8.4 Prediction
8.4.2 Experiments
8.4 Prediction
The data dissemination rate has improved within the broadcast message. The actual max-imum latency according to each system Interface Control Document is smaller than 100 min for Galileo, twice a day uploads for GLONASS and one upload per day for GPS. Upload rates higher than the declared may be employed to adapt the prediction to the clock performance, as in GPS where up to three uploads per day instead of one may be applied in the case of worse performing clocks [42].
Nowadays not only the broadcast navigation message is used for real or near real-time naviga-tion. A global GNSS user may employ full independent orbit and clock information, provided by third entities, such as public precise orbit determination centers as IGS or commercial ser-vices as Fugro [107]. These entities provide independent messages by diverse communication channels at different update rates, the almost real time being the current goal [28]. The actual limitation for IGS Ultra Rapid products is not the orbit but the clock prediction accuracy at 9 hours (3 latency plus 6 hours validity) and its robustness with respect to occasional outliers [139]. This limitation is expected to be overcome by IGS Real Time Service.
The update rate is linked to the fitting interval (item 2). Fitting intervals are chosen in accor-dance to the update rate achievable by the system. Shorter update rates require shorter fitting intervals. Prediction can be avoided for real time applications by using directly the last estima-tion, provided the clock is estimated at every epoch [67]. Longer update rates require longer intervals, for example, refitting based on several days using previous broadcast messages is ap-plied in some mass market receivers to perform long predictions in order to improve the time to first fix in a warm start [182].
Predictions or estimations can also be avoided. Expert users may compute their own satellites clocks as done in POD adjustments. Analogous to PPP on-line services, this solution is being simplified by internet applications as Magic online service [134] which allow the inclusion of user observations into a global POD solution with minimum interaction of the user.
Functional model
Once the model is estimated and the clock predicted, the error x(t) associated with the model can be computed as the difference between the clock prediction and the posterior clock estimation.
A standard reference period without events is selected for this analysis. As reference period one month from day 280 to 308 of the year 2009 has been selected. Main attention is given hereafter to the fitting model, the data intervals for the model and the adequacy of the 100 minutes maximum validity time foreseen for the Galileo system. The following strategies are tested and the results are summarized in Table 8.5 :
1. Quadratic fit to the last 24 hours (broadcast strategy in GIOVE-M)
2. Quadratic fit with different fitting intervals (1,6,12 hours) to each coefficient (a0,a1,a2).
3. Quadratic fit with different fitting intervals and two additional components (a3,a4) for the periodic component.
Periodic phase variations associated to clock estimations are a common feature in GNSS.
From strategy 2 results, it seems that the harmonic function cannot be neglected without increasing the error. As a consequence, two additional parameters are included in the fitting adjustment (Equation 8.2), where the harmonic period ω−1 is fixed to the orbit period.
4. Same as strategy 3, but the periodic terms are not transmitted to the user.
In order to remain within the 3 parameter model allowed by the broadcast message, the new strategy-4 is tested by fitting the prediction with the 5-parameter model and using only the 3 polynomial parameters (a0,a1,a2) to compute the prediction.
5. Different dt fitting intervals as multiples of the orbit period
Few information exists about the harmonic source and characteristics. It is not clear whether the inclusion of the two additional terms (a3-a4) in the model will be robust or could introduce outliers increasing the maximum error. Therefore, a simple approach is taken by selecting the fitting intervals (dt1, dt2) multiples of the harmonic/orbit period.
In this case, the GIOVE orbit period is around 14 hours and the selected fitting intervals aredt1=14 anddt2=28 hours for (a1,a2) respectively.
As summary a final check is performed by evaluating the error at 100 min maximum validity envisaged in the Galileo navigation message with all the strategies under test. Table 8.5 summa-rizes the results for each prediction strategy, where the first 4 columns represents the strategy applied and the last 4 the associated error observed for PHM and RAFS during the selected period at 100 minutes. For the strategy, the column ’#’ indicates the strategy number, column
8.4 Prediction
Strategy RAFS PHM
# dt(0,1,2) Estimation BRD rms max rms max 1 (24,24,24) a0-a2 a0-a2 7.31 12.62 0.38 1.46 2 (01,06,12) a0-a2 a0-a2 0.97 2.74 0.37 1.43 3 (01,06,12) a0-a4 a0-a4 0.70 2.13 0.33 1.73 4 (01,06,12) a0-a4 a0-a2 0.70 2.13 0.33 1.73 5 (01,14,28) a0-a2 a0-a2 1.48 4.48 0.27 1.34
Tab. 8.5: Clock prediction error at 100 min in nanoseconds with GIOVE clocks using different strategies
’dt’ the fitting intervals, column ’Estimation’ the parameters computed in the fitting and column
’BRD’ the parameters to be broadcast to the user.
Several conclusions can be extracted from the results. The main improvement with respect to the basic strategy (#1) is obtained due to the reduction of the fitting interval (#2). Inclusion of the harmonic terms (#3) improves slightly the prediction at 100 minutes, the improvement being better at longer intervals. The harmonic term mainly helps to stabilize the fitting error as the provision of the additional coefficients to the user has no effect on the final error solutions,
#3 and #4 being identical. Finally, the simple approach to use an integer multiple of the orbit period for the fitting interval (#5) provides the best accuracy for the PHM, while the error for the RAFS increases. This result is mainly due to the different σy at 6 and 14 hours for each clock. Obviously, the strategy which best suits one clock technology or unit may not be the best for another.
Stochastic model
The clock prediction objectives were twofold: first to reduce the prediction error in terms of standard deviation and maximum error, and second to assign a stochastic model to the clock prediction. An additional experiment is required for this second objective.
Equally important to have a good prediction is the possibility to associate a stochastic model to this prediction which can be used to provide a variance when computing the least squares adjustment. The stochastic model has been quantified following Equation 8.3:
σxp(t) =
σx2+σa20+ (σa1t)2+ (σa2t2)2+ (σyW F(t)t)2+ (σyFFln 2(t)t2)2 [8.3]
where :
• σxp(t), is the expected clock error
• σx is the clock phase estimation error computed from the 1-sigma distribution of the different estimation arcs. GIOVE estimation processing runs every hour estimating clocks and orbits with the last 48 hours of data. As a consequence, 48 different clock samples are available for the same instant. The average value obtained is 0.3 ns (1σ).
• σa0,σa1,σa2 are the a posteriori sigma of the least squares adjustment. The theoretical model for the clock prediction error is the one described in Equation 8.2. Such a formula is correct under the hypothesis of independent estimates of fit coefficients. If the coeffi-cient estimates are not independent some correlation terms appear and have to be taken into account in the uncertainty estimation. In order to eliminate such terms or to have at least negative correlations (which would not be a problem in the worst case analysis) baricentric coordinates have to be used in the polynomial fit estimate.
• σy2(t)is the Allan variance of the clock evaluated at the time of predictiont. The stochas-tic contribution on the uncertainty on clock prediction has been evaluated considering two types of noise: white noise and flicker noise. For the PHM on GIOVE-B the values of such noises have been taken from the specifications previously covered in Section 4.2 (1E-12 for WFN and 1E-14 for FFN). For GIOVE-A the scecified value of flicker noise (3E-14) has been taken while for the white noise an experimental value of 6E-12 has been considered, which is bigger than the value reported in the specs (5E-12).
The stochastic model defined in Equation 8.3 has been applied and the expected error named ase. Additionally, as the fitting a posteriori sigma (σa0,σa1,σa2) could not be representative of the adjustment an alternative approach has also been tested. The clock parametersa0,a1,a2 estimated over the moving windowt0−24h are stored at eacht0 and used to compute alterna-tive sigma values (σa0,σa1,σa2) by computing their standard deviation over the fitting interval.
Obviously, a converge period of 1 day is required to obtain the first full set ofa0−2values over the moving window. These alternative sigmas (σa0,σa1,σa2) are used in Equation 8.3 instead of (σa0,σa1,σa2) and the expected error named ase.
The grey lines in Figure 8.12 represent the instantaneous error for every single prediction.
The root mean square (rms) and the standard deviation (std) for the prediction error are com-puted at each prediction time. Both values present a good overlap indicating a zero mean unbi-ased distribution. The modified modelefollows with a better agreement the standard deviation (std) of the prediction for both strategies #2 and #5. On the contrary, the theoretical stochas-tic model e seems to underestimate the real error diverging for prediction times over 6 hours for strategy #2. This is most likely due to optimistic sigma values for the clock parameters (a0,a1,a2) without taking into account the orbit period. It has to be also remarked how strategy
#5 considerably reduces the error at 1 day.
In order to translate the clock predictions for all GNSS satellites, a strategy has to be selected.
The inclusion of the harmonic term could make the fitting unstable in the case that no harmonic exists. Strategy 2 is applied to the complete constellation of GNSS satellites for the three years (selected operation period from 2008 till 2011). A prediction at 100 minutes is selected to
8.4 Prediction
0 5 10 15 20
−10
−8
−6
−4
−2 0 2 4 6 8 10
hours from t0
nanoseconds
PRN: E16 DATE: 15/10/2009 − 02/11/2009 REJECTED: 0/4992 (>2.5ns) ZAOD: 0.06ns
rms std e e"
error
(a)Strategy #2
0 5 10 15 20
−10
−8
−6
−4
−2 0 2 4 6 8 10
hours from t
0
nanoseconds
PRN: E16 DATE: 15/10/2009 − 02/11/2009 REJECTED: 0/4992 (>2.5ns) ZAOD: 0.06ns
rms std e e"
error
(b)Strategy #5
Fig. 8.12: Clock prediction for GIOVE-B in PHM mode over 24 hours