## 2 Methods  
### Undifferenced Uncombined Observation Equation  
The undifferenced and uncombined (UDUC) observation equation for the \(k\)-th epoch is as follows:  
\[E\left(\Delta p_{r,i}^{s,g}\right)=H\cdot x_{r}+c\cdot\left[dt_{r}-dt^{s,g} \right]+m_{r}^{s,g}\cdot T_{r}^{s,g}+u_{i}\cdot I_{r,i}^{s,g}+B_{r,i}-B_{i}^{s,g} \tag{1}\]  
\[E\left(\Delta\sigma_{r,i}^{s,g}\right)=H\cdot x_{r}+c\cdot\left[dt_{r}-dt^{s,g} \right]+m_{r}^{s,g}\cdot T_{r}^{s,g}-u_{i}\cdot I_{r,i}^{s,g}+b_{r,i}-b_{i}^{s,g }+\lambda_{i}\cdot N_{r,i}^{s,g} \tag{2}\]  
In the formula, \(r\), \(s\), \(g\), and \(i\) (\(i=1,2,3\)) represent the GNSS receiver, satellite, constellation, and observation frequency points, respectively; \(E(\cdot)\) represents expectation; and \(\Delta p_{r,i}^{s,g}(k)\) and \(\Delta\sigma_{r,i}^{s,g}(k)\), respectively, represent the observed minus calculated (OMC) values. The observed values include code and phase observations, while the computed values are the sums of GNSS corrections. These corrections incorporate the a priori zenith tropospheric delay (ZTD) values and the geometric distance derived from the known satellite and station coordinates, with the Sagnac effect, solid earth tide, polar tides, antenna phase center offsets (PCOs), antenna phase center variations (PCVs) corrections, and phase wind-up corrections applied; here, \(x_{r}\) (\(x_{r}=\begin{bmatrix}dx&dy&dz\end{bmatrix}\)) is the three-dimensional increment of the k-epoch coordinates; H is the receiver-to-satellite unit direction vector; \(c\) is the speed of light in vacuum; \(dt_{r}(k)\) and \(dt^{s}(k)\) represent the \(k\) epoch clock offset parameters at the receiver and satellite ends, respectively; and \(T\) represents the residual delay in the troposphere corrected by prior models. Due to the high accuracy of prior zenith hydrostatic delay (ZHD), the remaining part is mainly composed of zenith wet delay (ZWD); \(m\) is the mapping function of ZWD; \(I_{r,i}^{s,g}\) is the first-order ionospheric delay at the \(i\)-th frequency; \(u_{i}=f_{1}^{2}/f_{i}^{2}\) is the ionospheric delay conversion coefficient at the \(i\)-th frequency; \(B_{r,i}\) and \(B_{i}^{s}\) are the receiver and satellite code hardware biases, respectively; \(b_{r,i}\) and \(b_{i}^{s,g}\) are the receiver and satellite carrier phase hardware biases, respectively; \(\lambda_{i}\) is the wavelength of frequency \(i\); and \(N_{r,i}^{s,g}\) is the ambiguity of frequency \(i\). The form of the random model is as follows: \(\sigma^{2}=diag\left(\sigma_{p}^{2},\sigma_{Z}^{2}\right)\).  
By utilizing products such as precise satellite orbits, clock bias, uncalibrated phase delay (UPD), or observable-specific signal bias (OSB), errors in the observation equations can be corrected to estimate accurate integer ambiguity parameters. Once the reference station ambiguities are resolved, precise estimates of the ionosphere and troposphere can be obtained. Due to corrections in the tropospheric model, it is necessary to incorporate the prior tropospheric model values and partially restore the model to its original state. The ionospheric data can then be converted to various frequencies through frequency relationships to achieve accurate ionospheric observations.