## 4 POD with Ambiguity Resolution
### Mathematical Models
#### 4.1.2 Single Receiver Ambiguity Resolution  
The float IF ambiguity can be rewritten as the combination of integer WL and float NL ambiguities as the following [57]:  
\[\begin{array}{l}\overline{N}_{r,\text{IF}}^{s}=\frac{f_{1}f_{2}}{f_{1}^{2}- f_{2}^{2}}N_{r,\text{WL}}^{s}+\frac{f_{1}}{f_{1}+f_{2}}\overline{N}_{r,\text{NL}}^{s}\\ \underline{N}_{r,\text{NL}}^{s}=N_{r,\text{NL}}^{s}+d_{r,\text{NL}}-d_{NL}^{ s}\end{array} \tag{5}\]  
where \(N_{r,\text{WL}}^{s}\) and \(N_{r,\text{NL}}^{s}\) are integer WL and NL ambiguities, \(\overline{N}_{r,\text{NL}}^{s}\) is float NL ambiguity, \(d_{r,\text{NL}}\) and \(d_{NL}^{s}\) are the NL PCBs of the receiver and satellite.  
The integer WL and NL ambiguities are fixed with two steps. Firstly, the WL ambiguity is resolved using the Hatch-Melbourne-Wubbena combination [60; 61; 62]:  
\[\overline{N}_{r,\text{WL}}^{s}=(\frac{f_{1}L_{r,1}^{s}-f_{2}L_{r,2}^{s}}{f_{1} -f_{2}}-\frac{f_{1}P_{r,1}^{s}+f_{2}P_{r,2}^{s}}{f_{1}+f_{2}})/\lambda_{WL}=N_ {r,\text{WL}}^{s}+d_{r,\text{WL}}-d_{WL}^{s} \tag{6}\]  
where \(d_{r,\text{WL}}\) and \(d_{WL}^{s}\) are receiver and satellite WL FCBs, \(\lambda_{WL}\) is WL wavelength. After correction of the WL FCBs, the integer WL ambiguity can be determined. Then integer NL ambiguity can be derived with the fixed WL ambiguity, the estimated float IF ambiguity, and the associated NL FCBs using Equation (5).  
Once the WL and NL ambiguities are resolved, the IF ambiguity can be reconstructed according to Equation (5). Usually satellite dependent WL and NL FCBs can be estimated with ground network stations or provided by GNSS analysis institutions [26; 55; 63]. However, receiver dependent FCBs are not available and cannot be removed from Equations (5) and (6). SD operation between satellites is suggested to cancel the receiver FCBs. The SD integer WL and NL ambiguities can be calculated using Equation (7):  
\[\begin{array}{l}\Delta N_{r,\text{WL}}^{s_{1},s_{2}}=\Delta\overline{N}_{r, \text{WL}}^{s_{1},s_{2}}+\Delta d_{NL}^{s_{1},s_{2}}\\ \Delta N_{r,\text{NL}}^{s_{1},s_{2}}=\Delta\overline{N}_{r,\text{NL}}^{s_{1}, s_{2}}+\Delta d_{NL}^{s_{1},s_{2}}\end{array} \tag{7}\]  
where \(\Delta\) is single difference operator, \(\Delta\overline{N}_{r,\text{WL}}^{s_{1},s_{2}}\) and \(\Delta\overline{N}_{r,\text{NL}}^{s_{1},s_{2}}\) are float SD WL and NL ambiguities of satellite \(s_{1}\) and \(s_{2}\) while \(\Delta N_{r,\text{WL}}^{s_{1},s_{2}}\) and \(\Delta N_{r,\text{NL}}^{s_{1},s_{2}}\) are integer ambiguities. \(\Delta d_{WL}^{s_{1},s_{2}}\) and \(\Delta d_{NL}^{s_{1},s_{2}}\) are SD hardware biases between satellites. Once SD WL and NL ambiguities are fixed, the SD IF ambiguity can be recovered as follows:  
\[\Delta\overline{N}_{r,\text{IF}}^{s_{1},s_{2}}=\frac{f_{1}f_{2}}{f_{1}^{2}-f_{ 2}^{2}}\Delta N_{r,\text{WL}}^{s_{1},s_{2}}+\frac{f_{1}}{f_{1}+f_{2}}(\Delta N _{r,\text{NL}}^{s_{1},s_{2}}-\Delta d_{NL}^{s_{1},s_{2}}) \tag{8}\]  
The recovered SD IF ambiguity is taken as pseudo-observation to constrain the undifferenced IF ambiguity parameters in the POD estimation using Equation (9):  
\[\Delta\overline{N}_{r,\text{IF}}^{s_{1},s_{2}}=\overline{N}_{r,\text{IF}}^{s_{ 1}}-\overline{N}_{r,\text{IF}}^{s_{2}},\quad W_{r}^{s_{1},s_{2}} \tag{9}\]where \(W_{r}^{s_{1},s_{2}}\) is the weight of the pseudo-observation and will be assigned with a large value to put a strong confidence.