## 2 Materials and Methods
### Real-Time Filtering Method  
A significant problem in real-time orbit determination for low Earth orbit (LEO) satellites is choosing appropriate estimation parameters and methodologies depending on the underlying dynamical and observational models. This work uses a simplified dynamic orbit determination approach for real-time applications. The method combines the dynamical model with geometric GNSS observations to enable the estimate of model parameters and empirical accelerations using observation data. These empirical accelerations help to correct flaws in the dynamic model. In situations where GNSS readings are interrupted or deteriorated in quality, orbit propagation via numerical integration can continue to deliver orbital forecasts, maintaining excellent stability. As a result, the reduced dynamic technique is naturally less sensitive to both model and observational mistakes. This technique builds on the satellite. This approach extends the previously introduced satellite dynamical equations and the GNSS observation model given in Section 2.1. The state vector \(\overset{\rightarrow}{X}\), reflecting the estimated parameters in onboard GNSS real-time orbit determination, can be expressed as follows:  
\[\overset{\rightarrow}{X}=(\overset{\rightarrow}{r},\overset{\rightarrow}{r},\overset{\rightarrow}{r},C_{R},C_{D},a_{emp,T},a_{emp,N},\delta t,N_{IF}^{1}, \ldots N_{IF}^{m}) \tag{15}\]  
In this expression, \(\overset{\rightarrow}{r}_{t}\) and \(\overset{\rightarrow}{r}_{t}\) are the position and velocity of the satellite's center of mass; \(C_{R}\) and \(C_{D}\) are the scale factors for solar radiation pressure and air drag, respectively; \(a_{emp,T}\) and \(a_{emp,N}\) are the empirical accelerations in the along-track and cross-track directions within the orbital frame; \(N_{IF}\) are the ambiguity parameters of the ionosphere-free (IF) combined carrier phase measurements; and m is the number of GNSS satellites involved in the solution. Using the defined set of estimated parameters, the carrier phase observation equation in Equation (1) is enlarged using a first-order Taylor series approximation, yielding the linearized observation equation:  
\[\begin{bmatrix}Z_{i,P}\\ Z_{i,L}\end{bmatrix}=\begin{bmatrix}A_{P}\\ A_{L}\end{bmatrix}\rightarrow\begin{bmatrix}\omega_{i,P}\\ \omega_{i,L}\end{bmatrix} \tag{16}\]  
In the equation, \(i\) represents the epoch; \(Z_{i,P}\) and \(Z_{i,L}\) denote the residuals between the observed and computed pseudorange and carrier phase measurements, respectively; \(\omega\) is the observation noise; \(A_{P}\) and \(A_{L}\) are the linearized observation matrices for pseudo-range and carrier phase, respectively; and \(\overrightarrow{x}_{i}=\overrightarrow{X}_{i}-\overrightarrow{X}_{i}^{*}\) denotes the initial state vector. Incorporating the dynamical model, the functional model of the filter can be expressed as:  
\[\begin{cases}\overset{\rightarrow}{X}_{i}=\Phi(i,i-1)\overset{\rightarrow}{X }_{i-1}+W_{i}\\ \overset{\rightarrow}{z}_{i}=A\overrightarrow{x}_{i}+\omega_{i}\end{cases} \tag{17}\]  
In the equation, \(W\) and \(\omega\) denote the process noise.  
In real-time data processing, filtering is commonly used to estimate parameters. For the functional model mentioned above, we use the Square Root Information Filter (SRIF) as an estimator. The JPL created SRIF, an improved form of the Kalman filter, which provides greater computing efficiency and numerical stability. Householder transformations are used during the filtering process, which is an important component of SRIF ([PERSON], 1977) [43]. The real-time orbit determination process based on SRIF is described in Figure 1.  
Figure 1: Real-time orbit determination procedure based on SRIF.