# Evaluating the Vulnerability of Several Geodetic GNSS Receivers under Chirp Signal L1/E1 Jamming
## 2 GNSS Use in Geodesy and Jamming
### Receiver Position and Dispersion of Coordinates While Jamming  
A qualitative parameter used to measure the effect of jamming on the position determination should initially be defined. The distance between the real position and the measured one was a sensible choice. It is well known that the accuracy of GNSS receivers is much better in a horizontal rather than a vertical direction. For this reason, the projections of the distance to the horizontal plane and to the vertical line were calculated instead of the total distance itself. The first step was to determine the real position from all the samples: northing (\(n_{n}\)), easting (\(e_{n}\)) and height (\(v_{n}\)). Since some of them were jammed, and therefore it was not a good idea to simply calculate the mean value of all of them; instead, the average was calculated using the weighted coefficients:  
\[\overline{a}=\frac{1}{N}\sum_{n=1}^{N}w_{n}a_{n}, \tag{8}\]  
where \(a\) is any of the quantities (\(n\), \(e\) or \(v\)), \(w_{n}\) the weighted function and \(N\) the number of samples. Similarly, the standard deviation could be obtained from:  
\[\sigma_{a}^{2}=\frac{1}{N-1}\sum_{n=1}^{N}w_{n}(a_{n}-\overline{a})^{2}. \tag{9}\]  
The weighting function should ideally be chosen such that it would be 1 for non-jammed samples and 0 otherwise. Although the jamming time was well known during the acquisition, an algorithm was used to determine the jamming presence in order to keep the procedure independent of the measurements. For this purpose, \(w_{n}\) was defined iteratively. All \(w_{n}\) were initially set to 1 in order to get a rough estimate of \(\overline{a}\) and \(\sigma_{a}\). In the next step, all samples that had at least one of the three quantities out of the 3\(\sigma_{a}\) region were considered jammed and \(w_{n}\) was set to 0 for them. The new weighting function improved the accuracy of \(\overline{a}\) and \(\sigma_{a}\). The procedure was then repeated until no change in \(w_{n}\) was observed. This typically occurred in five to six steps. It should be stressed that the weighting function obtained in this way is not exact. There are some samples outside the 3\(\sigma_{a}\) region that are not jammed and there are some jammed samples within it. The question is how they affect the determination of \(\overline{a}\) and \(\sigma_{a}\). In order to get an estimate of the former, a normal distribution of the position is assumed for each of the three quantities when the signal is not jammed. This basically means that if there were no jammed samples, the procedure used for the purposes of this research would yield the solution \(s\) of the equation:  
\[\frac{1}{\sqrt{2\pi}\,\sigma}\int\limits_{-3s}^{3s}x^{2}e^{-\frac{x^{2}}{2 \sigma^{2}}}dx=us^{2} \tag{10}\]  
instead of \(\sigma_{a}\), where \(u\) is the normalisation factor (due to weighting):  
\[u=\frac{1}{\sqrt{2\pi}\,\sigma}\int\limits_{-3s}^{3s}x^{2}e^{-\frac{x^{2}}{2 \sigma^{2}}}dx. \tag{11}\]The above equation can be easily solved numerically, resulting in the ratio \(s/\sigma=0.98306\), meaning this effect can be simply ignored. For the effect of jammed samples within the interval of interest to the increased \(\sigma\) estimation, the following assumptions will be made. The overall position distribution is composed of two normal distributions, each with its own deviation: \(\sigma_{N}\) for non-jammed samples and \(\sigma_{I}\) for jammed, where the former is under one centimetre while the latter is above one metre. Where \(r\) is the proportion of the jammed samples, the mean square of the weighted samples can be calculated using:  
\[us^{2}=\frac{1-r}{\sqrt{2\pi}\sigma_{N}}\int\limits_{-3s}^{3s}x^{2}e^{-\frac{ x^{2}}{2\sigma_{N}^{2}}}dx+\frac{r}{\sqrt{2\pi}\ \sigma_{I}}\int\limits_{-3s}^{3s}x^{2}e^{-\frac{x^{2}}{2\sigma_{I}^{2}}}dx, \tag{12}\]  
with \(u\) again being the normalisation factor. Since \(s\) should obviously be greater than \(0.98306\sigma\approx\ \sigma\) and \(99.7\%\) of non-jammed samples fall within an \(3\sigma\) interval, the first integral can be extended to infinity.