# Performance Analysis of Undifferenced NRTK Considering Time-Varying Characteristics of Atmosphere
## 2 Positioning Model of BDS Undifferenced NRTK
### Extracting Atmospheric Variation Trends from Observations  
By adjusting the PSD of the atmospheric parameter, the performance of ambiguity fixing for reference stations can be improved. From the combination of the original observations, the atmospheric observations can be extracted to reflect the atmospheric variations, and the PSD can be obtained to reasonably constrain the atmospheric parameters. According to Equation (1), the ionospheric observations (IOBS) obtained from original observations at frequencies m and n can be expressed as:  
\[\Delta t^{s}_{r}(t)=-\bigg{(}\frac{{{f_{m}}^{2}{f_{n}}^{2}}}{{{f_{1}}^{2}({f_{n }}^{2}-{f_{1}}^{2})}}\bigg{)}\{\Delta L^{s}_{\text{GF,mm}}(t)-\big{(}\lambda_{m} \cdot\Delta N^{s}_{r,m}-\lambda_{n}\cdot\Delta N^{s}_{r,n}\big{)}-\Delta\delta_{ r,mn}+\Delta\delta^{s}_{mn}+\varepsilon(\Delta L^{s}_{r,mn})\} \tag{36}\]  
where \(\Delta L^{s}_{\text{GF,mm}}(t)=\Delta L^{s}_{r,m}-\Delta L^{s}_{r,n}\) represents the geometric free (GF) combination of the original observation, and \(\Delta\delta_{r,mm}\Delta\delta^{s}_{mn}\) represent the hardware delay of the receiver and the satellite by GF combination, respectively. The ionosphere obtained through Equation (36) includes the effects of real ionospheric variations and observation noise, as well as phase ambiguity. Define IOBS by:  
\[\Delta I^{s}_{\text{GF,mm}}(t)=-\frac{{{f_{m}}^{2}{f_{n}}^{2}}}{{{f_{1}}^{2}({ f_{n}}^{2}-{f_{m}}^{2})}}\Delta L^{s}_{\text{GF,mm}}(t)=K_{mn}\cdot\Delta L^{s}_{ \text{GF,mm}}(t) \tag{37}\]  
Defining the epoch difference ionospheric (EDI) observation, the ionospheric delay after time difference is:  
\[\Delta t^{s}_{r}(\Delta t,i)=\Delta I^{s}_{\text{GF,mm}}(\Delta t,i)-K_{mn} \cdot\varepsilon_{mn} \tag{38}\]  
From Equation (38), it can be seen that the EDI includes the observation noise. The key to obtaining ionospheric variations through ionospheric observations is the weakening of observation noise. According to Equation (1), the ionosphere free (IF) combination can be obtained:  
\[\
abla\Delta L^{sq}_{IF,mn}(t)=\
abla\Delta T(t)+\Big{(}k_{m}\cdot\lambda_{m} \cdot\Delta N^{sq}_{r,m}+k_{m}\cdot\lambda_{n}\cdot\Delta N^{sq}_{r,n}\Big{)}- \Delta\delta_{r,mn}+\Delta\delta^{s}_{mn}-\varepsilon(\
abla\Delta L^{sq}_{IF, mn}) \tag{39}\]  
where \(\
abla\Delta L^{sq}_{IF,mn}=k_{m}\cdot\
abla\Delta L^{sq}_{r,m}+k_{n}\cdot \
abla\Delta L^{sq}_{r,n}\); \(k_{m}=\frac{\theta_{2}^{2}}{f_{m}^{2}-f_{n}^{2}}\), \(k_{n}=-\frac{\theta_{2}^{2}}{f_{m}^{2}-f_{n}^{2}}\) is the coefficient of the IF combination; \(\Delta\delta_{r,mn}\) and \(\Delta\delta^{s}_{mn}\) represent the hardware delay of the receiver and the satellite by IF combination, respectively. The tropospheric observations (TOBS) are defined as:  
\[\
abla\Delta T^{sq}_{IF,mn}(t)=\
abla\Delta L^{sq}_{IF,mn}(t) \tag{40}\]  
The epoch difference tropospheric (EDT) observation can be obtained as follows:  
\[\
abla\Delta\tau(\Delta t,i)=\
abla\Delta L^{sq}_{IF,mn}(\Delta t,i)- \varepsilon_{mn} \tag{41}\]  
Similar to the IOBS, the key to extracting the tropospheric variations is also to weaken the observation noise. In the following, we will weaken the observation noise to obtain the real atmospheric variations, and then obtain the PSD of the real atmospheric delay.