1. How to constrain the origin, orientation and scale
of a daily/weekly TRF
without perturbing the loading signal?
X. Collilieux, Z. Altamimi, T. van Dam, J. Ray

2. Outline Goal: Outline:
INPUT : time series of GPS SINEX solutions
OUTPUT : Position time series with respect to ITRF frame.
i.e. apparent geocenter motion removed
Outline:
How to constrain the origin, orientation and scale of a daily/weekly TRF without perturbing the loading signal?
Application to synthetic data. What is the best approach?
Application to Real data. Which strategy improves the fit with respect to a loading model?
Recommendations

3. I Problem : aliasing What is loading aliasing ? Station displacements
t+Δt
Estimated translation
Weekly
solution
Apparent translation
Related to displacement
(Aliasing!)
7-parameter
Transformation
translation
rotation
scale
Frame definition
ITRF
solution

4. I Problem : suggestions
i) Compute the transformation parameters (translation, rotations and/or scale) over a well-distributed network of stations
ii) Downweight height. (σup ≈ 10 . σnorth). In the standard case, (σup ≈ 3 . σnorth)
iii) Use a loading model conjointly to estimate the transformation parameters (w or w/o scale). Two-step approach
Input position of station i
Loading displacement
Estimated 7-parameters .
More similar shape if the loading model is correct

5. II Application to synthetic data : synthetic data
Loading model
Atmopshere (6h) NCEP
Ocean (12h) ECCO
Hydrology (1 month) GLDAS
Synthetic SINEX files
X(t) = XITRF2008(t_0) + VITRF2008(t-t_0)
+ dXloading
+ Σdiscont
+ dX_spatial
Ex: synthetic data
GRAZ

6. II Application to synthetic data : results
RMS to the true value is not the right statistic to investigate network effect
since noise amplitude is higher than the aliasing effect magnitude (expect if scale is estimated)
Two criteria :
GPS residual synth A
Annual signal is the most relevant criteria
s(t) = A cos ωt + B sin ωt
Loading model A
GPS residual synth B
Loading model B

7. II Application to synthetic data : results
7-parameters Whole network – standard strategy
Horizontal and height signals are badly recovered
Cosinus
Sinus

8. II Application to synthetic data : results
7-parameters Whole network estimated using the loading model
If scale is estimated, the use of the loading model is the only method that could handle scale problem!
Cosinus
Sinus
This method is of course the best with synthetic data

9. II Application to synthetic data : results
6-parameters Whole network (wihtout scale)
Horizontal still badly recovered but height is fine
Cosinus
Sinus

10. II Application to synthetic data : results
6-parameters whole network downweighted height (σup ≈ 10 . σnorth)
Better agreement in horizontal. Inphase and Outphase North term are slightly biased
Cosinus
Diagonal weight
matrix
Sinus

11. II Application to synthetic data : results
6-parameters whole network downweighted height (σup ≈ 10 . σnorth)
Better agreement in horizontal. Outphase North term is slightly biased
NB: Discrepancies increase if height is deweighted too much
Cosinus
Full weight matrix
to Preserve correlations
Sinus

12. II Application to synthetic data : results
6-parameters sub-network (wihtout scale)
Better agreement in horizontal. Inphase North term is slightly biased
Cosinus
Sinus

13. II Application to synthetic data : results
6-parameters sub-network downweighted height (σup ≈ 5 . σnorth)
Cosinus
Full weight matrix
to Preserve correlations
Sinus

14. II Application to synthetic data : conclusion
If the scale factor is stable and loading effects dominate, estimating the scale degrades height positioning except if a loading model is used to model station motion
Deweighting height has a positive effect but do not deweight height too strongly. 10 factor in standard deviation is sufficient
Using correlation or not when downweighting height gives a significantly different result
A well distributed network sub-network improves drastically the fit
The network influence is negligeable here if height is downweighted with 10 factor
Application to real data:
Is the scale necessary?
We retain 3 solutions to use with real data
sub-network 6 parameters
use loading model to estimate the 6 parameters
sub-network + downweighting height 6 parameters

15. III Application to real data
Do not estimate the scale with reprocessed data?
Loading model
mi1
GPS reprocessed
Scale factor mm
gf1
co1
es1
Difference with the loading model
Cause of these variations:
Deficiency in the loading model
Draconitic period (harmonics detected in the power spectra)
Thermal effects mm
Collilieux X., Métivier L., Altamimi Z., Van Dam T., Ray, J., Quality assessment of GPS reprocessed Terrestrial Reference Frame, GPS solutions, submitted

16. III Application to real data
MI1 GPS station position annual signals vs. Loading model annual signals
6-parameters sub-network
Cosinus
Sinus

17. III Application to real data
MI1 GPS station position annual signals vs. Loading model annual signals
6-paramaters loading corrections
Using the loading model to estimate the 6-parameters does not seem to help improving the fit
Cosinus
Sinus

18. III Application to real data
MI1 GPS station position annual signals vs. Loading model annual signals
6-paramaters sub-network downweighted height (σup ≈ 5 . σnorth)
Cosinus
Bias reduction
With respect to the model in the in phase term
But bias increase in the out phase term.
Sinus
Full weight matrix
to Preserve correlations

19. IV Conclusion
Do not estimate the scale or constrain the annual signal in the scale to 0 if you have checked the scale non-linear behavior beforehand
A constraint to nullify scale factor annual signal can be added on the scale estimates but the whole set of TRF is managed in one run
mi1 GPS solution scale factor mm
Exemple : mi1 network
2) 3 solutions with similar performances
Use a uniform sub-network and estimate 6 parameters (no scale)
Use a loading model to estimate the 6 parameters (no scale)
Downweigh height (σup ≈ σnorth) and estimate 6 parameters
3) Differences between GPS and loading model evaluated with those best strategies have WRMS of 0.4, 0.5 and 1.5 mm in East, North and Height.

20. Additional Slides

21. II Application to synthetic data : results
Averaged East, North and height signal over the network
Downweighting height with full covariance weight smooths the averaged north signal
Loading model
Truth
6-parameters
6-parameters subnetwork
6-parameters downweight H 10 full
6-parameters downweight H 10 diagonal

22. II Application to synthetic data : results
Median WRMS(residual vs model) and maximum WRMS(residual vs model)
RMS Annual (In phase & out of phase terms)
Correlation
Mean East, North and Height
East
North Up
Max RMS 2D
Max RMS Up
Load vs. residual
7-param
(0.6, 1.1)
(0.6, 1.3)
(2.0, 3.9)
0.45
1.32
(64%,75%,46%)
6-param
(1.6, 4.1)
0.51
0.57
(64%,73%,94%)
Sub-net 7-param
(0.4, 1.0)
(0.5, 1.3)
(2.0, 4.1)
0.20
1.44
(78%,82%,29%)
Sub-net 6-param
(0.5, 1.2)
(1.5, 4.2)
0.23
0.25
(79%,80%,95%)
Corr. Load 7-param
(0.4, 1.3)
(1.5, 4.1)
0.04
0.21
(95%,92%,94%)
Corr. Load 6-param
0.05
0.19
Deweigh height 7-param
(1.9, 4.0)
0.22
1.29
(48%,37%,50%)
Deweigh height 6-param
(1.5, 3.9)
(2.3, 7.4)
0.28
0.26
0.42
(55%,13%,93%)
(76%,2%,87%)
Deweigh height 6-param/subnet
(69%,14%,94%)

23. III Application to real data
Median WRMS(residual vs model) and maximum WRMS(residual vs model)
RMS Annual (In phase & out of phase terms)
Correlation
Mean East, North and Height
East
North Up
Max RMS 2D
Max RMS Up
Load vs. residual
Sub-net 7-param-annual
(1.5, 5.0)
(1.4, 5.1)
(4.0, 10.3)
0.9
1.6
6-param
(1.6, 4.8)
(1.6, 5.4)
(4.0, 10.0)
1.2
(30%,63%,75%)
Sub-net 6-param
(1.5, 4.9)
(4.1, 10.0)
(25%,60%,76%)
Corr. Load 6-param
(1.6, 4.9)
(1.5, 5.1)
Deweigh height 6-param
(1.4, 4.8)
(4.2, 10.0)
0.8
1.8
(10%,25%,79%)
Deweigh height 6-param/subnet

24. II Application to synthetic data : results
6-parameters Whole network (wihtout scale)
Horizontal still badly recovered but height is fine
Identity weight matrix

25. 7-paramaters sub-network
Horizontal signal is better restored but still height problem

26. 7-paramaters whole network height deweighted

27. II Application to synthetic data : results
6-parameters whole network downweighted height (σup ≈ σnorth)
If height deweigthed, this strategy dominates with respect to the network selection

28. III Application to real data
MI1 GPS station position annual signals vs. Loading model annual signals
6-paramaters sub-network downweighted height (10)
Bias reduction
With respect to the model in the in phase term
But bias increase in the out phase term.

29. I Problem : suggestion i
i) Compute the 6 or 7-parameters using a subset of stations
Selection criteria:
80% of points in the period of data
Discontinuity period with at least 20% of points 1)
First selection 2)
Remove stations in dense areas according to repeatability, annual signal amplitude etc…
Add stations in empty area trying to fullfill as maximum as possible the first selection criteria
Result for GPS mi1 network:

30. I Problem : suggestion ii
ii) Deweigh Height in the 6 or 7-parameter computation
Procedure used in GLOBK sofware (Herring …)
We implemented it following
* Guo et al., « Robust estimation for correlated observations: two local sensivity-based downweighting strategies», JoG 2010
Goal : to preserve the correlations but by increasing uncertainty on height
Varlocal = Λ . Corr .Λ
σ1east
σ2north
σ1up
σ2east
σ2up
. .
Correlation matrix
σ1east
σ2north
α σ1up
σ2east
α σ2up
. .
Λ= diag
Varnewlocal = Λnew . Corr .Λnew
Λnew= diag
We use α such that σup = 10 . σnorth

31. I Problem : suggestion iii
iii) Compute the 6 or 7-parameters using a loading model to mitigate the aliasing effect
Input position of station i
Estimated 7-parameters
Loading displacement .
More similar shape if the loading model is correct
2-step procedure:
Estimate the 6 or 7 parameters using the loading model
Apply the estimated 6 or 7 parameter transformation to input data

32. III Application to real data
BUT if the estimation of the scale is needed,
A constraint can be added to the scale estimates if the whole set of TRF is managed in one run
mi1 GPS solution scale factor
Exemple : mi1 sub-network mm

33. (mm)
Potsdam U
(mm)
Bahrein N