## 3 Methods
### Network and User Processing
#### 3.2.2 Ambiguity Resolution  
After every measurement update, reparameterized ambiguities are created via an admissible Z-Transformation. Admissible Z-Transformations [22] are found for every receiver-satellite link individually using the MLAMBDA algorithm [23; 24]. The reason why such a reparameterization of the ambiguities is introduced is explained hereafter and demonstrated in Example 1. Suppose we have a float estimate \(\hat{a}_{k}= \begin{bmatrix}a_{1}&\cdots&a_{n}\end{bmatrix} ^{T}\) and its VCM \(P_{h_{k}h_{k}}\) of the ambiguities at epoch \(k\). With a suitable Z-Transformation of \(\hat{a}_{k}\), i.e., \(\hat{z}_{k}= \begin{bmatrix}z_{1}&\cdots&z_{n}\end{bmatrix} ^{T}=Z\cdot\hat{a}_ {k}\), it could be possible that we can fix one (or more) of the Z-Transformed ambiguities \(z_{1},\ldots,z_{n}\) to their respective integer value with high confidence, while none of the original ambiguities \(a_{1},\ldots a_{n}\) can be fixed. For the \(f\) float ambiguities belonging to one satellite-receiver link, it is generally possible to find an admissible Z-Transformation where \((f-1)\) ambiguities can be fixed to integers at the first epoch they are observed or shortly after, when multi-frequency GNSS observations are processed with the functional model defined by Equation (2). Those are, in the scope of this work, denoted as wide-lane ambiguities. The remaining ambiguity, which usually takes significantly longer to fix to an integer, is denoted as the narrow-lane ambiguity. The terms wide-lane and narrow-lane are commonly used in GNSS processing and refer to the effect a linear combination has on the combined wavelength of the phase observations. In a wide-lane combination, the resulting wavelength is larger than the largest original wavelength in a narrow-lane combination, the combined wavelength is shorter than any of the original wavelengths [25]. Ambiguities can be understood to be fixed to integer easier as their wavelength increases. This wavelength interpretation cannot be directly applied to the Z-Transformed ambiguities. Unlike wide- and narrow-lane linear combinations, which are based on the wavelength of the combined observations, decorrelating Z-Transformations are entirely based on the VCM of the ambiguities. The wavelengths of the \((f-1)\) ambiguities obtained by the Z-Transformation that can be fixed to integers relatively easily do not necessarily fit the original definition of wide-lane ambiguities but share the same characteristic of being easy to fix to integer, which is why they will be denoted as wide-lane ambiguities in the scope of this work. The same applies to the remaining narrow-lane ambiguity. The wavelength of this Z-Transformed ambiguity is not necessarily shorter than any of the original ambiguities but remains significantly harder to be fixed to an integer than the other reparameterized ambiguities, which is why it is denoted as the narrow-lane ambiguity in this work. **Example 1**. : _Reparameterization of ambiguities._  
Given are the highly correlated L1, L2, and L5 float ambiguities and their VCM of a GPS satellite, as computed at the first epoch of user processing:  
\[\hat{a}_{1}= \begin{bmatrix}17.9401\\ 19.7336\\ 31.7038\end{bmatrix} ,P_{h_{1}h_{1}}= \begin{bmatrix}10.0195&10.0001&10.0588\\ 10.0001&10.0444&10.1137\\ 10.0588&10.1137&10.1858\end{bmatrix} . \tag{7}\]  
With the MLAMBDA algorithm, we obtain a decorrelating Z-Transformation with  
\[Z= \begin{bmatrix}-9&69&-59\\ 1&-5&4\\ 0&1&-1\end{bmatrix}  \tag{8}\]and the reparameterized ambiguities and their respective standard deviation read:  
\[\hat{z}_{1}= \begin{bmatrix}-670.37\pm 2.51\\ 46.09\pm 0.15\\ -11.97\pm 0.05\end{bmatrix} . \tag{9}\]  
It can be seen that two ambiguities can be fixed to integers with high confidence instantaneously. Those are denoted as wide-lane ambiguities in the scope of this research. One ambiguity remains challenging to fix to an integer, which is here denoted as the narrow-lane ambiguity.