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

Observing geocenter motion with GNSS

2

• 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?

3

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?

4

(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

• 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)

• 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)

6

Mathematical difficulties

7

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

201

coordinate geocenter Z the with correlated most the is which

parametersof ncombinatio linearxx 12

Practical diagnosis

8

« 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

9

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)

2nd order geocenter signature

10

δZgc = 1 cm δXgc = 1 cm

• 2nd issue: collinearity with station parameters– Positions, clock offsets, tropospheric parameters

11

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

12

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:

13

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).

14

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…

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

15

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

16

Zgc collinearity issue in SLR

17

• 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