# A Tri-Satellite Interference Source Localization Method for Eliminating Mirrored Location
## 4 Improved Tri-Satellite Localization Method  
Consider that it is difficult to directly solve the nonlinear localization equations formed by the measurement results at multiple moments. The [PERSON] iterative method is used to solve it. The above equations can be expressed as follows:  
\[\left\{\begin{array}{l}f_{11}=\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}\\ f_{12}=\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  
\[\mathbf{F}^{\prime}\left(x,y,z\right)=\left[ \begin{array}{ccc}\frac{\partial f_{ \text{f11}}}{\partial x}&\frac{\partial f_{\text{f11}}}{\partial y}&\frac{ \partial f_{\text{f11}}}{\partial z}\\ \frac{\partial f_{\text{f22}}}{\partial x}&\frac{\partial f_{\text{f2}}}{ \partial y}&\frac{\partial f_{\text{f2}}}{\partial z}\\ \vdots&\vdots&\vdots\\ \frac{\partial f_{\text{f12}}}{\partial x}&\frac{\partial f_{\text{f12}}}{ \partial y}&\frac{\partial f_{\text{f12}}}{\partial y}\\ \frac{\partial f_{\text{f22}}}{\partial x}&\frac{\partial f_{\text{f22}}}{ \partial y}&\frac{\partial f_{\text{f22}}}{\partial z}\\ \frac{\partial f_{\text{f}}}{\partial x}&\frac{\partial f_{\text{f}}}{ \partial y}&\frac{\partial f_{\text{f}}}{\partial z}\\ \end{array} \right]=\left[ \begin{array}{ccc}\frac{x-x_{\text{f11}}}{r_{ \text{f11}}}-\frac{x-x_{\text{f01}}}{r_{\text{f01}}}&\frac{y-y_{\text{f11}}}{ r_{\text{f0}}}&\frac{z-z_{\text{f11}}}{r_{\text{f11}}}-\frac{z-z_{\text{f0 }}}{r_{\text{f0}}}\\ \frac{x-x_{\text{f2}}}{r_{\text{f2}}}-\frac{x-x_{\text{f0}}}{r_{\text{f0}}} &\frac{y-y_{\text{f2}}}{r_{\text{f2}}}-\frac{y-y_{\text{f01}}}{r_{\text{f0}}} &\frac{z-z_{\text{f22}}}{r_{\text{f2}}}-\frac{z-z_{\text{f0}}}{r_{\text{f0}}} \\ \vdots&\vdots&\vdots\\ \frac{x-x_{\text{f12}}}{r_{\text{f1}}}-\frac{x-x_{\text{f02}}}{r_{\text{f0}}} &\frac{y-y_{\text{f01}}}{r_{\text{f1}}}-\frac{y-y_{\text{f02}}}{r_{\text{f0} }}&\frac{z-z_{\text{f12}}}{r_{\text{f01}}}-\frac{z-z_{\text{f02}}}{r_{\text{ f0}}}\\ \frac{x-x_{\text{f22}}}{r_{\text{f22}}}-\frac{x-x_{\text{f02}}}{r_{\text{f0}}} &\frac{y-y_{\text{f02}}}{r_{\text{f22}}}-\frac{y-y_{\text{f02}}}{r_{\text{f0 }}}&\frac{z-z_{\text{f22}}}{r_{\text{f02}}}-\frac{z-z_{\text{f02}}}{r_{\text{ f0}}}\\ \frac{2x}{N^{2}}&\frac{2y}{N^{2}}&\frac{2z}{\left[N\left(1-z\right)\right]^{2} }\end{array} \right] \tag{15}\]  
If the \(k\) th interference source coordinate in this iteration is \(\mathbf{P}^{k}=\left(x^{k},y^{k},z^{k}\right)\), it can be iterated according to the following formula:  
\[\mathbf{P}^{k+1}=\mathbf{P}^{k}-\left[\mathbf{F}^{\prime}\left(x^{k},y^{k},z^ {k}\right)\right]^{-1}\mathbf{F}\left(x^{k},y^{k},z^{k}\right) \tag{16}\]  
The process of the tri-satellite interference source localization method based on multiple moments is summarized as follows:  
Step 1: Calculate the distance difference between the corresponding three satellites and the interference source \(\left(r_{p1}-r_{p0}\right)\) and \(\left(r_{p2}-r_{p0}\right)\) according to the TDOA at \(P\) moments;  
Step 2: Initialization, set the number of iterations \(k=0\), and the initial position of the interference source \(\mathbf{P}^{0}\);  
Step 3: Calculate the Hessian matrix \(F^{\prime}\left(x^{k},y^{k},z^{k}\right)\) in the process of this iteration, and obtain the estimation result of the interference source position \(\mathbf{P}^{k+1}\) in the next iteration according to formula (16);  
Step 4: Define the position error \(\delta=\sqrt{\left(x^{k+1}-x^{k}\right)^{2}+\left(x^{k+1}-x^{k}\right)^{2}+ \left(x^{k+1}-x^{k}\right)^{2}}\) of the two iterations as the cost function in the iteration process. If it is less than a certain minimum threshold, stop the iteration and output the positioning result; otherwise, let \(k=k+1\), and return to step 3 to continue the iteration. Step 5: Precise localization moment by moment.