## 2 Hybrid Source Localization Based on TOA and TDOA  
### Hybrid Source Localization Algorithm Model  
As shown in Figure 1, we consider three-station TOA and four-station TDOA localization systems in three-dimensional space, where the TOA system is composed of three observation stations denoted as \(S_{4}\), \(S_{5}\), and \(S_{6}\), and the TDOA system is composed of four observation stations denoted as \(D_{0}\), \(D_{1}\), \(D_{2}\), and \(D_{3}\). The position coordinates of each station are denoted as \(D_{i}=\left[x_{i}\,y_{i}\,z_{i}\right]^{T}\), \(i=0\sim 3\), \(S_{i}=\left[x_{i}\,y_{i}\,z_{i}\right]^{T}\), \(i=4\sim 6\), and the coordinate of the signal source is \(T=\left[x\,y\,z\right]^{T}\). The distance and difference in distance between each station in the TDOA positioning system and signal source \(T\) are given as follows  
\[r_{i}=\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}+\left(z-z_{i} \right)^{2}},\;i=0\sim 3 \tag{1}\]  
\[r_{TDOA,\;i}=r_{i}-r_{0}=\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{ 2}+\left(z-z_{i}\right)^{2}}\]  
\[-\sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right) ^{2}},\;i=1\sim 3 \tag{2}\]  
Similarly, the distance between each station in the TOA positioning system and signal source \(T\) are stated as follows:  
\[r_{TOA,\;i}=\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}+\left(z- z_{i}\right)^{2}},\;i=4\sim 6 \tag{3}\]  
Combining Equations (2) and (3) yields the function matrix \(\mathbf{f}(\mathbf{x})\) as  
Figure 1: Systematic scenario of hybrid source localization technique. \[\mathbf{f}(\mathbf{x})= \begin{pmatrix}\sqrt{\left(x-x_{4}\right)^{2}+\left(y-y_{4}\right)^ {2}+\left(z-z_{4}\right)^{2}}\\ \sqrt{\left(x-x_{5}\right)^{2}+\left(y-y_{5}\right)^{2}+\left(z-z_{5}\right)^ {2}}\\ \sqrt{\left(x-x_{6}\right)^{2}+\left(y-y_{6}\right)^{2}+\left(z-z_{6}\right)^ {2}}\\ \sqrt{\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}+\left(z-z_{1}\right)^ {2}}\\ \sqrt{\left(x-x_{2}\right)^{2}+\left(y-y_{2}\right)^{2}+\left(z-z_{2}\right)^ {2}}\\ \sqrt{\left(x-x_{3}\right)^{2}+\left(y-y_{3}\right)^{2}+\left(z-z_{3}\right)^ {2}}\end{pmatrix} - \begin{pmatrix}0\\ 0\\ 0\\ 0\\ \sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^ {2}}\\ \sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^ {2}}\\ \sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^ {2}}\\ \sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^ {2}}\end{pmatrix}  \tag{4}\]  
The measured values of TOA and TDOA are represented as \(\tilde{r}_{TOA}\) and \(\tilde{r}_{TDOA}\), respectively. The complete measurement result is given as \(\tilde{\mathbf{\tau}}=[\tilde{r}_{TOA},\tilde{\mathbf{\tau}}_{TOA},\tilde{\mathbf{\tau}}_{ TOA,6}\tilde{\mathbf{\tau}}_{TDOA,1}\tilde{\mathbf{\tau}}_{TDOA,2}\)\(\tilde{r}_{TDOA,3}]^{T}\). The TOA and TDOA measurement positioning models can then be expressed as  
\[\tilde{\mathbf{\tau}}=\mathbf{f}(\mathbf{x})+\mathbf{n} \tag{5}\]  
where \(\mathbf{n}\) is the measured noise vector and satisfies the random probability distribution. Solving Equation (5) yields the position vector \(\mathbf{x}\). Typically, there are two types of solution algorithms: linear and nonlinear. Nonlinear approaches are more computationally intensive but more accurate, and often use least squares or maximum likelihood estimators to estimate the position vector \(\mathbf{x}\). There are two types of least squares methods, weighted and unweighted, when the measurement noise \(\mathbf{n}\) conforms to the zero-mean Gaussian distribution, weighted least squares are identical to the maximum likelihood estimation. Weighted least squares employ the inverse matrix of the measurement noise's covariance matrix to weigh the square of the error. Although this algorithm's accuracy performance has increased, it is less stable than least squares and is prone to large errors.