## 3 Time-Frequency Compensation and Measurement Model
### Unidirectional Orbit-Determination Motion and Observation Equation
Usually, inter-satellite measurements use bidirectional measurements, and subtracting the bidirectional pseudorange measurements at the same time can eliminate satellite orbit information and leave only satellite clock error information. The sum of bidirectional pseudorange measurements obtained simultaneously can eliminate satellite clocks. This study proposes an observation equation based on unidirectional measurement and then performs error correction on the observed values, and linearizes and constructs the observation equation. After estimating the parameters of the observation equation, the initial state of the satellite and the satellite clock error parameters are obtained, and the optimal estimation of the parameters is obtained through residual editing and iteration. Let \(Y=\rho_{0}\) and \(G(X,t)=\rho_{C}\); then, the observation equation is as follows:  
\[Y_{i}=G(X_{i},t_{i})+\varepsilon_{i},i=1,\cdots,n. \tag{11}\]  
In Equation (11), \(X_{i}\), \(Y_{i}\) and \(\varepsilon_{i}\) represent the satellite state vector, observation value and observation noise at time \(t_{i}\), respectively. The state Equation (9) and the observation Equation (11) represent the two fundamental relationships for determining the trajectory. Assuming that the reference state \(X^{*}\) is sufficiently close to the actual trajectory, Equations (7) and (9) can be expanded at \(X^{*}\) using a Taylor series, neglecting second-order and higher terms, resulting in the linearized fundamental relationships as follows:  
\[\left\{ \begin{array}{c}\dot{x}=\dot{X}(t)-\dot{X}^{*}(t)=B(t)x,x(t_{0})=x_{0 },\\ y_{i}=Y_{i}-Y\left(X_{i}^{*},t_{i}\right)=\dot{H}_{i}x+\varepsilon_{i}i=1, \cdots,n.\end{array} \right. \tag{12}\]  
In Equation (12), combining the state equation and observation equation yields the following:  
\[\left\{ \begin{array}{c}B(t)=\left.\frac{\partial F}{\partial X}\right|_{X^{ *}},\\ \dot{H}_{i}(t)=\left.\frac{\partial G}{\partial X}\right|_{X_{i}^{*}}.\end{array} \right. \tag{13}\]  
According to the numerical integration method, the solution to the first equation in (12) is as follows:  
\[x(t)=\Phi(t,t_{0})x_{0}. \tag{14}\]  
In this Equation (14), \(\Phi(t,t_{0})\) is the state transition matrix, which is the solution to the following system of matrix differential equations:  
\[\left\{ \begin{array}{c}\dot{\Phi}(t,t_{0})=B(t)\cdot\Phi(t,t_{0}),\\ \Phi(t_{0},t_{0})=I.\end{array} \right. \tag{15}\]  
The state variable \(X\) is propagated from time \(t_{0}\) to time \(t\) using the state transition matrix. Substituting Equation (14) into the second equation of (12) yields the linearized observation equation:  
\[y=Hx_{0}+\bar{P}_{0}. \tag{16}\]In Equation (16), \(\varepsilon\) is the observation noise and \(\bar{P}_{0}\) is the covariance moment:  
\[y=\left( \begin{array}{c}y_{1}\\ \vdots\\ y_{n}\end{array} \right),H=\left( \begin{array}{c}\bar{H}_{1}\Phi(t_{1},t_{0}) \\ \vdots\\ \bar{H}_{n}\Phi(t_{n},t_{0})\end{array} \right),\bar{P}_{0}=\left( \begin{array} []{c}\varepsilon_{1}\\ \vdots\\ \varepsilon_{n}\end{array} \right). \tag{17}\]  
The best estimate of \(x_{0}\) can be obtained from Equation (16) using the least squares method:  
\[\hat{x}_{0}=\left(H^{T}WH\right)^{-1}H^{T}Wy. \tag{18}\]  
In this Equation (18), \(W\) is the weight matrix of the observed data. Given the prior estimated parameter \(\hat{x}_{0}\) and the prior covariance matrix \(\bar{P}_{0}\), the estimate at time \(t\) is as follows:  
\[\hat{x}=\left(H^{T}WH+\bar{P}_{0}^{-1}\right)^{-1}\left(H^{T}Wy+\bar{P}_{0}^{- 1}\hat{x}_{0}\right). \tag{19}\]  
The improved trajectory is then obtained [30]:  
\[\hat{X}=X^{*}+\hat{x}. \tag{20}\]  
The above estimation is obtained based on linearization, which requires the initial value to be close to the true value; however, this is difficult to ensure in practice. Therefore, an iterative method is required to repeatedly solve the process, with the most recent estimate used as the reference value for linearization in each iteration, continuing until the convergence criterion is met.