## 2 Spectral Analysis of HY-2B Empirical Accelerations
### Frequency Characteristics  
Non-gravitational modeling is challenging because it does not perfectly describe the real in-flight environment encountered by a spacecraft and often relies on a variety of external products [23; 24; 45]. In addition to the scaling parameter, a pre-defined number of empirical accelerations, such as piecewise periodic or constant accelerations, were applied to cope with the remaining deficiencies of the satellite's prior model [21; 22]. Accordingly, the piecewise constant and periodic empirical accelerations can be obtained by:  
\[\tilde{r}=a_{0}+a_{1}\cos(k\cdot v)+a_{2}\sin(k\cdot v),\ k=1,2\cdots \tag{1}\]  
where \(\tilde{r}\) is the empirical acceleration used in POD, \(a_{0}\) is the constant acceleration, \(a_{1}\) and \(a_{2}\) are the coefficients of periodic acceleration, and \(k\) is the frequency of periodic acceleration. Based on a high-precision force model, the unmodeled dynamic model generally occurs at a frequency of one cycle per revolution (CPR) (i.e., 1 CPR empirical acceleration), which causes empirical acceleration, which is in the form of the orbital period [45]. For a refined PEA model, the fast Fourier transform (FFT) was used in this study to identify the potential periodicity of the satellite's empirical acceleration, and the frequency of PEA was reset for orbit determination and analysis.  
To fulfill our aims, an optimized periodic PEA model was proposed and implemented for the HY-2B mission. The processing flow was roughly divided into two steps. First, the piecewise constant acceleration strategy (i.e., only a constant PEA is used in Formula (1)) was used in POD to obtain a priori information for the residual acceleration and an analysis of their spectra was performed to detect the potential periodic signals. Second, a frequency test was carried out for comparison and verification. The piecewise constant accelerations of HY-2B POD on Day 8 of the year 2019 are depicted in Figure 1, clearly indicating the time variability of the remaining accelerations. Note that the periodic signals of the PEA for three directions are generally difficult to identify (see Figure 1), and thus the FFT was used in this study.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **Description** \\ \hline
**SLR validation** & \\ \hline Station coordinates & SLRF2014 [41] \\ Ocean tide loading & FES 2004 [36] \\ Tropospheric delay & Mendes and Pavlis [42] \\ Relativity & Space-time curvature correction \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._As a starting point, estimated constant accelerations as well as reduced dynamic orbit solutions were obtained for a 14-day period, in which the acceleration value was used for spectral analysis. A power spectrum analysis of the residual accelerations in three directions is shown in Figure 2. The along-track direction and orbital radial direction show a clear periodic signal, while the results of the cross-track direction are complicated. For the along-track direction, 4 CPR is the dominant periodic signal. There is a noticeable signal of 2 CPR in the radial direction. However, a mixed spectrum is observed in the cross-track direction, and it is hard to identify the periodicity of the acceleration.  
Figure 1: Piecewise constant accelerations of the reduced-dynamic orbit determination on 8 January 2019 (day of year DOY 8). These accelerations are described as constant values in the along-track (**top**), cross-track (**middle**), and radial (**bottom**) directions at consecutive intervals of 5 min.