Research on GPS Satellite Clock Bias Prediction Algorithm Based on the Inaction Method

Highlights

What are the main findings?

• NTFT spectrum shows that GPS satellite clock bias contains pronounced periodic components whose amplitudes and frequencies vary significantly over time.
• IM-SAM uses the inaction method to obtain instantaneous periodic parameters at the prediction boundary, overcoming the limitations of least-squares fitting at the data edge.

What are the implications of the main findings?

• IM-SAM enables more accurate and more stable short-term satellite clock prediction under time-varying periodic behavior.
• The clock bias predicted by IM-SAM provides higher PPP positioning accuracy compared with IGU-P.

Abstract

Satellite clock bias exhibits complex, time-varying periodic characteristics due to environmental disturbances. Accurate modeling and prediction of periodic terms play a crucial role in improving the precision and stability of short-term predictions. Traditional models such as spectral analysis model (SAM) estimate the frequency, amplitude, and phase of periodic terms through global fitting, which limits their ability to adapt to abrupt changes at the prediction boundary. To address this limitation, this paper proposes an improved spectral analysis model (IM-SAM) based on the inaction method (IM). The model employs IM to extract the instantaneous frequency, amplitude, and phase parameters of periodic terms precisely at the data endpoint, and utilizes the parameters of periodic terms at the data endpoint for prediction, effectively suppressing periodic fluctuations in prediction errors. Experimental results based on real GPS clock bias data demonstrate that the root mean square (RMS) of IM-SAM prediction errors is reduced by 19.14%, 14.39%, and 10.48% for 3 h, 6 h, and 12 h prediction tasks, respectively, compared with SAM. Furthermore, a kinematic precise point positioning experiment was performed using IM-SAM-predicted clock products and compared with the predicted half of IGS ultra-rapid clock products. The RMS of position error was reduced by 14.3%, 12.6%, and 7.9% in the east, north, and up directions, respectively. These results demonstrate the practical effectiveness and accuracy of IM-SAM in real-time clock prediction and GPS positioning applications.

1. Introduction

Satellite clock bias prediction is a foundational technology in Global Navigation Satellite System (GNSS). The accuracy of this technology directly influences the reliability of navigation, positioning, and high-precision timing services [1,2,3,4,5,6,7,8,9]. Studies have shown that, in addition to linear trends and random noise, satellite clock bias data exhibit pronounced periodic fluctuations caused by orbital dynamics, thermal effects, and satellite payload behavior. These periodic signatures are evident not only in GPS but also in BDS and Galileo precise clock products, as widely documented in multi-GNSS analyses [2,6,7,10,11,12,13,14,15]. Recognizing this behavior, many scholars have incorporated explicit periodic components—such as harmonic expansion or sinusoidal terms—into clock models, and have reported improved short-term prediction performance as a result [3,5,7]. In operational practice, the International GNSS Service (IGS) Ultra-Rapid clock products employ an empirical polynomial-plus-harmonic prediction model, in which the first 24 h arc is observation-based and the subsequent 24 h arc is extrapolated using fitted periodic terms [3,5]. In scenarios like these, achieving accurate estimation of periodic term parameters and high-precision extrapolation has become a key challenge in improving the accuracy of short-term clock bias prediction [2,6,7].

The traditional spectral analysis model (SAM) uses the Fourier transform to identify primary periodic components and perform global fitting on clock bias data to determine the frequency, amplitude, and phase parameters of each periodic term [2,3,5,6,7,15]. However, this method has notable limitations. In practical applications, the amplitude and frequency of periodic terms often fluctuate due to abrupt environmental changes [14,16,17,18]. Global fitting parameters essentially represent the statistical characteristics of the data over the entire time span, making them incapable of capturing the dynamic variations in periodic terms near the data endpoint, that is, the prediction starting point [6,7]. This lagging nature of global fitting parameters severely limits the accuracy and stability of short-term predictions [19].

In recent years, time-frequency analysis techniques have provided new methods for extracting dynamic parameters from time-varying periodic signals [11,14,20,21,22,23,24,25]. The inaction method (IM), based on the normal time-frequency transform (NTFT), is a harmonic signal extraction technique [11,23]. The IM directly extracts the instantaneous frequency, amplitude, and phase of signals along the spectral ridges in the NTFT time-frequency spectrum, enabling adaptive tracking of dynamic signal changes over time [21,24,26]. Compared to the Fourier transform and least-squares fitting, IM does not rely on preset parameters or global data fitting, demonstrating superior robustness and accuracy under complex operating environments and random noise interference [11,21].

To address the above challenges, this paper proposes an improved spectral analysis model (IM-SAM) based on the IM. The IM-SAM integrates IM into SAM. First, IM is employed to extract the parameters of periodic terms at the endpoints from satellite clock bias data. Then, an extrapolation model is constructed based on the strong physical correlation between the periodic term parameters of the data endpoint and the prediction interval. Unlike the global fitting parameters used in SAM, IM-SAM utilizes the parameters of periodic terms at the data endpoint for prediction, effectively suppressing periodic fluctuations in prediction errors and significantly enhancing the accuracy and stability of short-term clock bias predictions. The advantages of IM-SAM are validated through a series of experiments, providing a reliable theoretical foundation and practical reference for the refined modeling and prediction of satellite clock bias.

2. Normal Time-Frequency Transform and Inaction Method

2.1. Normal Time-Frequency Transform

For the target signal $f\left(t\right)$, its NTFT is defined as [23,24]:

$${\Psi }_f\left(\tau ,\varpi \right)=\begin{array}{cc}{\int }_{-\infty }^{+\infty }f\left(t\right)\overline{\psi \left(t-\tau ,\varpi \right)}dt& \tau ,\varpi \in \mathbb{R}\end{array}$$

(1)

where $\tau$ and $\varpi$ denote time and frequency, respectively; ${\Psi }_f$ represents the NTFT time-frequency spectrum of $f\left(t\right)$; ^¯ indicates the complex conjugate; $\mathbb{R}$ is the real number set; and $\psi \left(t,\varpi \right)$ is the kernel function of NTFT. The Fourier transform of $\psi \left(t,\varpi \right)$, denoted as $\widehat{\psi }\left(\omega ,\varpi \right)$, must satisfy the following conditions:

$$\left\{\begin{array}{cc}\widehat{\psi }\left(\omega ,\varpi \right)=1& \omega =\varpi \\ \left|\widehat{\psi }\left(\omega ,\varpi \right)\right|<1& \omega \ne \varpi \end{array}\right.$$

(2)

where | | denotes the modulus, and ^{^} represents the Fourier transform operator. A typical expression for the NTFT kernel function is:

$$\psi \left(t,\varpi \right)=\begin{array}{cc}\left|\mu \left(\varpi \right)\right|w\left(\mu \left(\varpi \right)t\right)\mathit{exp}\left(i\varpi t\right)& \mu \left(\varpi \right)\in \mathbb{R}\end{array}$$

(3)

where $\mu \left(\varpi \right)$ is a time-frequency resolution regulator, which can theoretically be set to any value or expression except zero. When analyzing clock bias data, $\mu \left(\varpi \right)$ is typically set to $\varpi$. $w\left(t\right)$ is referred to as the standard window function, and its Fourier transform $\widehat{w}\left(\omega \right)$ must satisfy:

$$\left\{\begin{array}{cc}\widehat{w}\left(\omega \right)=1& \omega =0\\ \left|\widehat{w}\left(\omega \right)\right|<1& \omega \ne 0\end{array}\right.$$

(4)

The standard window function used in this study is defined as:

$$w\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{2}{\sqrt{2\pi }\sigma }}exp\left(-{\displaystyle \frac{t^2}{2{\sigma }^2}}\right)& t\le 0\\ 0& t>0\end{array}\right.$$

(5)

where $\sigma$ is the window width, which can be set to $\sigma =180$ for clock bias data analysis.

2.2. Inaction Method

For a complex signal $h\left(t\right)=Aexp\left(i\theta \left(t\right)\right)=Aexp\left(i\left(\beta t+\phi \right)\right)$, where $A$, $\beta$, and $\phi$ represent amplitude, angular frequency, and initial phase, respectively, and $\theta \left(t\right)$ denotes the instantaneous phase. Applying NTFT to $h\left(t\right)$, the following relationships can be derived [23,24]:

$$\left\{\begin{array}{cc}\begin{array}{l}\widehat{\beta }\left(\tau \right)=\underset{\varpi }{\mathit{argmax}}\left|{\Psi }_h\left(\tau ,\varpi \right)\right|\\ \widehat{h}\left(\tau \right)={\Psi }_h\left(\tau ,\widehat{\beta }\left(\tau \right)\right)\\ \widehat{A}\left(\tau \right)=\left|{\Psi }_h\left(\tau ,\widehat{\beta }\left(\tau \right)\right)\right|\\ \widehat{\theta }\left(\tau \right)=arg\left({\Psi }_h\left(\tau ,\widehat{\beta }\left(\tau \right)\right)\right)\end{array}& \forall \tau \in \mathbb{R}\end{array}\right.$$

(6)

where $\tau$ and $\varpi$ denote time and frequency, respectively; ${\Psi }_h$ is the NTFT time-frequency spectrum of $h\left(t\right)$; $\widehat{\beta }\left(\tau \right)$ is the frequency $\varpi$ that maximizes $\left|{\Psi }_h\left(\tau ,\varpi \right)\right|$ for a fixed $\tau$; $\widehat{h}\left(\tau \right)$ represents the reconstructed time series from the spectral ridge of the NTFT time-frequency spectrum; and $\widehat{A}\left(\tau \right)$ and $\widehat{\theta }\left(\tau \right)$ represent the amplitude and phase of $\widehat{h}\left(\tau \right)$ at time $\tau$, respectively.

The mathematical properties described in Equation (6) are referred to as the inaction principle of NTFT, which describes how NTFT can accurately extract the time-frequency characteristics of a signal without requiring inverse transformation [11,21]. The instantaneous frequency $\widehat{\beta }\left(\tau \right)$ is determined by the location of the maximum value in the time-frequency spectrum, while the instantaneous amplitude $\widehat{A}\left(\tau \right)$ and phase $\widehat{\theta }\left(\tau \right)$ are given by the modulus and phase of the time-frequency spectrum, respectively [11,21,23,24]. For $h\left(t\right)$ with time-varying amplitude and frequency, the extracted $\widehat{\beta }\left(\tau \right)$, $\widehat{A}\left(\tau \right)$, and $\widehat{\theta }\left(\tau \right)$ along the spectral ridge can be regarded as approximations of $\beta \left(t\right)$, $A\left(t\right)$, and $\theta \left(t\right)$. This method of extracting $h\left(t\right)$ from the NTFT time-frequency spectrum is referred to as the inaction method in [11,21].

For periodic or quasi-periodic signals $H\left(t\right)=Acos\left(\theta \left(t\right)\right)=Acos\left(\beta t+\phi \right)$, where $A$, $\beta$, and $\phi$ represent amplitude, angular frequency, and initial phase, respectively, and $\theta \left(t\right)$ is the instantaneous phase. Using Euler’s formula, $H\left(t\right)$ can be expressed as:

$$H\left(t\right)=h\left(t\right)/2+\overline{h\left(t\right)}/2=Aexp\left(i\theta \left(t\right)\right)/2+Aexp\left(-i\theta \left(t\right)\right)/2$$

(7)

Applying NTFT to $H\left(t\right)$, the following relationships can be derived [21]:

$$\left\{\begin{array}{cc}\begin{array}{l}\widehat{\beta }\left(\tau \right)=\underset{\varpi >0}{\mathit{argmax}}\left|{\Psi }_H\left(\tau ,\varpi \right)\right|\\ \widehat{H}\left(\tau \right)=\mathfrak{R}\left\{2{\Psi }_H\left(\tau ,\widehat{\beta }\left(\tau \right)\right)\right\}\\ \begin{array}{l}\widehat{A}\left(\tau \right)=\left|2{\Psi }_H\left(\tau ,\widehat{\beta }\left(\tau \right)\right)\right|\\ \widehat{\theta }\left(\tau \right)=arg\left({\Psi }_H\left(\tau ,\widehat{\beta }\left(\tau \right)\right)\right)\end{array}\end{array}& \forall \tau \in \mathbb{R}\end{array}\right.$$

(8)

where $\widehat{\beta }\left(\tau \right)$ is the frequency $\varpi$ that satisfies $\varpi >0$ and maximizes $\left|{\Psi }_h\left(\tau ,\varpi \right)\right|$; $\mathfrak{R}\left\{ \right\}$ denotes the real part; and other symbols have the same meanings as in Equation (6).

Equation (8) shows that NTFT can extract the instantaneous characteristics of periodic and quasi-periodic signals, where the instantaneous frequency $\widehat{\beta }\left(\tau \right)$ is determined by the maximum value in the time-frequency spectrum, while the instantaneous amplitude $\widehat{A}\left(\tau \right)$ and phase $\widehat{\theta }\left(\tau \right)$ are determined by the modulus and phase of the spectrum, respectively. According to the inaction principle, IM extracts periodic and quasi-periodic signals along the spectral ridge of the NTFT time-frequency spectrum. Thus, IM can be considered a “line-passing” filter in the field of time-frequency filtering. For spaceborne atomic clocks operating in complex environments with various random noise influences, IM enables the concise and precise extraction of time-varying periodic terms from satellite clock bias data [11].

3. Clock Bias Prediction Algorithm Based on the Inaction Method

3.1. Construction of the Clock Bias Model

For clock bias data of length $m$, denoted as ${\left\{x\left(t_i\right)\right\}}_{i=1}^m$, the quadratic polynomial model (QPM) can be expressed as [8]:

$$x\left(t_i\right)=a_0+a_1\left(t_i-t_1\right)+a_2{\left(t_i-t_1\right)}^2+\epsilon \left(t_i\right)$$

(9)

where $x\left(t_i\right)$ is the clock bias at time $t_i$; $a_0$, $a_1$, and $a_2$ represent the clock offset, frequency offset, and frequency drift at time $t_1$, respectively; and $\epsilon \left(t_i\right)$ is the fitting residual. When there are more than three clock bias data points, the parameters $a_0$, $a_1$, and $a_2$ can be estimated using the least squares method.

For spaceborne atomic clocks, clock bias data are often influenced by the satellite’s orbital environment and exhibit significant periodic fluctuations. Incorporating periodic terms into the satellite clock bias model can effectively improve the accuracy and stability of short-term predictions. The SAM, which adds periodic terms to the quadratic polynomial, is expressed as [5,6,7]:

$$x\left(t_i\right)=a_0+a_1\left(t_i-t_1\right)+a_2{\left(t_i-t_1\right)}^2+\sum _{k=1}^p{A}_k\mathit{cos}\left[{\beta }_k\left(t_i-t_1\right)+{\phi }_k\right]+\epsilon \left(t_i\right)$$

(10)

where the instantaneous phase ${\theta }_k\left(t_i\right)={\beta }_k\left(t_i-t_1\right)+{\phi }_k$; $p$ is the number of periodic terms; $A_k$, ${\beta }_k$, and ${\phi }_k$ represent the amplitude, angular frequency, and initial phase of the $k$-th periodic term, respectively, which can be determined using spectral analysis combined with least squares fitting [5,6,7].

As noted in [3,5,6,7,8], the SAM has been widely used in satellite clock bias prediction. However, SAM struggles to adapt to time-varying periodic terms because its parameters of periodic terms rely on global fitting of historical data and must be extrapolated for future intervals during prediction. If the amplitude or frequency of the periodic terms fluctuates or changes abruptly at the data endpoint due to environmental changes, the SAM cannot capture these dynamics in time, leading to discrepancies between the extrapolated results and actual values [19]. This mismatch between globally fitted parameters and the actual dynamic environment causes an endpoint effect, significantly reducing the model’s accuracy and stability in short-term prediction.

To address this issue, this paper introduces the IM, embedding it into the SAM to form the IM-SAM. IM, based on the principle of NTFT, can directly extract the instantaneous frequency, amplitude, and phase of periodic terms along the ridge of the NTFT time-frequency spectrum without requiring global fitting of historical data. Its main advantages include: (1) IM can accurately extract parameters of periodic terms at the data endpoint, reducing parameter mismatch caused by global fitting. (2) IM does not rely on preset parameters, enabling it to accurately extract signal features even in noisy environments, demonstrating greater robustness and precision.

The IM-SAM is implemented as follows:

• Step 1: Use the least squares method to fit the clock bias data ${\left\{x\left(t_i\right)\right\}}_{i=1}^m$, obtaining the trend term $x_{trend}\left(t_i\right)=a_0+a_1\left(t_i-t_1\right)+a_2{\left(t_i-t_1\right)}^2$. Subtract the trend term to obtain the residual sequence containing periodic terms and noise, ${\epsilon }_{res}\left(t_i\right)=x\left(t_i\right)-x_{trend}\left(t_i\right)$.
• Step 2: Perform time-frequency analysis on the residual sequence ${\epsilon }_{res}\left(t_i\right)$ using NTFT to obtain the NTFT time-frequency spectrum $\Psi {\epsilon }_{res}\left(\tau ,\varpi \right)$.
• Step 3: Using IM, extract the instantaneous parameters of multiple periodic terms along the ridge of the NTFT time-frequency spectrum, including the instantaneous frequency ${\widehat{\beta }}_k\left(\tau \right)$, instantaneous amplitude ${\widehat{A}}_k\left(\tau \right)$, and instantaneous phase ${\widehat{\theta }}_k\left(\tau \right)$.
• Step 4: According to Equation (10), ${\theta }_k\left(t_i\right)={\beta }_k\left(t_i-t_1\right)+{\phi }_k$. Taking $t_i=t_m$, ${\beta }_k={\widehat{\beta }}_k\left(t_m\right)$, and ${\theta }_k\left(t_i\right)={\widehat{\theta }}_k\left(t_m\right)$, then ${{\widehat{\theta }}_k\left(t_m\right)=\widehat{\beta }}_k\left(t_m\right)\times \left(t_m-t_1\right)+{\phi }_k$, from which the initial phase ${\phi }_k$ of the periodic term at the final time $t_m$ can be obtained.
• Step 5: Substitute the time-varying parameters of periodic terms ${\widehat{\beta }}_k\left(t_m\right)$, ${\widehat{A}}_k\left(t_m\right)$, and ${\phi }_k$ extracted at $t_m$ into Equation (10) to predict clock bias for future periods.

By leveraging the dynamic information at the data endpoint, the IM-SAM overcomes the parameter mismatch inherent in SAM under global fitting, significantly enhancing the accuracy and stability of short-term predictions.

3.2. Simulation Example

In this section, we simulate the periodic term $p_1$ with slight frequency fluctuations but stable amplitude and the periodic term $p_2$ with stable frequency but slight amplitude variation to verify the effectiveness of IM [11]. After superimposing atomic clock noise and a trend term, the simulated clock bias $x\left(t\right)$ can be expressed as:

$$\left\{\begin{array}{l}{p}_1\left(t\right)={10}^{-9}\times \mathit{cos}\left({\displaystyle \frac{2\pi t\left(1+t/\mathrm{12,000,000}\right)}{\mathrm{21,600}}}\right)\\ p_2\left(t\right)={10}^{-9}\times \left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\sin \left({\displaystyle \frac{2\pi t}{\mathrm{1,200,000}}}\right)\right)\mathit{cos}\left({\displaystyle \frac{2\pi t}{\mathrm{43,200}}}\right)\\ x\left(t\right)=a_0+a_{1}t+a_2{t}^2+p_1\left(t\right)+p_2\left(t\right)+\epsilon \left(t\right)\end{array}\right.$$

(11)

where the sampling interval is 30 s, and the sampling duration is 96 h. The coefficients $a_0$, $a_1$, and $a_2$ are 0, ${10}^{-12}$, and $-{10}^{-21}$, respectively. $\epsilon \left(t\right)$ represents the linear superposition of white phase modulation noise, white frequency modulation noise, and random walk frequency modulation noise, with noise diffusion coefficients of ${10}^{-19}$, ${10}^{-24}$, and ${10}^{-33}$, respectively. As shown in Figure 1, Figure 1a–c represent the time domain images of the simulated periodic term, simulated noise term, and simulated clock bias data, respectively.

The IM-SAM and SAM were used to analyze $x\left(t\right)$ from 0 to 84 h to reconstruct the 6-h and 12-h periodic terms and to predict these terms for the 85–96 h period. The results are shown in Figure 1d, Figure 2 and Figure 3. Figure 1d shows the NTFT time-frequency spectrum obtained by applying NTFT to the first 84 h of $x\left(t\right)$ after removing the trend term. The horizontal axis represents time, and the vertical axis represents the period (reciprocal of frequency). The black solid lines at periods of 6 h and 12 h indicate the ridges of the NTFT time-frequency spectrum. Using IM along these ridges, it is possible to extract time-varying periodic terms with fluctuating frequency and amplitude. The reconstruction results for these periodic terms are shown as red solid lines in Figure 2a and Figure 3a. In Figure 2 and Figure 3, the green solid line represents the simulated periodic terms. In Figure 2a and Figure 3a, the blue and red solid lines represent the reconstructed periodic terms extracted from the simulated clock bias using SAM and IM-SAM, respectively, while the blue and red dashed lines represent the predicted periodic terms extrapolated by SAM and IM-SAM, respectively, based on the modeling results. In Figure 2b and Figure 3b, the solid lines represent the residuals (reconstruction error) obtained by subtracting the reconstructed periodic terms from the simulated periodic terms, while the dashed lines represent the residuals (prediction error) obtained by subtracting the predicted periodic terms from the simulated periodic terms.

As shown in Figure 2b and Figure 3b, the reconstruction error of the periodic term is larger at the data endpoint and smaller at the data center due to the edge effect of the least squares method used by SAM. In contrast, the IM used by IM-SAM can accurately extract the instantaneous amplitude, instantaneous frequency and instantaneous phase of the periodic term at any moment. Therefore, in the reconstruction stage of the periodic term, i.e., in the range of 0–84 h, the reconstruction errors of IM-SAM are smaller than those of SAM, especially at the two ends of the data, which indicates that IM exhibits higher accuracy than the least squares method in tracking the frequency and amplitude of the periodic term.

In the prediction phase, IM-SAM utilizes the frequency, amplitude, and phase parameters of the reconstructed periodic terms at the endpoint to generate the predicted periodic terms. This approach fully leverages an intuitive fact: the closer the data are to the prediction starting point, the more accurately they reflect the parameters of periodic terms, making them more valuable for prediction. As shown by the red dashed lines in Figure 2 and Figure 3, IM-SAM effectively reduces prediction error by capturing the latest dynamic information at the data endpoint, significantly enhancing short-term prediction accuracy. In contrast, the SAM employs a global fitting method to derive the parameters for the predicted periodic terms. However, this approach fails to adequately reflect the time-varying characteristics at the data endpoint, resulting in extrapolation results that deviate from actual values.

4. Experiments and Analysis

The final precise GPS satellite clock bias data (2 June 2023 to 17 June 2023) provided by the Center for Orbit Determination in Europe (CODE) was utilized to validate the effectiveness of IM-SAM. A total of 32 satellite clocks with continuous data and distinct periodic terms were selected, with key information presented in

Table 1. The status information of GPS satellite in orbit.

Satellite Type	Atomic Clock Types	PRN
IIR	Rb	G02 G13 G16 G19 G20 G21 G22
IIRM	Rb	G05 G07 G12 G15 G17 G29 G31
IIF	Rb	G01 G03 G06 G09 G10 G25 G26 G27 G30 G32
IIF	Cs	G08 G24
IIIA	Rb	G04 G11 G14 G18 G23 G28

. Prior to the prediction experiments, outliers in the clock bias data were detected and removed daily using the median absolute deviation method, and missing data were interpolated using a linear interpolation method [27]. During the prediction experiments, the predicted values were compared to the true values published by CODE. The root mean square (RMS) and standard deviation (STD) were calculated using Equation (12) to evaluate the model’s accuracy and stability.

$$\left\{\begin{array}{l}RMS=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{∆}_i^2}\\ STD=\sqrt{{\displaystyle \frac{1}{N-1}}{\displaystyle \sum _{i=1}^N}{\left({∆}_i-\overline{∆}\right)}^2}\end{array}\right.$$

(12)

where ${∆}_i$ represents the difference between the predicted value and the true value, also known as the prediction error. $N$ is the number of predicted values. $\overline{∆}$ is the average value of ${∆}_i$ in the single-day prediction.

4.1. Time-Frequency Characteristics of Satellite Clock Bias

To analyze the dynamic characteristics of periodic terms in satellite clock bias, NTFT was applied to the detrended clock bias data of 32 GPS satellite clocks over a 16-day period (DOY 153–168, 2023). The resulting NTFT time-frequency spectra are shown in Figure 4 and Figure 5, where each subfigure corresponds to a specific satellite clock (G01–G32). The horizontal axis represents time (in days), the vertical axis represents the oscillation period (in hours), and the color scale indicates the normalized spectral energy.

The following significant features can be summarized from the figures: ① In the time-frequency spectrum of most satellite clocks, periodic terms can be clearly identified as concentrated around 6 h, 12 h, and 24 h, indicating that these frequencies are the main periodic components of the clock bias. The periodic terms of some satellite clocks (such as G09, G05, G26, and G29) have relatively stable frequency distributions in the time-frequency spectrum and exist stably. The periodic terms of other satellite clocks (such as G06, G08, G20, and G31) show obvious frequency drift, that is, the frequency changes slightly or even significantly over time, reflecting the time-varying characteristics of their frequency components. ② From the changes in color intensity, it can be seen that there are obvious differences in the amplitude of periodic terms of different satellite clocks at different times. The 12-h or 24-h periodic terms of some satellite clocks (such as G06, G14, G17, and G30) have enhanced amplitudes (red areas) at certain times, indicating that the periodic fluctuation components are more significant during these times. Other satellite clocks, such as G11, G13, and G28, have weaker overall spectral energy, indicating that their periodic term amplitudes are small or that the periodic terms are not obvious due to the influence of noise terms.

In summary, periodic terms in GPS satellite clock bias generally exhibit obvious frequency drift and amplitude fluctuation characteristics, showing significant non-stationarity and time variability. Therefore, traditional methods that use fixed frequency and amplitude modeling (such as SAM) are difficult to accurately characterize their dynamic behavior. The IM-SAM proposed in this paper better adapts to such dynamic changes by extracting the instantaneous frequency and amplitude information of the data endpoints, thereby improving the prediction accuracy.

4.2. Prediction Error Analysis of Different Prediction Models

In order to analyze the performance of different models in satellite clock bias prediction, this paper selected four representative GPS satellites (IIR G21, IIR-M G17, IIF G25, and IIIA G11), and employed five models—the gray model (GM), Kalman filter model (KFM), QPM, SAM, and IM-SAM—to conduct a 14-day prediction experiment (DOY 155–168, 2023). Among them, the SAM and IM-SAM were used to model and predict the main periodic terms (6, 12, and 24 h) in GPS satellite clock bias.

In this study, a sliding time-window strategy is employed. For each iteration, three consecutive days of satellite clock bias data are selected; however, only the first 2.5 days are used for model fitting and parameter extraction, while the remaining 0.5 day serves as the prediction interval. Since clock bias is generally smooth within a single day but may exhibit discontinuities at day boundaries, this strategy ensures that the prediction segment primarily lies within one day rather than crossing the boundary between days. In this way, the potential influence of inter-day clock jumps on prediction evaluation is avoided, while sufficient data length is retained to capture long-period components (such as 24-h term). The time window is then shifted backward by one day, and the modeling and prediction procedure is repeated. Finally, the prediction error curves of each model within the 12-h prediction duration are plotted, as shown in Figure 6. Each subplot represents the prediction error curve of a specific satellite under different models.

As shown in Figure 6, all five prediction models exhibit a trend of prediction errors accumulating over time. The GM, KFM, and QPMs use different modeling methods, so there are differences in the prediction results for different satellites and different time periods (e.g., GM has high prediction accuracy for G17, while KFM has high prediction accuracy for G11). These differences reflect the adaptability and limitations of each method in handling satellite clock bias. However, although the specific prediction results of each model are different, their prediction errors show obvious periodic fluctuations. Compared with GM, KFM, and QPM, IM-SAM and SAM effectively suppress periodic fluctuations in prediction errors. Additionally, compared with SAM, IM-SAM significantly reduces short-term prediction errors, with prediction errors during the 3-h prediction duration notably lower than those of other models, demonstrating superior prediction performance.

Further analysis reveals that the time-varying characteristics of periodic term parameters greatly influence prediction accuracy. QPM only fits trend terms and does not consider periodic terms, resulting in obvious periodic oscillations in the prediction errors that impact short-term prediction accuracy. SAM incorporates periodic terms on the basis of QPM, significantly reducing periodic fluctuations in short-term prediction errors compared to QPM, as shown in Figure 6 for G21 and G25. However, due to the global fitting approach, SAM struggles to capture the dynamic changes in periodic terms near the data endpoint. This limitation causes the predicted periodic terms to mismatch the actual periodic terms in frequency, phase, and amplitude, resulting in prediction errors exceeding those of QPM in certain time periods, as shown by the G11 prediction error for DOY158 in Figure 6. In contrast, IM-SAM precisely extracts time-varying parameters of periodic terms through IM and effectively utilizes the parameters at the data endpoint for prediction, thereby reducing error accumulation and extreme deviations. The 3-h and 6-h prediction errors of IM-SAM for G21, G17, G25, and G11 are consistently lower than those of SAM and QPM, particularly for G25. Moreover, periodic oscillations in prediction errors are more effectively suppressed.

4.3. Prediction Accuracy and Stability Analysis

In order to avoid the influence of particularly good or poor prediction results from a single satellite clock on the evaluation of the prediction model performance, clock bias data from all 32 GPS satellite clocks were selected for experimental analysis. Based on the prediction process in Section 4.2, the RMS and STD of the model’s prediction error were used as indicators of prediction accuracy and stability. The average RMS of satellite clocks at prediction durations of 3, 6, and 12 h are shown in Figure 7, where the averaging is performed over all days of year (DOYs). The average RMS and STD of the five models at different prediction durations are shown in Figure 8 and Figure 9, where the averaging is computed over both DOYs and satellites. The statistical results of STD for different types of satellite clocks at a prediction duration of 12 h are shown in

Table 2. STD statistics of different types of satellite clocks in 12 h prediction duration.

STD (ns)	IIR	IIR-M	IIF-Rb	IIF-Cs	IIIA	Average
GM	0.4011	0.4643	0.5934	1.3711	0.5690	0.5671
KFM	0.3753	0.4786	0.4022	1.1638	0.0994	0.4039
QPM	0.4058	0.5216	0.3662	1.1853	0.1204	0.4140
SAM	0.3889	0.4519	0.3410	1.3141	0.1170	0.3946
IM-SAM	0.3312	0.4213	0.2948	1.1844	0.0790	0.3456
Average	0.3804	0.4675	0.3995	1.2437	0.1970

. The statistical analysis and comparison of RMS results for different models are presented in

Table 3. RMS statistics and comparative analysis results of different models.

Statistical Indicators	Model	3 h	6 h	9 h	12 h
RMS (ns)	GM	0.4415	0.6076	0.7746	0.9812
	KFM	0.3452	0.5050	0.6144	0.7082
	QPM	0.3399	0.4844	0.5933	0.7028
	SAM	0.2906	0.4480	0.5809	0.7009
	IM-SAM	0.2349	0.3835	0.5022	0.6274
Improvement rate (%)	GM vs. IM-SAM	46.78	36.87	35.15	36.05
	KFM vs. IM-SAM	31.93	24.05	18.25	11.41
	QPM vs. IM-SAM	30.87	20.81	15.34	10.72
	SAM vs. IM-SAM	19.14	14.39	13.53	10.48

.

A detailed analysis of Figure 7, Figure 8 and Figure 9 reveals significant differences in prediction accuracy among different types of satellite clocks, as follows: ① The GM exhibits relatively small prediction errors on IIR and IIR-M satellite clocks but shows significantly poorer prediction performance on other types such as IIF and IIIA, indicating limited model generalization capability. ② The SAM, which can suppress periodic fluctuations in prediction errors to some extent and performs slightly better than the QPM in short-term predictions. However, in long-term prediction tasks, the overall accuracy of SAM and QPM is similar. Even for some satellite clocks (such as G02, G13, and G30), the long-term prediction error of SAM is significantly greater than that of QPM, indicating that its global fitting of periodic terms has limitations when faced with frequency drift or sudden changes. ③ Due to the short time span of the clock bias data used in modeling, the KFM cannot fully utilize the advantages of Kalman filtering in clock bias modeling, resulting in poor overall prediction performance. However, it is worth noting that KFM performs relatively well in long-term predictions for IIIA, IIR, and IIR-M satellite clocks, indicating that it still has certain application value under specific types and stable conditions. ④ As shown in Figure 8 and Figure 9, the IM-SAM demonstrates the best short-term prediction performance on most satellite clocks except for IIF-Cs, and its average RMS is superior to other models across different prediction durations, highlighting its broad adaptability and stability. ⑤ As shown in Figure 7, except for G03, G08, and G10, the prediction accuracy of most IIF satellite clocks is generally better than that of IIR and IIR-M satellite clocks. Further analysis shows that the G03 satellite underwent a frequency tuning operation performed by the control segment during the analyzed period, which led to larger prediction errors. This tuning event occurred only for G03 within the analyzed time span. The prediction errors of G08 and G10 are mainly attributed to white frequency modulation noise within their atomic clocks.

Table 2 shows the STD statistics of different types of satellite clocks (IIR, IIR-M, IIF-Rb, IIF-Cs, IIIA) under various prediction models (GM, QPM, SAM, KFM, IM-SAM) for a 12-h prediction duration, which are used to evaluate the prediction stability of each model. ① Analysis of the prediction results of each model shows that IM-SAM has the lowest average STD on satellite clocks, at 0.3456 ns, demonstrating the best prediction stability. In contrast, GM has the highest average STD, at 0.5671 ns, and the worst prediction performance. Among traditional models, KFM (0.4039 ns), QPM (0.4140 ns), and SAM (0.3946 ns) show similar performance on satellite clocks. ② From the perspective of different types of satellite clocks, the IIF-Cs satellite clocks generally have higher STD in all models, indicating that their clock bias prediction is more difficult and their stability is poor. The IIIA satellite clocks have the lowest STD, especially under IM-SAM, which is only 0.0790 ns, indicating that its atomic clock performance is better and it has high predictability.

The statistical analysis results of Figure 8 and Table 3 indicate that the RMS of each prediction model shows significant differences under different prediction durations. The specific analysis is as follows: ① From the RMS metric, IM-SAM achieved the best prediction accuracy across all prediction durations, with average RMS values of 0.2349 ns (3 h), 0.3835 ns (6 h), 0.5022 ns (9 h), and 0.6274 ns (12 h), significantly outperforming other traditional models. This indicates that IM-SAM has a clear advantage in suppressing periodic fluctuations in errors and improving prediction stability. ② In terms of improvement rates compared to other models, IM-SAM shows the most significant improvement relative to the GM, particularly in the 3-h prediction, where the prediction accuracy improves by as much as 46.78%. Additionally, compared to the KFM, QPM, and SAM, IM-SAM achieved improvement rates of 31.93%, 30.87%, and 19.14% in 3-h predictions, and 24.05%, 20.81%, and 14.39% in 6-h predictions, respectively. These results fully demonstrate the strong adaptability and excellent generalization capability of the IM-SAM in short-term predictions. ③ Compared with the QPM, SAM showed a certain improvement in short-term predictions (3 h and 6 h), reflecting its partial ability to suppress prediction errors after introducing periodic terms into the model. However, in medium- and long-term predictions (9 h and 12 h), the prediction accuracy of SAM failed to improve continuously and even declined, indicating that its periodic term modeling has certain limitations. In contrast, although IM-SAM is similar to SAM in terms of modeling strategy, it significantly improves the time sensitivity and dynamic response capability of the model by introducing a periodic term prediction mechanism based on endpoint instantaneous parameters, enabling it to maintain leading prediction performance in all prediction durations, further verifying the effectiveness and robustness of IM-SAM in time series modeling.

4.4. Precise Point Positioning Experimental Analysis

To evaluate the practical applicability of the proposed IM-SAM method in real-time precise point positioning (PPP), a kinematic PPP experiment was conducted. Observational data from six IGS stations (BUCU, IISC, LCK4, LHAZ, PBR4, and URUM) on DOY 168, 2023, were selected, each with a sampling interval of 30 s and a continuous 9-h observation span. The IGS Ultra-Rapid product provides orbit and clock solutions in two parts: the observed part (IGU-O) and the predicted part (IGU-P). In this study, the IGU-P orbit was used for PPP processing, while the clock input was alternated between the IGU-P clock and the IM-SAM-predicted clock for comparison. The IM-SAM-predicted clock was generated by modeling the IGU-O clock and predicting 9 h ahead. The processing strategy follows the methods described in [28,29,30,31,32].

Figure 10 shows the time series of positioning errors in the East, North, and Up components for representative stations (BUCU, IISC, and LCK4), and

Table 4. Statistics of kinematic PPP positioning results for all measurement stations.

Station Name	IM-SAM E/N/U (cm)			IGU-P E/N/U (cm)			IM-SAM vs. IGU-P E/N/U (%)
BUCU	6.207	7.330	14.741	6.222	8.807	16.791	0.2	16.7	12.2
IISC	6.168	5.787	17.424	7.857	6.439	17.812	21.4	10.1	2.1
LCK4	6.233	5.647	10.260	8.446	6.488	10.387	26.1	12.9	1.2
LHAZ	11.002	7.253	15.736	11.953	8.431	18.984	7.9	13.9	17.1
PBR4	5.322	6.271	12.416	7.002	7.106	12.614	23.9	11.7	1.5
URUM	10.003	7.680	10.430	10.655	8.585	12.022	6.1	10.5	13.2
Average	7.489	6.661	13.501	8.689	7.643	14.768	14.3	12.6	7.9

summarizes the corresponding RMS positioning errors derived from these results. The errors were computed as the differences between the estimated station coordinates and the official IGS reference coordinates. The results indicate that the Up component exhibits larger RMS errors than the East and North components, which is consistent with the general characteristics of PPP. IM-SAM outperformed IGU-P at most measurement stations, with average RMS reductions of 14.3%, 12.6%, and 7.9% in the East, North, and Up directions, respectively. These results demonstrate the effectiveness of the proposed IM-SAM method for real-time PPP applications.

5. Conclusions

To address the endpoint effects caused by global fitting of periodic term parameters in traditional SAM, this study proposes the IM-SAM. IM-SAM employs NTFT to perform time-frequency analysis of clock bias data and utilizes IM to extract the parameters of periodic terms at the data endpoint, enabling a more accurate characterization of the dynamic variations in time-varying periodic terms during the extrapolation process. Through comparative analysis with simulation data and GPS precise clock bias data, the following main conclusions are drawn:

①

IM-SAM significantly outperformed traditional models in terms of RMS and STD metrics for all prediction durations (3 h, 6 h, 9 h, and 12 h). Among all tested models, IM-SAM achieved the lowest average RMS value, with improvement rates exceeding 30% compared to GM, KFM, and QPM for short-term predictions. IM-SAM reduces prediction error by 19.14% compared to SAM with a similar model structure in 3 h predictions and maintains its advantage in longer prediction intervals. Additionally, a kinematic PPP experiment was performed using IM-SAM-predicted clock products and compared with the IGU-P. The positioning results from six IGS stations revealed that IM-SAM clock products led to a notable improvement in positioning accuracy, with average RMS reductions in all coordinate directions.

②

Traditional models (GM, KFM, and QPM) do not model periodic terms, so their prediction errors exhibit significant periodic fluctuations. Although SAM partially mitigates these fluctuations through global fitting, its performance deteriorates in the presence of frequency drift or sudden changes. In contrast, IM-SAM is able to adaptively capture the latest periodic dynamics at the edge of the prediction boundary, significantly reducing prediction errors and making them more stable.

In summary, IM-SAM enables more accurate descriptions of the time-varying characteristics of satellite clock bias periodic terms and achieves higher precision and better stability in short-term predictions. Future research could further optimize the extrapolation algorithm of IM-SAM to accommodate longer-term clock bias prediction requirements and explore its applicability in other navigation systems.