An Analytical Formulation of the Correlation of GNSS-R Signals

We present an analytical formulation of the correlation coefficient of the electromagnetic fields scattered at near-specular direction by a rough or gently undulating surface and measured at two spatially separated positions occupied by a moving receiver at slightly different times. This allows us obtaining an explicit expression of the correlation time of the received signal in terms of radar and surface parameters. This work provides a contribution to the discussion, currently ongoing in the Global Navigation Satellite System Reflectometry (GNSS-R) scientific community, about the behavior of received signal fluctuations, especially when surface profile variations are such that the scattering is neither coherent nor completely incoherent. The scattering surface is here modeled as randomly rough, and the Kirchhoff approximation (KA) or the first-order small slope approximation (SSA1) is employed to compute the scattered field. In fact, the expression of the correlation coefficient is the same for both approximations. The obtained closed-form expression shows that as the surface correlation length increases, the degree of coherence smoothly increases from the value obtained with the expression already available in the literature for very rough surfaces to a value close to unity for gently undulating surfaces. The obtained behavior of correlation time as a function of surface parameters, system resolution, and observation geometry is in agreement with numerical simulations available in the literature. In general, obtained analytical results are in agreement with the observed behavior of GNSS-R signals over flat land surfaces.