1. Systematic errors in IGS terrestrial frame products Paul Rebischung, Zuheir Altamimi (with contributions from Ralf Schmid, Jim Ray & Wei Chen) IGS Workshop 2017, Paris, 3-7 July 2017
Good morning. So I’m going to talk about systematic errors in the daily IGS terrestrial fram solutions, which as you know, are the IGS input to the ITRF.

2. Outline GNSS-derived geocenter motion GNSS-derived terrestrial scale
GNSS station position time series
Draconitics
Fortnightly signals
Equipment-related biases
Bonus:
ITRF2014 post-seismic deformation models: 2.5 years after
Comparison of I/D/JTRF2014 polar motion series with geophysical excitation data
We’ll first review the geocenter motion time series derived from these IGS daily solutions, then their scale. And then I’ll talk about errors in the individual station position time series. And finally, I have two topics that could be of interest for this workshop, but I’m not sure I’ll have time to cover them in 15 minutes:
The predictive quality of the ITRF2014 post-seismic deformation models after 2 years and a half of extrapolation,
And a comparison of the I, D and JTRF2014 polar motion series with geophysical data.

3. GNSS-derived geocenter motion
So let’s start with the geocenter motion time series derived from the daily solutions of the IGS 2nd reprocessing, repro2. So the cyan dots in the left plots show the daily X, Y and Z geocenter coordinates from the daily IGS repro2 solutions. The blue curves are smoothed versions of the same time series that emphasize their annual and lower frequency parts. And finally, the red curves are similarly smoothed geocenter time series derived from the SLR contribution to ITRF2014. And the right plots show amplitude spectra of the same IGS repro2 and SLR geocenter time series.
The grey vertical lines in these plots show harmonics at the GPS draconitic year, and it’s clear that the 3 components of the repro2 geocenter time series present spurious signals at these harmonics, which explain quite a lot of their variability, especially for the Z component. Now if we look at the annual signals on the left, we can see that they’re pretty much in phase with SLR in X and Y, although the amplitude of the IGS annual signal is globally underestimated along X and overestimated along Y, compared to SLR, and also shows clear variations with time. For the Z component, we can also see such variations of the amplitue of the annual signal, and what’s most striking there, is that the IGS annual signal is mostly out of phase SLR, which might be explained by alternatively constructive and destructive interferences between the real annual geocenter motion and a spurious draconitic signal.
― Daily IGS repro2 combined geocenter coordinates
― Smoothed repro2 combined geocenter coordinates
― Smoothed geocenter coordinates derived from the SLR contribution to ITRF2014
Amplitude spectra of the: ― IGS repro2
― SLR-derived geocenter time series shown in the left plots

4. GNSS-derived geocenter motion
Offset at [mm]
Rate [mm/yr]
Annual amp [mm]
Annual phi [deg]
Semi-ann amp [mm]
Semi-ann phi [deg]
WRMS [mm]
XGC
1.6
0.28
1.5 / 2.6
41 / 48
0.9 / 0.8
271 / 282
3.5 / 2.9
YGC
2.6
-0.40
3.6 / 2.8
310 / 320
0.2 / 0.3
336 / 159
ZGC
7.0
-0.18
3.8 / 5.9
181 / 26
1.3 / 1.4
223 / 202
6.4 / 5.2
Results from trend + annual + semi-annual fits to: ― the repro2 combined geocenter time series
― geocenter time series derived from the SLR contribution to ITRF2014
Non-negligible offsets & rates wrt ITRF2008 (inter-AC agreement at the level of 3 mm; 0.3 mm/yr)
Annual geocenter motion:
Underestimated along X
Overestimated along Y
Out-of-phase with SLR along Z
Obvious contamination by draconitic signals
OK, so the amplitudes and phases of the annual and semi-annual signals from the previous slide are summarized in this table, in which I also reported the offsets and rates of the IGS repro2 geocenter time series with respect to ITRF2008. And we can see that they’re not negligible, with offsets of several mm, up to 7 mm in Z, and rates of several tenths of a millimeter per year.

5. GNSS-derived geocenter motion
GNSS still far from contributing to the definition of the ITRF origin
Why?
Nearly perfect correlation between geocenter coordinates and satellite & station clock offsets + tropospheric parameters + network shape in standard GNSS analyses
Modelling errors (most likely orbit modelling errors)
Prospects
Satellite clock modelling (Galileo?)
Inclusion of LEOs in GNSS analyses
Orbit modelling improvements
So to summarize, I’d say that the quality of GNSS-derived geocenter motion still leaves too much to be desired to consider a contribution of GNSS to the definition of the ITRF origin. But why is it so? And how could we improve things?
Well, the problem is actually due to the combination of two things. First, there’s the fact that in a standard GNSS analyis, there exists a nearly perfect correlation between geocenter coordinates on one hand, and a bunch of other parameters on the other hand, including satellite & station clock offsets, tropospheric parameters and the shape of the ground station network. And this nearly perfect correlation makes geocenter coordinates prone to contamination by modelling errors, and in that case, most likely orbit modelling errors.
So how could we improve the situation? Well, one possible way to break this correlation would be to model satellite clock offsets by other means than independent epoch-wise offsets. And this becomes possible with the ultra-stable hydrogen maser clocks on board the Galileo satellites, although there are still quite important orbit modelling errors to be solved for Galileo at the moment. But maybe one day, satellite clock modeling could help for the GNSS geocenter determination. Another possibility would be the inclusion of low Earth orbiters in GNSS analyses, because have a receiver clock that’s not attached to the ground also helps to reduce that correlation. And there has been several such studies by JPL, where we can see a clear influence of including a LEO on GNSS geocenter coordinates. But the results are still not very satisfactory in my opinion. And finally, of course, we need to improve the modelling of the GNSS satellite orbits.

6. GNSS-derived terrestrial scale
Mean scale (determined by igsyy.atx satellite z-PCOs):
igs08.atx: –0.3 ppb bias wrt ITRF2008 scale due to recent orbit modeling changes
igs14.atx: coincides with ITRF2014 scale at epoch
Scale rate (determined by the use of constant satellite z-PCOs, i.e. ”intrinsic GNSS scale rate”):
closer to ITRF2008 scale rate (–0.004 ppb/yr) than to ITRF2014 scale rate ( ppb/yr)
Non-linear scale variations:
Similar seasonal variations (i.e. network effect) with IGb08/igs08.atx and IGS14/igs14.atx
Non-linear, non-seasonal variations less scattered with igs14.atx thanks to improved z-PCOs, esp. for recently launched satellites
Ok, so let’s now look at the scale of the daily IGS solution. So the cyan dots in the top plot here are scale factors estimated between the daily repro2, then operational solutions and IGb08. The straight blue line is a linear fit. The other one a linear + annual + semi-annual fit. And the red dots are the residuals of the fit.
For the bottom figure, I generated special daily combined solutions, in which I fixed the satellite phase center offsets to their igs14.atx values instead of their igs08.atx values, so that these special daily solutions have their scale determined by the igs14.atx satellite z-PCOs. And I compared their scale to the IGS14 reference frame.
So what can we see? First, the mean scale, which is determined is both case by the mean value of the adopted satellite phase center offsets, and is therefore purely conventional. In the top figure, there’s an offset of about -0.3 ppb between repro2 and the ITRF2008 scale, which is well explained by recently introduced orbit modeling changes. While the offset in the bottom plot is approximately zero at epoch 2010, which is expected given the way in which we chose to define the igs14.atx satellite z-PCOs.
Now if we look at the scale rate, which is in both cases defined by the same assumption that the GNSS satellite phase center offsets are constant with time. That’s what was called the intrinsic GNSS scale rate by Xavier Collilieux and Ralf Schmid in their 2013 GPS Solutions paper. So we can see that this intrinsic GNSS scale rate is clearly closer to the ITRF2008 scale rate than to the ITRF2014 scale rate, which is annoying, but that’s how things are.
Then the seasonal variations are pretty much the same in both cases, and are due to the aliasing of the seasonal deformations of the station network into the daily estimated scale factors.
And finally, what’s interesting to see here is that the non-linear, non-seasonal scale variations of the IGS solutions are less scattered when using the igs14 satellite phase center offsets, especially in recent years. And that’s because we have more precise satellite phase center offset values in igs14.atx, especially for recently launched satellites.
Cyan: Scale factors between IGb08/igs08.atx- (resp. IGS14/igs14.atx-) based daily solutions and IGb08 (resp. IGS14) Blue: linear trend [+ annual and semi-annual signals] Red: residuals of the fits, shifted by -1.5 ppb

7. GNSS-derived terrestrial scale
The 0.03 ppb/yr scale rate difference between ITRF2014 and ITRF2008 clearly shows up when confronting the IGS daily solutions with both RFs.
The “intrinsic GNSS scale rate” based on constant z-PCO values is closer to the ITRF2008 than to the ITRF2014 scale rate.
The scale of igs14.atx-based GNSS solutions matches the ITRF2014 scale at epoch , but progressively diverges with time.
Schematic representation of the scale and scale rate differences between ITRF2008, ITRF2014 and GNSS solutions based on either igs08.atx or igs14.atx

8. GNSS-derived terrestrial scale
Contribution of GNSS to future ITRF mean scale might become possible with the availability of phase center calibrations for the Galileo satellites.
Contribution of GNSS to future ITRF scale rate could be considered,
but requires careful (re-)assessment of satellite z-PCO stability, hence of the precision of the intrinsic GNSS scale rate.
≈ 5 mm/yr & 0.25 mm/yr, respectively, according to Collilieux & Schmid (2013)
GNSS may also contribute to future ITRF non-linear (e.g. seasonal) scale variations,
once non-linear motion discrepancies at co-location sites are better understood.
Now to conclude with the scale part, and possible future contributions of GNSS to the ITRF scale definition.
A contribution of GNSS to the definition of the ITRF mean scale is currently not possible because the IGS uses purely conventional satellite phase center offset values, and actually relies on the ITRF scale to determine those values. But, last December, ESA released metadata for some Galileo satellites, including their phase center calibrations. And this could be the base for a contribution of GNSS to the scale of future ITRFs.
Regarding the scale rate, a contribution of GNSS to the ITRF scale rate could already be considered, based on the assumption that satellite phase center offsets are constant with time. But I think this would first need a careful re-assessment of this assumption, hence of the precision of the intrinsic GNSS scale rate.
And finally, if we move to a non-linear representation of the ITRF station coordinates in the future, for instance including seasonal displacements, then I think that GNSS could contribute to the definition of such ITRF non-linear scale variations. But this will be possible only after the discrepancies between the non-linear motions observed at co-location sites are better understood. And I refer to Zuheir’s presentation later this week for that subject.

9. Station position time series: spectra
Now let’s move to individual station position time series, and first have a look at their averaged spectra. So to generate these figures, I computed long-term linear frames based on the daily repro2 solutions of 9 different analysis centers. Then I computed Lomb-Scargle periodograms of the station position residual time series from each of those stackings, and I computed an averaged spectrum in East, North and Up, for each AC.
So we can observe this well-known background noise in GNSS station position time series which follows a power-law at low frequencies, and becomes white at high frequencies. Then their are clear peaks at the annual and semi-annual frequencies. As geophysically expected, the annual peaks are well centered around the annual frequency. But when looking closely at the semi-annual peaks, it seems that there’s actually more power at the 2nd draconitic harmonic than at the real semi-annual period. Then we can observe lots of higher draconitic harmonics that we would like not to see here. And a bit further in the spectra, there are a couple of lines that clearly appear for most ACs in the fortnightly band, and I’ll come back on that later.
Averaged Lomb-Scargle periodograms of the station position residual time series from the long-term stackings of the AC repro2 contributions
― results from white + power-law noise (i.e., a + b/fα) fits • crossover frequencies between white & power-law noises

10. Station position time series: draconitics
Strong spatial correlations for low harmonics
→ Main orbit-related source
Maps generally similar amongst ACs up to the 4th harmonic
→ Possibly common modeling errors
But let’s first come back to the draconitics. To better understand the source of these spurious signals, I fitted draconitic signals to the station position residual time series of the AC-specific long-term stackings, and I drew amplitude/phase maps of these signals. So here’s the example of the signals estimated at the 2nd draconitic harmonic, in the East component. The sizes of the dots represent the amplitudes of the draconitic signals, and the colors the phases. So two observations can be made here. First, those maps show very clear large-scale spatial correlations, which seems to imply a main orbit-related source, rather than something station-specific. And the second thing is that the maps look pretty much the same for all ACs, which seems to indicate common modelling errors. And that’s quite puzzling…

11. Station position time series: 14d signals
Now here’s a zoom on the fortnighly band of the square-root of the spectra that I showed two slides ago. So we can observed 3 different peaks there, and the first one is days, which is the alias of the M2 tide through a 24h sampling. So except for MIT and ULR, for which I think there’s a specific issue in the GAMIT software, we can see that those peaks are only present in the horizontal station position residuals of all ACs, which seems consistent with a rotational origin, for instance errors in the IERS subdaily EOP tide model. So I hope that JPL or ESA will see those peaks disappear in their tests of the newly available EOP tide models.
Square-root of the averaged Lomb-Scargle periodograms shown in slide 9: zoom on the fortnightly band
14.76 days (M2 alias):
Spectral peaks visible in the horizontal residuals of all ACs
 consistent with a rotational origin (e.g. error in the IERS subdaily EOP tide model)
More pronounced for GFZ/GTZ (?) and MIT/ULR (absence of ocean tide geopotential corrections in GAMIT orbit integrator?)

12. Station position time series: 14d signals
The situation is pretty much the same for the second peak at days, which is the alias of the O1 tide, except that the peaks are in overall smaller than for the M2 alias.
Square-root of the averaged Lomb-Scargle periodograms shown in slide 9: zoom on the fortnightly band
14.19 days (O1 alias):
Small spectral peaks discernable in the horizontal residuals of some ACs
 consistent with a rotational origin (e.g. error in the IERS subdaily EOP tide model)
More pronounced for MIT/ULR (absence of ocean tide geopotential corrections in GAMIT orbit integrator?)

13. Station position time series: 14d signals
And finally, the peaks to the right actually cover two different tide frequencies: the Mf tide at days, and a smaller companion tide at days. In this case, the lines are visible in all 3 components of the residuals. And something quite puzzling here is that there seems to be consistently more power at days, although this is the period of a minor tide. Strange…
Square-root of the averaged Lomb-Scargle periodograms shown in slide 9: zoom on the fortnightly band
13.66 days (Mf tide) / days (75,565 tide):
Both lines visible in (almost) all 3 components of all ACs
13.63 day lines generally larger, although they correspond to a minor tide!?
Much lower peaks for JPL: low-frequency error absorbed by JPL’s stochastic orbit parameters?
Vertical peaks larger for MIT/ULR

14. Equipment-related biases
Biases in IGS station positions are evidenced by equipment-related discontinuities:
What is the magnitude of equipment-related discontinuities in the IGS repro2 station position time series?
What does it tell about the accuracy of IGS station positions?
Do antenna change discontinuities get smaller with igs14.atx?
antenna change
receiver changes
antenna changes
Distribution of identified discontinuity causes in IGS repro2 station position time series
So this is all I had to say about periodic errors in IGS station position time series. My next topic is equipment-related biases in IGS station positions, but I don’t know if I have enough time left??
So biases in IGS station positions are evident because of equipment-related discontinuities. If there’s a jump in your time series when you change the antenna, or the receiver, it’s either because the position before, or after, or both are biased. So the questions I wanted to address here are:
MAS1 (Maspalomas, Spain) Cyan: Detrended repro2 station position time series Blue: piecewise linear + annual + semi-annual fit

15. Equipment-related biases
What is the magnitude of equipment-related discontinuities in the IGS repro2 station position time series?
Histograms and empirical cumulative distribution functions (CDF) of 985 equipment-related discontinuities estimated during the long-term stacking of the daily IGS repro2 solutions
The answer to the first question is in these histograms, which show the distributions of the sizes of equipment-related discontinuities estimated during the long-term stacking of the daily repro2 solutions. And the curves on the right are the cumulative distribution functions. So this is not normally distributed.

16. Equipment-related biases
What is the magnitude of equipment-related discontinuities in the IGS repro2 station position time series?
Histograms and empirical cumulative distribution functions (CDF) of 985 equipment-related discontinuities estimated during the long-term stacking of the daily IGS repro2 solutions
Best fitting Student’s t-distributions:
East: ν = 2.02; σ = 2.85 mm
North: ν = 2.10; σ = 2.74 mm
Up: ν = 1.88; σ = 7.45 mm
→ Discontinuity sizes aren’t normally distributed, but show "heavy tails".
But you can get very nice fits using Student’s t-distributions. This is a sort of heavy-tailed distribution, used to model processes where values far from the mean occur more frequently than in the normal case.
I tried to identify different sub-categories of equipment changes, with different sub-distributions, but without success. So I’ll stick to overall distributions hereafter.
Tried to identify different sub-categories:
antenna changes / receiver changes,
uncalibrated radomes,
problematic antenna types… but without success.

17. Equipment-related biases
What does it tell about the accuracy of IGS station positions?
Discontinuity sizes = station position biases – station position biases ? ? = * ? ? = *
Second question: what does it tell us about the accuracy of IGS station positions? Well the size of a discontinuity is basically the difference between the position bias before and the position bias after, which means, in terms of probability distribution, that the distribution of the discontinuities is the convolution of the bias distribution with itself. So which distribution, convolved with itself, gives a Student t distribution? I don’t have a theoretical answer to that. ? ? = *

18. Equipment-related biases
What does it tell about the accuracy of IGS station positions?
Student’s t ≈ Student’s t * Student’s t ≈ * ≈ *
But it turns out that numerically, another Student t distribution works not perfectly, but very very very well. ≈ *

19. Equipment-related biases
What does it tell about the accuracy of IGS station positions?
Equipment-related biases in IGS station positions follow heavy-tailed distributions (modelled here as Student’s t-distributions).
→ Confidence intervals grow « fast » with confidence levels.
Student's t params
confidence intervals [mm] ν
σ [mm]
90%
95%
99%
East
1.75
1.61
± 5.2
± 8.0
± 20.4
North
1.80
1.56
± 4.9
± 7.5
± 18.7 Up
1.66
4.17
± 14.1
± 22.0
± 59.0
From which we can conclude that the equipment-related biases in IGS station positions follow heavy-tailed distributions themselves. So you can see the distribution parameters I got in this table. The degree of freedom nu is above 1, but below 2, which means that the mean is well-defined – actually I assumed it was zero – but that the variance is undefined because their are many values far from the mean. In terms of confidence intervals, it means that you need huge intervals to capture most of the biases. For instance, to capture 99% of the station position biases, you need intervals as large as +/- 2 cm in horizontal and 6 cm in vertical.

20. Equipment-related biases
Do antenna change discontinuities get smaller with igs14.atx ?
Subset of 346 antenna change discontinuities involving at least one antenna with updated calibration from igs08 to igs14.atx
Apply PPP-derived position corrections to compute expected discontinuity sizes with igs14.atx
→ « igs08.atx to igs14.atx » position corrections have marginal impact on discontinuity sizes.
→ Errors in type-mean antenna calibrations do not seem to be a major contributor to current IGS station position biases.
igs08.atx (before correction)
igs14.atx (after correction)
And the last question was: do antenna change discontinuities get smaller with igs14.atx? To answer it, I chose a subset of antenna change discontinuities involving at least one antenna with an updated calibration in igs14.atx, of which you can see the distributions on the left. Then I used PPP-derived postion corrections to compute what would have been the discontinuity sizes using igs14.atx instead of igs08.atx. And well, you can’t really see a change in the distributions from left to right, which means that the antenna calibration updates have marginal impact on discontinuity sizes, hence that errors in our type-mean calibrations do not seem to be a major contribution to current IGS station position biases. And I mean errors within the type mean calibrations, not errors due to using type-mean calibrations instead of individual calibrations.

21. ITRF2014 PSD models: 2.5 years after
Compare IGS daily station position time series with propagated ITRF2014 coordinates: two examples
— daily IGS station position time series
— propagated ITRF coordinates
— ± 3σ confidence intervals end of ITRF input data
So the question I wanted to address here is: how well do the ITRF2014 post-seismic deformation models predict current station positions, after 2 and a half years of extrapolation. You can see two example here. In blue the daily IGS station position time series. And in red, the propagated ITRF2014 coordinates. The black vertical lines mark the end of the ITRF2014 input data, from which extrapolation goes on and on. In Fairbanks, the ITRF2014 model does a good job. But you can see in the East component of Faleolo that the model is diverging from the actual time series.

22. ITRF2014 PSD models: 2.5 years after
PSD model prediction errors
depend on each individual case, but generally on:
age of the last earthquake,
amplitude of the post-seismic deformations,
(presence of other non-linear deformations)
are mostly comparable with prediction errors of classical linear ITRF2014 coordinates:
end of ITRF2014 input data
Differences between IGS daily station position time series and propagated ITRF2014 coordinates
— Stations without PSD model
— Stations with PSD model

23. ITRF2014 PSD models: 2.5 years after
PSD model prediction errors
depend on each individual case, but generally on:
age of the last earthquake,
amplitude of the post-seismic deformations,
(presence of other non-linear deformations)
are mostly comparable with prediction errors of classical linear ITRF2014 coordinates,
especially for IGS14 stations (whose selection took the PSD model formal errors into account):
end of ITRF2014 input data
And this is especially true for the IGS14 stations whose selection took the formal errors of the PSD Models into account, precisely to avoid fast-growing prediction errors. So in short: if you use IGS14 stations with PSD models, you’re fine for now. But if you use other ITRF2014 stations with PSD models, be more careful.
Differences between IGS daily station position time series and propagated ITRF2014 coordinates
— IGS14 stations without PSD model
— IGS14 stations with PSD model

24. Inter-comparison of I/D/JTRF2014 polar motion series
Detrended polar motion differences
Power spectra of polar motion differences

25. Inter-comparison of I/D/JTRF2014 polar motion series
Background noise in PM differences approximately flicker
JTRF yp series shows spectral peaks at cpy (7d) and harmonics, due to using weekly ig2 solutions as input
JTRF yp [semi-]annual signals different from both others, due to filter approach used
JTRF different from both other series at low frequencies (<5 cpy), due to filter approach used
DTRF slightly different from both other series at mid frequencies (5-70 cpy)
ITRF slightly different from both other series at high frequencies (> 70 cpy)
[µas]
DTRF – ITRF
JTRF – ITRF
JTRF - DTRF xp yp
total RMS
9.904
11.791
16.839
21.676
15.262
17.252 d
5.637
3.959
7.074
14.031
6.238
11.330 d
2.390
1.399
1.535
5.366
2.183
4.056 d
0.090
0.130
0.172
1.078
0.237
1.027 d
0.122
0.056
0.159
0.599
0.088
0.551 d
0.132
0.042
0.041
0.436
0.126
0.427
< 5 cpy
5.499
9.542
14.552
14.939
12.823
11.197
5-70 cpy
4.551
4.531
3.227
3.302
4.423
4.491
> 70 cpy
3.099
3.127
2.977
2.888
2.267
2.398
RMS of the total pairwise polar motion difference series, of the periodic signals fitted to each difference series, and of their low-, mid- and high-frequency residuals

26. I/D/JTRF2014 polar motion series vs. geophysical excitation data
Convert polar motion series into “geodetic excitation” series by the “gain adjustment” method of Chen et al. (2013)
Compare geodetic excitation series with geophysical (LDC) excitation series derived from GRACE, SLR, plus atmospheric, oceanic and hydrological models (Chen et al., 2013)
Differences between “X – LDC” and “ITRF – LDC” excitation difference power spectra, offset by multiples of 50 mas²/d² for clarity
Negative values mean: closer to LDC than ITRF
Positive values mean: further from LDC than ITRF

27. I/D/JTRF2014 polar motion series vs. geophysical excitation data[]
ig2 – LDC
IG2 – LDC
IG2_s – LDC
ITRF – LDC
DTRF – LDC
JTRF – LDC
[mas/d] χ1 χ2
corr coeff
0.971
0.984
total RMS
8.072
8.101
8.100
8.073
8.075
8.112
8.081
8.108 d
1.223
0.513
1.220
0.516
1.218
0.517
0.515
1.221
0.526 d
1.149
1.039
1.041
1.147
1.037
1.151
1.042
1.158 d
1.408
1.046
1.404
1.043
1.401 d
0.212
0.110
0.211
0.116
0.219
0.114
0.235
0.112
< 5 cpy
3.421
3.134
3.422
3.135
3.419
5-70 cpy
4.713
4.784
4.712
4.787
4.790
4.715
> 70 cpy
5.132
5.521
5.135
5.517
5.136
5.519
5.141
5.532
5.145
5.528
Correlation coefficients between geodetic and LDC excitation series; RMS of the “geodetic – LDC” excitation difference series, of the periodic signals fitted to each difference series, and of their low-, mid- and high-frequency residuals
Total RMS significantly larger for DTRF (χ2) and JTRF
ITRF & IG2_s globally show best agreement with LDC at [semi-]annual periods; JTRF globally worst
ITRF shows better agreement with LDC than DTRF & JTRF at high frequencies

28. Thank you for your attention!