## 2 Materials and Methods
### Solving for Ambiguity  
Currently, the solution of ambiguity is usually fixed using the least-squares ambiguity decorrelation adjustment (LAMBDA), which is a search method based on the ambiguity domain using mainly the principle of integer least-squares estimation. The integer least-squares principle can be written in the following form:  
\[y=Aa+Bb+\zeta \tag{4}\]  
where \(y\) is the vector of double-difference observations, \(A\) is the coefficient matrix of the ambiguity parameter, \(B\) is the coefficient matrix of the non-ambiguity parameter, \(b\) is the vector of other non-ambiguity parameters, \(a\) is the ambiguity parameter vector (\(a\) is an integer), and \(\zeta\) is the noise vector. When solving Equation (4) using the least-squares principle, the solution process is often referred to as integer least squares estimation since it contains both integer and real parameters to be solved, and in general, the solution has no analytic form. The integer least squares estimation is usually solved according to the following three processes [13], as shown in the processing flow illustrated in Figure 1.  
In the first step, the floating-point solution of the ambiguity parameter and its covariance matrix are first calculated without considering the ambiguity as an integer:  
\[\left[ \begin{array}{c}\widetilde{a}\\ \widetilde{b}\end{array} \right]_{,}\left[ \begin{array}{cc}Q_{\widetilde{a} \widetilde{a}}&Q_{\widetilde{a}\widetilde{b}}\\ Q_{\widetilde{b}\widetilde{a}}&Q_{\widetilde{b}\widetilde{b}}\end{array} \right] \tag{5}\]  
Among them, the superscript \"\(\sim\)\" means floating point. Figure 1: Flow chart of ambiguity resolution steps of network RTK reference station. In the second step, after the floating-point solution vector \(\widetilde{a}\) and the integer vector \(a\) are obtained, the integer solution of the fuzzy degree is found by minimizing the distance between the floating-point solution vector and the integer vector and by searching for the fuzzy degree parameter \(a\):  
\[\min_{a}(\widetilde{a}-a)^{T}Q_{\widetilde{a}\widetilde{a}}^{-1}(\widetilde{a}-a) \tag{6}\]  
In the third step, after obtaining the integer solutions of the whole-period ambiguity \(\widetilde{a}\), the fixed solutions of the non-ambiguity parameters and their covariance matrices are obtained by the following two equations:  
\[\widehat{b}=\widetilde{b}-Q_{\widetilde{b}\widetilde{a}}Q_{\widetilde{a} \widetilde{a}}^{-1}(\widetilde{a}-\widetilde{a}) \tag{7}\]  
\[Q_{\widetilde{b}\widetilde{b}\widetilde{b}}=Q_{\widetilde{b}\widetilde{b}}-Q_ {\widetilde{b}\widetilde{a}}Q_{\widetilde{a}\widetilde{a}}^{-1}Q_{\widetilde{ a}\widetilde{b}} \tag{8}\]  
The LAMBDA method specifies the following search space when performing the search in the second step [14; 15; 16]:  
\[(\widetilde{a}-a)^{T}Q_{\widetilde{a}\widetilde{a}}^{-1}(\widetilde{a}-a)<T \tag{9}\]  
where \(T\) is the threshold, the integers in the above search space are all search objects, and a set of integer vectors can satisfy Equation (6). The strong correlation between the fuzzy degrees leads to a relatively low efficiency of the LAMBDA algorithm search. Therefore, it is necessary to first reduce the correlation between the covariance matrix and the fuzzy degree parameters using a \(Z\) transformation [17], and then search them using the LAMBDA algorithm with the following procedure:  
\[\widetilde{z}=Z^{T}\widetilde{a} \tag{10}\]  
\[Q_{\widetilde{z}\widetilde{z}}=Z^{T}Q_{\widetilde{a}\widetilde{a}}Z \tag{11}\]  
The reduced correlation matrix \(Z\) should satisfy (1) full rank, not only forward transformable but also inverse transformable; (2) \(|detZ|~{}=~{}1\); (3) integer property to ensure that the fixed ambiguity is still an integer after the inverse transformation.