Appendix M
Accuracy of a 14-Satellite Ensemble Versus a 24-Satellite Ensemble

Below is a comparison of the accuracy of a 14-satellite ensemble clock versus a 24-satellite ensemble.

Case A

Assume that all satellites have clocks equal to Block IIR cesium clocks. (Block IIR rubidiums are a factor of two more stable.) A 14-satellite ensemble is used. Consider synchronization error between two satellites whose ensembles have the minimum overlap of four. Note that these satellites are on opposite sides of the earth, and would probably never be used in the same stand-alone solution, so this is the worst case scenario.

Analysis

For T = 15 minutes, Δf/f = 10-12 Allan variance slope is -1/2. Autonomous navigation ranging error is 1 ns, measured each 15 minutes.

To determine the optimum clock averaging interval if (1) T = 15 minutes; (2) ranging error is 1 ns/N1/2; (3) N is the number of 15 minute ranging epochs used for averaging; and (4) the error due to clock instability is [(10-12)(1/N1/2)(N) intervals x 900 s/interval], the optimum is about 15 minutes, where measurement error and clock instability each contributes about 1 ns of error. The produces a combined (RSS) error of 1.4 ns or 0.4 meters.1

Given that a 14-satellite ensemble is quite adequate for the case in which all clocks are well-behaved atomic standards (rubidium or cesium), it seems evident that an ensemble of all the clocks is better. First, it will have marginally smaller error, by (14/24)1/2 = 0.76. Second, it will compare all satellite clocks at each autonomous navigation measurement,

1  

1 nanosecond times the speed of light = 30 centimeters



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--> Appendix M Accuracy of a 14-Satellite Ensemble Versus a 24-Satellite Ensemble Below is a comparison of the accuracy of a 14-satellite ensemble clock versus a 24-satellite ensemble. Case A Assume that all satellites have clocks equal to Block IIR cesium clocks. (Block IIR rubidiums are a factor of two more stable.) A 14-satellite ensemble is used. Consider synchronization error between two satellites whose ensembles have the minimum overlap of four. Note that these satellites are on opposite sides of the earth, and would probably never be used in the same stand-alone solution, so this is the worst case scenario. Analysis For T = 15 minutes, Δf/f = 10-12 Allan variance slope is -1/2. Autonomous navigation ranging error is 1 ns, measured each 15 minutes. To determine the optimum clock averaging interval if (1) T = 15 minutes; (2) ranging error is 1 ns/N1/2; (3) N is the number of 15 minute ranging epochs used for averaging; and (4) the error due to clock instability is [(10-12)(1/N1/2)(N) intervals x 900 s/interval], the optimum is about 15 minutes, where measurement error and clock instability each contributes about 1 ns of error. The produces a combined (RSS) error of 1.4 ns or 0.4 meters.1 Given that a 14-satellite ensemble is quite adequate for the case in which all clocks are well-behaved atomic standards (rubidium or cesium), it seems evident that an ensemble of all the clocks is better. First, it will have marginally smaller error, by (14/24)1/2 = 0.76. Second, it will compare all satellite clocks at each autonomous navigation measurement, 1   1 nanosecond times the speed of light = 30 centimeters

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--> giving improved potential for autonomous fault detection and system stability characteristics in the presence of anomalous behavior. Case B If quartz oscillators with Δf/f = 10-11 are used with 900 s inter-satellite link ranging updates, a 14-satellite ensemble would allow significant differences (few ns) to exist among the ensemble clocks of different satellites. If a 14-satellite ensemble is used, consider synchronization error between two satellites whose ensembles have no overlap. (Only because this is easier to analyze. The real case is not this bad). Again, note that these satellites are on opposite sides of the Earth, and would probably never be used in the same stand-alone solution. Analysis (1) For a 14-satellite ensemble: error per clock (~10-11)(900 s)N = (9 ns)N where: N is the number of 15-minute intervals that this minimum overlap occurs. For a 4-hour period, N = 16. When averaged over 14 clocks, the error would be reduced to: (9ns)(16/14 1/2) = 38 ns. Also, the 38 ns would not only show up as an offset from UTC, but would add to the UERE and, thus, affect the stand-alone position solution. Although as mentioned above, the real case would not be this bad. (2) For a full constellation 24-satellite ensemble: The clock error of the full constellation would drift by [{(10-11)(3600)(4)}/24 -1/2 = 29 ns] over the same 4-hour period. While this 29 ns drift would show up as an offset from UTC, it would be a common clock error for the entire constellation, and would not significantly affect the stand-alone position solution. In summary, the main reason for a 24-satellite clock ensemble is to enable use of more reliable, lower mass and power quartz oscillators in most of the satellites. Atomic clocks would be used in four satellites to provide redundant steering of the ensemble to UTC.