# Performance Assessment of Multi-GNSS PPP Ambiguity Resolution with LEO-Augmentation
## 3 Methods
### Ppp-Ar  
Owing to the complexity of the frequency division multiple access system, only the code division multiple access system was considered for the ambiguity-fixed solution. In Equation (13), the reparametrized real-value ambiguity of the IF combination can be expressed as a combination of wide-lane ambiguity and narrow-lane ambiguity [4]:  
\[\overline{N}_{r,\text{IF}}^{s}=\frac{f_{1}}{f_{1}+f_{2}}\overline{N}_{r,\text {NL}}^{s}+\frac{f_{1}f_{2}}{f_{1}^{2}-f_{2}^{2}}\overline{N}_{r,\text{WL}}^{s}, \tag{17}\]  
The wide-lane ambiguity \(\overline{N}_{r,\text{WL}}^{s}\) can be calculated using the Hatch-Melbourne-Wubbena (HMW) [33; 34; 35] combined observation, which is defined as the difference between the wide-lane carrier and the narrow-lane pseudororange:  
\[\overline{N}_{r,\text{WL}}^{s}=\frac{L_{r,\text{WL}}^{s}-P_{r,\text{NL}}^{s}}{ \lambda_{\text{WL}}}=\widetilde{N}_{r,\text{WL}}^{s}+f_{r,\text{WL}}-f_{\text {WL}}^{s}+\varepsilon(\text{HMW}), \tag{18}\]  
In this formula, \(\widetilde{N}_{r,\text{WL}}^{s}\) indicates the wide-lane integer ambiguity with the whole-week effect of wide-lane hardware delay. \(f_{r,\text{WL}}\) and \(f_{\text{WL}}^{s}\) represent the fractional cycle bias (FCB) at the receiver and satellite ends, respectively. \(\varepsilon(\text{HMW})\) represents the measurement noise error of the HMW combination. Owing to the large measurement noise of the HMW combination, it is necessary to smooth the wide-lane ambiguity between epochs to reduce the influence of the measurement noise and the multipath effect. Assuming the ground observation network consists of \(n\) stations, and each station can observe \(m\) satellites, the following virtual observation equation can be obtained according to Equation (18):  
\[ \begin{bmatrix}\overline{N}_{1,\text{WL}}^{1}-\widetilde{N}_{1,\text{WL}}^{1 }\\ \vdots\\ \overline{N}_{1,\text{WL}}^{m}-\widetilde{N}_{1,\text{WL}}^{m}\\ \vdots\\ N_{2,\text{WL}}^{m}-\widetilde{N}_{2,\text{WL}}^{m}\\ \vdots\\ N_{n,\text{WL}}^{1}-\widetilde{N}_{n,\text{WL}}^{1}\\ \vdots\\ N_{n,\text{WL}}^{m}-\widetilde{N}_{n,\text{WL}}^{m}\\ \end{bmatrix} = \begin{bmatrix}1&&&-1&&\\ \vdots&&&\ddots&\\ 1&&&-1\\ \vdots&&&\ddots&\\ 1&&&-1\\ &\ddots&&&\vdots&\\ &1&-1&\\ &&1&-1&\\ &&\vdots&\ddots&\\ &&&1&&&-1\\ \end{bmatrix}\begin{bmatrix}f_{1,\text{WL}}\\ \vdots\\ \vdots\\ f_{n,\text{WL}}\\ f_{\text{WL}}^{1}\\ \vdots\\ f_{\text{WL}}^{m}\\ \vdots\\ f_{\text{WL}}^{m}\\ \end{bmatrix} , \tag{19}\]  
where \(\overline{N}_{n,\text{WL}}^{m}-\widetilde{N}_{n,\text{WL}}^{m}\) is the virtual observation, that is, the fractional part of the wide-lane ambiguity. The normal equation of the above equation is rank deficient. To solve the equation, the FCB = 0 of a receiver or satellite can be selected as the reference and taken as the virtual observation, and wide-lane FCBs estimation can be performed. After the wide-lane FCBs are obtained, the fixed wide-lane ambiguity is substituted into Equation (17) to obtain the narrow-lane ambiguity:  
\[ \begin{array}{ll}\widetilde{N}_{r,\text{NL}}^{s}&=\frac{f_{1}+f_{2}}{f_{1}} \overline{N}_{r,\text{IF}}^{s}-\frac{f_{2}}{f_{1}-f_{2}}\widetilde{N}_{r,\text {WL}}^{s}\\ &=\widetilde{N}_{r,\text{NL}}^{s}+\left(f_{r,\text{NL}}+\frac{f_{2}}{f_{1}-f_{ 2}}f_{r,\text{WL}}\right)-\left(f_{\text{NL}}^{s}+\frac{f_{2}}{f_{1}-f_{2}}f _{\text{WL}}^{s}\right)\,\\ &=\widetilde{N}_{r,\text{NL}}^{s}+\widetilde{f}_{r,\text{NL}}-\overline{f}_{ \text{NL}}^{s}\\ \end{array}  \tag{20}\]In the above formula, the narrow-lane FCB absorbs the wide-lane FCB. Narrow-lane FCBs can be obtained using an estimation process similar to that of wide-lane FCBs. After accurate estimation of WL and NL FCBs, PPP-AR can be realized. According to Equation (20), the IF combination ambiguity is reversed, and the intersatellite single difference is used to eliminate the narrow-lane FCB at the receiver end to obtain Equation (21).