## 2 Modeling and Analysis of Carrier-Phase Differential Positioning
### Analysis of Ill-Conditioned Problems for the Identification Matrix  
Fast positioning requires that the positioning calculation be completed in a relatively short time. But the spatial structure of satellites changes very little in a short time, which makes the identification matrix seriously ill-conditioned and the inversion matrix unstable. By least squares spectral decomposition for Equation (10), Equation (11) can be obtained. \[\hat{Y}=\sum_{i=1}^{k+3}\frac{u_{i}^{T}\hat{L}}{\delta_{i}}v_{i}=\sum_{i=1}^{k+ 3}\frac{u_{i}^{T}\hat{L}}{\delta_{i}}v_{i}+\sum_{i=1}^{k+3}\frac{u_{i}^{T} \varepsilon}{\delta_{i}}v_{i} \tag{11}\]  
where \(u_{i}\) and \(v_{i}\) are the left and right singular value vectors of the singular value decomposition of the coefficient matrix, \(\delta_{i}\) is the decreasing singular value, \(\hat{L}\) and \(\hat{Y}\) are the true values. Although the spectral decomposition value \(u_{i}^{T}\varepsilon\) is small, its rate of convergence towards 0 is much lower than that of the singular value \(\delta_{i}\). In fast positioning, there are three singular values that are close to 0 in the integer ambiguity parameter identification matrix. When \(\delta_{i}\to 0\), then \(\left|u_{i}^{T}\varepsilon/\delta_{i}\right|\rightarrow\infty\). Thus, observation noise and other errors will cause instability of the least squares solution. According to Tikhonov regularization, the estimation criteria used for the observation Equation (10) are as follows:  
\[\min\Bigl{\{}\|A\hat{Y}-L\|+\alpha\hat{Y}^{T}R\hat{Y}\Bigr{\}} \tag{12}\]  
where \(\alpha\) is the regularization parameter; \(R\) is the regularization matrix, and \(A\) is the Euclidean 2-norm. The key to Tikhonov regularization to deal with ill-conditioned problems is how to select the regularization parameter \(\alpha\) and the regularization matrix \(R\). The process of constructing the regularization matrix \(R\) in this paper is as follows [23]:  
Step 1: Solving the singular value decomposition of matrix \(A\):  
\[A=U_{k\times(k+3)}D_{(k+3)\times(k+3)}V_{(k+3)\times(k+3)} \tag{13}\]  
Step 2: Decomposing matrices \(D\) and \(V\):  
\[ \begin{array}{c}D_{k\times 3}\\ (k+3)\times(k+3)\end{array} =\left[ \begin{array}{cc}D_{1}&0\\ k\times k&\\ 0&D_{2}\\ 3\times 3\end{array} \right]^{\prime}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
Step 3: Calculating matrix \(S\):  
\[ \begin{array}{c}S\\ 3\times(\hat{k}+3)=D_{2}^{1/2}\\ 3\times 3\end{array} \left[ \begin{array}{cc}v_{11}^{\rm T}&0\\ 3\times 3\end{array} \right] \tag{15}\]  
Step 4: Calculating matrix \(R\) with matrix \(S\):  
\[R=S^{\rm T}S \tag{16}\]  
where \(R\) is a \((k+3)\times(k+3)\) singular matrix, which is the regularization matrix. Combined Equation (12) Equation(10) can be rewritten as:  
\[\hat{Y}=\left[ \begin{array}{c}X\\ N\end{array} \right]=\left(A^{\rm T}PA+R\right)^{-1}A^{\rm T}PL \tag{17}\]  
Compared with \(N_{0}\) in Equation (10), the addition of \(R\) makes the inverse of \(\left(A^{\rm T}PA+R\right)\) normal, the essence of which is to add minor constraints to the part corresponding to the baseline component to overcoming the ill-conditioned of the identification matrix. The regularization leads to a more accurate floating-point solution of ambiguity, which uses \(\left(A^{\rm T}PA+R\right)^{-1}\) replace the covariance matrix to determine the search space of integer ambiguity. **3. Identification of the Integer Ambiguity Parameter**  
3.1. Modeling of the Integer Ambiguity Parameter Identification Based on LAMBDA  
In the double-difference observation Equation (17), \(N\) is an integer. Thus, the solution of integer ambiguity can be treated as an integer least squares solution problem, which can be performed in three steps [24]. First, ignoring the integer characteristics of \(N\), we can obtain the estimated value \(\hat{X}\) of \(X\), the floating-point solution \(\hat{N}\) of \(N\), and the covariance matrix \(Q_{\hat{N}}\) of \(\hat{N}\).