## 3 Multiple RFI Sources Geolocation Method
### RFI Sources Geolocation Estimation
It is worth noting that since almost all RFI sources are within a certain range from the ground, we can approximate that the RFI sources exist on an ellipsoid centered at the center of the earth. This will serve as a constraint for the above optimization problem  
Figure 5: Schematic diagram of RFI source location estimation. The figure shows the vector direction from the SAR system to the ground RFI sources at the 1-st, the i-th, and the N-th azimuth time obtained by using the RFI direction-of-arrival (DOA) estimation results. The dot-and-dash line represents the flight trajectory of the SAR, and the dotted line in three colors represents the estimated vector \(\mathbf{\tilde{v}}\) from the SAR to the three RFI sources. In the earth-centered, earth-fixed (ECEF) coordinate system, a point on the earth can be expressed as:  
\[\left\{ \begin{array}{l}\left(P_{j}^{k}\right)^{T}\mathbf{\Lambda}^{-1}P_{j}^{k}=1 \\ \mathbf{\Lambda}=diag\big{(}\begin{array}{cc}R_{a}{}^{2}&R_{a}{}^{2}&R_{b}{}^{2} \end{array}\end{array} \right)\quad, \tag{45}\]  
where \(R_{a}\) and \(R_{b}\) are respectively the semi-major axis and semi-minor axis of the earth, so the constraint condition is defined as:  
\[\left\{ \begin{array}{l}1\leq\left(\mathbf{H}P_{j}^{k}\right)^{T}\mathbf{ \Lambda}^{-1}\mathbf{H}P_{j}^{k}\leq\chi^{2}\\ \mathbf{\Lambda}=diag\big{(}\begin{array}{cc}R_{a}{}^{2}&R_{a}{}^{2}&R_{b}{}^{ 2}\end{array}\end{array} \right)\quad, \tag{46}\]  
where \(\mathbf{H}\) represents the transformation matrix from the ENU coordinate system to the ECEF coordinate system, \(\chi\) represents the range ratio. In this paper, we assume that the height of the RFI source does not exceed 10 km, so it should satisfy \(\chi\in(1,1.0016]\). The optimization problem for solving the coordinate position of the \(k\)-th RFI source is expressed as:  
\[\left\{ \begin{array}{l}\min\sum\limits_{i=1}^{N_{e}}\left\|P_{j}^{k}-_{i}- \left[P_{j}^{k}-_{i}\right]\cdot\widehat{\mathbf{v}}_{i}^{k}\right\|_{2}\\ s.t.\ 1\leq\left(\mathbf{H}P_{j}^{k}\right)^{T}\mathbf{\Lambda}^{-1}\mathbf{H}P_{j}^ {k}\leq\chi^{2}\\ 1\leq i\leq N\&i\leq N\&i\end{array} \right.\quad. \tag{47}\]  
The optimization problem in Equation (47) is non-convex and needs to be solved in its nonlinear constraints, the particle swarm optimization is applied to solve it. As a method with the ability to solve non-convex problems and global search capabilities, the PSO algorithm is easy to be implemented and the computational cost is low in memory requirements [27]. It is also a derivative-free method that can achieve convergence rapidly. Not only that, it can find a good solution in a complex search range based on the experience of particle swarms [27]. It first randomly initializes a group of particles, and then continuously searches for suitable function values in the domain. In this process, these particles share information, which enables them to constantly adjust their own optimization direction and finally find the optimal solution in the complex search space [27]. In the solution process, we define the size of the particle swarm as \(N_{PSO}\) and use the optimization function and constraint conditions in Equation (47) as the fitness function and search range of the PSO algorithm. As mentioned above, the solution of this optimization problem is a three-dimensional vector, so during the t-th iteration the position of the j-th particle in the search space can be represented by a three-dimensional vector \(\rho_{j}^{t}=\left(\rho_{j1}^{t},\rho_{j2}^{t},\rho_{j3}^{t}\right)\), and its velocity is \(\mu_{j}^{t}=\left(\mu_{j1}^{t},\mu_{j2}^{t},\mu_{j3}^{t}\right)\). The shared information of the particle swarm during the t-th iteration calculation is: a) Individual optimal position: the best position that the j-th particle has ever been to is \(\delta_{j}^{t}=\left(\delta_{j1}^{t},\delta_{j2}^{t},\delta_{j3}^{t}\right)\); (b) Group optimal position: the global best position of the group discovered so far is expressed as \(\delta_{g}^{t}=\left(\delta_{g1}^{t},\delta_{g2}^{t},\delta_{g3}^{t}\right)\).