# GNSS Jamming Observed on Sounding Rocket Flights from Northern Scandinavia +
## 3 Rough Estimation of the Jammer Location  
The location of the jammer can at least be roughly estimated from the different altitudes at which the jamming signal appeared during the ascent and disappeared during the descent of the various sounding rocket trajectories. In particular, the different altitudes at  
Figure 4: Spectrogram of the baseband signal recorded by the Novatel OEM719 on MAIUS\(-\)2. Figure 3: Periodogram of the baseband signal recorded by the Novatel OEM719 on MAIUS\(-\)2.  
which the jamming appeared or disappeared during the flights of SOAR launched from ASC and MAIUS-2 launched from Esrange make it possible to geometrically analyze the origin of the jamming signal if the following assumptions are made:  
* The jammer that affected the flight of the SOAR sounding rocket is the same jammer that affected the flight of the MAIUS-2 sounding rocket;
* The jammer had not moved between the flights of the SOAR sounding rocket on 13 November 2023 and the MAIUS-2 sounding rocket on 2 December 2023;
* There is only one jammer that is located somewhere on the ground;
* The Earth's surface is modelled by the WGS84 ellipsoid, and the terrain is not considered;
* The lobe of the jamming signal, whose shape and orientation are unknown, is limited on the lower side by a minimum elevation angle, which can be positive, zero or even negative if the bending of the jamming signals by the troposphere is considered. Navigation signals in the L-band which are emitted horizontally with zero elevation are bended by the troposphere up to 0.5 deg. The Earth's ellipsoid is described by the function \[f(\boldsymbol{r})=\boldsymbol{r}^{T}\boldsymbol{M}\boldsymbol{r}-1=0\] (1) with the matrix \[\boldsymbol{M}= \begin{pmatrix}a^{2}&0&0\\ 0&a^{2}&0\\ 0&0&b^{2}\end{pmatrix} ^{-1}\] (2) where \[\boldsymbol{r}\] is the Cartesian position vector of a point on the ellipsoid, and \[a\] and \[b\] are the semi-major and semi-minor axes of the ellipsoid, respectively, as defined by WGS84. The minimum elevation mask can be described by a cone whose principal axis is perpendicular to the Earth's ellipsoid and whose apex is at the location of the jammer on the Earth's ellipsoid. The opening angle of the cone is \[\theta=\pi-2\varepsilon\] with \[\varepsilon\] being the minimum elevation angle. The equation of the cone is given with \[g(\boldsymbol{r})=\boldsymbol{n}^{T}\big{(}\boldsymbol{r}-\boldsymbol{r}_{j} \big{)}-\|\boldsymbol{n}\|\big{\|}\boldsymbol{r}-\boldsymbol{r}_{j}\big{\|} \sin\varepsilon=0\] (3) where \[\boldsymbol{r}\] is the Cartesian position vector of a point on the cone, \[\boldsymbol{r}_{j}\] is the Cartesian position vector of the jammer on the Earth's ellipsoid and \[\boldsymbol{n}\] is the normal vector, which is the gradient vector of the Earth's ellipsoid at \[\boldsymbol{r}_{j}\], \[a\]0. Using (1), (3) and (4), and assuming that the \(m\) points of the sounding rocket flight paths marking the start and end of interference, \(\boldsymbol{r}_{1},\boldsymbol{r}_{2},\ \cdots,\ r_{m}\), lie on the cone of the minimum elevation mask, a quadratic equation system consisting of \(m+1\) equations can be set up. \[h\big{(}\boldsymbol{r}_{j}\big{)}=\begin{pmatrix}\boldsymbol{r}_{j}^{T} \boldsymbol{M}\boldsymbol{r}_{j}-\boldsymbol{r}_{1}^{T}\boldsymbol{M} \boldsymbol{r}_{j}+\big{\|  
The start and end points of the interference \(\mathbf{r}_{1},\mathbf{r}_{2},\ \cdots,\ \mathbf{r}_{m}\) are identified from the C/N\({}_{0}\) ratios as well as the MAIUS-2 spectrogram. The quadratic equation system is solved for the unknown jammer location \(\mathbf{r}_{j}\) using MATLAB's ([[https://www.mathworks.com/products/matlab.html](https://www.mathworks.com/products/matlab.html)]([https://www.mathworks.com/products/matlab.html](https://www.mathworks.com/products/matlab.html))) fminsearch function. The used cost function is \(J\left(\mathbf{r}_{j}\right)=\mathbf{h}^{T}\left(\mathbf{r}_{j}\right)\mathbf{h}\left(\mathbf{r}_ {j}\right)\).