## 4 POD with Ambiguity Resolution
### Mathematical Models  
#### 4.1.1 GNSS Observation Model  
GNSS code and carrier phase observations between a satellite and a receiver are usually described by the following equations [35; 57]:  
\[\begin{array}{l}P^{s}_{r,j}=\rho^{s}_{r}+c(dt_{r}-dt^{s})+I^{s}_{r,j}+b_{r,j} -b^{s}_{j}\\ I^{s}_{r,j}=\rho^{s}_{r}+c(dt_{r}-dt^{s})-I^{s}_{r,j}+\lambda_{j}(N^{s}_{r,j}+B_ {r,j}-B^{s}_{j})+\lambda_{j}\omega^{s}_{r}\end{array} \tag{3}\]  
where \(P^{s}_{r,j}\) and \(L^{s}_{r,j}\) are code and carrier phase observations in meters, \(r\) and \(s\) represent the receiver and GNSS satellite, \(j\) is the signal frequency, \(\rho^{s}_{r}\) is the geometry distance between receiver and satellite, \(c\) is the speed of light, \(dt_{r}\) and \(dt^{s}\) are receiver and satellite clock errors, \(I^{s}_{r,j}\) is ionospheric delay [58], \(b_{r,j}\) and \(b^{s}_{j}\) are receiver and satellite hardware biases of pseudorange, \(B_{r,j}\) and \(B^{s}_{j}\) are hardware biases of the carrier phase observation, \(\lambda_{j}\) is signal wavelength and \(N^{s}_{r,j}\) is the integer ambiguity. \(\omega^{s}_{r}\) is the phase wind-up error and can be corrected by model [53], in the remaining part, this item will be omitted.  
Both ionosphere free and ionosphere float models can be used for POD. The first order ionospheric delay can be eliminated by the ionosphere free model, while in the ionosphere float model, the ionospheric delay is estimated as an unknown parameter. The ionosphere free approach eliminates the ionospheric delay twice and leads to smaller redundancy. The ionosphere free method owns fewer unknown parameters and it also has the drawback of lacking flexibility for further model strengthening [59]. In this investigation, the ionosphere free model is employed.  
Following the IGS convention, the IF pseudorange biases will be assimilated into the receiver and satellite clock offsets. IF observations can be formulated as:  
\[\begin{array}{l}P^{s}_{r,IF}=\rho^{s}_{r}+c(\overline{d}t_{r}-\overline{d} t^{s})\\ L^{s}_{r,IF}=\rho^{s}_{r}+c(\overline{d}t_{r}-\overline{d}t^{s})+\lambda_{1} \overline{N}^{s}_{r,IF}\\ \overline{N}^{s}_{r,IF}=N^{s}_{r,IF}+(B_{r,IF}-b_{r,IF}/\lambda_{1})-(B^{s}_{ IF}-b^{s}_{IF}/\lambda_{1})\end{array} \tag{4}\]  
where \(P^{s}_{r,IF}\) and \(L^{s}_{r,IF}\) are IF code and carrier phase observations, \(\overline{d}t_{r}\) and \(\overline{d}t^{s}\) are receiver and satellite clock errors, including the pseudorange hardware biases. \(b_{r,IF}\) and \(b^{s}_{IF}\) are the IF pseudorange hardware biases while \(b_{r,IF}\) and \(B^{s}_{IF}\) are the carrier phase biases. \(\lambda_{1}\)  
Figure 4: Carrier phase residuals of GRCC satellite derived from (**a**) DD AR solution and (**b**) SD AR solution with application of PCV corrections.  
is the wavelength of L1. \(\overline{N}_{r,\text{IF}}^{s}\) is the IF ambiguity, which includes the hardware biases in both pseudorange and carrier phase observations, \(N_{r,IF}^{s}\) is a combination of integer L1 and L2 ambiguities.  
In order to fix the ambiguities, [PERSON] presents a sequential integer ambiguity resolution method to resolve the DD ambiguities of long baseline. The integer DD WL ambiguities are firstly determined with a combination of pseudorange and carrier phase observations. Then the DD narrow-lane (NL) ambiguities are fixed using the estimated IF ambiguities and already fixed integer WL ambiguities [57]. For a single receiver integer ambiguity resolution, GNSS satellites SD observations are used to cancel the receiver based phase bias. [PERSON] et al. provide a detailed description about the model to realize single receiver AR [35]. These methods are employed in this contribution to fix the SD and DD ambiguities.