## 2 Methods
### Functional Models for the Determination of the Satellite Acceleration
#### 2.1.2 Carrier Phase Differentiation Method  
The ionosphere-free carrier phase combination observation between the GPS satellite and LEO satellite receiver is given as follows [21]:  
\[\varphi_{r}^{S}=\varphi_{r}^{S}+c\left(dt_{r}-dt^{S}\right)+rel_{r}^{S}+N_{r} ^{S}+\omega_{r}^{S}+\varepsilon_{r}^{S} \tag{4}\]  
where \(\varphi_{r}^{S}\) is the ionosphere-free carrier phase combination measurement; \(\rho_{r}^{S}\) is the geometric range between the GPS satellite and the LEO satellite receiver antenna; \(dt_{r}\) and \(dt^{S}\) are the LEO satelliteand GPS satellite clock offsets, respectively; \(rel_{r}^{S}\) is the relativistic correction; \(N_{r}^{S}\) is the carrier phase ambiguity; \(\omega_{r}^{S}\) is the wind-up effect; and \(\epsilon_{r}^{S}\) is the remaining unidentifiable noise. The data pre-processing of raw carrier phase observations focused on the error corrections in Equation (4). The accurate error corrections (\(rel_{r}^{S}\) and \(\omega_{r}^{S}\)) terms were modeled referring to [PERSON] et al. [27] and the satellite receiver clock offsets \(dt^{S}\) were interpolated with GPS precise clock products released by International GNSS Service(IGS) [28]. After pre-processing the data, we could construct an item \(\hat{\phi}_{r}^{S}=\varphi_{r}^{S}+cdt^{S}-rel_{r}^{S}-\omega_{r}^{S}\), and introduce it into Equation (4). We have:  
\[\hat{\phi}_{r}^{S}=\rho_{r}^{S}+cdt_{r}+N_{r}^{S}+\epsilon_{r}^{S} \tag{5}\]  
Differentiating Equation (5) yields [24]:  
\[\hat{\phi}_{r}^{S}=\hat{\rho}_{r}^{S}+c\dot{dt}_{r}+\dot{N}_{r}^{S}+\dot{ \epsilon}_{r}^{S} \tag{6}\]  
Given that \(\dot{N}_{r}^{S}=0\) when no cycle slip occurs with \(\dot{\epsilon}_{r}^{S}\) omitted, we have:  
\[\dot{\phi}_{r}^{S}=\dot{\rho}_{r}^{S}+c\dot{dt}_{r} \tag{7}\]  
Differentiating Equation (7) then gives:  
\[\ddot{\phi}_{r}^{S}=\ddot{\rho}_{r}^{S}+c\ddot{dt}_{r} \tag{8}\]  
According to the inter-satellite geometric relationship between the GPS satellite and LEO satellite [29], the equations of the LEO's satellite velocity and acceleration are represented as [24]:  
\[\left\{ \begin{array}{l}\dot{\rho}_{r}^{S}-\mathbf{e}_{r}^{S}\cdot\ddot{\mathbf{x}}^{ S}=-\mathbf{e}_{r}^{S}\cdot\dot{\mathbf{x}}_{r}\\ \ddot{\rho}_{r}^{S}-\mathbf{e}_{r}^{S}\cdot\ddot{\mathbf{x}}^{S}-\frac{1}{\rho_{r}^{S} }\left[\dot{\mathbf{x}}_{r}^{S}\right]^{2}-\left(\dot{\rho}_{r}^{S}\right)^{2} \right]=-\mathbf{e}_{r}^{S}\cdot\ddot{\mathbf{x}}_{r}\end{array} \right. \tag{9}\]  
where \(\mathbf{e}_{r}^{S}\) is the unit direction vector between the GPS satellite and LEO satellite receiver; \(\dot{\mathbf{x}}^{S}\) and \(\ddot{\mathbf{x}}^{S}\) are the GPS satellite velocity and acceleration; and \(\dot{\mathbf{x}}_{r}\) and \(\ddot{\mathbf{x}}_{r}\) are the LEO satellite velocity and acceleration, respectively, which are considered unknown parameters to be determined. Introducing Equation (7) and Equation (8) into Equation (9), the expression of the carrier phase-based acceleration is derived as follows:  
\[\left\{ \begin{array}{l}-\mathbf{e}_{r}^{S}\cdot\dot{\mathbf{x}}_{r}+c\dot{dt}_{r}= \dot{\phi}_{r}^{S}-\mathbf{e}_{r}^{S}\cdot\ddot{\mathbf{x}}^{S}\\ -\mathbf{e}_{r}^{S}\cdot\ddot{\mathbf{x}}_{r}+c\ddot{dt}_{r}=\ddot{\phi}_{r}^{S}-\mathbf{e }_{r}^{S}\cdot\ddot{\mathbf{x}}^{S}-\frac{1}{\rho_{r}^{S}}\left[\dot{\mathbf{x}}_{r}^{ S}\right]^{2}-\left(\dot{\rho}_{r}^{S}\right)^{2}\right]\end{array} \right. \tag{10}\]  
where \(\dot{\mathbf{x}}^{S}\) and \(\ddot{\mathbf{x}}^{S}\) are provided by IGS precise ephemeris; \(\rho_{r}^{S},\dot{\rho}_{r}^{S}\) and \(\mathbf{e}_{r}^{S}\) are well known from the study of the line-of-sight accelerations [23]; and \(\dot{\phi}_{r}^{S},\dot{\phi}_{r}^{S}\) are derived from the GPS carrier phase observations by the sixth-order seven-point differentiation operator, which are separately listed as follows [26]:  
\[ \begin{array}{l}\dot{\phi}_{r}^{S}=\ \mathbf{g}^{T}\cdot\dot{\phi}_{r}^{S}\\ \ddot{\phi}_{r}^{S}=\ \hbar^{T}\cdot\dot{\phi}_{r}^{S}\end{array}