1. GNSS Observations of Earth Orientation Jim Ray, NOAA/NGS 1. Polar motion observability using GNSS – concepts, complications, & error sources – subdaily considerations 2. Performance of IGS polar motion series – compare Final, Rapid, & Ultra-rapid products – assess random & systematic errors 3. Utility of IGS length-of-day (LOD) – assess value for combinations with VLBI UT1 4. Impact of errors in subdaily EOP tide model – effects on orbits, EOPs, & other IGS products Wuhan University, May 2013

2. EOPs are the five angles used to relate points in the Terrestrial & Celestial Reference Frames: [CRF] = P · N(ψ, ε) · R(UT1) · W(x p, y p ) · [TRF] –Precession-Nutation describes the motion of the Earth’s rotation axis in inertial space –Rotation about axis given by UT1 angle –Wobble of pole in TRF given by terrestrial coordinates of polar motion (x p, y p ) But only three angles, not five, are independent –this conventional form is used to distinguish excitation sources: Nutation ↔ driven by gravitational potentials outside Earth system Polar Motion ↔ driven by internal redistributions of mass/momentum –separation of Nutation & Polar Motion estimates given by convention Earth Orientation Parameters (EOPs) 02 (x p, y p )

3. Motions defined in frequency domain –note that diurnal retrograde motion in TRF is fixed in CRF: -1.0 cycle per sidereal day (TRF) = 0.0 cycles per sidereal day (CRF) Because GNSS cannot observe CRF (quasar frame), it does not measure precession-nutation or UT1 –but GNSS can sense nutation-rate & UT1-rate (LOD) changes –GNSS is superb for Polar Motion due to robust global tracking network –pole position is essentially an unmarked point in the TRF Separation of Nutation & Polar Motion 03 ← polar motionpolar motion → precession nutation frequency in Terrestrial Frame frequency in Celestial Frame

4. Suppose a priori pole position has some unknown error: Due to diurnal Earth spin, PM error causes sinusoidal apparent motion for all TRF points as viewed from GNSS satellite frame –(x p, y p ) partials are simple diurnal sine waves –amplitude & phase depend only on station XYZ location –quality of PM estimates depends mostly on Earth coverage by GNSS stations –IGS formal errors: σ x,y = 5 µas Observability of Polar Motion (PM) 04 actual pole positionassumed pole position Signature of PM error in GNSS Observations ← 1 solar day →

5. GPS satellites have period of ~0.5 sidereal day –ground tracks repeat every ~1 sidereal day –differs from 1 solar day by only ~4 minutes –other GNSS constellations have longer or shorter periods –any common-mode near-diurnal orbit errors can alias into PM estimates Any other net diurnal sinusoidal error in GNSS orbits will also alias into PM estimates –main error comes from model for 12h/24h EOP tides –mostly caused by EOP effect of ocean tidal motions –current IERS model has errors at < ~20% Other common mode effects could also be important: –diurnal temperature effects (e.g., heights of GNSS stations) –diurnal troposphere modeling errors –various other tidal modeling errors –local station multipath signatures due to ground repeat period Some Observability Complications 05

6. First, “subdaily” polar motion is not a well-defined concept –overlaps with nutation band in retrograde sense –inseparable from a global rotation of satellite frame –so constraint normally applied to block diurnal retrograde frequencies –this is effectively a filter with poor response for GNSS arcs of ~1 day [D. Thaller et al., J. Geodesy, 2007] Second, observability is reduced for intervals <1 solar day –partial diurnal sinusoidal cannot be separated from other parameters –so parameter continuity is required for direct subdaily estimates –most common approach (Bern group) is to use 1 hr continuous segments –this operates as another filter, but with other disadvantages (next slides) So subdaily results are easily affected by spurious effects On “Subdaily" Polar Motion 06 ← polar motionpolar motion → precession nutation frequency in Terrestrial Frame subdaily prograde PM →← subdaily retrograde PM

7. Compare offset + rate to continuous linear segments (CLS) –IGS requests daily PM estimates as mid-day offsets + rates –but some Analysis Centers prefer CLS approach –results are not equivalent near Nyquist frequency –CLS results are non-physical at high freqs Consider cosine wave at Nyquist freq –φ = π –CLS & offset + rate give exactly same estimates for this phase Now shift cosine by -90° –φ = π/2 –CLS estimates are all 0.0 –but offset + rate estimates are not zero & not constant Effects of “Continuity Filter” (1/3) 07 CLS estimation Offset + rate estimation

8. CLS attenuates Nyquist signal amplitudes by factor of 2 –power reduced by factor of 4 at Nyquist frequency –power starts dropping at ~0.6 x Nyquist frequency & higher Filter effect clearly seen in IGS PM results –most Analysis Centers follow f -4 power law for sub-seasonal periods, e.g., GFZ (below right, during 11 Mar 2005 – 29 Dec 2007) –but CODE used CLS parameters & had strong high-freq smoothing Effects of “Continuity Filter” (2/3) 08 Smoothed PSD for Reprocessed CODE PMSmoothed PSD for Reprocessed GFZ PM

9. CLS method is not a simple smoothing filter –it distorts signal content by attenuating certain phases over others –causes all parameters to be strongly correlated at all times –should not be used when signals of interest are near Nyquist sampling Unfiltered IGS daily PM can be extrapolated to estimate subdaily PM variance (non-tidal) –sub-seasonal PSD follows f -4 power law (integrated random walk process) –fits to GFZ PSD over 0.1 to 0.5 cpd: PSD x (f) = (48.11 µas 2 /cpd) * (f/cpd) -4.55 PSD y (f) = (64.21 µas 2 /cpd) * (f/cpd) -4.10 –if valid at f > 0.5 cpd, then integrate over 1 cpd → infinity: σ 2 x (subdaily) = 13.55 µas 2 σ 2 y (subdaily) = 20.73 µas 2 –much too small to be detectable Effects of “Continuity Filter” (3/3) 09 Smoothed PSD for Reprocessed GFZ PM

10. Three methods probably feasible: –Kalman filter use normal deterministic PM parameters for daily offset + rate add stochastic model (f -4 integrated random walk) to estimate deviations probably can be done with JPL’s GIPSY, but I know of no results –CLS only method used till now but problems noted above are serious & probably gives unreliable results –invert from overlapping daily fits in principle, probably could invert normal daily offset + rate fits but use overlapping data arcs (highly correlated estimates) would probably need to add f -4 integrated random walk model to inversion not known to be tried could be tested using IGS Ultra-rapid PM series (24 hr arcs with 6 hr time steps) Subdaily PM (non-tidal) power is so small, no clear reason to try to measure –but filling band from 0.5 to 1.0 cpd could aid excitation studies (e.g., using IGS Ultras) Estimating “Subdaily" PM 10