Natural Fractal Surface

Natural surfaces exhibit scale-invariance properties and power-law spectra over a wide range of scales. For this reason, they are well described in the frame of fractal geometry and, in particular, using fractional Brownian motion (fBm) 2-D statistical processes. In this letter, we present the derivation of the correlation of Global Navigation Satellite System reflectometry (GNSS-R) signals scattered by a fBm surface. We here show that this correlation depends on a parameter related to the root-mean-square (rms) surface slope as measured at the electromagnetic wavelength scale. When this parameter increases, the correlation smoothly decreases from a value close to unity reaching the value of the roughness-independent expression already available in the literature. In our experiments, based on roughness measurements available in the literature, we first show how the description of the roughness of natural surfaces can be conveniently obtained via fBm parameters. Then, we illustrate the behavior of the correlation time for different roughness regimes. For a high-altitude airborne receiver, values ranging from about 6 ms (as predicted by the expression already available in the literature) to several tens of milliseconds are obtained.