The Effect of Observation Discontinuities on LEO Real-Time Orbital Prediction Accuracy and Integrity

2.1 RP: Reduced-Dynamic POD Including Estimated Stochastic Accelerations

The RP strategy is based on well-established gravitational perturbation models and compensates for certain non-gravitational forces, like SRP or air drag effects, through estimable dynamic parameters. These parameters, along with the Keplerian elements at the initial epoch, are solved through BLS processing.

The observation model for the ionosphere-free (IF) code and phase observations can be expressed as:

$E\left(\Delta p_{\text{IF}}^s\left(t_i\right)\right)=A_K^s\left(t_i\right)X_K+A_{\text{SRP}}^s\left(t_i\right)a_{\text{SRP}}+A_{\text{Sto}}^s\left(t_i\right)a_{\text{Sto}}+c\times \Delta t_r\left(t_i\right)$1

$E\left(\Delta {\phi }_{\text{IF}}^s\left(t_i\right)\right)=A_K^s\left(t_i\right)X_K+A_{\text{SRP}}^s\left(t_i\right)a_{\text{SRP}}+A_{\text{Sto}}^s\left(t_i\right)a_{\text{Sto}}+c\times \Delta t_r\left(t_i\right)+{\lambda }_{\text{IF}}\times N_{\text{IF}}^s$2

where E(·) denotes the expectation operator, and $\Delta p_{\text{IF}}^s\left(t_i\right)$ and $\Delta {\phi }_{\text{IF}}^s\left(t_i\right)$ represent the code and phase observed-minus-computed (O-C) terms, respectively, at epoch i for GNSS satellite s. The matrices $A_K^s,A_{\text{SRP}}^s$, and $A_{\text{Sto}}^s$ are design matrices containing the partial derivatives of the code and phase observations with respect to the unknown parameters X_K, a_SRP, and a_Sto, respectively. Of the unknowns, X_K represents the six Keplerian elements at the initial epoch t₀; a_SRP denotes the constant SRP parameters; and a_Sto is the piece-wise constant stochastic acceleration, which reset every few minutes. Δt_r is the estimable clock error of the LEO satellite receiver, and c is the speed of light. Of the two IF terms, λ_IF is the IF wavelength, and $N_{\text{IF}}^s$ represents the IF ambiguity for satellite s.

By combining Equations (1) and (2), the dynamic orbital parameters can be estimated through BLS adjustment. The Cartesian orbits can then be numerically integrated at each epoch within the POD arc (Beutler 2005). The orbits from the most recent 4 h of the POD arc are then used to fit orbit parameters and generate the Keplerian elements X_K and the nine solar radiation pressure parameters a_SRP,all, which are later used for orbit prediction $\left({\hat{X}}_{\text{pre}}\right)$. The vector a_SRP,all of the nine SRP parameters can be expressed as:

$a_{\text{SRP},\text{all}}={\left(a_{R0},a_{T0},a_{A0},a_{\text{RS}},a_{\text{TS}},a_{\text{AS}},a_{\text{RC}},a_{\text{TC}},a_{\text{AC}}\right)}^T$3

where a_R0, a_T0 and a_A0 denote the constant SRP terms in the radial (R), along-track (T), and cross-track (A) directions, respectively. a_RS, a_TS, a_AS are their sine counterparts, and a_RC, a_TC, a_AC are the corresponding cosine counterparts. These parameters form the SRP accelerations in the three directions (a_SRP,R, a_SRP,T, a_SRP,A) as follows:

$a_{\text{SRP},R}=a_{R0}+a_{\text{RS}}\times \sin \left(u\right)+a_{\text{RC}}\times \cos \left(u\right)$4

$a_{\text{SRP},T}=a_{T0}+a_{\text{TS}}\times \sin \left(u\right)+a_{\text{TC}}\times \cos \left(u\right)$5

$a_{\text{SRP},A}=a_{A0}+a_{\text{AS}}\times \sin \left(u\right)+a_{\text{AC}}\times \cos \left(u\right)$6

where u refers to the argument of latitude.

2.2 CP: Reduced-Dynamic POD Without Estimating Stochastic Accelerations

The CP strategy follows the RP strategy but with two key differences: first, it estimates all nine SRPs a_SRP,all in the POD process instead of only the three constant terms (see Eqs. (1) and (2)); and second, it does not estimate any piece-wise constant stochastic accelerations in the POD process.

For this strategy, the code and phase IF O-C terms can be expressed as follows:

$E\left(\Delta p_{\text{IF}}^s\left(t_i\right)\right)=A_K^s\left(t_i\right)X_K+A_{\text{SRP},\text{all}}^s\left(t_i\right)a_{\text{SRP},\text{all}}+c\times \Delta t_r\left(t_i\right)$7

$E\left(\Delta {\phi }_{\text{IF}}^s\left(t_i\right)\right)=A_K^s\left(t_i\right)X_K+A_{\text{SRP},\text{all}}^s\left(t_i\right)a_{\text{SRP},\text{all}}+c\times \Delta t_r\left(t_i\right)+{\lambda }_{\text{IF}}\times N_{\text{IF}}^s$8

where $A_{\text{SRP},\text{all}}^S$ contains the partial derivatives of the code and phase observations with respect to a_SRP,all.

2.3 BP: Combination of the RP and CP Strategies

The BP strategy combines the strengths of both the RP and CP methods for orbit determination. The RP orbits $\left({\hat{X}}_{\text{RP}}\right)$ are determined within the non-gap period, and the CP orbits $\left({\hat{X}}_{\text{CP}}\right)$ are determined within the gap period. These segments are then merged before orbit prediction (see Figure 2). The combined orbit $\left({\hat{X}}_{\text{BP}}\right)$ is expressed as:

${\hat{X}}_{\text{BP}}\left(T_{\text{all}}\right)=\left\{{\hat{X}}_{\text{RP}}\left(t_{\text{start}}\sim t_{\text{gap,start}}\right),{\hat{X}}_{\text{CP}}\left(T_{\text{gap}}\right),{\hat{X}}_{\text{RP}}\left(T_{\text{tail}}\right)\right\}$9

where T_all, T_gap, and T_tail represent all the time points within the POD arc, the observation data gap, and the tail segment, respectively. t_start and t_gap,start are the starting time points of the POD arc and the GNSS observation gap, respectively. Note that, in this notation, lowercase t denotes a specific time point (epoch), while uppercase T indicates a time interval.

2.4 EP: Reduced-Dynamic POD Discarding Tail Data

The EP strategy follows the RP method but discards the gap and the tail observation data, at the cost of extending the prediction time. The EP prediction time T_EP,pre consists of the following parts:

$T_{\text{EP},\text{pre}}=l_{\text{gap}}+l_{\text{tail}}+T_{\text{pre}}$10

where l_gap, l_tail, and T_pre refer to the length of the observation gap, the length of the tail data, and the scheduled prediction time (1 h for other strategies), respectively (see Figure 2).

4.1 Accuracy of the Predicted Orbits

When the GNSS observations are complete (i.e. no gaps), POD using high-precision real-time GNSS products can achieve centimeter-level accuracy. However, data gaps lasting several hours or more can significantly decrease this accuracy. For example, Figure 5 shows the POD results for a representative 24-hour arc for Sentinel-6A on DOY 033, 2022. This arc started at 02:55 in GPS time (GPST) with a 9-h data gap followed by a 1-h tail segment. The top panel of Figure 5 shows the along-track orbital errors for the RP, CP, and EP strategies, which are typically larger than the errors in the radial and cross-track directions. In the RP strategy (red line in the top panel), the POD accuracy during the observation gap degrades rapidly from a few centimeters to the decimeter level. In the EP strategy (green line in the top panel), the shortened 14-h processing arc maintains POD quality, but then prediction accuracy during the much longer 10-h prediction window decreases significantly. The CP approach (blue line in the top panel), which does not involve estimating the stochastic accelerations, shows worse overall POD results than the RP and EP strategies but more limited accuracy loss during the gap. The bottom panel of Figure 5 shows the results from the BP strategy, which combines the RP orbit prior to the gap and the CP-derived orbits during the gap to achieve a high-precision orbit solution.

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FIGURE 5

Along-track POD errors with the RP, CP, and EP (top) and BP (bottom) strategies for Sentinel-6A on DOY: 033, 2022. The arc starts at 02:55 in GPST, with a 9 h data gap followed by a 1 h tail of data. In the bottom panel, the BP strategy integrates the non-gap period of the RP results and the gap period of the CP results.

Figure 6 illustrates the orbital errors of the BP strategy in all three directions for the same arc. During the period without data gaps, the orbital errors in all three directions were below 5 cm. During the gap, the along-track errors increased to 1 dm, while the cross-track errors remained around 5 cm, and the radial errors stayed below 5 cm.

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FIGURE 6

POD errors of the BP strategy in the radial (R), along-track (T), and cross-track (A) directions. The arc starts at 02:55 in GPST, with a 9-h data gap followed by a 1-h tail of data.

Based on the POD results from the four proposed strategies, Figure 7 shows the 1-h prediction errors in the along-track direction for the same arc. The BP strategy (yellow line), which combines the strengths of the RP and CP orbits, generally delivers the highest prediction accuracy. In contrast, the EP strategy results in the largest errors due to the long prediction period. However, this outcome is specific to the tested case of a 9-h data gap with a 1-h tail segment; the relative prediction performance under other combinations of gap and tail lengths will be discussed later in this section.

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FIGURE 7

Along-track orbital prediction errors for the four tested strategies over the 1-h prediction window for real-time orbital prediction errors (representing the period bounded by gray dashed lines in Figure 2). The POD arc starts at 02:55 in GPST, with a 9-h data gap followed by a 1-h tail.

As mentioned above, this study conducted 1440 processing rounds of 24-h orbit determination and 1-h orbit prediction for Sentinel-6A. For ground users, a key accuracy metric is the root mean square (RMS) of the OURE, denoted σ_OURE, which represents the average projection of the orbital errors towards the Earth (Wang, El-Mowafy, & Yang, 2022). This value can be calculated as follows:

${\sigma }_{\text{OURE}}=\sqrt{{\omega }_R^2{\sigma }_R^2+{\omega }_{\text{TA}}^2\left({\sigma }_T^2+{\sigma }_A^2\right)}$11

where σ_R, σ_T, and σ_A represent the LEO satellite orbital errors in the radial, along-track, and cross-track directions, respectively, and ω_R and ω_TA are projection coefficients toward the Earth’s surface that depend on the satellite’s orbital height. For Sentinel-6A flying at an altitude of approximately 1340 km, ω_R and ω_TA are about 0.6395 and 0.5432 (Chen et al. 2013), respectively.

Figure 8 shows the mean σ_OURE values for the four orbital prediction strategies over the first 0–10 min of the prediction interval for gap lengths of 3, 5, 7, and 9 h.

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FIGURE 8

Mean σ_OURE of the four tested strategies over a 0-10 min prediction interval. The length of the tail data is 15 min (left panel) and 60 min (right panel).

Results are shown for a 15-min tail segment in the left panel and a 60-min tail in the right panel. As shown in the figure, the RP, CP, and BP methods exhibit relatively stable prediction results across different gap lengths. Of these, the BP strategy generally delivers the best prediction results over the first 10 min, achieving mean σ_OURE values of about 7.7 and 3.5 cm for tails of 15 and 60 min, respectively. Comparing the two panels of Figure 8 shows that, for a given gap length, increasing the tail length from 15 to 60 min improves the 0–10 min prediction accuracy for all strategies except EP, as evidenced by an approximately 3 cm reduction in the mean σ_OURE for the RP, CP, and BP methods. This result indicates that shorter tails (i.e., gaps near the end of the POD arc) decrease the prediction accuracy of the RP, CP, and BP methods. In contrast, the EP strategy shows a dramatic decrease in accuracy when the gap length increases and when the tail length is extended from 15 min (left panel) to 60 min (right panel). This degradation is caused by the longer prediction intervals required for the EP approach.

Table 2 shows the mean σ_OURE when the left panel of Figure 8 (with a 15-min tail segment) is extended to a prediction time of 10–20 min. Overall, the BP strategy continues to deliver the best prediction performance, except in the case of short gaps (3 h), where the EP strategy begins to outperform the others. This shift occurs because shortening the gap length significantly reduces the prediction interval for the EP strategy, while extending the prediction time by 10 min has little additional influence on the EP results because the prediction times are already long.

To provide a more general overview of the prediction behavior of the four strategies across different gap and tail lengths, the mean σ_OURE of the BP strategy over a 1-h prediction interval is compared with the results for the EP, CP, and RP strategies in Figures 9, 10, and 11, respectively. Each figure is structured with four panels from left to right representing gap lengths of 3, 5, 7, and 9 h, respectively. Within each panel, different line colors illustrate results for different tail lengths, and results for the EP and BP methods are shown by solid and dashed lines, respectively. For reference, the prediction results without a data gap (using the RP strategy) are given in grey.

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FIGURE 9

Mean σ_OURE of the 1-h real-time orbital prediction results from the BP (dashed lines) and EP strategies (solid lines) for different observation gap and tail lengths. For reference, the gray dotted line represents the prediction results of the RP strategy without any observation gaps.

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FIGURE 10

Mean σ_OURE of the 1-h real-time orbital prediction results from the BP (dashed lines) and CP strategies (solid lines) for different observation gap and tail lengths. For reference, the gray dotted line represents the prediction results of the RP strategy without any observation gaps.

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FIGURE 11

Mean σ_OURE of the 1-h real-time orbital prediction results from the BP (dashed lines) and RP strategies (solid lines) for different observation gap and tail lengths. For reference, the gray dotted line represents the prediction results of the RP strategy without any observation gaps.

Based on Figure 9 (comparing the BP and EP strategies), the following observations can be made:

• 1) The EP method shows superior prediction accuracy for short gaps with short tails (red lines in the left panel of Figure 9). However, this advantage diminishes as the tail length increases. For longer tails (e.g., 60 min; blue lines in the left panel of Figure 9), the BP method demonstrates superior prediction accuracy.

• 2) The prediction accuracy of the EP method deteriorates progressively with increasing gap and tail lengths due to the longer prediction interval. With a gap length of 3 h, the mean σ_OURE for the EP strategy is below 1 dm, whereas with a gap length of 9 h, the mean σ_OURE exceeds 2 dm.

• 3) In contrast, the prediction accuracy of the BP method is insensitive to the gap length but improves with longer tail lengths. In other words, gaps that approach the end of the processing arc (i.e., with shorter tails) cause more damage to the BP prediction results.

• 4) The BP method combines RP results in the non-gap period with the CP estimates during the gap. The “jumps” observed in some of the BP solutions (dashed lines) are likely caused by this arc concatenation and the application of a 4.42-sigma outlier exclusion threshold.

Figure 10 compares the results of the CP and BP strategies across different combinations of gap and tail lengths. Like the BP strategy, the CP results are generally insensitive to the gap length but highly sensitive to tail length, with both strategies benefiting from longer tails. In general, the BP strategy (solid lines) achieves better prediction accuracy than CP (dashed lines) for shorter prediction intervals, whereas for prediction intervals over 45 min, the BP strategy becomes comparable to or slightly worse than CP.

Figure 11 compares the 1-h prediction results of the RP and BP approaches, showing that the BP strategy consistently outperforms RP across a variety of gap and tail lengths. Specifically, the RP strategy with a short tail length of 15 min (solid red lines) exhibits a rapid degradation in the prediction accuracy over time, with the mean σ_OURE exceeding 3 dm for predictions beyond 30 min. In contrast, the BP method maintains a prediction accuracy of approximately 1.5 dm (red dashed lines) under the same conditions. Like the CP and BP strategies, the RP results are insensitive to the gap length but sensitive to the tail data length, with a particularly pronounced decline in accuracy as the tail length decreases from 30 min (solid green lines) to 15 min (solid red lines). However, the 60-min tail results do not outperform the 45-min tail results for the RP strategy, likely because the RP strategy estimates stochastic accelerations even with large gaps. This estimation strategy strongly affects the quality of the POD results during large gaps (as illustrated by the poor RP results during the gap period in the top panel of Figure 5), and these disturbances presumably overshadow any marginal gains due to increasing tail data lengths in the conducted tests.

In summary, the BP strategy generally provides the best prediction accuracy among the four tested methods. Its accuracy is insensitive to gap length, but this accuracy may degrade significantly when the tail data length decreases. The only exception to this overall trend is under conditions of short data gaps (e.g., 3 h) and tail lengths (e.g., 15 min), in which case the EP strategy outperforms the BP strategy, especially for prediction times over 30 min.

4.2 Integrity of the Predicted Orbits

In addition to prediction accuracy, many applications also require high real-time orbit integrity. This section accordingly addresses the ability of each strategy to limit large prediction errors. Figure 12 shows the 68.27% (1σ) and 99.9% percentile lines of all computed OURE values across all four strategies, based on a scenario with a 9-h data gap and 15-min tail. Based on the 1σ percentile lines (top panel of Figure 12), the BP strategy consistently demonstrates superior performance, whereas the RP strategy exhibits a significant increase in prediction error as the prediction time increases. These results generally align with the orbit accuracy results described in Section 4.1. In contrast, when considering the 99.9% percentile lines (bottom panel), the BP strategy performs poorly, and the EP strategy instead shows the lowest OURE values. Despite the reduced prediction accuracy and extended prediction time caused by discarding the tail data in the EP strategy, the 99.9% percentile lines of its OURE remain at the sub-meter level. While the BP strategy (which merges the orbits from different strategies) maintains a superior 1σ percentile performance at 1 to 2 dm, its 99.9% percentile performance decreases to the meter level. Overall, in this long-gap and short-tail scenario, the CP strategy (green lines) provides the best combination of accuracy and integrity for predictions within 30 min.

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FIGURE 12

68.27% (top) and 99.9% (bottom) percentile lines of the OURE for the four tested strategies over a 1-h prediction interval. The data gap and tail lengths are set to 9 h and 15 min, respectively.

Figure 13 shows the 99.9% percentile lines of the OURE for predicted orbits under the four most extreme scenarios: (i) short-gap/short-tail (top left); (ii) short-gap/ long-tail (top right); (iii) long-gap/short-tail (bottom left); and (iv) long-gap/long-tail (bottom right). In the bottom two panels (representing long-gap scenarios), the red lines for the RP strategy are obscured by the yellow lines for the BP strategy. In general, the EP strategy (blue lines) exhibits the best performance across all four scenarios, with short gaps and short tails (left panels) being more favorable due to the shorter prediction time they require. The CP strategy (green lines) ranks second, though its prediction accuracy with short tails (top panels) decreases significantly with longer prediction times. The RP and BP strategies both show worse prediction accuracy in the 99.9% percentile lines than the EP and CP strategies.

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FIGURE 13

99.9% percentile lines of the OURE for the four tested strategies over a 1-h prediction interval. Percentile lines are shown for four extreme scenarios defined by the minimum and maximum observation and tail lengths.

For integrity monitoring of the prediction solution, a protection level (PL) that serves as a cap on possible solution errors can be computed (Blanch et al. 2012). The first step in computing the PL involves assessing the distribution of the prediction errors. Our results show that the prediction errors from different rounds at each prediction time point are not Gaussian distributed. A two-step method (Blanch et al. 2019) is therefore used to compute the mean values (m_q) and standard deviations (σ_q) that define a Gaussian distribution that overbounds the cumulative distribution function (CDF) of the empirical error distribution. The PLs for the OUREs at prediction time t_i are then calculated as follows:

${\text{PL}}_q\left(t_i\right)=K\times {\sigma }_q\left(t_i\right)+m_q\left(t_i\right),\text{with} q=1,2,3$12

where q = 1, 2, 3 corresponds to the radial, along-track, and cross-track directions. The parameter K is defined as:

$K=C^{-1}\left(1-\frac{P}{2}\right)$13

where C^–1(·) denotes the inverse CDF of a standard normal distribution, with the probability P of an incorrect orbital prediction contained in the parentheses. With a probability of 1 – P, the orbital prediction errors at the prediction time t_i are expected to be bounded by PL_q(t_i). The three PL_q(t_i) values corresponding to the three directions are then used to derive the overall PL for the OURE, denoted as PL_OURE :

${\text{PL}}_{\text{OURE}}=\sqrt{{\omega }_R^2{\text{PL}}_1^2\left(t_i\right)+{\omega }_{\text{TA}}^2\left({\text{PL}}_2^2\left(t_i\right)+{\text{PL}}_3^2\left(t_i\right)\right)}$14

Figure 14 shows the PLs of the OURE for the predicted orbits with the risk threshold of P = 10⁻⁵ for the same four scenarios shown in Figure 13. As in Figure 13, the EP strategy (blue lines) provides the tightest PLs in all four scenarios, with PLs of about 0.4, 1.14, 0.45, and 1.31 m in the short-gap/short-tail, long-gap/short-tail, short-gap/long-tail, and long-gap/long-tail scenarios, respectively. Shorter data gaps and tails (left panels of Figure 14), which correspond to shorter prediction times, are essential for achieving smaller PLs with the EP strategy. Among the remaining three strategies, the CP strategy (green lines) generally delivers more stable and reliable PLs than the RP or BP strategies.

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FIGURE 14

PLs (P = 10⁻⁵) of the OUREs for the predicted orbits from the four tested strategies

Notably, the CP strategy exhibits a clear divergence in PL performance depending on the tail data length, with longer tails resulting in considerably smaller PLs. In this experiment, the last four hours of the POD arc were used to fit the dynamic parameters for orbit prediction. As shown in Figure 15, a key difference between the 15 min and 60 min tail cases is the continuity and amount of available data within these four hours that can be used to fit orbital prediction parameters. For example, in the case of a 3-h gap with a 15-min tail (first row in Figure 15), the 4-h fitting arc is split into two short, discontinuous segments, whereas in the case of a 3-h gap followed by a 60-min tail (second row in Figure 15), a complete 1-hour arc is available. Similarly, in the cases with 9 h gaps (third and fourth rows in Figure 15), the lengths of the available data differ based on the tail data length (i.e., 15 and 60 min). The fitting arcs that contain a complete 60-min segment within the 4-h window (second and fourth rows of Figure 15) resulted in improved POD performance, more accurate prediction parameter fitting, and, as a result, better prediction outcomes.

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FIGURE 15

Data availability within the last 4 h of the POD arc

Figure 16 provides an overview of the PLs for the EP strategy, assuming tail data lengths of 15 and 60 min (left and right panels), varying gap lengths (colored lines), and a more relaxed risk threshold for incorrect orbit prediction of P = 10⁻³. For both tested tail lengths, the PLs increase with increasing gap length; for example, in the left panel, the average PL increases from approximately 3.1 dm at a gap length of 3 h to 8.7 dm at a gap length of 9 h. Based on the greater fluctuation and overall larger PLs in the right relative to the left panel, PL stabilities over the prediction period decrease with increasing tail length. Averaged PLs for the EP method over the first prediction hour are summarized in Table 3. In general, lower PLs reflect better integrity monitoring. For example, with a preset alert limit of 1 m, the availabilities for a gap length of 9 h (blue lines) would be 99.17% and 57.85% for the 15-min and 60-min tail scenarios, respectively. For gap lengths of 3–7 h (red, green and yellow lines), availabilities remain at 100% under both tested tail lengths.

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FIGURE 16

PLs (P = 10^–3) of the OUREs for orbits predicted by the EP strategy for tail data lengths of 15 min (left panel) and 60 min (right panel)