## 3 Methods
### Network and User Processing
#### 3.2.2 Ambiguity Resolution
The wide-lane ambiguities in this example can be fixed in the reparameterized ambiguity vector \(\hat{z}_{1}\), while no ambiguities in the original ambiguity vector \(\hat{a}_{1}\) can be fixed. As a result, this reparameterization can decrease the number of stochastic parameters and increase the accuracy of the solution, as will be shown later. In most cases, the integer values of the original ambiguities belonging to one receiver-satellite link remain unknown until all reparameterized ambiguities are fixed. Being able  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Item** & **Description** & **Item** & **Description** \\ \hline \multicolumn{4}{c}{**Observations and Corrections**} \\ \hline Observations & Raw pseudo-range and carrier phase, signals and channels as listed in Table 1 & Loading Effects & Solid Earth Tide, Pole Tide corrected [26] & Solid Earth Tide, Pole Tide corrected [26] \\ \hline Observation weighting * & \(\sigma(\varepsilon)=\sigma_{0}+2\cdot\sigma_{0}\cdot\exp^{(\frac{\varepsilon}{ -2\cdot 2\cdot 2\cdot 2})}\) & Relativistic Clock Effects & Shapiro Effect and orbit eccentricity corrected \\ \hline Elevation Mask & 5\({}^{\circ}\) & Station Coordinates & Network: Fixed to Reference Frame solution [15] \\ \hline Phase Center Offset \& Variations & Corrected with IGS14.atx model [28] & Satellite Orbits & Fixed to final Code orbit [16] \\ \hline Phase Wind-Up & Corrected [29] & Tropospheric Mapping, Hydrostatic Zenith Delay & GPT3 [30] \\ \hline Satellite Attitude & Nominal Yaw-Steering [31] & Earth Rotation Parameters (ERP) & Final Code ERP [16] \\ \hline \multicolumn{4}{c}{**Parameter Estimation ** and Process Noises \(\sigma^{2}\)**} \\ \hline Receiver/Satellite Clocks & \(\sigma^{2}=10^{5}\)\(\frac{\text{m}^{2}}{\text{s}}\) (“epoch-wise”) & Zenith Tropospheric Delay & \(\sigma^{2}=4\)\(\frac{\text{mm}^{2}}{\text{h}}\) \\ \hline Receiver/Satellite Clocks & Constant parameters & Ionospheric Slant Delays & \(\sigma^{2}=10^{5}\)\(\frac{\text{m}^{2}}{\text{s}}\) (“epoch-wise”) \\ \hline Receiver/Satellite Phase Biases & Constant or one reference phase clock & & Constant along one continuous satellite arc, reset when phase observation is detected as an outlier \\ \hline \multicolumn{4}{c}{**\(\sigma_{0}=0.3\) m (\(\sigma_{0}=0.003\) m) for code (phase) observations. Formula based on [32] with changed parameters. **} \\ \multicolumn{4}{c}{**\(\text{* Predicted parameters at the first epoch are initialized with a high variance, to not significantly influence any following estimates. }\)} \\ \multicolumn{4}{c}{**\(\text{* Predicted parameters at the first epoch are initialized with a high variance, to not significantly influence any following estimates. to instantaneously or near-instantaneously fix the \((f-1)\) wide-lane ambiguities for every receiver-satellite link reduces the number of parameters in the measurement update (6) and improves the accuracy of the state estimate compared to when no wide-lane ambiguities are introduced and fixed. When introducing a reparameterization into the functional model, here a reparameterization of the ambiguities \(\hat{z}_{k}=Z_{k}\cdot\hat{a}_{k}\), the functional model of the observation needs to be adjusted with  
\[L_{k}+v_{k}= \begin{bmatrix}A_{k}^{b}&A_{k}^{a}\cdot Z_{k}^{-1}\end{bmatrix}  \cdot \begin{bmatrix}\hat{b}_{k}\\ Z_{k}\cdot\hat{a}_{k}\end{bmatrix} = \begin{bmatrix}A_{k}^{b}&A_{k}^{a}\cdot Z _{k}^{-1}\end{bmatrix} \cdot \begin{bmatrix}\hat{b}_{k}\\ \hat{z}_{k}\end{bmatrix} , \tag{10}\]  
with \(A_{k}^{a}\) being the design matrix corresponding to the ambiguities \(\hat{a}_{k}\) and \(A_{k}^{b}\) corresponding to all other parameters \(\hat{b}_{k}\). Instead of solving for \(\hat{a}_{k}\), we have changed the functional model to directly solve for \(\hat{z}_{k}\). Ambiguities at the user level are attempted to be fixed using the Ratio Test Integer Aperture estimation approach according to the algorithm as described in [13], using a fixed fail rate \(P_{f}=0.5\%\).