## 3 Proposed Tracking Algorithm
### Strong Tracking Kalman Filter  
The source coordinates and their rate of change are taken as the state parameters, \(\mathbf{X}_{cv,k}=\left[ \begin{array}{cccc}x_{k}&z_{k}&v_{x}&v_{z}\end{array} \right]\). Assume that the state equation is given by:  
\[\mathbf{X}_{cv,k}=\mathbf{F}\mathbf{X}_{cv,k-1}+w_{cv,k} \tag{19}\]\[\text{where }\mathbf{F}=\left[ \begin{array}{cccc}1&0&\Delta T&0\\ 0&1&0&\Delta T\\ 0&0&1&0\\ 0&0&0&1\end{array} \right],\Delta T\text{ is the data output interval expressed in seconds, and}\]  
\(w_{cv,k}\) is the system noise at the current moment, whose covariance matrix is \(Q_{cv,k}\). The measurement information represents directly output positioning information, so the measurement equation is given by:  
\[\mathbf{Z}_{cv,k}=\mathbf{H}\mathbf{X}_{cv,k}+\mathbf{V}_{cv,k} \tag{20}\]  
where \(\mathbf{Z}_{cv,k}=\left[ \begin{array}{cccc}x_{k}^{R}&z_{k}^{R}\end{array} \right]\), \(\mathbf{H}_{cv,k}=\left[ \begin{array}{cccc}1&1&0&0\end{array} \right]\), \(x_{k}^{R}\) is directly output location information at the current time; \(\mathbf{V}_{cv,k}\) is the measurement noise at the current moment, and its covariance matrix is \(\mathbf{R}_{cv,k}\). In the strong tracking kalman filter, the gain matrix \(\mathbf{K}_{k}\) needs to be determined online to satisfy the following conditions:  
\[E[x_{k}-\hat{x}_{k}][x_{k}-\hat{x}_{k}]^{T}=\min \tag{21}\]  
\[E[\mathbf{r}_{k}^{T}\mathbf{r}_{k+j}]=0,\quad k=0,1,2,\cdots,j=1,2\cdots \tag{22}\]  
where \(\mathbf{r}_{k}=\mathbf{Z}_{k}-\mathbf{H}_{k}\mathbf{X}_{k,k-1}\) represents an innovation vector, whose covariance matrix is \(\mathbf{N}_{k}=\mathbf{H}_{k}\mathbf{P}_{k,k-1}\mathbf{H}_{k}^{T}\). Equation (22) requires \(\mathbf{r}_{k}\) to be orthogonal, which actually represents an index of the residual sequence that is orthogonal on the basis of the index of minimum variance of the residual. If the signal is not affected by environmental factors, such as refraction, reflection, and multipath effect, it can be considered that \(\mathbf{r}_{k}\) obeys the Gaussian distribution with the zero mean. In this case, Equation (22) satisfies. Therefore, in that case, the strong tracking kalman filter represents the standard extended kalman filter. However, uncertain influencing factors in the external environment will lead to the deviation of the filter state estimate from the system state. In this situation, to realize the system state tracking by the filter, the gain matrix needs to be adjusted online to make the residual sequence still be mutually orthogonal. Based on Equations (19) and (20), the strong tracking filtering algorithm can be expressed as follows:  
\[\left\{ \begin{array}{l}\mathbf{X}_{k,k-1}=\mathbf{F}\mathbf{X}_{k-1}\\ \mathbf{P}_{k,k-1}=\mathbf{L}(k)\mathbf{P}_{k-1}\mathbf{F}^{T}+\mathbf{Q}_{cv,k-1}\\ \mathbf{K}_{k}=\mathbf{P}_{k,k-1}\mathbf{H}_{k}^{T}(\mathbf{H}_{k}\mathbf{P}_{k,k-1}\mathbf{H}_{k}^{T} +\mathbf{R}_{cv,k})^{-1}\\ \mathbf{P}_{k}=(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{P}_{k,k-1}\\ \mathbf{X}_{k}=\mathbf{X}_{k,k-1}+\mathbf{K}_{k}(\mathbf{Z}_{k}-\mathbf{H}_{k}\mathbf{X}_{k,k-1})\\ \mathbf{L}(k)=diag[\lambda_{1(k)},\lambda_{2(k)},\cdots\lambda_{n(k)}]\end{array} \right. \tag{23}\]  
where \(\mathbf{X}_{k,k-1}\) is the predicted state vector, \(\mathbf{P}_{k,k-1}\) is the covariance matrix of the predicted state, \(\mathbf{K}_{k}\) is the Kalman gain matrix; \(\mathbf{X}_{k-1}\) and \(\mathbf{X}_{k}\) represent the estimated state matrices at times (\(k-1\)) and \(k\), and \(\mathbf{P}_{k-1}\) and \(\mathbf{P}_{k}\) are their respective covariance matrices, respectively; \(\mathbf{Z}_{k}\) indicates the measured value at the current time; \(\mathbf{I}\) is the identity matrix; \(\mathbf{L}(k)\) is the time-varying elimination matrix; \(\lambda_{i(k)}\geq 1(i=1,2,\cdots,n)\) is the \(n\) time-varying fading factor, which can be expressed as follows:  
\[\lambda_{i(k)}= \begin{cases}\lambda_{0(k)}&\lambda_{0(k)}>1\\ 1&\lambda_{0(k)}\leq 1\end{cases}