## 2 System Model
### GDOP Criterion
In Equation (4), \(\mathbf{1}^{(k)}\) denotes the unit vector pointing from the receiver's initial estimated position to the \(k^{th}\) satellite, and each component of \(\mathbf{1}^{(k)}\) is the direction cosine of the vector from the receiver's estimated position to the satellite direction. \[\mathbf{1}^{(k)}=\frac{1}{\left\|\mathbf{x}^{(k)}-\mathbf{x}_{0}\right\|}\Big{(}x^{(k)}-x_ {0},y^{(k)}-y_{0},z^{(k)}-z_{0}\Big{)}^{\text{T}} \tag{5}\]  
The \(K\) linear Equation (6) can be expressed in matrix notation as  
\[\delta\rho=\left[ \begin{array}{c}\delta\rho^{(1)}\\ \delta\rho^{(2)}\\ \vdots\\ \delta\rho^{(K)}\end{array} \right]=\left[ \begin{array}{c}\left(-\mathbf{1}^{(1) }\right)^{\text{T}}&1\\ \left(-\mathbf{1}^{(2)}\right)^{\text{T}}&1\\ \vdots\\ \left(-\mathbf{1}^{(K)}\right)^{\text{T}}&1\end{array} \right]\left[ \begin{array}[ ]{c}\delta\mathbf{x}\\ \delta b\end{array} \right]+\tilde{\epsilon}_{\rho} \tag{6}\]  
There are four unknowns in the \(K\) (at least four) linear equations: \(\delta\mathbf{x}\) and \(\delta b\). Moreover, Equation (6) can be reduced to  
\[\delta\rho=\mathbf{G}\left[ \begin{array}{c}\delta\mathbf{x}\\ \delta b\end{array} \right]+\tilde{\epsilon}_{\rho} \tag{7}\]  
\[\mathbf{G}=\left[ \begin{array}{c}\left(-\mathbf{1}^{(1)}\right)^{\text{T}}&1\\ \left(-\mathbf{1}^{(2)}\right)^{\text{T}}&1\\ \vdots\\ \left(-\mathbf{1}^{(K)}\right)^{\text{T}}&1\end{array} \right],\mathbf{k}_{K}=\left[ 1\ \ \ldots\ \ 1\right]^{\text{T}} \tag{8}\]  
Ideally, measurement errors in positioning systems can be characterized by their mean vector and covariance matrix. This statistical approach allows for calculating the minimum mean square error position estimate and determining position error variance. However, precisely describing measurement errors is typically challenging. For practical purposes, researchers often simplify the model by assuming satellite measurement errors are zero-mean, uncorrelated, and homoscedastic. Under these simplified conditions, Equation (1) can be expressed as follows:  
\[ \begin{split}\text{E}\big{(}\tilde{\epsilon}_{\rho}\big{)}=0\\ \text{Cov}\big{(}\tilde{\epsilon}_{\rho}\big{)}=\text{E}\Big{(}\tilde{ \epsilon}_{\rho}\tilde{\epsilon}_{\rho}^{\text{T}}\Big{)}=\sigma_{\text{URE}}^ {2}I\end{split}  \tag{9}\]where \(\mathrm{E}(\cdot)\) denotes the mean or expected value, \(\mathrm{Cov}(\cdot)\) represents the covariance, \(I\) is the identity matrix, and \(\sigma_{\mathrm{URE}}\) is the standard deviation of the user range error (URE) common to each satellite, which is simplified as \(\sigma\). URE encompasses errors from satellite ephemeris and clock inaccuracies, signal propagation effects (ionospheric and tropospheric residuals), and receiver-induced uncertainties (multipath and measurement noise), collectively impacting the precision of satellite-based positioning systems. The quality of position estimation is described based on the receiver-to-satellite geometry matrix \(\mathbf{G}\). The covariance matrix is  
\[\mathrm{Cov}\left[ \begin{array}{c}\Delta\mathbf{x}\\ \Delta b\end{array} \right]=\mathrm{Cov}\left[ \begin{array}{c}\mathbf{\hat{x}}\\ \hat{b}\end{array} \right]=\sigma^{2}\Big{(}\mathbf{G}^{\mathrm{T}}\mathbf{G} \Big{)}^{-1} \tag{10}\]  
The variances of the position errors in \(x\), \(y\), and \(z\) directions are denoted by \(\sigma_{x}^{2}\), \(\sigma_{y}^{2}\) and \(\sigma_{z}^{2}\), respectively, with the variance of the clock bias estimates represented by \(\sigma_{b}^{2}\). Let \(\mathbf{H}=\Big{(}\mathbf{G}^{\mathrm{T}}\mathbf{G}\Big{)}^{-1}\), where \(H_{ii}\) represents the \(i^{th}\) element on the diagonal of the matrix \(\mathbf{H}\), providing a systematic approach to characterizing positional uncertainty and measurement error covariance. Then, GDOP is defined as follows (Figure 1):  
\[\mathrm{GDOP}=\sqrt{H_{11}+H_{22}+H_{33}+H_{44}}=trace(\mathbf{H}) \tag{11}\]