# The Symmetry of GPS Orbit Ascending Nodes
## 2 Materials and Methods
In this study, the robust estimation method ([PERSON], 2017) with the Huber damping function ([PERSON], 2017) was used. According to the M-estimation theory, it is a generalization of the maximum likelihood method consisting of the determination of M-estimators minimizing the objective function \(\varphi\)([PERSON] and [PERSON], 2017):  
\[\phi(\mathbf{y};\mathbf{X})=\sum\limits_{i=1}^{n}\rho(y_{i};\mathbf{X})=\sum\limits_{i=1}^ {n}\rho(v_{i}) \tag{1}\]  
where \(\mathbf{y}=[y_{1},\ldots,y_{n}]^{T}\in\mathcal{R}^{n}\) is an observation vector, \(\mathbf{X}\in\mathcal{R}^{r}\) is a vector of the determined parameters, \(\rho(y_{i};\mathbf{X})\) is a function not necessarily resulting from the probabilistic properties of the variables, and \(\mathbf{v}=[v_{1},\ldots,v_{n}]^{T}\in\mathcal{R}^{n}\) is a vector of theoretical observational corrections. Assuming that the function \(\rho(y_{i};\mathbf{X})\) is differentiable, the optimization criterion \(\sum_{i=1}^{n}\rho(v_{i})=min\) is solved by the equation:  
\[\frac{\partial\phi(\mathbf{y};\mathbf{X})}{\partial\mathbf{X}}=0 \tag{2}\]  
Determining the gradient of the objective Function (1)  
\[ \begin{split}\mathbf{g}(\mathbf{y};\mathbf{X})=\left[\frac{\partial\phi(\mathbf{ y}\mathbf{X})}{\partial\mathbf{X}}\right]^{T}=\left[\frac{\partial}{\partial\mathbf{X}} \sum\limits_{i=1}^{n}\rho(v_{i})\right]^{T}=\left[\sum\limits_{i=1}^{n}\frac{ \partial\rho(v_{i})}{\partial v_{i}}\frac{\partial v_{i}}{\partial\mathbf{X}} \right]^{T}=\\ =\left[\sum\limits_{i=1}^{n}\psi(v_{i})\frac{\partial v_{i}}{ \partial\mathbf{X}}\right]^{T}=A^{T}[\psi(v_{1},\ldots,v_{n})]^{T}=A^{T}\mathbf{\psi}( \mathbf{v})\end{split}  \tag{3}\]  
it can be written that the M-estimators are \(\mathbf{\hat{X}}\) values that satisfy the following system of equations:  
\[\mathbf{A}^{T}\mathbf{\psi}(\mathbf{v})=0 \tag{4}\]  
Expressions \(\psi(v_{i})\) in formulas (3) and (4) can be equated as a proportional function to the influence function \(IF(y_{i};\mathbf{X})\) described by [PERSON] (2017). Due to the existing proportionality of the \(IF(y_{i};\mathbf{X})\) function and \(\psi(v_{i})\), the \(\psi(v_{i})\) function is referred to as the influence  
Figure 2: Orbits of GPS satellites in the Earth-centered inertial (ECI) reference frame for the epoch 13 September 2020, 12:41:54 UTC. function [17]. Based on the influence function, a specific form of the weight function is determined which plays a particularly important role in the robust estimation context:  
\[w(v_{i})=\frac{\partial\rho(v_{i})}{\partial v_{i}^{2}}=\frac{\psi(v_{i})}{2v_{i}} \tag{5}\]  
In this least-squares method, with the objective function \(\phi_{LS}(\mathbf{y};\mathbf{X})=\sum_{i=1}^{n}p_{i}v_{i}^{2}\), the weight function takes the form \(w_{LS}=p_{i}\), where \(\mathbf{P}=diag[p_{1},\dots,p_{n}]=diag\Big{[}\sigma_{y_{1}}^{-2},\dots,\sigma_{y_ {n}}^{-2}\Big{]}\) is the matrix of weights created based on the assumed errors of the mean observations. In the study, an iterative calculation method was used, where the weight function in each repetition defined the modification of the original weights of observations \(y_{i}\). At each iteration step, a new observation weight was determined in relation to the absolute value of the estimator \(\hat{v}_{i}\) using the Huber weight function that eliminates the influence of gross errors on the estimation results [27]:  
\[w(v_{i})=\left\{ \begin{array}{ccc}p_{i}&for&|v_{i}|\leq t\\ \frac{t}{|v_{i}|}p_{i}&for&|v_{i}|>t\end{array} \right. \tag{6}\]  
The variable \(t\) is interpreted as a control parameter with a significant impact on the length and efficiency of the iterative process. In the weight Function (6), the parameter \(t\) defines the acceptable range \(\Delta_{v_{i}}=<-t;t>\) for standardized observational corrections \(v_{i}=\hat{v}_{i}/\sigma\).