## 2 Methodology
### Variance Component Estimation  
Observation methods characterized by different technologies and background devices (e.g., electronic systems) vary in measurement accuracy and noise or presence of systematic errors. In some cases, the use of a priori adopted values for weighting might not be beneficial. When combining several observation types for the further estimation of parameters, it is worth considering the relationship between the observation errors, which are probably distinct between each observation technique [57]. As a result, variance estimation methods have emerged to optimally estimate variance and covariance components. VCE is an algorithm that recognizes the relative contribution of measurements of each type or selected partial results in the final solution [58]. Figure 2: Location of the GNSS ground stations used in the simulations, shown on the two maps for better readability: (**a**) 44 and 16 stations placed globally, and (**b**) 16 stations in Europe. The VCE is used in the calculation of the gravity field models for weighting individual satellite passes or monthly solutions of gravimetric missions [59]. The VCE is also exploited in the kinematic orbit determination, where posteriori variances are obtained for different types of GNSS observations to eliminate erroneous measurements [60]. In estimating precise orbits, the VCE method allows the detection of outliers, and the weighting factors can be obtained for individual satellites in each epoch [59]. The VCE method is also used in static and kinematic single point positioning [61; 62; 63]. The VCE method can also be applied in a combined solutions used for the realization of a terrestrial reference system [64; 65]. The obtained weighting coefficients are used in scaling the covariance matrices of individual solutions before their combination. In addition, VCE can be used in data fusion during parameter estimation and satellite orbit determination using ISL measurements [15]. In our research, the WLS method is performed iteratively to determine orbit. The VCE method is usually applied in the last iteration of WLS and is also performed iteratively until the convergence criterion of the VCE algorithm is met [61; 62; 66]. A convergence criterion can be set for given parameters or for the weighting factor values obtained by the VCE. There are several approaches to the implementation of the VCE method, but in this study, we focus on the [PERSON] approach [61; 62; 64; 67] and [PERSON] approach [57; 59; 61]. The VCE algorithm was applied as follows:  
Step 1. In iteration \(k=0\), normal equations matrices for GNSS measurements (\(\textbf{N}_{GNSS}\)) and ISL (\(\textbf{N}_{ISL}\)) are computed with the assignment of weighting coefficients \(S_{GNSS}^{(k)}\) and \(S_{ISL}^{(k)}\) equal to a priori measurement noise \(\sigma_{GNSS}\) and \(\sigma_{ISL}\), respectively:  
\[\textbf{N}_{GNSS}=\textbf{A}_{GNSS}^{T}\textbf{P}_{GNSS}\textbf{A}_{GNSS}, \tag{4}\]  
\[\textbf{N}_{ISL}=\textbf{A}_{ISL}^{T}\textbf{P}_{ISL}\textbf{A}_{ISL}, \tag{5}\]  
\[iteration\ k=0\ \rightarrow\ S_{GNSS}^{(k)}=\sigma_{GNSS},\ S_{ISL}^{(k)}= \sigma_{ISL}. \tag{6}\]  
Step 2. The sum of weighted normal matrices is computed:  
\[\textbf{N}^{(k)}=\frac{1}{S_{GNSS}^{2\ (k)}}\textbf{N}_{GNSS}+\frac{1}{S_{ ISL}^{2\ (k)}}\textbf{N}_{ISL}, \tag{7}\]  
Step 3. In iteration \(k=k+1\) new weighting coefficients \(S_{GNSS}^{(k+1)}\) and \(S_{ISL}^{(k+1)}\) are determined according to the selected approach, where \(n_{i}\) means number of observations for each respective measurement technique:  
1. Helmert approach \[ \begin{pmatrix}\textbf{v}_{GNSS}^{T}\textbf{P}_{GNSS}\textbf{v}_{GNSS}\\ \textbf{v}_{ISL}^{T}\textbf{P}_{ISL}\textbf{v}_{ISL}\end{pmatrix} = \begin{pmatrix} s_{11}&s_{12}\\ s_{21}&s_{22}\end{pmatrix}\begin{pmatrix}S_{GNSS}^{2(k+1)}\\ S_{ISL}^{2(k+1)}\end{pmatrix} ,\] (8)  
\[s_{ii}=n_{i}-2\times trace \begin{pmatrix}\textbf{N}^{-1}\textbf{N}_{i}\\ \end{pmatrix} +trace \begin{pmatrix}\textbf{N}^{-1}\textbf{N}_{i}\textbf{N}^{-1 }\textbf{N}_{i}\end{pmatrix} , \tag{9}\]  
\[