## 3 Multiple RFI Sources Geolocation Method
### RFI Sources Geolocation Estimation  
By means of applying the RFI DOA estimation algorithm in Section 3.3 to the echo signal at each azimuth time when the SAR is interfered with, the positional relationship between the interference signal of K RFI sources and the radar array at each azimuth time can be calculated. We define \(\mathbf{U}\) represents the DOA matrix at each azimuth time, and \(\mathbf{u}_{i}\) is DOA estimation of K RFI signals at the i-th azimuth time:  
\[\left\{\begin{array}{l}\mathbf{U}=\left[ \begin{array}{cccc}\mathbf{u}_{ 1}&\mathbf{u}_{2}&\cdots&\mathbf{u}_{N}\\ \mathbf{u}_{i}=\left[\begin{array}{cccc}\mathbf{u}_{1}^{1}&\mathbf{u}_{2}^ {2}&\cdots&\mathbf{u}_{i}^{K}\end{array}\right]\\ \mathbf{u}_{i}^{k}=\left(\begin{array}{cccc}\hat{\theta}_{i}^{k}&\hat{\phi}_ {i}^{k}\end{array}\right)^{T}\\ 1\leq i\leq N\&i&\leq N^{*},1\leq k\leq K\&k\&\in N^{*}\end{array} \right.. \tag{41}\]  
Thus, at each azimuth time, a vector \(\mathbf{\tilde{V}}\) from the radar array to each interference source can be calculated by the DOA estimation result, its computation method approximates to Equation (41):  
\[\left\{ \begin{array}{l}\mathbf{\tilde{V}}=\left[\begin{array}{cccc}\mathbf{ \tilde{v}}_{1}&\mathbf{\tilde{v}}_{2}&\cdots&\mathbf{\tilde{v}}_{N}\end{array} \right]\\ \mathbf{\tilde{v}}_{i}=\left[\begin{array}{cccc}\mathbf{\tilde{v}}_{i}^{1}& \mathbf{\tilde{v}}_{i}^{2}&\cdots&\mathbf{\tilde{v}}_{i}^{K}\end{array}\right] \\ \mathbf{\tilde{v}}_{i}^{k}=\left(\begin{array}{cccc}1,\text{sec}\,\delta_{i }^{k}\cdot\text{cot}\,\hat{\phi}_{i}^{k},\text{tan}\,\hat{\theta}_{i}^{k} \end{array}\right)^{T}\\ 1\leq i\leq N\&i&\in N^{*},1\leq k\&\&\in N^{*}\end{array} \right., \tag{42}\]where \(\tilde{\mathbf{v}}_{i}^{k}\) represents the vector direction from the SAR system to the \(k\)-th RFI source at the i-th azimuth time calculated from the RFI DOA estimation result, and \(\mathbf{\tilde{V}}\) is the matrix of these estimated vector directions of all azimuth time. The schematic diagram of solving the RFI source location is shown in Figure 5.  
We define \({}_{i}=\left(x_{i},y_{i},z_{i}\right)^{T}\)(\(1\leq i\leq N\&i\in N^{*}\)) as the spatial coordinate of the radar array at the i-th azimuth time, and \(P_{j}^{k}=\left(x^{k},y^{k},z^{k}\right)^{T}\) denotes the spatial coordinates of the \(k\)-th RFI source. Accordingly, the vector \(\mathbf{V}\) connecting the \(k\)-th RFI source and the radar array at all azimuth time is written as:  
\[\left\{\begin{array}{l}\mathbf{V}=\left[ \begin{array}{cccc}\mathbf{v}_{1}& \mathbf{v}_{2}&\cdots&\mathbf{v}_{N}\\ \mathbf{v}_{i}^{1}&\mathbf{v}_{i}^{2}&\cdots&\mathbf{v}_{i}^{k}\\ \mathbf{v}_{i}^{k}=\Upsilon_{i}-P_{j}^{k}\\ 1\leq i\leq N\&i\in N^{*},1\leq k\leq K\&k\&\end{array} \right]\right., \tag{43}\]  
where \(\mathbf{v}_{i}^{k}\) denotes the real vector direction from the SAR system to the \(k\)-th RFI source at the i-th azimuth time. Obviously, if DOA estimates are accurate enough, \(\mathbf{\tilde{V}}\) and \(\mathbf{V}\) are parallel, which can be written as \(\mathbf{\tilde{V}}\parallel\mathbf{V}\). At the i-th azimuth time, this parallel relationship can be expressed as:  
\[\frac{x_{i}-x^{k}}{1}=\frac{y_{i}-y^{k}}{\sec\tilde{\theta}^{k}(i)\cdot\cot \tilde{\phi}^{k}(i)}=\frac{z_{i}-z^{k}}{\tan\tilde{\theta}^{k}(i)}. \tag{44}\]  
Theoretically, a straight line equation like Equation (44) can be obtained at each azimuth time, and the intersection of all straight lines is the position of the RFI source, so the position of the RFI source can be solved by combining the linear equations at all azimuth time. However, the inevitable noise and systematic errors will cause DOA estimation errors, which will make the equations difficult to solve. Hence, we propose to transform this problem into an optimization problem, which is to find a point in the space coordinate system so that the sum of the distances from the point to each straight line is the smallest.