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

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 12 points/day

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

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

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

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 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

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

Smoothness Can Count Too

Response to TW Step

Response to PPP Step

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

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

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

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

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

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

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

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

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

Backups Mostly for the TWSTT WG meeting

Is there a seasonal variation?

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USNO diurnals getting better

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