# Analytical Computation of the Spatial Resolution in GNSS-R and Experimental Validation at L1 and L5
## 1 Introduction  
Global navigation satellite systems reflectometry (GNSS-R) is a relatively new remote sensing technique that uses navigation signals as signals of opportunity in a multi-static radar configuration [1].  
The spatial resolution of GNSS-R instruments was first analyzed in [2], and it was estimated from the dimensions of the area determined by the intersection of the first iso-delay and iso-Doppler lines. In this case, for a Low Earth Orbit (LEO) satellite at 700 km, the predicted spatial resolution for an instrument using the GPS L1 C/A (Coarse/Acquisition) code would be in the range of 36 to 54 km, for elevation angles from 90\({}^{\circ}\) to 40\({}^{\circ}\).  
Later on, the definition of spatial resolution was revised and \"an effective spatial resolution derived as a function of measurement geometry and delay-Doppler (DD) interval, and as a more appropriate representation of resolution than the geometric resolution previously used in the literature\" was proposed in [3]. This effective spatial resolution was defined following an approach similar to the definition of effective bandwidth used in filter theory. An increase in the effective spatial resolution is observed for increasing incidence angle, receiver altitude, and delay-Doppler window. With this definition, the effective spatial resolution is estimated to range from 25 to 37 km for a 700 km height receiver (Figure 4, [3]).  
The above definitions implicitly assume that the scattering is incoherent, as the radar equation used is the one that corresponds to surface scattering. However, the development of advanced GNSS-R receivers, capable of processing the coherent component and separating it from the incoherently scattered component, showed that much finer features can be distinguished [4; 5]. In fact, when coherent scattering occurs, the GNSS reflectivity values have a very high spatial resolution, as most of the power is mainly coming from the first Fresnel zone [6], an ellipse of semi-major axis \(a\) (Equation (1a)) and semi-minor axis \(b\) (Equation (1b)):  
\[a =\sqrt{\lambda\frac{R_{T}\cdot R_{R}}{R_{T}+R_{R}}} \tag{1a}\] \[b =\frac{a}{cos(\theta_{i})}, \tag{1b}\]  
where \(\lambda\) is the electromagnetic wavelength, \(R_{T}\) and \(R_{R}\) are the distances from the transmitter and receiver to the specular reflection point, and \(\theta_{i}\) is the incidence angle. This means that even from an LEO satellite, the spatial resolution can be as small as \(\sim\)400-500 m.  
In [7] the spatial resolution under coherent scattering was theoretically analyzed in detail, including the effect of the different Fresnel zones. It was shown that up to \(\sim\)45\({}^{\circ}\) most of the power is coming from an area determined by the first Fresnel zone, although a power equivalent to the one received in free-space conditions is actually coming from a region \(\sim\)0.58-0.62 times the first Fresnel zone. However, significant contributions from regions farther away than the first Fresnel zone also contribute to the received power, making its interpretation more difficult. This region was \(\sim\)800 m away at a 60\({}^{\circ}\) elevation angle for an LEO satellite at 500 km height.  
In this work, we have gone one step further and analyzed the spatial resolution from a theoretical point of view, using known principles of electromagnetism. The use of these results makes it possible to compute in a very compact form the expression of the ringings and oscillations observed when the specular reflection point transitions from the land into the ocean or vice versa. Analytical results are validated with airborne experimental data gathered at L1 and L5 with the Microwave Interferometric Reflectometer (MIR) instrument when crossing the coastline [8].