-
Journal of Geodesy manusript No.(will be inserted by the
editor)Strategies to mitigate aliasing of loading signals
whileestimating GPS frame parametersXavier Collilieux � Tonie van
Dam � Jim Ray � David Coulot � LaurentM�etivier � Zuheir
AltamimiReeived: date / A
epted: dateAbstrat Although GNSS tehniques are
theoretiallysensitive to the Earth enter of mass, it is often
prefer-able to remove intrinsi origin and sale informationsine they
are known to be a�eted by systemati er-rors. This is usually done
by estimating the parame-ters of a linearized similarity
transformation whih re-lates the quasi-instantaneous frames to a
seular framesuh as the International Terrestrial Referene
Frame(ITRF). It is well known that non-linear station mo-tions,
not-a
ounted for in the seular ITRF, an par-tially alias into these
parameters. We disuss in thispaper some proedures that may allow
reduing thesealiasing e�ets in the ase of the GNSS tehniques,mainly
GPS. The options inlude the use of well dis-tributed sub-networks
for the frame transformation es-timation, the use of site loading
orretions, a modi�a-tion of the stohasti model by down-weighting
heights,or the joint estimation of the low degrees of the
defor-mation �eld. We on�rm that the standard approahonsisting of
estimating the transformation over thewhole network is partiularly
harmful for the loadingsignals if the network is not well
distributed. Down-weighting the height omponent, using a uniform
sub-network, or estimating the deformation �eld performX.
Collilieux, Z. Altamimi, L. M�etivierIGN/LAREG et GRGS, 6-8 av.
Blaise Pasal, 77455 Marne LaVall�ee edex 2, FraneTel.:
+33-164-153138Fax: +33-164-153253E-mail: xavier.ollilieux�ign.frT.
van DamUniversity of Luxembourg, 162a, avenue de la Faenerie,
L-1511LuxembourgJ. RayNOAA National Geodeti Survey, 1315 East-West
Hwy, SilverSpring, Maryland 20910
similarly in drastially reduing the aliasing e�et am-plitude.
The appliation of these methods to repro-essed GPS terrestrial
frames permits an assessmentof the level of agreement between GPS
and our loadingmodel, whih is found to be about 1:5 mm in heightand
0:8 mm WRMS in the horizontal at the annual fre-queny. Aliased
loading signals are not the main soureof disrepanies between
loading displaement modelsand GPS position time series.Keywords
Loading e�ets � Terrestrial RefereneFrame � GNSS1 IntrodutionGlobal
Navigation Satellite System (GNSS) tehniquesare used to a
urately monitor ground deformations,from a few minutes to
deades. The most a
urate pro-essing strategies onsist in proessing GNSS data
re-eived at a wide set of global stations simultaneouslyusing the
most urrent and onsistent models. All thephenomena that a�et the
GNSS observables need tobe modeled, espeially if their time sales
of variationare shorter than the sampling rate of the estimated
pa-rameters. This is the ase for solid Earth tides, poletides, and
oean tidal loading e�ets whih are wellmodeled (MCarthy and Petit,
2004). Non-tidal load-ing e�ets, whih inlude the e�et of the
atmosphere,oean irulation, and hydrologial loading are still un-der
investigation. Correlations have been noted withspae geodeti
results, either from GNSS (van Damet al, 1994, 2001), Very Long
Baseline Interferometry(VLBI) (van Dam and Herring, 1994; Petrov
and Boy,2004), Satellite Laser Ranging (SLR) and Doppler
Or-bitography Integrated on Satellites (DORIS) (Mangia-rotti et al,
2001), but non-tidal loading e�ets are not
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2yet reommended for operational GNSS data proessing(Ray et al.,
2007; see http://www.bipm.org/utils/en/events/iers/Conv_PP1.txt ).
Comparisons withspae geodeti results are still needed to validate
themodels and to develop optimal strategies to attenuatesystemati
loading e�ets without introduing exessivemodelling errors.GPS, SLR,
VLBI and DORIS position time seriesare expeted to show similar
variations if position timeseries are omputed in the same referene
frame and ifthe loading signatures are signi�ant ompared to
mea-surement errors. However, tehnique-spei� systematierrors limit
the empirial orrelations so far. For exam-ple, GPS apparent
geoenter motion is not in agree-ment with expeted values (Lavall�ee
et al, 2006). A sim-ple frame transformation is ommonly used to
removeglobal biases that a�et all the station positions. A
tri-dimensional similarity expresses station positions withrespet
to an external referene frame, usually the In-ternational
Terrestrial Referene Frame (ITRF) or a re-lated frame, by (where
negligibly small non-linear termshave been dropped):X i(t) = T (t)
+ (1 + �(t)) � [X ir(t0) + _X ir � (t� t0)℄+R(t) � [X ir(t0) + _X
ir � (t� t0)℄ + Æistat (1)where X i is the estimated position of
station i at theepoh t, X ir and _X ir are its position and veloity
in thereferene frame expressed at the epoh t0, Æistat is thenoise
term and T , R and � are the transformation pa-rameters,
respetively, the translation vetor, the anti-symmetri rotation
matrix and the sale fator at theepoh t. This transformation is also
the basis for theminimum onstraint equations that are sometimes
usedto regularize, with minimum information, station oor-dinates
estimated with spae geodeti tehniques. In-deed, orientation should
normally be onstrained forall tehniques, as well as the origin for
VLBI.It is known that applying suh a transformationa�ets non-linear
variations of the estimated time se-ries of GPS station oordinates
(Blewitt and Lavall�ee,2000; Tregoning and van Dam, 2005;
Collilieux et al,2009). Indeed, as the ITRF is a seular frame
(Altamimiet al, 2007), the station position seasonal variationsan
partly alias into the transformation parameters.This e�et is not
desired and an be problemati formany appliations: omparison with
loading models, in-version to estimate loading mass density
distributions,or omparison of spae geodeti results from
di�erenttehniques. The aim of this paper is to
quantitativelydesribe this aliasing e�et and to review and
evalu-ate proedures that ould be used to redue it. Setiontwo
desribes the syntheti data that are onstruted
to evaluate various suggested proedures. Setion threeassesses
the results of the tests arried out on the syn-theti data to show
the performanes of the methods.And �nally setion four applies the
proedures to realGPS solutions in order to ompare GPS
displaementswith loading models.2 Strategy2.1 Method to ompute
position time seriesThis setion realls the most general method that
anbe used to ompute position time series in an homoge-neous
referene frame from a set of daily/weekly solu-tions.Firstly, a
seular referene frame is needed. It is re-ommended to reompute
seular positions and veloi-ties for every station or for a subset
of reliable stationsfrom its own set of solutions. At this step,
disontinu-ities should be identi�ed in the position time series
andmodeled in the estimated seular frame Xref (t). Theestimated
long term oordinates should be referred tothe adopted seular
referene frame, for example, theITRF, using stations showing the
same disontinuitylist. This step is neessary to avoid possible
errors inthe adopted seular frame or inonsistenies with theinput
dataset whih may a�et transformed positiontime series. Seondly, the
transformation parametersshould be estimated between eah
daily/weekly solutionand the estimated seular oordinates of the
epoh us-ing equation (1). The next setion will disuss
di�erentstrategies for this purpose. Finally, detrended residualsdX
i(t) an be omputed as follow:dX i(t) = X i(t)�X iref (t)�[T̂ (t) +
(R̂(t) + �̂(t) � I3) �X i0(t)℄ (2)where X i0(t) are approximated
oordinates of station iwhereas trended residuals tX i(t) an be
omputed asfollow:tX i(t) = X i(t)� [T̂ (t) + (R̂(t) + �̂(t) � I3)
�X i0(t)℄ (3)The �rst two steps an be merged into one single step
asdone in the CATREF software (Altamimi et al, 2007).However, less
exibility is allowed for the estimationof the transformation
parameters. We will spei�allydisuss here the seond step whih
onsists in estimat-ing transformation parameters. The di�erenes
betweenthe various methods will be highlighted using
synthetidata.
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32.2 Syntheti data and testsWe have simulated GPS weekly station
position sets asfollowsX i(t) = X iitrf2008(t)+(t�t0)� _X
iitrf2008+�iload(t)+Æi(t)(4)where X i(t) is the position vetor of
the station i atepoh t, �iload(t) is the loading displaement in
theCenter of Figure (CF) frame and Æi(t) is a spatiallyorrelated
noise term. Station positions have been gen-erated from 1998.0 to
2008.0, omprising 512 weeks.The real GPS network of the
Massahusetts Instituteof Tehnology (MIT) analysis enter (MI1
reproessedsolution), whih is the most inlusive of all the
repro-essed GPS solutions, has been adopted and stationpositions
have been simulated only when station pa-rameters were available in
their SINEX �les. The fullnetwork is omposed of 748 stations with
many stationsonentrated in North Ameria and Europe. The ve-tor of
spatially orrelated noise Æ(t) has been simulatedfrom the full
ovariane matries of the MI1 solutions.The loading displaement
model�iload(t) has been om-puted as the sum of three loading
displaement mod-els. The �rst inludes the e�et of the atmosphere
ata 6-hour sampling rate a
ording to the model of theNational Center for Environmental
Predition surfaepressure. The seond is derived from the ECCO
OeanBottom Pressure model at a sampling rate of 12 hours(JPL,
2008). The third predits the hydrologial e�etat monthly intervals
(Rodell et al, 2004). These modelshave been averaged or
interpolated to weekly spaingbefore being merged. They spei�ally
show power atthe seasonal frequenies and espeially the annual
(Rayet al, 2008).2.3 Desription of testsSyntheti data sets omputed
from equation 4 havebeen analyzed as if they were real data. We
estimatethe transformation parameters between the position setof
week t and a seular referene frame expressed withrespet to ITRF2008
preliminary solution (Altamimi,Z. and Collilieux, X. and M�etivier,
L., 2010). With realdata, estimated transformation parameters are
non-zerodue to apparent geoenter motion (ombination of Cen-ter of
Mass (CM) displaement with respet to CFdue to loads and systemati
errors), onventional ori-entation of the weekly frame, GPS sale
dependenywith the satellite and ground antenna phase enter o�-sets
and variations, noise, and aliasing e�ets related to
loading. No frame error has been introdued in equa-tion 4 to
onstrut the syntheti data, whih meansthat estimated translation,
rotation, and sale parame-ters from syntheti solutions only reet
noise and alias-ing terms. Figure 1 shows the transformation
parame-ters estimated from the syntheti data in the hereafteralled
standard approah: all the transformation pa-rameters are estimated
with all available stations. Sig-ni�ant aliased annual signals an
be seen, espeially inthe X and Z translation omponents, in the sale
fator,and also in the rotations. The extra noise variations in2006
is related to the large variations of the varianes ofsome point
positions at the time of an Earthquake; Sta-tion SAMP (Indonesia)
is the most a�eted. We haveheked that this extra-noise does not
hange the on-lusions shown here.The olumns of Table 1 enumerate the
strategiesthat are tested here to redue the aliasing error.
In-stead of using the whole set of available stations to es-timate
the transformation parameters, strategy subnetonsists in using a
well distributed subset of stations toompute the transformation
parameters. Indeed, load-ing e�ets are spatially orrelated and
extrating a sub-set of stations is useful to avoid over-weighting
those ar-eas with a high station density, whih a
entuates thealiasing e�et. Stations of the sub-network are
hosento have at least 80% of the full 11-year period overedby data
and a limited set of disontinuities with seg-ments longer than 20%.
We followed the approah sug-gested by Collilieux et al (2007) to
remove stations indense areas and ensure a globally uniform
distribution.However, we had to preserve some stations in
poorlyovered areas that did not exatly math the above ri-teria,
spei�ally in the southern hemisphere.It is also worth noting that
the loading signals havelarger amplitude in the height than in the
horizon-tal omponents (Farrell, 1972). With respet to the
7-parameter transformation, loading e�ets an be on-sidered as
biases sine they are not modeled, and thisbias is more important in
the vertial. As a onse-quene, down-weighting the height
measurements hasbeen suggested to redue the aliasing e�et (T.
Her-ring, personal ommuniation, 2009). This approah isalready
implemented in the Globk software (Herring,2004). There are several
ways to implement this down-weighting. We tested two approahes.
First, we hose touse the inverse of the diagonal ovariane matrix of
thesolution to weight the transformation but we modi�edthe height
formal error by a saling fator. This ap-proah is named hereafter
downdiag. However, the o�diagonal terms of the ovariane matries
ontain sig-ni�ant statistial information, whih is important
topreserve. So we also implemented the down-weighting
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4of heights while preserving the orrelation terms of theovariane
matries following Guo et al (2010), alleddownfull.Another way to
handle this problem is to hangethe frame transformation model to
inlude informationabout the loading displaements:X i(t) = T (t) +
(1 + �(t)) � [X ir(t) +�iload(t)℄+R(t) � [X ir(t) +�iload(t)℄ +
Æistat (5)It allows a
ounting for the non-linear variations of thereferene frame. This
approah is hereafter named load-mod. It has been shown to be
equivalent to orretingdaily/weekly station positions by the model
prior toestimating the transformation parameters (Collilieuxet al,
2010a).Finally, we test the degree-1 deformation approahsuggested
by Lavall�ee et al (2006), equation (A6-A7),whih onsists in
estimating the low degree spherialharmonis of the load mass density
that generates thedeformation �eld simultaneously with the
transforma-tion parameters, hereafter alled loadest. Those
authorswere however interested in the degree 1 terms of theload
surfae density whereas we fous here on the trans-formation
parameters. Please note that there is an errorin equation (A7): (
3h+2l � 1) should be replaed with1=(h+2l3 � 1).It is worth noting
that in all these methods, theframe sale fator, �, in equation 1 an
be estimated ornot.2.4 Evaluation of the alias errorThe outputs of
all the proessing methods desribedabove are the transformation
parameters. One they areomputed, they an be inorporated into
equation (2)or (3) to ompute the residuals of the station
positionsfor every station. When syntheti data are proessed, itis
possible to quantify the e�etiveness of all the meth-ods by heking
how lose the estimated transforma-tion parameters are to zero. As
their e�et on stationpositions is di�erent from one site to
another, we alsoompare the station position residuals to the
loadingdisplaements that have been used to reate the syn-theti
data.Note that theWeighted Root Mean Squares (WRMS)of the di�erenes
for station positions are dominated bynoise. As a onsequene, these
statistis are not inter-esting to evaluate the aliasing e�ets. As
the loadinge�ets have a large signal at the annual frequeny
(seeFigure 1), we hoose to evaluate eah method on itsability to
properly reover the loading signal at the an-nual frequeny in the
station position time series.
3 Evaluation of the methods3.1 SaleNot estimating the sale in
the frame transformationhas been reommended by several authors
(Tregoningand van Dam, 2005; Lavall�ee et al, 2006). Indeed, asan
been noted in Figure 1, a large annual signal is ob-served in the
sale when it is estimated in the standardapproah. Figure 2a) shows,
for every stations with suf-�ient data (more than three years), the
omparisonbetween the in phase and out of phase terms of theannual
signals estimated in the station position timeseries residuals
(omputed by applying veloities andtransformation parameters only)
and the annual sig-nals estimated in the loading models used to
generatethe data. The more losely the points are loated on
thediagonal, the more satisfatory the transformation is. Itis
interesting to note the systemati behavior of the an-nual signal.
Most of the terms are over-estimated usingthe standard method,
exept the East omponent. Asould be expeted the height omponent is
the most af-feted, espeially the out of phase term whih is biasedby
about 1 mm. If the sale is not estimated, see Figure2b), the piture
is almost unhanged for the horizontalomponents but the height
annual signals are obviouslybetter reovered. Figure 3a-b) shows the
aliased load-ing signal in the translations and sale fators for
thesetwo ases. The translation parameters are almost un-hanged when
the sale fator is not estimated sine theGPS network overs almost
the whole globe.Collilieux et al (2010b) showed that the annual
vari-ations observed in the newly reproessed GPS sale fa-tor an be
partly explained by our loading model butnot ompletely. However,
the sale behavior is quite sta-ble in time so that it is reasonable
to estimate one on-stant sale fator for the whole period of time.
Thisan be done in a one-step run if all the transformationparameter
time series are estimated simultaneously orin a two-step approah by
applying in a seond stepthe mean sale fator to the referene
solution. A 6-parameter transformation (no sale) an then be
esti-mated. Unfortunately, most of the methods presentedabove annot
fully �x the problem of aliasing in thesale fator. The method
onsisting in inorporatingthe loading model in the transformation
(loadmod) per-forms niely, see Figure 2) and Figure 3), but only
ifthe loading perfetly �ts the GPS data (see setion 4for
disussion). It is possible to redue the annual sig-nal in the sale
when using a well distributed networkof stations for the frame
transformation, as disussedby Collilieux et al (2007). However, the
performane ofthe method is variable and depends strongly on the
sub-
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5network. Indeed, we did not notie a redution in theannual sale
amplitude when studying our MIT welldistributed sub-network either
with syntheti or realdata. Only the estimation of the deformation
�eld, asalready disussed by Lavall�ee et al (2006), seems to
de-rease signi�antly the sale fator annual signal (seesetion 3.4)
but does not nullify it.As a onsequene, we will disuss the
following re-sults in the ase of a 6-parameter frame
transformation.The sale issue will be disussed further for
strategyloadest only.3.2 Down-weighting heightTransformation
parameters obtained with the standardapproah have been omputed
using the full ovari-ane matrix of the solutions. It is worth
noting thatGPS height determinations are about 3 times less pre-ise
than the horizontals, whih means that the heightomponent is
naturally down-weighted in the standardapproah (�up � 3 �
�north).Figure 2d-f) shows the results obtained when thewhole
network of stations is used to ompute the trans-formation
parameters while applying down-weightingof the heights (no sale
estimated here). The only dif-ferene with 2b) is the weighting.
Three di�erent down-weighting strategies are shown. Height
unertainties weremultiplied by 1:5 (�up � 5 ��north) or by 3:0 (�up
� 10 ��north) but the orrelations were preserved, Figure 2d-e).
Correlations were anelled in Figure 2f) (�up � 10 ��north). It an
be learly notied that down-weightingheight has a positive e�et for
the horizontal ompo-nents. When the height weight is slightly
dereased, thepattern of the annual is losed to the standard asebut
the error has been signi�antly mitigated. Eventhe height agreement
is improved with estimated or-relations larger than 99% and mean
deviations smallerthan 0.3 mm for the in phase and out of phase
terms.When the height weight dereases again (Figure 2e)),the
agreement gets better. Indeed, the translation pa-rameter along the
x and y axes beome smaller, see Fig-ure 3d-e). However, there is a
di�erene depending onwhether the orrelations are used or not in the
weight-ing. The reovered annual term in the North omponentis
di�erent, ompare Figure 2e) and Figure 2f), whihis related to the
di�erenes in the x- and z-translations,see Figure 3e) and Figure
3f). The out of phase termis generally under-estimated in the
full-weighting asewhereas the in phase term is slightly
over-estimated.However, the error is reasonable for both methods
whensyntheti data are studied. We also tried to derease theheight
weight even more but the general level of agree-
ment between the residuals and the true values did notimprove
signi�antly.3.3 Using a sub-networkRestriting the transformation to
a subset of stations isthe most natural way to proeed. This is what
is om-monly done when some station oordinates that areweakly
determined are rejeted from the transforma-tion estimation (with a
simple outlier rejetion test).Additionally, using a well
distributed sub-network sig-ni�antly redues the transformation
parameter biases.Our network is omposed by 77 stations, seleted
fol-lowing the riteria de�ned above. Figures 2g) and 3g)show the
performane of the method. The biggest biasin the residual position
time series is observed in thein phase annual term of the north
omponent and inthe out of phase term of the east omponent. The
av-erage error at the annual frequeny is within 0.2 mmWRMS for this
term whih shows that the approah isreliable to mitigate aliasing
e�ets. Aliased annual sig-nals are reasonably small in the
translation parametersalthough signal along the z axis is still
visible. Whenthe height was down-weighted onjointly by 1:5,
thealiasing errors have slightly dereased but only by 0.1mm annual
WRMS in the in phase terms of the annualsignal for the north and
height omponents. For thispartiular weighting, it performs better
to use a sub-set of stations than the full network. When the
heightunertainty is multiplied by 3:0, the e�et of the
down-weighting tends to dominate whih means that usingeither the
sub-network or the full network give similarresults.3.4 Estimating
the deformation �eldWe have seen above that using a loading model
in thetransformation is e�etive. The main limitation is theloading
model a
uray and possible GPS systematierrors but another limitation is
the availability of theloading model itself. Estimating the
displaements ausedby the loading of the Earth's rust is an
alternative.However, due to the spatial distribution of GPS
sites,it is only possible for the longest wavelengths of
thedeformation �eld. Following Wu et al (2003), we onlyestimated
the load surfae density oeÆients up to thedegree �ve. We also paid
attention to model the defor-mation �eld in the CF frame, by
adopting degree 1load Love numbers in the CF (Blewitt, 2003), in
orderto estimate a translation whih relates ITRF origin toGPS frame
origin. Indeed, modeling the deformation�eld in the Center of
Network frame (Wu et al, 2002)
-
6would have had no e�et to redue aliasing and model-ing the
deformation �eld in he CM would have removedthe geoenter motion
ontribution from the estimatedtranslation, whih is not desired.For
omparison with the other approahes, we �rstplotted the results when
the sale was �xed to zero. Al-though only low degrees are
estimated, it an be notiedon Figure 2h) and 3h), built with a
trunation degreeequal to �ve, that the method is e�etive to redue
thealiasing e�et. It performs better than any other in
thehorizontal and is as e�etive in the vertial. And here,the full
ovariane matrix has been used with no mod-i�ation of the stohasti
model. The full network ofstations is also used, exept those that
have been iden-ti�ed as outliers in the least squares estimation
proess.We also estimated the sale fator in the frame
trans-formation as a test. The aliasing e�et depends on
thetrunation degree of the spherial harmoni expansionof the load
density. We notied using the syntheti datathat the sale fator
annual signal amplitude beomessmaller than 0.2 mm for degree three
up to degree six.Figure 3i) shows for example the estimated sale
fa-tor for a trunation degree of �ve. The variations ofsale,
ompared to Figure 3a) are drastially reduedbut inter-annual
variations are not removed. We noteda larger annual signal in the
sale fator estimated fromreal data with an amplitude of 0.6�0.1 mm
for a trun-ation degree equal to �ve. This is however muh
smallerthan the amplitude estimated in the standard approahwhih is
1.6mm. If the estimation of the sale is needed,this approah is
relevant but does not fully solve thealiasing issue, espeially at
the inter-annual frequenies.3.5 NNR-onditionWe disussed above the
7-parameter transformation whihis used to onstrain the frame
origin, orientation, andsale. However, GPS is theoretially
sensitive to the ori-gin and sale of the frame so that only the
orientationmust be de�ned in priniple. Constraining a normal
ma-trix with the standard minimum onstraint approah isequivalent to
performing a non-weighted transforma-tion between a onventionally
oriented frame and theoutput frame. As a onsequene, this onstraint
shoulda�et the loading signal as well, but to a lesser extentsine
only orientation is onsidered. We performed thesame omputation as
above but estimating only the ro-tation parameters when using the
full network of sta-tions (standard) or a well distributed
sub-network (sub-net). Aliased loading e�ets in the rotation
parametersshow repeatabilities smaller than 5.6 �as in any
ases,whih is about 0.15 mm. However, the annual signalamplitude in
the X and Y omponent is divided by
about 2 to reah 2:3 and 3:8 �as respetively whena sub-network is
used. As a onsequene, the impaton the position time series is quite
small. The worstdetermined term is the in phase annual term in
theNorth omponent for both standard and subnet strate-gies. While
the orrelation and WRMS of the in phaseNorth term with respet to
the true values are 86%and 0.2 mm for the standard ase, it is
however 96%and 0.1 mm when a well distributed sub-network is
se-leted for the NNR ondition. As a onsequene, a well-distributed
network is required for applying the NNR-ondition and to reover
annual signals in the horizontalat the level of 0.1 mm WRMS.4
Appliation to real dataWe applied here the approahes desribed above
to realGPS position time series to see if the agreement be-tween
the GPS position time series and our loadingmodel is improved
ompared to the standard approah.Figure 4 is similar to Figure 2,
exept that the annualsignal plotted in the y axis omes from the
analysisof the MI1 GPS data. The x axis still shows the an-nual
signal estimated in the loading model over thesame epohs of
observations. Suh a plot represents thelevel of agreement between
GPS produts and the load-ing model at the annual frequeny,
depending on theapproah adopted to de�ne the frame origin,
orienta-tion and sale. Figure 4a) shows the standard approahwhen
the sale is estimated, as a referene. A lear biasan be observed,
espeially for the out of phase terms inthe height, as seen with the
syntheti data. As a on-sequene, for all the results that are shown
next, thesale fator is not estimated. We also show on Figure5a) the
translations and sale fator estimated for thestandard approah.
Figure 5b-f) shows the di�erenesbetween the estimated translations
for alternative ap-proahes and the standard approah.Using the
loading model in the transformation (load-mod), f. Figure 4b), does
not show better agreementbetween GPS and the loading model than any
of theother methods: using a sub-network for the frame
trans-formation (subnet), Figure 4), down-weighting
height(downfull), Figure 4d-e) or estimating the deformation�eld
(loadest), Figure 4f). This shows that disrepaniesbetween GPS
displaements and the loading models arenot related to the aliasing
e�ets. It an be notied onFigure 5 that translation di�erenes with
the standardapproah reah about 1 mm at the annual frequeny inthe x
and z axes. As observed with syntheti data, theagreement between
the GPS North omponent annualterm and the loading model is better
when the alias-ing is redued. The loadest approah seems to
perform
-
7slightly better than any of the other approahes. In-deed, the
WRMS of the di�erenes of annual signals (inphase and out of phase
terms) and their orrelations aresmaller in all the omponents exept
the out of phaseterm in the North omponent. However, the �t withthe
loading model is satisfatory for the three other ap-proahes
although aution should be addressed to in-terpret the results of
the downfull strategy. The Northannual out of phase terms reovered
when the heightunertainty is multiplied by three seem to be
under-evaluated ompared to any other approahes, whih isnot the ase
when the height unertainty is multipliedby 1.5. This was not so
obvious for the syntheti dataalthough visible. We notie that this
e�et is relatedto larger di�erene in the z translation annual
signal,see Figure 5e). As a onsequene, using an unertaintysaling
fator of 1.5 or using diagonal weight only ispreferred. We think it
is always better not to a�et thestohasti model, whih is why we
favor using either awell distributed network for the frame
transformationor estimating the low degree oeÆients of the
defor-mation �eld simultaneously.Lavall�ee et al (2006) suggested
two approahes toestimate the low degrees of the load surfae
density.We adopted here the so-alled degree-1 deformation ap-proah
sine we wanted to remove the absorbed loadingsignal in the
translations while preserving the geoen-ter motion in these
parameters. The CM-approah on-sists in modelling the deformation
�eld in the CM frameand estimating rotation parameters only sine
GPS istheoretially sensitive to CM. The two approahes leadto two
distint estimates of the deformation �eld. Asan illustration, Table
2 supplies the annual signals es-timated in the geoenter motion
time series omputedfrom the degree-1 oeÆients. Di�erenes may reah
upto 2.2 mm in the amplitudes and may exeed one monthin phase.
Forward loading model from Collilieux et al(2009) is supplied as a
omparison but any of the two es-timations really seem to �t better
to the loading model.However, the aliasing e�et redution using
these twodistint deformation �elds is similar whih validates
thedegree-1 deformation �eld used for the purpose of alias-ing
mitigation in this study. We also reported in Table 2the opposite
of the translation estimated with degree-1deformation approah. It
an be observed that apparentreproessed GPS geoenter motion is still
not reliable.Thanks to the di�erent tests we did, we are nowable to
onlude about the level of agreement betweenthe GPS position time
series and the loading model.Indeed, we an reasonably exlude the
aliasing e�etas being a major soure of disrepanies. When look-ing
at �gure 4f), the following general omments anbe formulated. The
agreement of the annual signal in
the Height omponent is good in average sine all thepoints are
loated along the diagonal. The mean dis-repany is 1.6 mm for the in
phase term and 1.5 mmfor the out phase term. In the horizontal, the
loadingmodel generally shows a smaller amplitude than theGPS and
the agreement for the in phase and out ofphase term of the annual
signals is less than 0:8 forboth omponents. These results are
enouraging butdisrepanies are still quite important. The
horizontalomponent signals of the GPS stations should be
in-vestigated further in the future studies, espeially byomparing
GPS results with di�erent loading modelsand the results of the
Gravity Reovery and ClimateExperiment (GRACE) to better understand
the originof the disrepanies shown here.5 Conlusion and
reommendationsWe reviewed the proedures that an be used to mod-ify
the origin, orientation, and sale of a time series ofGPS frames. We
paid attention to disuss the trans-formations whih preserve the
loading signals that areinherently ontained in the station
oordinates. This isespeially important in order to interpret
orretly thenon-linear variations in the station position time
series.We showed using syntheti data that the standard ap-proah
onsisting in using the largest set of stations inthe frame
transformation is not optimal whether thesale is estimated or not.
The sale parameter shouldbe de�nitively �xed to a onstant value
over time orits seasonal variations �xed to zero. But a rigorous
ap-proah is possible only if all the frame time series areanalyzed
in one unique estimation proess. The bene-�t of using an
alternative approah is espeially impor-tant for the horizontal
omponent annual signal. Down-weighting height, restriting the
station set to a welldistributed sub-network, or estimating the low
degreesof the load surfae density all perform well. The well
dis-tributed network approah is the easiest to implementwhereas the
height down-weighting may seem diÆultto omprehend due to the
modi�ation of the stohastimodel. A slight advantage is given to the
third methodonsisting in estimating the deformation �eld, whih
isalmost free of any systemati bias a
ording to our sim-ulations. Thanks to this study, we were able
to onludethat aliasing e�et is not the main soure of
disrepanybetween GPS position time series and the loading mod-els.
Annual signals are shown to agree at the 1:5 mmlevel in the height
and the 0:8 mm level in the hori-zontal. Further studies are needed
to understand thesoures of the remaining inonsistenies.
-
8-1
0
1
-1 0 1
a)
COS SIN COS SIN COS SIN East North Height
(83%,0.3) (76%,0.3) (90%,0.4) (91%,0.2) (98%,1.0) (99%,1.3)
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-1
0
1
-1 0 1
b) (81%,0.4) (69%,0.3) (89%,0.5) (86%,0.3) (97%,0.6)
(98%,0.5)
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-1
0
1
-1 0 1
c) (99%,0.0) (99%,0.0) (99%,0.0) (99%,0.0) (100%,0.2)
(100%,0.2)
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-1
0
1
-1 0 1
d) (90%,0.2) (85%,0.2) (93%,0.2) (96%,0.1) (99%,0.3)
(99%,0.2)
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-1
0
1
-1 0 1
e) (95%,0.1) (97%,0.1) (93%,0.1) (97%,0.3) (100%,0.2)
(99%,0.3)
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-1
0
1
-1 0 1
f) (90%,0.2) (96%,0.1) (93%,0.2) (96%,0.2) (99%,0.2)
(99%,0.3)
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-1
0
1
-1 0 1
g) (96%,0.1) (89%,0.1) (97%,0.2) (96%,0.1) (99%,0.3)
(99%,0.2)
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-1
0
1
-1 0 1
h) (97%,0.1) (97%,0.1) (98%,0.1) (98%,0.1) (100%,0.2)
(99%,0.3)
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6Fig. 2 Comparison of the annual signal estimated in
the residuals of the frame transformation as de�ned in setion 2.3
(y axis) appliedto syntheti data, and the annual signal estimated
in the loading model that has been used to generate the syntheti
data (x axis)in millimeters. In phase term (COS) and out of phase
term (SIN) are presented for the East, North and Height omponents
from theleft to the right for the di�erent strategies presented in
setion 2.2. a) standard, sale estimated; b) standard, sale not
estimated; )loadmod, sale estimated; d) downfull height standard
deviations multiplied by 1.5; e) downfull height standard
deviations multipliedby 3.0; f) downdiag height standard deviations
multiplied by 3.0; g)subnet ; h)loadest. Sale fators have not been
estimated for d) toh). Numbers inside the brakets for eah plot are
the orrelation oeÆient and the WRMS of the di�erenes in
millimeters.
-
9-3-2-10123
2000 2004 2008
a)
(mm
)
2000 2004 2008 2000 2004 2008 2000 2004 2008
b)
(mm
)
-3-2-10123
-3-2-10123
c)
(mm
)
d)
(mm
)
-3-2-10123
-3-2-10123
e)
(mm
)
f)
(mm
)
-3-2-10123
-3-2-10123
g)
(mm
)
h)
(mm
)
-3-2-10123
-3-2-10123
2000 2004 2008
i)
(mm
)
2000 2004 2008 2000 2004 2008 2000 2004 2008Fig. 3 Translations
along the x, y and z axis in millimeters and sale fators in
millimeters (ppb value multiplied by 6.4) estimatedfrom syntheti
weekly frames and the seular frame. A di�erent strategy has been
used for eah row. See the legend of Figure 2 forrows a) to h).
Strategy used for row i) is idential to row h) exept that the sale
fator has been also estimated.Table 1 Strategies used in this study
to estimate the transformation between a weekly/daily frame and a
seular frame.Options standard subnet downfull downdiag loadmod
loadestStations used All Well distributed All All All
AllSub-networkWeight matrix Full Full Full, Diagonal, Full
FullHeight down-weighted Height down-weightedLoading Corretions No
No No No Yes Noload density parameters No No No No No Up to degree
5
-
10-2-1012
-2 -1 0 1 2
a)
COS SIN COS SIN COS SIN East North Height
(47%
,0.9
)
(38%
,0.8
)
(42%
,1.1
)
(54%
,0.7
)
(80%
,1.8
)
(82%
,1.9
)
-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-2-1012
-2 -1 0 1 2
b)
(39%
,0.8
)
(39%
,0.7
)
(33%
,0.9
)
(57%
,0.6
)
(84%
,1.6
)
(83%
,1.5
)
-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-2-1012
-2 -1 0 1 2
c)
(39%
,0.8
)
(36%
,0.7
)
(32%
,0.9
)
(57%
,0.6
)
(84%
,1.6
)
(83%
,1.5
)
-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-2-1012
-2 -1 0 1 2
d)
(42%
,0.9
)
(39%
,0.7
)
(33%
,0.9
)
(57%
,0.6
)
(83%
,1.6
)
(83%
,1.5
)
-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-2-1012
-2 -1 0 1 2
e)
(35%
,0.8
)
(38%
,0.6
)
(22%
,0.8
)
(57%
,0.6
)
(85%
,1.7
)
(83%
,1.6
)-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6
-2-1012
-2 -1 0 1 2
f)
(45%
,0.8
)
(42%
,0.7
)
(34%
,0.8
)
(53%
,0.6
)
(84%
,1.6
)
(83%
,1.5
)
-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-2-1012
-2 -1 0 1 2
-6-3036
-6 -3 0 3 6
-6-3036
-6 -3 0 3 6Fig. 4 Comparison of the annual signal estimated in
the residuals of the frame transformation as de�ned in setion 2.3
(y axis) appliedto real data, and the annual signal estimated in
the loading model (x axis) in millimeters. a) standard, sale
estimated; b) loadmod ; )subnet ; d) downfull height standard
deviations multiplied by 1.5; e) downfull height standard
deviations multiplied by 3.0; f) loadest.Sale fators have not been
estimated for b) to f). Same legend as �gure 2.Aknowledgements This
work was partly funded by the CNESthrough a TOSCA grant. All the
plots have been made withthe General Mapping Tool (GMT) software
(Wessel and Smith,1991). ReferenesAltamimi Z, Collilieux X, Legrand
J, Garayt B,Bouher C (2007) ITRF2005: A new release of
theInternational Terrestrial Referene Frame based ontime series of
station positions and Earth Orienta-tion Parameters. J Geophys Res
112(B09401), DOI
-
11-10-505
10
2000 2004 2008
a)
(mm
)
2000 2004 2008 2000 2004 2008
-10-505
10
2000 2004 2008
b)
(mm
)
-3-2-10123
-3-2-10123
c)
(mm
)
d)
(mm
)
-3-2-10123
-3-2-10123
e)
(mm
)
2000 2004 2008
f)
(mm
)
2000 2004 2008
-3-2-10123
2000 2004 2008Fig. 5 a) Translations along the x, y and z axis
in millimeters and sale fators in millimeters (ppb value multiplied
by 6.4) estimatedfrom mi1 weekly frames and the seular frame. b-f)
Di�erenes between translation parameters estimated with tested
strategies andthose derived from the standard method shown in a).
See the legend of Figure 4.Table 2 Estimations of the geoenter
motion (CM w.r.t. CF) from GPS mi1 solution with two di�erent
strategies. Phase are supplieda
ording to the model A � os(2� � (t � 2000:0) � �) with t in
year.X amplitude X phase Y amplitude Y phase Z amplitude Z phasemm
degrees mm degrees mm degreesdegree-1 deformation approah 3.8�0.3
48�4 2.0�0.2 3�5 7.9�0.5 38�3CM-approah 1.0�0.2 71�9 2.4�0.2 310�5
4.5�0.3 51�4opposite of the Translation from 0.4�0.2 165�0 3.6�0.3
302� 5 4.4�0.5 125�7degree-1 deformation approahLoading model
(Collilieux et al, 2009) 2.1�0.1 28 �2 2.1�0.1 338 � 2 2.7�0.1 48 �
210.1029/2007JB004949Altamimi, Z and Collilieux, X and M�etivier, L
(2010)ITRF2008: an improved solution of the
InternationalTerrestrial Referene Frame. Journal of Geodesy
in-press, DOI 10.1007/s00190-011-0444-4Blewitt G (2003)
Self-onsisteny in referene frames,geoenter de�nition, and surfae
loading of the solidEarth. J Geophys Res 108Blewitt G, Lavall�ee D
(2000) E�et of annually re-peating signals on geodeti veloity
estimates. TenthGeneral Assembly of the WEGENER Projet (WE-GENER
2000), San Fernando, Spain, Septembre 18-20Collilieux X, Altamimi
Z, Coulot D, Ray J, SillardP (2007) Comparison of very long
baseline inter-ferometry, GPS, and satellite laser ranging
heightresiduals from ITRF2005 using spetral and orre-lation
methods. J Geophys Res 112(B12403),
DOI10.1029/2007JB004933Collilieux X, Altamimi Z, Ray J, van Dam T,
Wu X(2009) E�et of the satellite laser ranging networkdistribution
on geoenter motion estimation. J Geo-phys Res 114:4402{+, DOI
10.1029/2008JB005727
-
12-3-2-10123
1998 2000 2002 2004 2006 2008
a)
(mm
)
-3-2-10123
b)
(mm
)
-3-2-10123
c)
(mm
)
-3-2-10123
d)
(mm
)
-90-60-30
0306090
e)
(mic
roas
)
-90-60-30
0306090
f)
(microas)
-90-60-30
0306090
1998 2000 2002 2004 2006 2008
g)
(mic
roas
)
Fig. 1 Transformation parameters estimated from synthetidata
omputed with the whole network of stations (standard).a)
x-translation b) y-translation ) z-translation d) sale fatorin
millimeter (value in ppb saled by 6.4) x-rotation f) y-rotationg)
z-rotation.Collilieux X, Altamimi Z, Coulot D, van Dam T, RayJ
(2010a) Impat of loading e�ets on determina-tion of the
International Terrestrial Referene Frame.Adv Spae Res 45:144{154,
DOI 10.1016/j.asr.2009.08.024Collilieux X, M�etivier L, Altamimi Z,
van Dam T, RayJ (2010b) Quality assessment of GPS
reproessedTerrestrial Referene Frame. GPS Solutions inpress,DOI
10.1007/s10291-010-0184-6Farrell WE (1972) Deformation of the Earth
by Sur-fae Loads. Reviews of Geophysis and Spae Physis10:761{+Guo
J, Ou J, Wang H (2010) Robust estimation fororrelated observations:
two loal sensitivity-baseddownweighting strategies . J Geodesy
84(4):243{250,DOI 0.1007/s00190-009-0361-yHerring T (2004) GLOBK:
Global Kalman �lter VLBIand GPS analysis program version 4.1. Teh.
rep.,Mass. Inst. of Tehnol., CambridgeJPL (2008) E
o oean data assimilation.http://e
ojplnasagov/
Lavall�ee DA, van Dam TM, Blewitt G, Clarke PJ(2006) Geoenter
motions from GPS: A uni�ed ob-servation model. J Geophys Res
111:5405, DOI10.1029/2005JB003784Mangiarotti S, Cazenave A,
Soudarin L, Cr�etaux JF(2001) Annual vertial rustal motions
preditedfrom surfae mass redistribution and observed byspae
geodesy. J Geophys Res 106:4277{4292,
DOI10.1029/2000JB900347MCarthy D, Petit G (2004) IERS Tehnial
Note32 - IERS Conventions (2003). Teh. rep., Ver-lag des Bundesamts
fur Kartographie und Geo-dasie, Frankfurt am Main, Germany, also
availableat http://maia.usno.navy.mil/onv2003.htmlPetrov L, Boy J
(2004) Study of the atmospheri pres-sure loading signal in very
long baseline interfer-ometry observations. J Geophys Res p 3405,
DOI10.1029/2003JB002500Ray J, Altamimi Z, Collilieux X, van Dam TM
(2008)Anomalous harmonis in the spetra of gps posi-tion estimates.
GPS solution pp 55{64, DOI 10.1007/s10291-007-0067-7Rodell M,
Houser PR, Jambor U, Gottshalk J,Mithell K, Meng CJ, Arsenault K,
Cosgrove B,Radakovih J, Bosilovih M, Entin JK, Walker JP,Lohmann D,
Toll D (2004) The Global Land Data As-similation System. Bull Amer
Meteor So 85(3):381{394Tregoning P, van Dam TM (2005) E�ets of
atmo-spheri pressure loading and seven-parameter trans-formations
on estimates of geoenter motion and sta-tion heights from spae
geodeti observations. J Geo-phys Res 110:3408, DOI
10.1029/2004JB003334van Dam TM, Herring TA (1994) Detetion of
atmo-spheri pressure loading using very long baseline
in-terferometry measurements. J Geophys Res 99:4505{5417van Dam TM,
Blewitt G, Hein MB (1994) Atmo-spheri pressure loading e�ets on
Global PositioningSystem oordinate determinations. J Geophys
Res99:23,939{23,950, DOI 10.1029/94JB02122van Dam TM, Wahr J, Milly
PCD, ShmakinAB, Blewitt G, Lavall�ee D, Larson KM (2001)Crustal
displaements due to ontinental water load-ing. Geophys Res Lett
28:651{654, DOI 10.1029/2000GL012120Wessel P, Smith WHF (1991) Free
software helpsmap and display data. EOS Transations 72:441{441,DOI
10.1029/90EO00319Wu X, Argus DF, Hein MB, Ivins ER, Webb FH(2002)
Site distribution and aliasing e�ets in theinversion for load
oeÆients and geoenter motionfrom GPS data. Geophys Res Lett
29:63{1, DOI
-
1310.1029/2002GL016324Wu X, Hein MB, Ivins ER, Argus DF, Webb
FH(2003) Large-sale global surfae mass variations in-ferred from
GPS measurements of load-indued de-formation. Geophys Res Lett
30:5{1, DOI doi:10.1029/2003GL017546
