## 3 System Model and Problem Formulation
### Problem Formulation
#### 3.2.3 Extrapolation Error of Single-Satellite Ephemeris  
To analyze the lower bound of the orbit extrapolation error based on single-satellite ephemeris information, the observed function \(D\) is defined. It denotes the operation of a time-series data process, consisting of the computation of a self-correlation with sliding window \(l\) for any interval [0, \(T_{D}\)] sequentially. The results are a set of data \(Z\), which is  
\[Z=\{d_{1},d_{2},\cdots,d_{n}\} \tag{9}\]  
where \(d_{n}\) represents the \(n\)-th group of data in the interval [0, \(T_{D}\)]. We separate the broadcast ephemeris extrapolation errors into three types: the RE, TE, and PE.  
The RE consists of unpredictable noise in the ephemeris data due to various external and internal factors, such as satellite hardware limitations, signal interference, and environmental conditions. These errors do not follow a specific pattern and vary randomly over time. The TE arises due to the accumulation of systematic biases in the ephemeris data over time. These errors are usually caused by deficiencies in the satellite dynamics model, including gravitational influences and orbital perturbations, and they accumulate gradually as the satellite's orbit continues. The PE is caused by the periodic nature of the satellite's motion around the Earth. These errors are usually caused by the gravitational effects of celestial bodies, the oblateness of the Earth, and other periodic factors.  
If the following conditions are met within \(Z\)  
\[\forall d_{i}\in Z,\ d_{i}<d_{t},\ i\
eq n \tag{10}\]  
this set of data time series can be considered to contain an RE \(\Delta eph(rand)\). Here, \(d_{t}\) in Equation (10) represents a threshold value. \(\Delta eph(rand)\) affects all ephemeris orbit extrapolation errors received by the UE at the current moment.  
If the following conditions are met within \(Z\)  
\[\left\{\begin{aligned} d_{i}\in Z,d_{i}>d_{t},n\
eq t\\ \forall d_{i},d_{j},d_{i}\
eq d_{j},i,j\leq n\end{aligned}\right. \tag{11}\]  
this set of data time series can be considered to contain a TE \(\Delta eph(trend)\). \(\Delta eph(trend)\) is caused by the characteristics of the SGP4 algorithm, in which ephemeris orbit extrapolation error is inevitable.  
If the following conditions are met within \(Z\)  
\[\left\{\begin{aligned} d_{i}\in Z,d_{i}>d_{t},n=t\\ d_{1}=d_{2}=\cdots=d_{n}\end{aligned}\right. \tag{12}\]  
this set of time-series data can be considered to exhibit a PE \(\Delta eph(period)\). \(\Delta eph(period)\) is caused by the elliptical motion of the satellite around the Earth, where some errors periodically change with the movement of the satellite.  
Then, we further separate these errors by defining three types of separation operators \(\mathbf{K}=\{K_{1},K_{2},K_{3}\}\). By orthogonally decomposing the extrapolation error component \(\Delta eph(jt)\) with the three operators, the different orbit extrapolation errors can be separated:  
\[\left\{\begin{aligned} K\!\cdot\!\Delta eph(rand)&= (0,a,b)\\ K\!\cdot\!\Delta eph(trend)&=(a,0,b)\\ K\!\cdot\!\Delta eph(period)&=(a,b,0)\end{aligned}\right. \tag{13}\]  
In Equation (13), \(a\) and \(b\) denote non-zero values.