## 2 The Geometric Dilution of Precision and Genetic Algorithm
### GDOP Criterion
For simplicity, the measurement errors from different satellites were assumed to be zero-mean, uncorrelated, and homoscedastic. Under such conditions, Equation (1) can be written as  
\[ \begin{array}{l}\mathrm{E}\big{(}\tilde{\varepsilon}_{\rho}\big{)}=0\\ \mathrm{Cov}\big{(}\tilde{\varepsilon}_{\rho}\big{)}=\mathrm{E}\Big{(}\tilde{ \varepsilon}_{\rho}\tilde{\varepsilon}_{\rho}^{\mathrm{T}}\Big{)}=\sigma_{ \mathrm{URE}}^{2}I\end{array}  \tag{11}\]  
where \(\mathrm{E}(\cdot)\) denotes the mean or expected value, \(\mathrm{Cov}(\cdot)\) denotes 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 \(\sigma\). Strictly speaking, this assumption has not been verified in many cases [39]. The quality of position estimation is described based on the receiver-to-satellite geometry matrix \(\mathbf{G}\). The covariance matrix is  
\[\mathrm{Cov}\bigg{[} \begin{array}{c}\Delta\mathbf{x}\\ \Delta b\end{array} \bigg{]}=\mathrm{Cov}\bigg{[} \begin{array}{c}\mathbf{\hat{x} }\\ \hat{b}\end{array} \bigg{]}=\sigma^{2}\Big{(}\mathbf{G}^{\mathrm{T}}\mathbf{G}\Big{)}^{-1} \tag{12}\]  
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, and the variance of the clock bias estimates is denoted by \(\sigma b^{2}\). Let \(\mathbf{H}=\Big{(}\mathbf{G}^{\mathrm{T}}\mathbf{G}\Big{)}^{-1}\). Using \(H_{ii}\) to denote the \(i^{th}\) element on the diagonal of the matrix \(\mathbf{H}\), Equation (12) can be written as  
\[\sigma_{x}^{2}=\sigma^{2}H_{11};\sigma_{y}^{2}=\sigma^{2}H_{22};\sigma_{z}^{2} =\sigma^{2}H_{33};\sigma_{b}^{2}=\sigma^{2}H_{44} \tag{13}\]  
Position error (Root Mean Square) \(=\sqrt{\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{z}^{2}}\)  
\[=\sigma\sqrt{H_{11}+H_{22}+H_{33}} \tag{14}\]  
Equations (13) and (14) show that the position estimation depends on two factors: (i) the magnitude of the variance of the user range error \(\sigma^{2}\) and (ii) a term that depends solely on the receiver-satellite geometry configuration matrix \(\mathbf{H}\), which is completely determined by matrix \(\mathbf{G}\). The Dilution Of Precision (DOP) is defined based on the above equation to describe the effect of the receiver-satellite geometry factor. \[ \begin{split}\text{PDOP}=\sqrt{H_{11}+H_{2}+H_{33}}\\ \text{GDOP}=\sqrt{H_{44}}\\ \text{GDOP}=\sqrt{H_{11}+H_{2}+H_{33}+H_{44}}=\text{trace}(\mathbf{H} )\end{split}  \tag{15}\]  
The DOP gives a simple description of the user-satellite geometric configuration. In general, the smaller the DOP and \(\sigma\), the higher the accuracy of the position computation. If the receiver is limited in the number of satellites it can track simultaneously, the user pays the price for larger root mean square positioning errors caused by larger DOP values. Early GPS receivers could track only four satellites at a time, and PDOP or GDOP were often used as the criteria for selecting the \"best four satellites\". The role of the accuracy factor in GPS positioning is often misunderstood: a smaller DOP value does not imply a smaller position error, and vice versa. The larger the GDOP, the more spread out the positioning error, and it is possible to obtain a position estimate with a larger error or a smaller error. The root mean square of the localization error and GDOP are linearly related.