# A Tri-Satellite Interference Source Localization Method for Eliminating Mirrored Location
## 3 Multi-Moment TDOA Localization Model  
As shown in Figure 1, the hyperboloid surfaces can be obtained based on the TDOA. The first hyperboloid surface is formed based on the TDOA measured from the main satellite and the first adjacent satellite at some moment. The second hyperboloid surface is formed based on the TDOA measured from the main satellite and the second adjacent satellite at some moment. At each moment, the two TDOA measurements of the three satellites including the main satellite and two adjacent satellite can be measured to form the TDOA hyperboloid surface 1. Under the assumption of zero elevation of the interference source, the location result of the interference source is the intersection of the two TDOA hyperboloid surfaces and the earth ellipsoid. The intersection is the result of tri-satellite location. In the situation shown in Figure 1, it can be found that the TDOA surface intersects at a straight line, and the straight line intersects with the earth's ellipsoid surface at two points. One point is the correct location of the interference source target, and the other is the mirrored position. Without any prior information, it is difficult to distinguish the correct location from the mirrored location only by the TDOA of a single moment. The following Figure 2 shows the intersection of the TDOA LOP when there are mirrored points in tri-satellite localization. It can be found that the LOP at multiple times gradually converge to one point at the real position, with the TDOA LOP at the mirrored point position intersect being more divergent. Therefore, the TDOA measurement results at multiple times can be used for localization to determine the true location of the interference source. Figure 2: The intersection of the TDOA LOP in multiple moments. Assuming that the coordinate of the interference source is \(\left(x,y,z\right)\), the coordinate of the main satellite at the first moment is \(\left(x_{p0},y_{p0},z_{p0}\right)\), the coordinate of the first adjacent satellite is \(\left(x_{p1},y_{p1},z_{p1}\right)\), and the second coordinate of the adjacent satellite is \(\left(x_{p2},y_{p2},z_{p2}\right)\). The measured TDOA between the interference source and the first adjacent satellite and second adjacent satellite at the first moment are \(\tau_{p1}\) and \(\tau_{p2}\), respectively, which can be given by:  
\[\tau_{p1} =\left(r_{p0}-r_{p1}\right)/c \tag{10}\] \[\tau_{p2} =\left(r_{p0}-r_{p2}\right)/c \tag{11}\]  
where \(c\) is the speed of light, \(r_{p0}\) is the distance between the interference source of the main satellite, \(r_{p1}\) is the distance between the interference source of the first adjacent satellite, \(r_{p2}\) is the distance between the interference source of the second satellite, and \(p\) represents the \(p\) th moment. According to the distance calculation method and the earth ellipsoid model, the tri-satellite localization equation set based on multiple moments can be given as follows:  
\[\left\{ \begin{array}{l}\sqrt{\left(x-x_{11}\right)^{2}+\left(y-y_{11}\right) ^{2}+\left(z-z_{11}\right)^{2}}-\sqrt{\left(x-x_{10}\right)^{2}+\left(y-y_{10 }\right)^{2}+\left(z-z_{10}\right)^{2}}=\tau_{11}\\ \sqrt{\left(x-x_{12}\right)^{2}+\left(y-y_{12}\right)^{2}+\left(z-z_{12} \right)^{2}}-\sqrt{\left(x-x_{10}\right)^{2}+\left(y-y_{10}\right)^{2}+\left(z -z_{10}\right)^{2}}=\tau_{12}\\ \vdots\\ \sqrt{\left(x-x_{p1}\right)^{2}+\left(y-y_{p1}\right)^{2}+\left(z-z_{p1} \right)^{2}}-\sqrt{\left(x-x_{p0}\right)^{2}+\left(y-y_{p0}\right)^{2}+\left( z-z_{p0}\right)^{2}}=\tau_{p1}\\ \sqrt{\left(x-x_{p2}\right)^{2}+\left(y-y_{p2}\right)^{2}+\left(z-z_{p2} \right)^{2}}-\sqrt{\left(x-x_{p0}\right)^{2}+\left(y-y_{p0}\right)^{2}+\left( z-z_{p0}\right)^{2}}=\tau_{p2}\\ \frac{x^{2}}{N^{2}}+\frac{y^{2}}{N^{2}}+\frac{z^{2}}{\left[\left[\left[\left(1 -z\right)\right]\right]^{2}}=1\end{array} \right.\right.