1. Vondrák Filtering for Timescales Demetrios Matsakis, Gianna Panfilo, and Gérard Petit USNO & BIPM

2. Who is he? Expert on Geodesy – Hipparcos PI for Earth Rotation – Time series for long-term polar motion/nutation First filter: A smoother – For any function: (X i, Y unsmoothed,i ) Data need not be equally spaced – Y i determined by compromise between 1.Fidelity to original data – Y i = Y unsmoothed,i 2.Smoothness – As defined by integral of the output’s squared third derivatives – Lagrange interpolation between the (X i, Y i ) points Third-order polynomials applied between adjacent X i Polynomials defined to exactly cross 4 adjacent (X i, Y i ) points – One parameter: ratio of desired smoothness to desired fidelity – Used by BIPM to smooth UTC(k) for Circular T See Feissel and Lewandowski, Bull. Geod., 58 (1984) Z. Jiang, internal memo on application to TW @ 12 points/day

3. Second Filter, for Combining Datasets Two input time series – One time series gives the phase (time) – Second time series gives the derivative (frequency) Like before: – Determines y i for each x i – Lagrangian interpolation between (x i,y i ) – Least squares is used to set the y i Vondrák and Cepek, 2000, Astron. Astrophs. Suppl. Ser. 147, pp 347-359

4. Determining the y i Weighted Least Squares compromise between – Smoothness of final curve Integral of third derivative squared – Fidelity to weighted phase data of one time series – Fidelity to weighted frequency data of second time series Two parameters 1.(Phase RMS)/(Output Smoothness) 2.(Freq. RMS)/(Output Smoothness) This work is uses BIPM’s FORTRAN code – 305 lines – Note: x i that are too close together can result in bad fits

5. BIPM’s “Combined Links” Combine TW with PPP – PPP is more precise =>gives frequencies – TW assumed to be more accurate => sets phases Diurnals require multiday smoothing for TW data Input: TW and PPP dataOutput: combined series

6. BIPM’s Weights & Filter’s Transfer Functions At one day, PPP (non-)diurnal dominates over TW diurnal – 60% transfer of 1-day component of phase series – Almost 100% transfer of 1-day component of frequency series Jiang and Petit, Metrologia 46 (2009) pp 305-314 F P BIPM weights = 100,000 phase (TW); 1,000,000 frequency (PPP); 1 smoothness Relative Costs: 1 ns phase =.32 ns/day freq = 1 ns/day/day/day drift

7. BIPM weights are pretty good … Contribution of 100 ps TW noise < 8 ps (smooth curve)

8. Smoothness Can Count Too

9. Response to TW Step

10. Response to PPP Step

11. Variable Diurnals (sudden onset) TW simulations are red Filter outputs are blue Noiseless PPPPPP with 30 ps white noise

12. Constant frequency difference between the two series not big problem (if reasonable and TW is right) PPP average frequency is zero TW slope = 0.1 ns/dayTW slope = 1000 ns/day

13. Red line is fitted value of signal after fitting to noisy diurnal A Kalman Filter can also remove a diurnal USNO routinely does so on AMC-USNO baseline SP gave a talk on this at the Taiwan meeting

14. Vondrák vs. Kalman Bad PPP Data 2ns TW cal change

15. a very simple smoother/averager can also work (though it filters out short-term clock behavior) boxcar smoothing: each TW point replaced by average of adjacent 24 points does not use PPP, but differences with Vondr á k (red) are small Raw TW data Vondr á k Averaged TW data

16. Comparison of Vondrák combination with TW Smooth For clock with large frequency offset that inverts itself. (PPP data not shown) Raw TW data Vondrák Smoothed TW

17. Vondrák Filters can combine clocks plot below: output = best of maser + best of cesium

18. A caution on accuracy assuming long-term stability does not insure it

19. Conclusions Vondrák smoothing does excellent job of eliminating diurnals Step functions would be spread over a few days – But their existence would violate the assumptions – As do drifts that have been observed in TW & PPP Other filters could perhaps do as good job – Tuning a Kalman filter would require care – Simple smoothing of TW data would not handle gaps Vondrák filters could also create timescales

20. Backups Mostly for the TWSTT WG meeting

21. Is there a seasonal variation?

23. t

24. USNO diurnals getting better

26. Output hardly depends on time-range 2 days trimmed off solution edges