## 2 Signal Analysis  
For the GPS dual-frequency scenario, the expected observed-minus-computed (O-C) vectors of the DD short-baseline phase (\(\Delta\Phi_{DD}\)) and code (\(\Delta P_{DD}\)) observations can be formulated as [25]:  
\[\mathrm{E}(y)=\mathrm{E} \begin{bmatrix}\Delta\Phi_{DD}\\ \Delta P_{DD}\end{bmatrix} =\underbrace{ \begin{bmatrix}e_{2}\otimes(D_{m}^{T}G)& \mathrm{diag}(\lambda_{1},\lambda_{2})\otimes I_{m-1}\\ e_{2}\otimes(D_{m}^{T}G)&0\end{bmatrix} }_{A}\cdot \begin{bmatrix}\Delta r\\ a\end{bmatrix}  \tag{1}\]  
where \(G=[u^{1},\cdots,u^{m}]^{T}\) contains the unit vector \(u^{s}\) from satellite \(s\) (\(s=1,\cdots,m\)) to the rover receiver, and the differencing operator \(D_{m}^{T}=[-e_{m-1},I_{m-1}]\) forms the between-satellite difference with \(e_{m-1}\) and \(I_{m-1}\) denoting the vector of ones with the length \(m-1\) and the identity matrix with size \((m-1)\times(m-1)\), respectively. \(m\) denotes the number of satellites with 1 representing here the reference satellite. \(\lambda_{j}\) stands for the wavelength on frequency \(j\), and \(\mathrm{diag}(\cdot)\) denotes the diagonal matrix with the diagonal elements contained in \((\cdot)\). \(\mathrm{E}(\cdot)\) is the expectation operator. The rover coordinate increment vector starting from an approximate position and the DD ambiguity vector are represented by \(\Delta r\) and \(a\), respectively. The operator \(\otimes\) denotes the Kronecker product. Apart from the a priori DD receiver-satellite geometric distance, the corrections of the antenna phase centre offset (PCO) and phase centre variation (PCV), and the a priori DD tropospheric delays computed based on the Saastamoinen model [26] and the Ifadis mapping function [27] are also corrected for the O-C terms. Assuming that the code and phase observations are uncorrelated, the variance-covariance matrix of the O-C terms, \(Q_{y}\) can be formulated as:  
\[Q_{y}= \begin{bmatrix}Q_{\phi DD}&0\\ 0&Q_{pDD}\end{bmatrix} = \begin{bmatrix}(I_{2}\otimes D_{m}^{T})Q_{\phi}(I_{2} \otimes D_{m})&0\\ 0&(I_{2}\otimes D_{m}^{T})Q_{p}(I_{2}\otimes D_{m})\end{bmatrix}  \tag{2}\]  
where the variance-covariance matrices of the between-receiver phase (\(Q_{\phi}\)) and code (\(Q_{p}\)) observations are diagonal matrices, given as:  
\[Q_{\phi} = \text{blkdiag}\Bigg{(}\sigma_{\phi,1}^{2}\times W_{\phi,1}^{-1}, \sigma_{\phi,2}^{2}\times W_{\phi,2}^{-1}\Bigg{)} \tag{3}\] \[Q_{p} = \text{blkdiag}\Bigg{(}\sigma_{p,1}^{2}\times W_{p,1}^{-1},\sigma_ {p,2}^{2}\times W_{p,2}^{-1}\Bigg{)} \tag{4}\]  
for which \(\sigma_{\phi,j}\) and \(\sigma_{p,j}\) represent the between-receiver phase and code standard deviations on frequency \(j\) in the zenith direction (and for a between-receiver C/N\({}_{0}\) of \(\infty\)), respectively. blkdiag(\(\cdot\)) forms the block diagonal matrix. The phase and code weighting matrices \(W_{\phi,j}\) and \(W_{p,j}\) are constructed as follows:  
\[W_{\phi,j} = \text{diag}(w_{\phi,j}^{1},\cdots,w_{\phi,j}^{m}) \tag{5}\] \[W_{p,j} = \text{diag}(w_{p,j}^{1},\cdots,w_{p,j}^{m}) \tag{6}\]  
for which the phase and code weighting functions \(w_{\phi,j}^{s}\) and \(w_{p,j}^{s}\) for satellite \(s\) on frequency \(j\) will be discussed later. For the purpose of signal analysis, and with the ground truth of the baseline coordinates known, the DD phase (\(\delta\phi_{DDj}^{\text{ls}}\)) and code (\(\delta p_{DDj}^{\text{ls}}\)) residuals can then be expressed as:  
\[\delta\phi_{DDj}^{\text{ls}} = \Delta\phi_{DDj}^{\text{ls}}-\Delta\rho_{DD}^{\text{ls}}-\lambda _{j}\times N_{j}^{\text{ls}} \tag{7}\] \[\delta p_{DDj}^{\text{ls}} = \Delta p_{DDj}^{\text{ls}}-\Delta\rho_{DD}^{\text{ls}} \tag{8}\]  
where \(\Delta\phi_{DDj}^{\text{ls}}\) and \(\Delta p_{DDj}^{\text{ls}}\) represent the DD O-C terms of the phase and code observations on frequency \(j\), respectively, and \(\Delta\rho_{DD}^{\text{ls}}\) denotes the DD geometry increment based on the ground truth and a priori coordinates of the rover.