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## Appendix MAccuracy 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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