# Performance Analysis of Undifferenced NRTK Considering Time-Varying Characteristics of Atmosphere
## 2 Positioning Model of BDS Undifferenced NRTK
### Determination of Atmospheric Power Spectral Density  
The expression for the variance of EDI observations can be obtained from Equation (38) as follows:  
\[\begin{array}{ll}D(\Delta I^{s}_{\text{GF,mm}})&=D(\Delta t^{s}_{r})+D( \varepsilon_{mn})=\sum\limits_{i}^{i-n}\Delta I^{s}_{\text{GF,mm}}(\Delta t,i)^ {2}/n\\ &\Rightarrow D(\Delta t^{s}_{r})=D(\Delta I^{s}_{\text{GF,mm}})-2K_{mn}{}^{2} \big{(}{\lambda_{m}}^{2}+{\lambda_{n}}^{2}\big{)}D(\varepsilon_{\phi})\end{array} \tag{42}\]  
In the equation, \(\varepsilon_{\phi}\) is the observation noise in phase units of the carrier phase observation, \(n\) is the number of time difference ionospheric observations used to calculate meanvalue and variance, \(n=l/\Delta t\), which is related to the data length \(l\) and sampling interval \(\Delta t\). If the time difference interval is \(\Delta t\), the calculation of the IPSD is as follows:  
\[q^{2}(\Delta t^{s}_{r})=q^{2}(\Delta I^{s}_{GF,mn})-2K_{mn}{}^{2}\Big{(}{ \lambda_{m}}^{2}+{\lambda_{n}}^{2}\Big{)}D(\varepsilon_{\phi})/\Delta t \tag{43}\]  
where \(q^{2}(\Delta I^{s}_{GF,mn})=D(\Delta I^{s}_{GF,mn})/\Delta t\) represents the observations power spectral density (OPSD), including the impact of observation noise. It can be seen from Equation (43) that the key to extracting IPSD from ionospheric observations is to weaken the noise. The noise impact level depends on the time interval \(\Delta t\), and the larger the time interval, the smaller the impact level of noise. However, an excessively large difference interval will ignore ionospheric variations within the time interval, resulting in a decrease in the temporal resolution and not reflecting the rapid ionospheric variations. It is very important to determine the difference interval that can reflect the ionospheric variations [36]. For this reason, we study the ionospheric variance in unit time, and define statistics \(\Big{(}\Delta I^{s}_{GF,mn}(\Delta t,i)\Big{)}^{2}/\Delta t\) as referring to the ionospheric variance in unit time. If the cumulative variance value is a stable process, it shows that noise plays a major role. If the difference interval can reflect the ionospheric variations, the random walk of the ionosphere will change the slope of the curve. When the differential time interval is obtained, the IPSD can be calculated by Equation (44):  
\[q^{2}(\Delta t^{s}_{r},i)=\sqrt{\frac{\sum\limits_{i=1}^{i=n1}\Big{(}\Delta I ^{s}_{GF,mn}(\Delta t,i)\Big{)}^{2}-\sum\limits_{i=1}^{i=n2}\Big{(}\Delta I^ {s}_{GF,mn}(\Delta t,i)\Big{)}^{2}}{(n1-n2)\cdot\Delta t}} \tag{44}\]  
Equation (44) represents the change in ionospheric variance within a certain time interval, that is, the change of PSD. When solving, we set the time span with the interval of \(n1-n2=60\)s and calculate the PSD within this time period. Similarly, the calculation formula of TPSD can be obtained:  
\[q^{2}(\
abla\Delta\tau,i)=\sqrt{\frac{\sum\limits_{i=1}^{i=n1}\Big{(}\
abla \Delta^{sq}_{IF,mn}(\Delta t,i)\Big{)}^{2}-\sum\limits_{i=1}^{i=n2}\Big{(} \
abla\Delta^{sq}_{IF,mn}(\Delta t,i)\Big{)}^{2}}{(n1-n2)\cdot\Delta t}} \tag{45}\]  
When solving the PSD, the choice of differential time interval is very important. It can be seen from Equation (43) that with the increase of differential time interval, the influence of observation noise will become smaller and smaller, but too large a differential interval will also reduce the time resolution of the PSD. It is not conducive to real-time parameter constraints, so we will study the minimum differential time interval that can reflect atmospheric variations.