# Assessment of Quad-Frequency Long-Baseline Positioning with BeiDou-3 and Galileo Observations
## 2 Materials and Methods
### Determination of the EWL Ambiguity  
Taking BDS-3 as an example, the EWL carrier phase measurements and the pseudo-range code measurements on B1C, B1I, B3I, and B2a can be used to form a set of observation equations, expressed as follows:  
\[\left[\begin{array}{c}\mathbf{v}_{P_{1}}\\ \mathbf{v}_{P_{2}}\\ \mathbf{v}_{P_{3}}\\ \mathbf{v}_{P_{4}}\\ \mathbf{v}_{EWL}\end{array}\right]=\left[\begin{array}{ccc}\mathbf{B}&\eta_{1}\mathbf{I }_{\mathbf{S}}&0\\ \mathbf{B}&\eta_{2}\mathbf{I}_{\mathbf{S}}&0\\ \mathbf{B}&\eta_{3}\mathbf{I}_{\mathbf{S}}&0\\ \mathbf{B}&\eta_{4}\mathbf{I}_{\mathbf{S}}&0\\ \mathbf{B}&-\eta_{EWL}\mathbf{I}_{\mathbf{S}}&\lambda_{EWL}\mathbf{I}_{\mathbf{S}}\end{array}\right] \left[\begin{array}{c}\mathbf{x}\\ \mathbf{ion}\\ \mathbf{N}_{EWL}\end{array}\right]-\left[\begin{array}{c}I_{P_{1}}\\ I_{P_{2}}\\ I_{P_{3}}\\ I_{P_{4}}\\ I_{EWL}\end{array}\right] \tag{7}\]  
where the symbols \(\mathbf{v}_{P_{1}},\mathbf{v}_{P_{2}},\mathbf{v}_{P_{3}},\mathbf{v}_{P_{4}}\) represent the DD pseudo-range observation residual vectors on B1C, B1I, B3I, and B2a, respectively; \(\mathbf{v}_{EWL}\) represents the residual vector of DD carrier phase observations of an EWL signal; \(\mathbf{B}\) is the design matrix of the baseline parameter \(\mathbf{x}\); \(\eta_{EWL}\) is the ionospheric scalar factor of an EWL signal; \(s\) represents the number of DD satellite pairs tracked in a current epoch; \(\mathbf{I}_{\mathbf{S}}\) is an identity matrix with dimension \(s\); \(\mathbf{ion}\) and \(\mathbf{N}_{EWL}\) represent an unknown DD ionospheric bias matrix and a DD EWL ambiguities matrix with dimension \(s\), respectively; and the last terms \(I_{\mathbf{P}_{1}}\), \(I_{\mathbf{P}_{2}}\), \(I_{\mathbf{P}_{3}}\), \(I_{\mathbf{P}_{4}}\), and \(I_{EWL}\) are observed minus the computed (OMC) vectors from their corresponding observations.  
Suppose that the combination coefficients of EWL are \(i^{\prime},j^{\prime},k^{\prime},m^{\prime}\), the covariance matrix of (9) can then be written as follows [21]:  
\[\mathbf{Q}_{DD}=\mathbf{T}\Big{(}2 diag\Big{(}\Big{[}\sigma_{P}^{2},\sigma_{\phi}^{2} \Big{]}\Big{)}\otimes\mathbf{I}_{\mathbf{4}}\otimes\mathbf{P}^{-1}\Big{)}\mathbf{T}^{T} \tag{8}\]  
\[\mathbf{T}=\left[\begin{array}{cc}\mathbf{I}_{\mathbf{4}\cdot s}&\mathbf{0}\\ \mathbf{0}&\mathbf{T}_{EWL}\otimes\mathbf{I}_{\mathbf{s}}\\ \mathbf{T}_{EWL}=[i^{\prime},j^{\prime},k^{\prime},m^{\prime}]\end{array}\right] \tag{9}\]  
where \(\sigma_{P}^{2}\) and \(\sigma_{\phi}^{2}\) are the basic variances between undifferenced pseudo-range and carrier phase observations in the zenith direction, respectively; \(\otimes\) denotes the Kronecker product; \(\mathbf{P}=\mathbf{Q}^{-1}\) is the DD weight matrix; \(\mathbf{Q}\) is the cofactor matrix that depends on the elevation of observations [22]; \(\mathbf{T}\) is a linear transformation matrix used to transform the original observation matrix into a combined observation matrix; and \(\mathbf{T}_{EWL}\) is a combination coefficient matrix of EWL. It is worth noting that, when calculating the long-baseline, Equation (8) needs to calculate the inter-satellite difference of the two stations separately and to then add them to obtain \(\mathbf{Q}_{DD}\).