impact of non-tidal loading applied at different levels in

Deutsches Geodätisches Forschungsinstitut (DGFI-TUM)Technische Universität München

M. Glomsda, M. Bloßfeld, M. Seitz, M. Gerstl, D. Angermann, F. Seitz

Impact of non-tidal loading applied at different levels in VLBI analysis

Deutsches Geodätisches Forschungsinstitut (DGFI-TUM)

Technische Universität München

Frontiers of Geodetic Science

Stuttgart, Germany, September 18, 2019

Deutsches Geodätisches Forschungsinstitut (DGFI-TUM) | Technische Universität München 2

Outline

Site displacements induced by

tidal loading

non-tidal loading

Application of site displacements at

observation level

solution level

normal equation level

Impact on estimated parameters

station coordinates

scale

Summary


Outline


tidal loading

non-tidal loading


observation level

solution level



station coordinates

scale

Summary


Tidal loading effects

Periodic short term displacements of sites on the Earth‘s crust are, among others, generated by…

… the gravitational forces of Sun and Moon:

solid Earth tide,

tidal atmospherical loading,

tidal ocean loading.

… the centrifugal forces of Earth rotation:

pole tide,

ocean pole tide.

For long term reference frames, instantaneous site positions are regularized by subtracting these

tidal (loading) effects and other displacements.


Non-tidal loading effects

There are other short term site displacements caused by rather local and irregular changes in

atmospheric pressure,

the mass redistribution of ocean water, or

the mass redistribution of land water (hydrology).

These non-tidal loading effects are generally not included in the regularization step and official

geodetic analyses.

Exception: contributions to the International VLBI Service for Geodesy and Astrometry (IVS) should

contain non-tidal atmospherical loading corrections.

Given recent advances in measuring and modelling, is it worthwhile adding non-tidal loading effects in

the analysis of VLBI observations?


Non-tidal loading: site displacements (1)

Usually computed from pressure anomalies with the Green‘s function approach,

based on Farrell (1972), formulas according to Petrov and Boy (2004):

vertical displacement: 𝑑𝑣 𝑝, 𝑡 = ∆𝑃 𝑞, 𝑡 𝐺𝑣 𝜓 cos 𝜑 𝑑𝜆 𝑑𝜑

𝐺𝑣 𝜓 =𝐺𝑅

𝑔2 𝑛=0∞ ℎ𝑛

′ 𝑃𝑛(cos(𝜓))

horizontal displacement: 𝑑ℎ 𝑝, 𝑡 = 𝑛 𝑝, 𝑞 ∆𝑃 𝑞, 𝑡 𝐺ℎ 𝜓 cos 𝜑 𝑑𝜆 𝑑𝜑

𝐺ℎ 𝜓 = −𝐺𝑅

𝑔2 𝑛=1∞ 𝑙𝑛

′ 𝜕𝑃𝑛(cos 𝜓 )

𝜕𝜓

𝑝 = site coordinates 𝜓 = spherical distance of 𝑝 and 𝑞 ℎ𝑛′ , 𝑙𝑛

′ = load Love numbers

𝑞 = pressure source coordinates 𝜑 = geocentric latitude for 𝑞 R = mean Earth’s radius

Δ𝑃 = pressure anomaly λ = longitude for 𝑞 𝑔 = mean surface gravity

𝑛 = a tangential unit vector 𝑃𝑛 = Legendre polynomial of degree 𝑛 G = gravitation constant



Usually computed from pressure anomalies with the Green‘s function approach,

based on Farrell (1972), formulas according to Petrov and Boy (2004):

vertical displacement: 𝑑𝑣 𝑝, 𝑡 = ∆𝑃 𝑞, 𝑡 𝐺𝑣 𝜓 cos 𝜑 𝑑𝜆 𝑑𝜑

𝐺𝑣 𝜓 =𝐺𝑅

𝑔2 𝑛=0∞ ℎ𝑛

′ 𝑃𝑛(cos(𝜓))

Green‘s function for continental crust based on Preliminary Reference Earth Model (PREM):



Site displacements for station WETTZELL

computed from non-tidal loading by the

Earth-System-Modelling group at GFZ

Potsdam (ESMGFZ, Dill & Dobslaw [2013]).


Outline


tidal loading

non-tidal loading


observation level

solution level



station coordinates

scale

Summary


Gauss-Markov Model

We use the common functional and stochastic model in the DGFI Orbit and Geodetic parameter

estimation Software (DOGS):

𝐴 Δ𝑥 = 𝑙 − 𝑣,

where

The best solution (least weighted sum of residuals) is provided by the normal equation:

𝐴𝑇𝑃𝐴 Δ𝑥 = 𝐴𝑇𝑃 𝑙 ⟼ 𝑁 + 𝑁𝐷 Δ𝑥 = 𝑦.

𝑙 = 𝑏 − 𝑓 𝑥0 ∈ ℝ𝑚 observed minus computed (OMC) vector,

𝑏 ∈ ℝ𝑚observation vector,

𝑓 𝑥0 ∈ ℝ𝑚 vector of functional values,

𝑥0 ∈ ℝ𝑛 vector of a-priori parameter values,

Δ𝑥 ∈ ℝ𝑛 vector of parameter corrections,

𝑣 ∈ ℝ𝑚 vector of residuals,

𝐴 = 𝜕𝑓𝑖 𝑥0 𝜕𝑥𝑗 𝑖𝑗∈ ℝ𝑚×𝑛 design (Jacobi) matrix.


Application of site displacements: levels

Observation level (OBS)

Modify the functional model and solve the new normal equation:

𝒇 𝒙𝟎 𝒕 ⟼ 𝒇 𝒙𝟎 𝒕 , 𝜹𝒙 𝒕 ,

𝑵 + 𝑵𝑫 𝜟 𝒙 = 𝒚.

Solution level (SOL)

Subtract session-wise (average) displacements from the original corrections, i.e. after the

application of datum-constraints:

𝜹𝒙 𝒕 = 𝜹𝒙 ∀ 𝒕 ∈ 𝟎, 𝑻 ,

𝜟 𝒙 = 𝜟𝒙 − 𝜹𝒙.

Normal equation level (NEQ)

Modify the right-hand-side with session-wise (average) displacements before the application of

datum-constraints, and solve the new normal equation:

𝑵 𝜟 𝒙 + 𝜹𝒙 = 𝒚

⇒ 𝑵 + 𝑵𝑫 𝜟 𝒙 = 𝒚 − 𝑵𝜹𝒙.


Application of site displacements: average values

For each session, we take the average of all original displacements (ESMGFZ) given for the

corresponding observation period:


Outline


tidal loading

non-tidal loading


observation level

solution level



station coordinates

scale

Summary


Station coordinates: per session

Change in parameter estimates when applying non-tidal atmospheric loading (ESMGFZ):

For OBS and NEQ, site displacements are applied before the parameter estimation with NNT and NNR

conditions. Hence, their estimates differ from those of SOL.


Station coordinates: WRMS

Change in WRMS for up-component when applying any or all non-tidal loading (ESMGFZ):

Stations

ordered by

descending

# sessions.

Again, OBS and NEQ provide similar results, and the WRMS is generally reduced. It is best to apply all

non-tidal loadings together. SOL seems to be rather inappropriate.


Helmert transformation onto DTRF2014: scale

As site displacements computed from non-tidal loading are largest for the vertical (up) component,

they are likely to affect the scale in a global VLBI network.

We perform session-wise 7-parameter Helmert transformations onto the DTRF2014 for each non-tidal

loading scenario (combination of applied loading type and application level).

In a frequency analysis of the resulting scale parameters, the annual signal disappears only

when hydrological loading is applied:


Outline


tidal loading

non-tidal loading


observation level

solution level



station coordinates

scale

Summary


Summary

The application of non-tidal loading in VLBI analysis can improve the estimated station parameters

in terms of an reduced WRMS with respect to the linearized station motion.

The best results are obtained when all non-tidal loading types (atmospheric, oceanic, hydrologic)

are considered jointly, since no type is dominant at every VLBI station.

For VLBI, it seems to hardly matter whether the site displacements are applied at the observation

or (by averaging over session periods) the normal equation level.

Application on solution level, however, was found to be inappropriate.

The annual signal of the scale in 7-parameter Helmert transformations is removed when hydrologic

loading is applied (on any level).


Thank you for your attention!

Are there any questions?


Bibliography

Dill R. and Dobslaw H. [2013], Numerical simulations of global-scale high-resolution hydrological

crustal deformations, J. Geophys. Res. Solid earth 118, doi:10.1002/jgrb.50353.

Farrell W. E. [1972], Deformation of the Earth by Surface Loads, Reviews of Geophysics and

Space Physics, Vol. 10, No. 3, pp. 761-797.

Petrov L. and Boy J.-P. [2004], Study of the atmospheric pressure loading signal in very

longbaseline interferometry observations, J. Geophys. Res., 109, B03405.


Backup slides.


Application of site displacements (1)

Let 𝛿𝑥 𝑡 ∈ ℝ𝑛 be the vector of site displacements at epoch 𝑡 as obtained from non-

tidal loading, for example. Entries are equal to zero for non-station parameters.

Application at observation level:

𝑓 𝑥0 𝑡 ⟼ 𝑓 𝑥0 𝑡 , 𝛿𝑥 𝑡 ,

which implies

𝐴 ⟼ 𝐴,

𝑙 ⟼ 𝑙 = 𝑏 − 𝑓 𝑥0 𝑡 , 𝛿𝑥 𝑡 ,

𝑦 ⟼ 𝑦 = 𝐴𝑇𝑃 𝑙,

and new equations

𝐴 Δ 𝑥 = 𝑙 − 𝑣,

𝑁 + 𝑁𝐷 Δ 𝑥 = 𝑦.





Application at solution level:

Assumption 1. Omitting non-tidal loading effects provides parameter corrections

consisting of the true correction plus an error term 𝑒:

Δ𝑥 = Δ 𝑥 + 𝑒.

Assumption 2. The site displacements are constant during a VLBI session:

𝛿𝑥 𝑡 = 𝛿𝑥 ∀ 𝑡 ∈ 0, 𝑇 .

Assumption 3. The error term is equal to the site displacement for each parameter:

𝑒 = 𝛿𝑥.

Approach. A-posteriori modification of the estimated corrections:

Δ𝑥 = Δ 𝑥 + 𝛿𝑥 ⇔ Δ 𝑥 = Δ𝑥 − 𝛿𝑥.





Application at normal equation level:

We make the same assumptions as for the solution level, i.e.:

𝛿𝑥 𝑡 = 𝛿𝑥 ∀ 𝑡 ∈ 0, 𝑇 ,

Δ𝑥 = Δ 𝑥 + 𝛿𝑥.

Then we insert this into the original (unconstrained) normal equation:

𝑁Δ𝑥 = 𝑦

⇔ 𝑁 Δ 𝑥 + 𝛿𝑥 = 𝑦

⇔ 𝑁Δ 𝑥 + 𝑁𝛿𝑥 = 𝑦

⇔ 𝑁Δ 𝑥 = 𝑦 − 𝑁𝛿𝑥.


Gauss-Markov Model: stochastic part

We use the common functional and stochastic model in the DGFI Orbit and Geodetic

parameter estimation Software (DOGS):

𝐴 Δ𝑥 = 𝑙 − 𝑣

and

𝐶 = 𝜎0

2𝑄 = 𝜎0

2𝑃−1,

𝐶 ∈ ℝ𝑚×𝑚 covariance matrix of observations 𝑏,

Q ∈ ℝ𝑚×𝑚 co-factor matrix of observations 𝑏,

𝜎0

2 variance factor,

P ∈ ℝ𝑚×𝑚 weight matrix of observations 𝑏.


Formal errors (OBS)

The formal errors (standard deviations) depend on the weighted square sum of residuals:

𝜎𝑗 = 𝜎0

2𝑁−1

𝑗𝑗 , 𝜎0

2=

𝑣𝑇𝑃 𝑣

𝑚 − 𝑛.

As 𝑣𝑇𝑃 𝑣 does not change significantly (and 𝑁 only slightly), neither do the formal errors:


Formal errors (NEQ)

The formal errors (standard deviations) depend on the weighted square sum of residuals:

𝜎𝑗 = 𝜎0

2𝑁−1

𝑗𝑗 , 𝜎0

2=

𝑣𝑇𝑃 𝑣

𝑚 − 𝑛.

As 𝑣𝑇𝑃 𝑣 does not change significantly (and 𝑁 not at all), neither do the formal errors:

impact of non-tidal loading applied at different levels in

Documents