# 100 Picosecond/Sub-10\({}^{-17}\) Level GPS Differential Precise Time and Frequency Transfer
## 2 Differential Precise Time and Frequency Transfer
### Single-Difference Ambiguity Estimation  
According to Equation (1), it is known that the DD carrier-phase ambiguity solution eliminates the receiver clock offset, which is the most crucial parameter in time and frequency transfer. To retain the clock offset, we perform inter-station SD resolution using the pseudorange and carrier-phase observations. The SD model is shown in Equation (4):  
\[\left\{ \begin{array}{ll}P_{f(m,n)}^{i}=\rho_{f(m,n)}^{i}+c\cdot t_{f(m,n)}^{j}+ eph_{f(m,n)}^{i}+T_{f(m,n)}^{i}+I_{f(m,n)}^{i}+\varepsilon_{P_{f(m,n)}^{i}}\\ L_{f(m,n)}^{i}=\rho_{f(m,n)}^{i}+c\cdot t_{f(m,n)}^{j}+eph_{f(m,n)}^{i}+T_{f(m,n )}^{i}-I_{f(m,n)}^{i}+\lambda_{f}\cdot N_{f(m,n)}^{i}+\varepsilon_{L_{f(m,n)}^{ i}}\\ \end{array} \right. \tag{4}\]  
where \(t_{rf(m,n)}^{i}\) represents the inter-station SD receiver clock offset and \(c\) represents the speed of light. For the SD carrier-phase observation, receiver clock offset and SD carrier-phase ambiguity are linearly dependent. To separate them, we introduce the initial value of the inter-station receiver clock offset through SD pseudorange observation and then obtain SD carrier-phase ambiguity. When DD carrier-phase ambiguities are fixed, we use the pseudorange and carrier-phase observations to calculate the float SD carrier-phase ambiguity, \(N_{f(m,n)}^{i}(t)\), at epoch \(t\). For short baselines of several kilometers, ignoring the SD ionospheric delay residual, the SD tropospheric delay residual, and the SD ephemeris residual, Equation (5) can be obtained:  
\[N_{f(m,n)}^{i}(t)=\frac{1}{\lambda_{f}}\cdot(L_{f(m,n)}^{i}(t)-P_{f(m,n)}^{i}( t)) \tag{5}\]  
Due to the influence of the noise and multipath of pseudorange, it is generally difficult to accurately determine the SD integer ambiguity using only one epoch of pseudorange measurement. It is known that the ambiguity remains continuous for each visible satellite once the receiver is correctly locked. Therefore, we smoothed Equation (5) for \(l\) epochs to reduce the effects of pseudorange noise and multipath and obtained the float SD carrier-phase ambiguity, which is shown in Equation (6). Usually, the pseudorange noise and multipath of the satellite at the highest elevation is the smallest, so the reference satellite can be selected by using the satellite elevations. \[\overline{N}_{f(m,n)}^{i}=\frac{1}{l}\sum_{t=1}^{t=l}N_{f(m,n)}^{i}(t)=\frac{1 }{l\cdot\lambda_{f}}\cdot\sum_{t=1}^{t=l}(L_{f(m,n)}^{i}(t)-P_{f(m,n)}^{i}(t)) \tag{6}\]  
When using Equation (6) to determine the SD integer ambiguity, the number of epochs required is closely related to the precision of pseudorange measurement. It is generally believed that integer ambiguity can be determined when the accuracy of the average value reaches \(\pm 6\) cm [24]. Then, we use the integer rounding method to obtain the SD integer carrier-phase ambiguity, \(N_{f(m,n)}^{i}\), of the reference satellite, \(i\), as shown in Equation (7). \[N_{f(m,n)}^{i}=\left[\overline{N}_{f(m,n)}^{i}\right]=\left[\frac{1}{l\cdot \lambda_{f}}\cdot\sum_{t=1}^{t=l}(L_{f(m,n)}^{i}(t)-P_{f(m,n)}^{i}(t))\right] \tag{7}\]  
where \([\cdot]\) denotes rounding to the nearest integer. After obtaining the fixed DD integer ambiguity by Equation (1) and the SD integer ambiguity of reference satellite \(i\) by Equation (7), it is easy to obtain SD integer ambiguity, \(N_{f(m,n)}^{j}\), of the visible satellite, \(j\), by using Equation (8). \[N_{f(m,n)}^{j}=N_{f(m,n)}^{i}+N_{f(m,n)}^{i,j} \tag{8}\]  
After obtaining L1 DD integer ambiguity and DD wide-lane integer ambiguities, \(N_{W(m,n)}^{i,j}\), by using Equations (2) and (3), IF combination SD ambiguity, \(N_{IF(m,n)}^{j}\), of the visible satellite, \(j\), can be obtained. \[N_{IF(m,n)}^{j} =N_{IF(m,n)}^{i}+N_{IF(m,n)}^{i,j} \tag{9}\] \[=N_{IF(m,n)}^{i}+\frac{f_{1}}{f_{1}+f_{2}}N_{f_{1}(m,n)}^{i,j}+ \frac{f_{1}\cdot f_{2}}{f_{1}^{2}-f_{2}^{2}}N_{W(m,n)}^{i,j}\]From Equation (6), it can be found that estimating the SD carrier-phase ambiguity of the reference satellite, \(i\), requires a convergence time.