## Appendix A  
Using the general relativity theory, we illustrate how time is defined in the phase model (Equation (A1)). The GPS time is defined for an observer to a clock on the earth geoid (expressed as absolute seconds since 6 January 1980, 00:00). We define a GPS time on the earth's geoid when the receiver phase is determined as GPS time t, the coordinate time. When we move the clock to the LEO satellite, the proper time becomes  
\[t^{\prime}\ =\ t+\delta^{R}(t)\] (A1)  
Thus, the observer on-ground time at \(t\),  
\[t\ =\ t^{\prime}-\delta^{R}(t)\ =\ t^{R}\big{(}t^{\prime}\big{)}-\Delta t^{R} \big{(}t^{\prime}\big{)}-\delta^{R}(t)\approx t^{R}\big{(}t^{\prime}\big{)}- \Delta t^{R}\big{(}t^{\prime}\big{)}-\delta^{R}\big{(}t^{\prime}\big{)}\] (A2)  
here, \(t^{R}(t^{\prime})\) is the receiver local time, \(\Delta t^{R}(t^{\prime})\) is the receiver time error, and \(\delta^{R}(t^{\prime})\) is the time correction due to the general relativity theory. The same holds for the GNSS satellite. \[t\ =\ t^{\prime\prime}-\delta^{T}(t)\ =\ t^{T}(t^{\prime\prime})-\Delta t^{T}(t^{ \prime\prime})-\delta^{T}(t)\approx t^{T}(t^{\prime\prime})-\Delta t^{T}(t^{ \prime\prime})-\delta^{T}(t^{\prime\prime})\] (A3)  
Here, \(t^{T}(t^{\prime\prime})\) is transmitter local time, \(\Delta t^{T}(t^{\prime\prime})\) is the transmitter time error, \(\delta^{T}(t^{\prime\prime})\) is the time correction from a local observer on earth geoid to proper time on the transmitter. One can think of the actual distance between the transmitter and receiver at coordinate time \(t\) on the ground, traveled by a GPS wave from time \(t-\tau\) to time \(t\), with \(\tau\) is the signal propagation time  
The actual distance between the LEO receiver and GNSS transmitter (phase center) can be written as:  
\[D\bigg{(}\overset{\
eg}{x}(t),\ \overset{\
eg}{x}^{T}(t-\tau)\bigg{)}\ =\ \|\overset{ \
eg}{x}^{R}(t)-\overset{\
eg}{x}^{T}(t-\tau)\|+\delta D\bigg{(}\overset{ \
eg}{x}^{R}(t),\ \overset{\
eg}{x}^{T}(t-\tau)\bigg{)}\] (A4)  
Finally, the phase model can be simulated at GPS time \(t\). \[\varnothing_{L}^{F}(t)\ =\ \varnothing_{G}^{F}(t-\tau)+D\bigg{(}\overset{ \
eg}{x}^{R}(t),\ \overset{\
eg}{x}^{T}(t-\tau)\bigg{)}+\Delta S\] (A5)However, we do not know the time \(t\) precisely. We only know time at the receiver clock \(t^{R}\) when the phase measurement was taken. We also need to estimate the signal transmitted time, expressed as some time delay \(\tau\) before the GPS time \(t\). Replace Equations (A2) and (A3) with Equation (A5) we can get. \[ \begin{split}\varnothing_{L}^{F}\big{(}t^{R}(t^{\prime})-\Delta t ^{R}(t^{\prime})-\delta^{R}(t^{\prime})\big{)}&\approx \varnothing_{G}^{F}\big{(}t^{\tau}(t^{\prime\prime}-\tau)-\Delta t^{T}(t^{ \prime\prime}-\tau)-\delta^{T}(t^{\prime\prime}-\tau)\big{)}\\ +D\bigg{(}\overset{\rightarrow}{x}(t),\ \overset{\rightarrow}{x} ^{T}(t-\tau)\bigg{)}+\Delta S\end{split} \] (A6)  
Recall that the GNSS signals are microwave signals. The phase is the continuous counting of the wavenumbers. Thus, the final phase model can be written as:  
\[ \begin{split}\varnothing_{L}^{F}\big{(}t^{R}(t^{\prime})\big{)}& -\varnothing_{G}^{F}\big{(}t^{T}(t^{\prime\prime}-\tau)\big{)}\ =\ \Delta t^{R}(t^{\prime})\cdot\mathbb{C}+\delta^{R}(t^{\prime})\cdot\mathbb{C}- \Delta t^{T}(t^{\prime\prime}-\tau)\cdot\mathbb{C}\\ -\delta^{T}(t^{\prime\prime}-\tau)\cdot\mathbb{C}+D\bigg{(} \overset{\rightarrow}{x}^{R}(t),\ \overset{\rightarrow}{x}^{T}(t-\tau)\bigg{)}+\Delta S+\in \end{split} \] (A7)  
\(\varnothing_{L}^{F}\big{(}t^{R}(t^{\prime})\big{)}\) is the local receiver simulated GPS signal phase. \(\varnothing_{G}^{F}\big{(}t^{T}(t^{\prime\prime}-\tau)\big{)}\) is the transmitted signal phase received by the receiver. An NCO replica is generally used to track the phase measurements \(\varnothing_{L}^{F}\big{(}t^{R}(t^{\prime})\big{)}-\varnothing_{G}^{F}\big{(} t^{T}(t^{\prime\prime}-\tau)\big{)}\). Generally, the phase \(\varnothing_{G}^{F}\big{(}t^{T}(t^{\prime\prime}-\tau)\big{)}\) is accompanied by an unknown phase ambiguity \(N\).