# The Symmetry of GPS Orbit Ascending Nodes
## 2 Materials and Methods
If the standardized correction \(v_{i}\) is outside the range of acceptable values of measurement errors, then in the subsequent iteration steps, the weight of the \(i\)-th observation \(p_{i}\) will be reduced to the value that eliminates the influence of the observation \(y_{i}\) the final estimation results. The value of the parameter \(t\) is generally taken as 1.5, 2, 2.5, or 3. Using the general form of the weight Function (6), Equation (4) can be written as  
\[\mathbf{A}^{T}\mathbf{w}(\mathbf{v})\mathbf{v}=\mathbf{0} \tag{7}\]  
where \(\mathbf{w}(\mathbf{v})\mathbf{v}\) is the diagonal matrix of the weight function. The solution to Equation (2) is the M-estimator of the parameter vector \(\mathbf{X}\) with the following form:  
\[\mathbf{X}^{j}=\Big{[}A^{T}\mathbf{w}\Big{(}\mathbf{v}^{j-1}\Big{)}A\Big{]}^{-1}\mathbf{A}^{T }\mathbf{w}\Big{(}\mathbf{v}^{j-1}\Big{)}\mathbf{y} \tag{8}\]  
where \(j=1\),..., \(c\) is the number of iterative steps in the process of determining the M-estimator. Formula (6) shows that the weight function values are modifications of the original observation weights from the least-squares method algorithm. Determined weight values were used as values initiating the iterative process of the robust M-estimators calculation. Moreover, the weight function of the \(i\)-th observation in the robust M-estimate can also be written using the damping function \(w(v_{i})=p_{i}t(v_{i})=p_{i}\), where \(p_{i}\) is the equivalent weight of the \(i\)-th observation in a given iteration step. Then, in the iterative process (8), there is an equivalent matrix of weights:  
\[\mathbf{P}=\left[ \begin{array}{cccc}p_{1}&0&\cdots&0\\ 0&p_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&p_{n}\end{array} \right] \tag{9}\]  
which is a modification of the weight function \(\mathbf{w}(\mathbf{v})=\mathbf{P}\) resulting from the application of the damping function of the Huber method:\[w(v_{i})=\left\{ \begin{array}{ccc}1&\textit{for}&|v_{i}|\leq t\\ \frac{t}{|v_{i}|}&\textit{for}&|v_{i}|>t\end{array} \right. \tag{10}\]  
The iterative process of determining the M-estimator (8) can be considered completed when in the \(c\)-th iteration step, the obtained M-estimator \(\hat{\textbf{X}}\) satisfies Equation (7) and the damping matrix is the identity matrix:  
\[\textbf{f}(\textbf{v})=\left[ \begin{array}{cccc}t(v_{1})&0&\cdots&0\\ 0&t(v_{2})&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&t(v_{n})\end{array} \right]=\left[ \begin{array}{cccc}1&0&\cdots&0 \\ 0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{array} \right] \tag{11}\]  
In practice, the standard deviation \(\sigma\) is unknown and is generally replaced by the error of the mean of the measured quantity. For this reason, also the standardization of corrections is carried out based on the determined errors of the mean corrections \(v_{i}=\hat{v}_{i}/\sigma_{\hat{v}_{i}}\). The covariance matrix of the vector \(\hat{\textbf{v}}\) has the following form [17]:  
\[\textbf{C}_{\hat{p}}=m_{0}^{2}[\textbf{P}^{-1}-\textbf{A}[\textbf{A}^{T} \textbf{P}\textbf{A}]^{-1}\textbf{A}^{T}] \tag{12}\]  
where the diagonal elements of the matrix are the squares of the mean errors and \(m_{0}^{2}=\textbf{v}^{T}\textbf{P}\textbf{v}/(n-m)\) is the estimator of the variance coefficient. The difference \((n-m)\) is the number of redundant observations calculated from the number of observation equations \(n\) and the number of parameters \(m\). The results of the robust estimation for the OP D subset are shown in Figure 3. The arithmetic mean of the satellites \(\Omega\) is not a valid position estimate of OP ANs and is marked in black. This is due to a significant anomalous separation of the SVN46 orbit from the remaining OP D satellites. A significantly greater value of the correction \(v_{SVN46}\) in iteration (8) resulted in a reduction in the observation weight \(p_{SVN46}\) by the Huber damping Function (6) to the value minimizing the impact of observations on the final value of the OP D AN position estimator.