## 2 The Geometric Dilution of Precision and Genetic Algorithm  
### GDOP Criterion  
The apparent distance \(\rho\) between the receiver and the satellite at a given instant of time is known as the pseudorange. The pseudorange \(\rho_{c}^{(k)}\) is eliminated by compensating for satellite clock error and residual errors as much as possible, as in the following equation. \[\rho_{c}^{(k)}=r^{(k)}+c\cdot\delta t_{u}+\xi_{\rho}^{(k)} \tag{1}\]  
\(\tilde{\epsilon}_{\rho}^{(k)}\) represents a comprehensive impact of residual errors. Let \(\mathbf{x}=(x,y,z)\) and \(\mathbf{x}^{(k)}=\left(x^{(k)},y^{(k)},z^{(k)}\right)\) be the vectors representing the location of the receiver and the \(k^{th}\) satellite at the time of measurement and signal emission, respectively, where \(k=1\),\(2\),\(\ldots\),\(K\). The true range between the receiver and the satellite is then given by  
\[ \begin{split} r^{(k)}&=\sqrt{\left(x^{(k)}-x \right)^{2}+\left(y^{(k)}-y\right)^{2}+\left(z^{(k)}-z\right)^{2}}\\ &=\left\|\mathbf{x}^{(k)}-\mathbf{x}\right\|\end{split}  \tag{2}\]  
Thus,  
\[\rho_{c}^{(k)}=\left\|\mathbf{x}^{(k)}-\mathbf{x}\right\|+b+\tilde{\epsilon}_{\rho}^{ (k)} \tag{3}\]  
where \(b\) is a parameter that accounts for the receiver clock bias \(c\cdot\delta t_{\rm u}\). Given \(K\) pseudorange measurements from different satellites, each of which is a nonlinear equation, as in Equation (2), we have four unknowns to solve: three components of \(\mathbf{x}\) and \(b\). Therefore, we need at least four equations to determine the unknowns. This implies that the receiver position can be computed only if pseudoranges from at least four satellites are simultaneously available, and in general, no additional information is needed to estimate the receiver clock bias. A straightforward way to solve the \(K\) nonlinear Equation (2) is to linearize it around an initial guess of the user position and then apply an iterative method. We use the Newton-Raphson method for this purpose. Suppose \(\mathbf{x}_{0}=(x_{0},y_{0},z_{0})\) and \(b_{0}\) are the initial estimates of the user position and receiver clock bias, respectively. Therefore, from Equation (3), we can obtain the corrected pseudorange measurement for the \(k^{th}\) satellite, denoted by \(\rho_{c}^{(k)}\). This is compared with the approximate pseudorange value \(\rho_{0}^{(k)}\) that corresponds to \(\mathbf{x}_{0}\) and \(b_{0}\). \[\rho_{0}^{(k)}=\left\|\mathbf{x}^{(k)}-\mathbf{x}_{0}\right\|+b_{0} \tag{4}\]  
We can write \(\mathbf{x}=\mathbf{x}_{0}+\delta\mathbf{x}\) and \(b=b_{0}+\delta b\), denoting \(\delta\mathbf{x}\) and \(\delta b\) as the corrections to the initial estimates. We can then set up a linear system of equations to solve for \(\delta\mathbf{x}\) and \(\delta b\). \[ \begin{split}\delta\rho^{(k)}&=\rho_{c}^{(k)}- \rho_{0}^{(k)}\\ &=\left\|\mathbf{x}^{(k)}-\mathbf{x}_{0}-\delta\mathbf{x}\right\|-\left\|\mathbf{ x}^{(k)}-\mathbf{x}_{0}\right\|+(b-b_{0})+\tilde{\epsilon}_{\rho}^{(k)}\\ &\approx-\frac{\left(\mathbf{x}^{(k)}-\mathbf{x}_{0}\right)}{\left\|\mathbf{ x}^{(k)}-\mathbf{x}_{0}\right\|}\cdot\delta\mathbf{x}+\delta b+\tilde{\epsilon}_{\rho}^{(k)}\\ &=-\mathbf{1}^{(k)}\cdot\delta\mathbf{x}+\delta b+\tilde{\epsilon}_{\rho }^{(k)}\end{split}  \tag{5}\]We use the Taylor series expansion to approximate the vector form of the equations. In Equation (5), \(\mathbf{1}^{(k)}\) 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. \[\mathbf{1}^{(k)}=\frac{1}{\|\mathbf{x}^{(k)}-\mathbf{x}_{0}\|}\Big{(}x^{(k)}-x _{0},y^{(k)}-y_{0},z^{(k)}-z_{0}\Big{)}^{\mathrm{T}} \tag{6}\]  
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[