Combined covariance reductions for Kalman filter composite clocks

The impact of combining two covariance matrix reduction methods on the resulting Kalman filter timescale estimates and covariances is addressed. The Brown and Greenhall reduction operations commute, and the combination of both reductions gives the same Kalman filter estimates as the Greenhall reduction alone. The benefit of combining both methods is that the Greenhall reduced covariances are further reduced in the positive semi-definite sense. A numerical example demonstrates the combined reduction by calculating the trace of the original and reduced covariance matrices. It is pointed out that the corrected clock offsets can follow the system time offset with different precisions, but the mutual disagreements of the corrected time offsets (called synchronization error) can be identical