Observing geocenter motion with GNSS
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Insensitivity of GNSS to geocenter motion through the network shift approach
Paul Rebischung, Zuheir Altamimi, Tim Springer
AGU Fall Meeting 2013, San Francisco, December 9-13, 20131
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Observing geocenter motion with GNSS
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• Degree-1 deformation approach (Blewitt et al., 2001):– Based on the fact that loading-induced geocenter motion is
accompanied by deformations of the Earth’s crust.– Gives satisfying results.– But can only sense non-secular, loading-induced geocenter motion.
• Network shift approach:– Weekly AC solutions theoretically CM-centered.– AC → ITRF translations should reflect geocenter motion.– But unlike SLR, GNSS have so far not proven able to reliably observe
geocenter motion through the network shift approach.– Why?
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Example of network shift results
– The translations of the different IGS ACs show various features.– But none properly senses the X & Z components of geocenter motion.
— SLR (smoothed)— GPS (ESA)— GPS (ESA, smoothed)
Annual signal missed
Spurious peaks at
harmonics of 1.04 cpy
Why?
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(Multi-) Collinearity
• Consider the linear regression model:y = Ax + v = Σ Aixi + v
– Ai = ∂y / ∂xi = « signature » of xi on the observations
• Collinearity = existence of quasi-dependenciesamong the Ai’s
• Consequences:– Some (linear combinations of) parameters cannot be reliably inferred,– are extremely sensitive to any modeling or observation error,– have large formal errors.
observations parameters residuals
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• Is the estimation of a particularparameter xi subject tocollinearity issues?– θi = angle between Ai and the hyper-
plane Ki containing all other Aj’s
– VIFi = 1 / sin²θi
– θi = π/2 (VIFi = 1) : xi is uncorrelated with any other parameter.
– θi → 0 (VIFi → ∞) : xi tends to be indistinguishable from the other parameters.
• If yes, why?– The orthogonal projection αi of Ai on Ki corresponds to the linear
combination of the xj’s which is the most correlated with xi. 5
Variance inflation factor (VIF)
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• Geocenter coordinates are not explicitly estimated parameters.– They are implicitly realized through station coordinates.→ Extend previous notions to such « implicit parameters ».
• There are perfect orientation singularities.→ Extend previous notions so as to handle singularities supplemented
by minimal constraints.
• The whole normal matrix is not available.– Clock parameters are either reduced or annihilated by forming
double-differenced observations.→ Practical collinearity diagnosis (next slide)
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Mathematical difficulties
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0)
1)
2)
– Simulate « perfect » observationsx0 → y0
– Introduce a 1 cm error on the Z geocenter coordinate:x1 = x0 + [0, 0, 0.01, …, 0, 0, 0.01, 0, …0]T
– Re-compute observations → y1
– Solve the constrained LSQ problem:
(How can the introduced geocenter error be compensated / absorbed by the other parameters?)
→ x2, y2
error geocenter the keeping whileyyminimize 02
VIFerror remainingerrorintroduced
yy
yy2
02
201
error introducederror absorbed""
yy
yyyy2
01
202
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coordinate geocenter Z the with correlated most the is which
parametersof ncombinatio linearxx 12
Practical diagnosis
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« Signature » of a geocenter shift
• From the satellite point of view:
GPS LAGEOS
δZgc = 1 cm
δXgc = 1 cm
· impact on a particular observation— epoch mean impact
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1st issue: satellite clock offsets
• Satellite clocks ↔ constant per epoch and satellite→ The epoch mean geocenter signature is 100% absorbable by
(indistinguishable from) the satellite clock offsets.→ The GNSS geocenter determination can only rely on a 2nd order signature.
• In case of SLR :– The epoch mean signatures of Xgc and Ygc are directly observable.
→ No collinearity issue for Xgc and Ygc (VIF ≈ 1)
– The epoch mean signature of Zgc is absorbable by the satellite osculating elements.→ Slight collinearity issue for Zgc (VIF ≈ 9)
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2nd order geocenter signature
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δZgc = 1 cm δXgc = 1 cm
• 2nd issue: collinearity with station parameters– Positions, clock offsets, tropospheric parameters
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So what’s left?
• δXgc = 1 cm:
From the point of view of a satellite… …and of a station
• VIF > 2000 for the 3 geocenter coordinates!(More than 99.96% of the introduced signal could be absorbed.)
· impact on an observation, before compensation · impact on an observation, after compensation
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Role of the empirical accelerations– The insensitivity of GNSS to geocenter
motion is mostly due to the simultaneous estimation of clock offsets and tropospheric parameters.
– The ECOM empirical accelerations only slightly increase the collinearity of theZ geocenter coordinate.
– This increase is due to the simultaneous estimation of D0, BC and BS:
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Conclusions (1/2)
• Current GNSS are barely sensitive to geocenter motion.– The 3 geocenter coordinates are extremely collinear with other GNSS
parameters, especially satellite clock offsets and all station parameters.
– Their VIFs are huge (at the same level as for the terrestrial scale when the satellite z-PCOs are estimated).
– The GNSS geocenter determination can only rely on a tiny 3rd order signal.
– Other parameters not considered here (unfixed ambiguities) probably worsen things even more (cf. GLONASS).
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Conclusions (2/2)
• The empirical satellite accelerations do not have a predominant role.– Contradicts Meindl et al. (2013)’s conclusions
• What can be done?– Reduce collinearity issues
(highly stable satellite clocks?)
– Reduce modeling errors(radiation pressure, higher-order ionosphere…)
– Continue to rely on SLR…
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Thanks for your attention!
For more:
Rebischung P, Altamimi Z, Springer T (2013) A collinearity diagnosis of the GNSS geocenter determination. Journal of Geodesy. DOI: 10.1007/s00190-013-0669-5
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Parameter response to δZgc = 1 cm
Network distortion:
→ Explains the significant correlations between origin & degree-1 deformations observed in the IGS AC solutions
ZWDs:(as a function of time, for each station)
And their means:(as a function of latitude)
Station clock offsets:(as a function of time, for each station)
And their means:(as a function of latitude)
Tropo gradients:(as a function of latitude)
N/S gradients
W/E gradients
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Zgc collinearity issue in SLR
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• δZgc = 1 cm:
– This slight collinearity issue probably contributes to the lower qualityof the Z component of SLR-derived geocenter motion.
– To be further investigated…
gcgc
gc
2
δZa
siniδe
2π
-ω:0e
δZasinωsini
δe
δZcosωsiniae
e1δM
:0e
· impact on an observation, before compensation · impact on an observation, after compensation— radial orbit difference
– The epoch mean signature of δZgc is compensated by a periodic change of the orbit radius obtained through:
→ VIF ≈ 9.0