# Integrity Monitoring of PPP-RTK Positioning: Part II: LEO Augmentation
## 3 PPP-RTK Processing Strategy
With the partial ambiguity resolution (PAR) [40] enabled, the ambiguities are resolved using the least-squares ambiguity decorrelation adjustment (LAMBDA) method [41] for each constellation. A formal ambiguity success rate of 99.99% is set to be the threshold for successful ambiguity fixing, and a ratio test with a threshold of 3 is set for the ambiguity validation [42]. After the convergence of the network solutions, satellite clocks, satellite phase biases and ionospheric delays (of the network stations) are provided to users. To achieve fast integer ambiguity resolution (IAR) of the user, it is essential to interpolate the user ionospheric delays, e.g., based on the least-squares collocation [43] and the best linear unbiased prediction (BLUP) method [44]. In general, the interpolated user ionospheric delay vector \(\hat{t}_{u}(t_{i})\) can be expressed as:  
\[\hat{t}_{u}^{\star}(t_{i})=F_{u}(t_{i})\Bigg{(} \begin{array}{c}\hat{\tilde{t }}(t_{i})\\ \hat{d}_{u^{\prime}r,\mathrm{GF}}\Big{(}t_{1|i-1}\Big{)}\end{array} \Bigg{)}, \tag{6}\]  
where \(\hat{\tilde{t}}\) denotes the estimated ionospheric vector of all network stations and satellites, \(\hat{d}_{u^{\prime}r,\mathrm{GF}}\Big{(}t_{1|i-1}\Big{)}\) is the geometry-free (GF) code bias at \(t_{1}\) between a reference network station \(u^{\prime}\) and the network station \(r\), which is estimated during the ionosphere interpolation of the previous epoch. \(F_{u}\) contains the interpolation coefficients for building the user ionospheric delays. The estimated satellite clocks (\(\hat{dt}^{s}\)), satellite phase biases (\(\hat{\delta}_{j,i}^{s}\)) and the interpolated user ionospheric delays (\(\hat{t}_{u}^{s}\)) are corrected in the user observations with:  
\[\mathrm{E}\Big{(}\Delta g_{u,j}^{s}(t_{i})+\hat{dt}^{s}(t_{i})+ \widetilde{\delta}_{j,i}^{s}(t_{i})+\mu_{i}^{ss}(t_{i})\Big{)}=(G^{s}(t_{i}))^ {\mathrm{T}}\Delta x_{u}(t_{i})+(g^{s}(t_{i}))^{\mathrm{T}}\Delta\tau_{u}(t_{ i})+\widetilde{d}\widetilde{u}_{u}(t_{i})+\widetilde{\delta}_{u,j}^{s}(t_{i})+ \lambda_{j}\widetilde{\lambda}_{u,j}^{s}(t_{i}), \tag{7}\] \[\mathrm{E}\Big{(}\Delta p_{u,j}^{s}(t_{i})+\hat{dt}^{s}(t_{i})- \mu_{i}^{ss}(t_{i})\Big{)}=(G^{s}(t_{i}))^{\mathrm{T}}\Delta x_{u}(t_{i})+(g^ {s}(t_{i}))^{\mathrm{T}}\Delta\tau_{u}(t_{i})+\mu_{j}\widetilde{d}_{u, \mathrm{GF}}(t_{i}), \tag{8}\]  
where the estimable user receiver clock (\(d\widetilde{\hat{t}}_{u}\)), user receiver phase bias (\(\widetilde{\delta}_{u,j}\)) and GF code bias (\(\widetilde{d}_{u,\mathrm{GF}}\)) vary for different constellations. Their forms can be found in [14]. \(G^{s}\) denotes the satellite-to-receiver unit vectors, and \(\Delta x_{u}\) denotes the positional increments to the a priori values. In addition to the corrections themselves, the variance-covariance (V-C) matrices of the combined network corrections, i.e., \(\hat{dt}^{s}(t_{i})+\hat{\delta}_{j,i}^{s}(t_{i})+\mu_{j}^{s}(t_{i})\) for phase and \(\hat{dt}^{s}(t_{i})-\mu_{j}^{ss}(t_{i})\) for code, need to be considered in the user processing to enable fast and correct IAR. For converged network solutions, the variance due to the user ionosphere interpolation plays a major role in the V-C matrix of the combined network corrections. With the combined network corrections and its V-C matrix computed, the user processing is performed with the PAR enabled using the same threshold for the ASR and the ratio test. More details of the PPP-RTK processing can be found in the first part of this series of papers [14].