J Geod (2012) 86:1–14DOI 10.1007/s00190-011-0487-6

ORIGINAL ARTICLE

Strategies to mitigate aliasing of loading signals while estimatingGPS frame parameters

Xavier Collilieux · Tonie van Dam · Jim Ray ·David Coulot · Laurent Métivier · Zuheir Altamimi

Received: 17 January 2011 / Accepted: 17 May 2011 / Published online: 8 June 2011© Springer-Verlag 2011

Abstract Although GNSS techniques are theoretically sen-sitive to the Earth center of mass, it is often preferable toremove intrinsic origin and scale information from the esti-mated station positions since they are known to be affectedby systematic errors. This is usually done by estimating theparameters of a linearized similarity transformation whichrelates the quasi-instantaneous frames to a long-term framesuch as the International Terrestrial Reference Frame (ITRF).It is well known that non-linear station motions can partiallyalias into these parameters. We discuss in this paper someprocedures that may allow reducing these aliasing effects inthe case of the GPS techniques. The options include the use ofwell-distributed sub-networks for the frame transformationestimation, the use of site loading corrections, a modifica-tion of the stochastic model by downweighting heights, orthe joint estimation of the low degrees of the deformationfield. We confirm that the standard approach consisting ofestimating the transformation over the whole network is par-ticularly harmful for the loading signals if the network isnot well distributed. Downweighting the height component,using a uniform sub-network, or estimating the deformation

X. Collilieux (B) · D. Coulot · L. Métivier · Z. AltamimiIGN/LAREG et GRGS, 6-8 av. Blaise Pascal,77455 Marne La Vallée Cedex 2, Francee-mail: xavier.collilieux@ign.fr

T. van DamUniversity of Luxembourg, 162a, avenue de la Faïencerie,1511 Luxembourg, Luxembourg

J. RayNOAA National Geodetic Survey, 1315 East-West Hwy,Silver Spring, MD 20910, USA

L. MétivierInstitut de Physique du Globe de Paris,4 place Jussieu, 75005 Paris, France

field perform similarly in drastically reducing the amplitudeof the aliasing effect. The application of these methods toreprocessed GPS terrestrial frames permits an assessment ofthe level of agreement between GPS and our loading model,which is found to be about 1.5 mm WRMS in height and 0.8mm WRMS in the horizontal at the annual frequency. Ali-ased loading signals are not the main source of discrepanciesbetween loading displacement models and GPS position timeseries.

Keywords Loading effects · Terrestrial Reference Frame ·GNSS · GPS · Geocenter motion

1 Introduction

Global Navigation Satellite System (GNSS) techniques,especially the Global Positioning System (GPS), are used toaccurately monitor ground deformations on timescales fromsub-second to decadal. The most accurate processing strate-gies consist of processing GNSS data received at a wide setof global stations simultaneously using the most current andconsistent models. All the phenomena that affect the GNSSobservables need to be parameterized or modeled, especiallyif their time scales of variation are shorter than the samplingrate of the estimated parameters. This is the case for solidEarth tides, pole tides, and ocean tidal loading effects whichare well modeled (McCarthy and Petit 2004). Non-tidal load-ing effects, which include the effect of the atmosphere, oceancirculation, and hydrological loading are still under investi-gation. Correlations have been noted with space geodeticresults, namely from GPS (van Dam et al. 1994, 2001), VeryLong Baseline Interferometry (VLBI) (van Dam and Herring1994; Petrov and Boy 2004), Satellite Laser Ranging (SLR)and Doppler Orbitography Integrated on Satellites (DORIS)

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(Mangiarotti et al. 2001), but non-tidal loading effects arenot yet recommended for operational GNSS data processing(Ray et al. 2007; see http://www.bipm.org/utils/en/events/iers/Conv_PP1.txt). Further comparisons with space geo-detic results are still needed to validate the models and todevelop optimal strategies to attenuate systematic loadingeffects without introducing excessive modeling errors.

GPS, SLR, VLBI, and DORIS position time series areexpected to show similar variations if position time seriesare computed in the same reference frame and if theloading signatures are significant compared to measurementerrors. However, technique-specific systematic errors limitthe empirical correlations so far. For example, GPS geocen-ter motion inferred from the network shift approach is notin agreement with expected values (Lavallée et al. 2006).A simple frame transformation is commonly used to removeglobal biases that affect all the station positions. A tri-dimen-sional similarity equation can be used to express station posi-tions with respect to an external reference frame, usually theInternational Terrestrial Reference Frame (ITRF) or a relatedframe, by (where negligibly small non-linear terms have beendropped)

Xi (t) = T (t) + (1 + λ(t)) · [Xir (t0) + Ẋ ir · (t − t0)]+R(t) · [Xir (t0) + Ẋ ir · (t − t0)] + δi (1)

where Xi is the estimated position of station i at the epocht, Xir is its position in the reference frame expressed at theepoch t0 and Ẋ ir is the corresponding velocity, δ

i is thenoise term and T, R and λ are the transformation parameters,respectively, the translation vector, the anti-symmetric rota-tion matrix and the scale factor at the epoch t. This transfor-mation is also the basis for the minimum constraint equationsthat are sometimes used to invert normal equation systemsin order to estimate station coordinates with space geodetictechniques: orientation should normally be constrained forall techniques, as well as the origin for VLBI.

It is known that applying such a transformation affectsnon-linear variations of the estimated time series of stationcoordinates (Blewitt and Lavallée 2000; Tregoning and vanDam 2005; Collilieux et al. 2009). Indeed, as the ITRF is along-term frame (Altamimi et al. 2007), the station positionseasonal variations can partly alias into the transformationparameters. This effect is not desired and can be problem-atic for many applications: comparison with loading models,inversion to estimate loading mass density distributions, orcomparison of space geodetic results from different tech-niques. The aim of this paper was to quantitatively describethis aliasing effect and to review and evaluate procedures thatcould be used to reduce it. Section 2 describes the syntheticdata that are constructed to evaluate various suggested pro-cedures. Section 3 assesses the results of the tests carriedout on the synthetic data to show the performances of the

methods. And finally Sect. 4 applies the procedures to realGPS solutions in order to compare GPS displacements withloading models.

2 Strategy

2.1 Method to compute position time series

This section recalls the most general method that can be usedto compute position time series in an homogeneous referenceframe from a set of daily/weekly solutions.

First, a long-term reference frame Xref(t) is needed. It isrecommended to recompute long-term positions and veloci-ties for every station or for a subset of reliable stations fromits own set of solutions and not to use directly an exter-nal reference frame designated by Xir (t0) + Ẋ ir · (t − t0)in Eq. (1). At this step, discontinuities should be identifiedin the position time series and modeled in the estimatedlong-term frame Xref(t). The estimated long-term coordi-nates should be referred to the adopted long-term referenceframe datum, for example, the ITRF, using stations showingthe same discontinuity list. This step is necessary to avoidpossible errors in the adopted long-term frame or inconsis-tencies with the input dataset which may affect transformedposition time series. Second, the transformation parame-ters should be estimated between each daily/weekly solutionand the estimated long-term coordinates of the epoch usingEq. (1). The next section will discuss different strategies forthis purpose. Finally, detrended residuals �d Xi (t) can becomputed as follows:

�d Xi (t) = Xi (t) − Xiref(t) − [T̂ (t) + (R̂(t)

+λ̂(t) · I3) · Xi0(t)] (2)where Xi0(t) are approximated coordinates of station iwhereas the position time series w.r.t. the external referenceframe Xiex(t) can be computed as follows:

Xiex(t) = Xi (t) − [T̂ (t) + (R̂(t) + λ̂(t) · I3) · Xi0(t)] (3)The first two steps can be merged into one single step as donein the CATREF software (Altamimi et al. 2007). However,less flexibility is allowed for the estimation of the transforma-tion parameters. We will specifically discuss here the secondstep which consists of estimating transformation parameters.The differences between the various methods will be high-lighted using synthetic data.

2.2 Synthetic data and tests

We have simulated GPS weekly station position sets asfollows:

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http://www.bipm.org/utils/en/events/iers/Conv_PP1.txthttp://www.bipm.org/utils/en/events/iers/Conv_PP1.txt

Strategies to mitigate aliasing of loading signals 3

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Fig. 1 Map of the MI1 network (dots). The well-distributed sub-network is shown with larger colored dots. The stations of that sub-network thatdid not fill the requirement of at least 80% of estimation positions over the 11 years are highlighted in pink

Xi (t)=Xiitrf2008(t0)+(t − t0) · Ẋ iitrf2008+�iload(t)+δi (t)(4)

where Xi (t) is the position vector of the station i at epocht, t0 = 2005.0 is the reference epoch of ITRF2008, �iload(t)is the loading displacement in the Center of Figure (CF) frameand δi (t) is a spatially correlated noise term. Station positionshave been generated from 1998.0 to 2008.0, comprising 512weeks. The real GPS network of the Massachusetts Instituteof Technology (MIT) analysis center (MI1 reprocessed solu-tion), which is the most inclusive of all the reprocessed GPSsolutions, has been adopted and station positions have beensimulated only when station parameters were available intheir SINEX files. The full network is composed of 748 sta-tions, see Fig. 1, with many stations concentrated in NorthAmerica and Europe. We restricted our simulations to the441 GPS stations of the ITRF2008. The vector of spatiallycorrelated noise δ(t) has been simulated from the full covari-ance matrices of the MI1 solutions. Time-correlated noiseprocesses are not mandatory here since we want to studysystematic effects, so they have been ignored. The loadingdisplacement model �iload(t) has been computed as the sumof three loading displacement models. They have been gen-erated using the Green’s function approach and the load Lovenumbers of Han and Wahr (1995). The first includes theeffect of the atmosphere at a 6-h sampling rate according tothe model of the National Center for Environmental Predic-tion surface pressure. The second is derived from the ECCOOcean Bottom Pressure (OBP) model at a sampling rate of12 h (Stammer et al. 2002). The third model, GLDAS, pre-dicts the hydrological effect at monthly intervals (Rodell et al.2004). These models have been averaged or interpolated toweekly spacing before being merged. They specifically showpower at the seasonal frequencies and especially the annual(Ray et al. 2008).

2.3 Description of tests

Synthetic data sets computed from Eq. (4) have been ana-lyzed as if they were real data. We estimated the transfor-mation parameters between the position set of week t anda long-term reference frame expressed with respect to theITRF2008 preliminary solution (Altamimi et al. 2011). Withreal data, estimated transformation parameters are non-zerodue to apparent geocenter motion (combination of Center ofMass (CM) displacement with respect to CF due to loads andsystematic errors), conventional orientation of the weeklyframe, GPS scale dependency with the satellite and groundantenna phase center offsets and variations, noise, and ali-asing effects related to loading. No frame error has beenintroduced in Eq. (4) to construct the synthetic data, whichmeans that estimated translation, rotation, and scale param-eters from synthetic solutions only reflect noise and alias-ing terms. Figure 2 shows the transformation parametersestimated from the synthetic data in what will hereafter bedesignated the standard approach: all the transformationparameters are estimated with all available stations. Signif-icant aliased annual signals can be seen, especially in theX and Z translation components, in the scale factor, andalso in the rotations. The additional noise variations in 2006are related to large variations of the variance of some pointpositions around the time of the 2004-12-26 Sumatra earth-quake; station SAMP (Indonesia) is the most affected. Wehave determined that this extra noise does not change theconclusions shown here.

The columns of Table 1 designate the strategies that aretested here to reduce the aliasing error. Instead of using thewhole set of available stations to estimate the transforma-tion parameters, strategy subnet consists of using a well-distributed subset of stations to compute the transformationparameters. Indeed, loading effects are spatially correlated

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Fig. 2 Transformationparameters estimated fromsynthetic data computed withthe whole network of stations(standard). a X-translation,b Y-translation, c Z-translation,d scale factor (value in ppbscaled by 6.4), e X-rotation,f Y-rotation, g Z-rotation

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Table 1 Strategies used in this study to estimate the transformation between a weekly/daily frame and a long-term frame

Options Standard Subnet Downfull Downdiag Loadmod Loadest

Stations used All Well distributed sub-network All All All All

Weight matrix Full Full Full, Diagonal, Full Full

Height downweighted Height downweighted

Loading corrections No No No No Yes No

Load parameters No No No No No Up to degree 5

and extracting a subset of stations is useful to avoid over-weighting those areas with a high station density, whichaccentuates the aliasing effect. Stations of the sub-networkare chosen to have at least 80% of the full 11-year periodcovered by data and a limited set of discontinuities with seg-ments longer than 20%. We followed the approach suggestedby Collilieux et al. (2007) to remove stations in dense areasthus ensuring a globally uniform distribution. However, we

had to preserve some stations in poorly covered areas thatdid not exactly match the above criteria, specifically in thesouthern hemisphere, see Fig. 1.

It is also worth noting that the loading signals have largeramplitude in the height than in the horizontal components(Farrell 1972). With respect to the 7-parameter transforma-tion, loading effects can be considered as biases, which aretherefore more important in the height, since they are not

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Strategies to mitigate aliasing of loading signals 5

modeled. As a consequence, downweighting the height mea-surements has been suggested to reduce the aliasing effect(T. Herring, personal communication, 2009). This approachis already implemented in the Globk software (Herring2004). There are several ways to implement this downweigh-ting. We tested two approaches. First, we chose to use theinverse of the diagonal covariance matrix of the solution toweight the transformation but we modified the height for-mal error by a scaling factor (1.5 or 3.0). This approach ishereafter designated as downdiag. However, the off-diagonalterms of the covariance matrices contain significant statisti-cal information, which is important to preserve. As a result,we also implemented the downweighting of heights whilepreserving the correlation terms of the covariance matricesfollowing Guo et al. (2010). We first compute the covari-ance Dl in the local frames from the covariance matrix D byDl = R · D · Rt where R is a block-diagonal matrix com-posed by the rotation matrices from the global to the localframe for every station. We note σ ie , σ

in, σ

iu the standard devi-

ations of the positions of station i along the East, North andUp component, and C the associated correlation matrix. Thenew downweighted covariance matrix Ddown is given by

Ddown = Rt · diag(σ 1e , σ 1n , α · σ 1u , . . . , σme , σmn , α · σmu ) · C·diag(σ 1e , σ 1n , α · σ 1u , . . . , σme , σmn , α · σmu ) · R

where α is the downweighting factor and m the number ofstations. This dataset will be referred to as downfull.

Another way to handle this problem is to change the frametransformation model to include information about the load-ing displacements:

Xi (t) = T (t) + (1 + λ(t)) · [Xir (t) + �iload(t)]+R(t) · [Xir (t) + �iload(t)] + δistat (5)

It allows accounting for the non-linear variations of thereference frame. This approach is hereafter named loadmod.It has been shown to be equivalent to correcting daily/weeklystation positions by the model prior to estimating the trans-formation parameters (Collilieux et al. 2010a).

Finally, we test the degree-1 deformation approach sug-gested by Lavallée et al. (2006), equation (A6–A7), whichconsists of estimating the low degree spherical harmonics ofthe load mass density that generates the deformation fieldsimultaneously with the transformation parameters, hereaf-ter called loadest. Those authors were, however, interestedin the degree-1 terms of the load surface density whereas wefocus here on the transformation parameters. Please note that

there is an error in equation (A7) of that paper:(

3h′+2l ′ − 1

)

should be replaced with 1/ (

h′+2l ′3 − 1

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It is worth noting that in all these approaches, the framescale factor, λ, in Eq. (1) may or may not be estimated.

2.4 Evaluation of the aliasing error

The outputs of all the processing methods described aboveare the epoch by epoch transformation parameters. Once theyare computed, they can be incorporated into Eqs. (2) or (3)to compute the residuals of the station positions for everystation. When synthetic data are processed, it is possible toquantify the effectiveness of each of the methods by check-ing how close the estimated transformation parameters are tozero. As their effect on station positions is different from onesite to another, we also compare the station position residualsto the loading displacements that have been used to create thesynthetic data.

Note that the Weighted Root Mean Squares (WRMS) ofthe differences for station positions are dominated by noise.As a consequence, these statistics are not useful for evaluatingthe aliasing effects. As the loading effects have a large signalat the annual frequency, see Fig. 2, we choose to evaluate eachmethod on its ability to properly recover the loading signalsat the annual frequency in the station position time series.

3 Evaluation of the methods

3.1 Scale

Not estimating the scale in the frame transformation has beenrecommended by several authors (Tregoning and van Dam2005; Lavallée et al. 2006). Indeed, as can been noted inFig. 2d, a large annual signal is observed in the scale whenit is estimated in the standard approach. Inter-annual varia-tions are also visible; they are mostly attributed to continentalwater loading (van Dam et al. 2001). Figure 3a shows, for allstations with sufficient data (more than three years), the com-parison between the in-phase and out-of-phase terms of theannual signals estimated in the station position time seriesresiduals [computed according to Eq. (2)] and the annualsignals estimated in the loading models used to generate thedata. The more closely the points are located on the diago-nal, the more satisfactory the transformation. It is interestingto note the systematic behavior of the annual signal. Mostof the terms are over-estimated using the standard method,except the East component. The bimodal distribution noticedin the East component can be explained by an aliasing in theX-translation for the out-of-phase term and an aliasing ofthe X-translation and rotations for the in-phase term. Theeffect of the rotations tends to increase the error in SouthAmerica and decrease the X-translation aliasing in Siberia.As could be expected, the height component is the mostaffected, especially the out-of-phase term which is biasedby about 1 mm. If the scale is not estimated, see Fig. 3b, thepicture is almost unchanged for the horizontal componentsbut the height annual signals are obviously better recovered.

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Strategies to mitigate aliasing of loading signals 7

� Fig. 3 Comparison of the annual signal estimated in the residuals ofthe frame transformation as defined in Sect. 2.3 applied to syntheticdata (Y-axis), and the annual signal estimated in the loading model thathas been used to generate the synthetic data (X-axis) in millimeters.In-phase term (COS) and out-of-phase term (SIN) are presented forthe East, North and Height components from the left to the right forthe different strategies presented in Sect. 2.2. a standard, scale esti-mated; b standard, scale not estimated; c loadmod, scale estimated; ddownfull height standard deviations multiplied by 1.5; e downfull heightstandard deviations multiplied by 3.0; f downdiag height standard devi-ations multiplied by 3.0; g subnet; h loadest. Scale factors have not beenestimated for d to h. Numbers inside the brackets for each plot are thecorrelation coefficient and the WRMS of the differences in millimeters

Figure 4a, b shows the aliased loading signal in the transla-tions and scale factors for these two cases. The translationparameters are almost unchanged when the scale factor is notestimated since the GPS network almost covers the wholeglobe.

Collilieux et al. (2010b) showed that the annual variationsobserved in the GPS scale factor can be partly explained byour loading model, but not completely. However, the scalebehavior is quite stable in time so that it is reasonable to esti-mate one constant scale factor for the whole period of time.This can be done in a one-step run if all the transformationparameter time series are estimated simultaneously or in atwo-step approach by applying first the mean scale factorto the reference solution. A 6-parameter transformation (noscale) can then be estimated.

Unfortunately, most of the methods presented above can-not fully solve the problem of aliasing in the scale factor ifone scale factor is estimated per solution. The method con-sisting of incorporating the loading model in the transfor-mation (loadmod) performs nicely, see Figs. 3c and 4c, butonly if the loading perfectly fits the GPS data (see Sect. 4for discussion). It is possible to reduce the annual signal inthe scale when using a well-distributed network of stationsfor the frame transformation, as discussed by Collilieux et al.(2007). However, the performance of the method is variableand depends strongly on the sub-network. Indeed, we didnot notice a reduction in the annual scale amplitude whenstudying our MI1 well-distributed sub-network either withsynthetic or real data. Only the estimation of the deformationfield, as already discussed by Lavallée et al. (2006), seemsto decrease significantly the scale factor annual signal (seeSect. 3.4) but does not nullify it.

As a consequence, we will discuss the following resultsin the case of a 6-parameter frame transformation. The scaleissue will be discussed further for the strategy loadest only.

3.2 Downweighting height

Transformation parameters obtained with the standardapproach have been computed using the full covariancematrix of the solutions. It is worth noting that GPS height

determinations are about 3 times less precise than the hori-zontals (σup ≈ 3 · σnorth), which means that the height com-ponent is naturally downweighted in the standard approach.

Figure 3d–f shows the results obtained when the wholenetwork of stations is used to compute the transformationparameters while applying downweighting of the heights (noscale estimated here). The only difference with Fig. 3b isthe weighting. Three different downweighting strategies areshown. Height uncertainties were multiplied by 1.5 (σup ≈5 · σnorth) or by 3.0 (σup ≈ 10 · σnorth) but the correlationswere preserved, Fig. 3d, e. Correlations were canceled in Fig.3f (σup ≈ 10 · σnorth). It can be clearly noticed that down-weighting the heights has a positive effect on the horizontalcomponents. When the height weight is slightly decreased,the pattern of the annual is close to the standard case but theerror has been significantly mitigated. Even the height agree-ment is improved with estimated correlations larger than 99%and mean deviations smaller than 0.3 mm for the in-phase andout-of-phase terms. When the height weight is decreased fur-ther, see Fig. 3e, the agreement improves. Indeed, the trans-lation parameters along the X- and Y-axes become smaller,see Fig. 4d, e. However, there is a difference depending onwhether the correlations are used or not in the weighting. Therecovered annual term in the North component is different,compare Fig. 3e and f, which is related to the differences inthe X- and Z-translations, see Fig. 4e and f. The out-of-phaseterm is generally under-estimated in the full-weighting casewhereas the in-phase term is slightly over-estimated. How-ever, the error is reasonable for both methods when syntheticdata are studied. We also tried to decrease the height weighteven more but the general level of agreement between theresiduals and the true values did not improve significantly.

3.3 Using a sub-network

Restricting the transformation to a subset of stations is themost natural way to proceed. This is what is commonlydone when some station coordinates that are weakly deter-mined are rejected from the transformation estimation (witha simple outlier rejection test). Additionally, using a well-distributed sub-network significantly reduces the transfor-mation parameter biases. Our sub-network, shown on Fig. 1,is composed by 77 stations, selected following the criteriadefined above. Figures 3g and 4g show the performance ofthe method. The biggest bias in the residual position timeseries is observed in the in-phase annual term of the northcomponent and in the out-of-phase term of the east compo-nent. The average error at the annual frequency is within 0.2mm WRMS for this term which shows that the approach isreliable for mitigating aliasing effects. Aliased annual signalsare reasonably small in the translation parameters althoughsignal along the Z-axis is still visible. When the height isdownweighted conjointly by 1.5, the aliasing errors decrease

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Fig. 4 Translations along the X-, Y- and Z-axis in millimeters and scalefactors in millimeters (ppb value multiplied by 6.4) estimated betweenthe synthetic weekly frames and the long-term frame. A different strat-

egy has been used for each row. See the legend of Fig. 3 for rows a to h.Strategy used for row i is identical to row h except that the scale factorhas been estimated also

but only by 0.1 mm annual WRMS in the in-phase terms ofthe annual signal for the north and height components. Forthis particular weighting, it is better to use a subset of stationsrather than the full network. When the height uncertainty ismultiplied by 3.0, the effect of the downweighting tends todominate which means that using either the sub-network orthe full network give similar results.

3.4 Estimating the deformation field

We have seen from the results above that using a loadingmodel in the transformation is effective. The main limitationis the loading model accuracy and possible GPS systematic

errors, but another limitation is the availability of the load-ing model itself. Estimating the displacements caused by theloading of the Earth’s crust is an alternative. However, dueto the spatial distribution of GPS sites, it is only possible forthe longest wavelengths of the deformation field. FollowingWu et al. (2003), we only estimated the load surface den-sity coefficients up to spherical harmonic degree five. Wealso paid attention to model the deformation field in the CFframe, by adopting the degree-1 load Love numbers in the CFframe (Blewitt 2003), in order to estimate a translation thatrelates the ITRF origin to the GPS frame origin. Indeed, mod-eling the deformation field in the Center of Network frame(Wu et al. 2002) would have had no effect on reducing the

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Strategies to mitigate aliasing of loading signals 9

Fig. 5 Estimated normalizedsurface mass density coefficientsof degree 1 and 2 using strategyloadest and a truncation degreeequal to 5 are shown in red withtheir 1-σ formal errors (no scaleestimated). These results havebeen obtained using syntheticGPS data. The true loadcoefficients are shown in black.Degree-1 coefficients have beenconverted into geocenter motionusing equation 1 of Collilieuxet al. (2009)

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aliasing. Further modeling the deformation field in the CMwould have removed the geocenter motion contribution fromthe estimated translation, which is not desired.

For comparison with the other approaches, we first plot-ted the results when the scale was fixed to zero. Figure 5shows the estimated surface mass density coefficients fromsynthetic data for a truncation degree equal to five. It canbe noticed that the estimated coefficients are consistent withthe expected values. Although only the low degrees are esti-mated, Figs. 3h and 4h show that the method is effective atreducing the aliasing effect. It performs better than any otherin the horizontal and is as effective in the height. Please notethat the full covariance matrix has been used with no modi-fication of the stochastic model. The full network of stationsis also used, except those that have been identified as outliersin the least squares estimation process.

We also estimated the scale factor in the frame transfor-mation as a test. The aliasing effect depends on the trun-cation degree of the spherical harmonic expansion of theload density. We noticed using the synthetic data that thescale factor annual signal amplitude becomes smaller than0.2 mm for degree three up to degree six, see Fig. 6. Figure 4ishows, for example, the estimated scale factor for a truncationdegree of five. The variations of scale, compared with Fig. 4aare drastically reduced, but inter-annual variations are notremoved. We noted a larger annual signal in the scale factorestimated from real data with an amplitude of 0.6 ± 0.1 mmfor a truncation degree equal to five. This is, however, muchsmaller than the amplitude estimated in the standard approachwhich is 1.6 mm. If the estimation of the scale is needed, thisapproach is relevant but does not fully solve the aliasing issue,especially at the inter-annual frequencies, see Fig. 4i.

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0 1 2 3 4 5 6

0

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rror

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Fig. 6 Aliased annual signal amplitudes in the translation and scalefactor (ppb value multiplied by 6.4) estimated using the strategy loa-dest as a function of the truncation degree of the deformation field.These results have been obtained using synthetic GPS data

3.5 NNR-condition

Above, we discussed the 7-parameter transformation that isused to constrain the frame origin, orientation, and scale.However, GPS is theoretically sensitive to the origin and scaleof the frame so that only the orientation must be defined inprinciple. Constraining a normal matrix with the standardminimum constraint approach is equivalent to performing auniformly-weighted transformation between a convention-ally oriented frame and the output frame. As a consequence,this constraint should affect the loading signal as well, but toa lesser extent since only orientation is considered. We per-formed the same computation as above but estimating onlythe rotation parameters when using the full network of sta-tions (standard) or a well-distributed sub-network (subnet).Aliased loading effects in the rotation parameters show rep-eatabilities smaller than 5.6 µas in any cases, which is about0.15 mm. However, the annual signal amplitude in the X andY component is divided by about 2 to reach 2.3 and 3.8 µas,respectively when a sub-network is used. As a consequence,the impact on the position time series is quite small. Theworst determined term is the in-phase annual term in theNorth component for both standard and subnet strategies.While the correlation and WRMS of the in-phase North termwith respect to the true values are 86% and 0.2 mm for thestandard case, they are, however, 96% and 0.1 mm when awell-distributed sub-network is selected for the NNR condi-tion. As a consequence, a well-distributed network is requiredfor applying the NNR-condition and to recover annual sig-nals in the horizontal at the level of 0.1 mm WRMS.

4 Application to real data

In this section, we applied the approaches described aboveto real GPS position time series to see if the agreement

between the GPS position time series and our loading modelis improved compared with the standard approach. Figure 7is similar to Fig. 3, except that the annual signal plotted onthe Y-axis comes from the analysis of the MI1 GPS data.The X-axis still shows the annual signal estimated in theloading model over the period 1998.0-2008.0. Such a plotrepresents the level of agreement between GPS products andthe loading model at the annual frequency, depending on theapproach adopted to define the frame origin, orientation andscale. Figure 7a shows the standard approach when the scaleis estimated, as a reference. A clear bias can be observed,especially for the out-of-phase terms in the height, as seenwith the synthetic data. As a consequence, for all the resultsthat are shown next, the scale factor is not estimated. We alsoshow in Fig. 8a the translations and scale factor estimatedfor the standard approach. Figure 8b–f shows the differencesbetween the estimated translations for alternative approachesand the standard approach.

Using the loading model in the transformation (load-mod), cf. Figure 7b, does not show any better agreementbetween GPS and the loading model compared with any ofthe other methods: using a sub-network for the frame trans-formation (subnet), Fig. 7c, downweighting height (down-full), Fig. 7d, e or estimating the deformation field (loadest),Fig. 7f. This shows that discrepancies between GPS displace-ments and the loading models are not related to the aliasingeffects. It can be noticed in Fig. 8 that translation differ-ences with the standard approach reach about 1 mm at theannual frequency in the X- and Z-axes. As observed withsynthetic data, the agreement between the GPS North com-ponent annual term and the loading model is better when thealiasing is reduced. The loadest approach seems to performslightly better than any of the other approaches. Indeed, theWRMS of the differences of annual signals (in-phase andout-of-phase terms) and their correlations are smaller in allthe components except the out-of-phase term in the Northcomponent. However, the fit with the loading model is satis-factory for the three other approaches. Nonetheless, cautionshould be used when interpreting the results of the down-full strategy. The North annual out-of-phase terms recov-ered when the height uncertainty is multiplied by three seemto be under-evaluated compared with any other approaches.This is not the case when the height uncertainty is mul-tiplied by 1.5. This was not so obvious for the syntheticdata although it was visible. We notice that this effect isrelated to larger differences in the Z-translation annual sig-nal, see Fig. 8e. As a consequence, using an uncertaintyscaling factor of 1.5 or using diagonal weight only is pre-ferred. We think it is always better to leave the stochas-tic model unaffected, which is why we favor using either awell-distributed network for the frame transformation or esti-mating the low-degree coefficients of the deformation fieldsimultaneously.

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Strategies to mitigate aliasing of loading signals 11

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Fig. 7 Comparison of the annual signal estimated in the residuals ofthe frame transformation as defined in Sect. 2.3 applied to real data(Y-axis), and the annual signal estimated in the loading model (X-axis) in millimeters. a standard, scale estimated; b loadmod; c subnet;

d downfull height standard deviations multiplied by 1.5; e downfullheight standard deviations multiplied by 3.0; f loadest. Scale factorshave not been estimated for b to f. Same legend as Fig. 3

5 Discussion

Lavallée et al. (2006) suggested two approaches for esti-mating the low degrees of the load mass surface den-sity. We adopted here the so-called degree-1 deformationapproach since we wanted to remove the absorbed load-ing signal in the translations while preserving the net-trans-

lation due to degree-1 in these parameters. In that case,translation and rotation parameters and the degree-1 of themass surface density are estimated simultaneously as well ashigher degree terms. The CM-approach consists of model-ing the deformation field in the CM frame instead of the CFframe and estimating rotation parameters only since GPS istheoretically sensitive to CM. The two approaches lead to two

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Fig. 8 a Translations along the X-, Y- and Z-axis in millimetersand scale factors in millimeters (ppb value multiplied by 6.4) esti-mated between the MI1 weekly frames and the long-term frame. b–f

Differences between translation parameters estimated with tested strat-egies and those derived from the standard method shown in a. See thelegend of Fig. 7

Table 2 Geocenter motion (CM w.r.t. CF) annual signal from the degree-1 mass surface density parameters estimated from the GPS MI1 solutionswith two different strategies (lines 1 and 2)

X amplitude X phase Y amplitude Y phase Z amplitude Z phase(mm) (degrees) (mm) (degrees) (mm) (degrees)

Degree-1 deformation approach 3.8 ± 0.3 48 ± 4 2.0 ± 0.2 3 ± 5 7.9 ± 0.5 38 ± 3CM-approach 1.0 ± 0.2 71 ± 9 2.4 ± 0.2 310 ± 5 4.5 ± 0.3 51 ± 4CM-approach (Fritsche et al. 2010) 0.1 ± 0.2 43 ± 93 1.8 ± 0.2 334 ± 11 4.0 ± 0.2 25 ± 3GPS/GRACE/OBP (Collilieux et al. 2009) 1.3 ± 0.3 6 ± 14 3.0 ± 0.3 338 ± 6 4.6 ± 0.2 23 ± 3Loading model (Collilieux et al. 2009) 2.1 ± 0.1 28 ± 2 2.1 ± 0.1 338 ± 2 2.7 ± 0.1 48 ± 2Opposite of the translation from 0.4 ± 0.2 165 ± 30 3.6 ± 0.3 302 ± 5 4.4 ± 0.5 125 ± 7

degree-1 deformation approach

Phase are supplied according to the model A · cos(2π · (t − 2000.0) − φ) with t in years. Independent geocenter motion estimates are also reported(lines 3–5) as well as some values obtained from translation estimates (line 6)

distinct estimates of the mass surface density coefficient esti-mates. As an illustration, Table 2, lines 1 and 2, provides theannual signals estimated in the geocenter motion time seriescomputed from the degree-1 coefficients (equation 1 of Colli-lieux et al. (2009)). Differences may reach up to 3.4 mm in theamplitudes and may exceed one month in phase. Our CM-approach solution agrees within 1 mm in amplitude and 1month in phase with the reprocessed solution of Fritsche et al.(2010). For comparison, we also supplied two distinct mod-els from Collilieux et al. (2009). The first is a forward loading

model and the second is the result of a global inversion usingGPS, Gravity Recovery and Climate Experiment (GRACE)and OBP. Either of our two degree-1 estimations seems toagree better with these two loading models. However, thealiasing effect reduction using these two distinct deforma-tion fields is only different at the level of 0.3 mm RMS foreach translation component without clear seasonal patterns,thus validating the degree-1 deformation field approach usedfor the purpose of aliasing mitigation in this study. We alsoreported in Table 2 the opposite of the translation estimated

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Strategies to mitigate aliasing of loading signals 13

with the degree-1 deformation approach (sum of Fig. 8a andf). It can be observed that apparent reprocessed GPS geocen-ter motion is still not reliable, even with the aliasing effectremoved.

Thanks to the different tests performed here, we are nowable to draw some conclusions about the level of agreementbetween the GPS position time series and the loading model.Indeed, we can reasonably exclude the aliasing effect as beinga major source of discrepancies. When looking at Fig. 7f, thefollowing general comments can be formulated: The agree-ment of the annual signal in the Height component is goodon average since all the points are located along the diago-nal. The discrepancy is 1.6 mm WRMS for the in-phase termand 1.5 mm WRMS for the out phase term. In the horizon-tal, the loading model generally shows a smaller amplitudethan the GPS and the agreement for the in-phase and out-of-phase terms of the annual signals is less than 0.8 mm for bothcomponents. These results are encouraging but the discrep-ancies are still quite important. The horizontal componentsignals of the GPS stations should be investigated further inthe future studies, especially by comparing GPS results withdifferent loading models and the results of the GRACE tobetter understand the origin of the discrepancies shown here.

6 Conclusion and recommendations

We reviewed the procedures that can be used to modify theorigin, orientation, and scale of a time series of GPS frames.We paid attention to discuss the transformations that preservethe loading signals that are inherently contained in the stationcoordinates. This is especially important in order to inter-pret correctly the non-linear variations in the station posi-tion time series. Using synthetic data, we showed that thestandard approach consisting of using the largest set of sta-tions in the frame transformation is not optimal of whetherthe scale is estimated or not. The scale parameter should bedefinitively fixed to a constant value over time or its sea-sonal variations fixed to zero. But a rigorous approach ispossible only if all the frame time series are analyzed in oneunique estimation process. The benefit of using an alternativeapproach is especially important for the annual signals in thehorizontal components. Downweighting height, restrictingthe station set to a well-distributed sub-network, or estimat-ing the low-degrees of the load-surface density all performwell. The well-distributed network approach is the easiestto implement whereas the handling of the covariance termsis still to be defined when downweighting height. A slightadvantage is given to the third method consisting of esti-mating the deformation field, which is almost free of anysystematic bias according to our simulations. Thanks to thisstudy, we were able to conclude that the aliasing effect is notthe main source of discrepancy between GPS position time

series and the loading models. Annual signals are shown toagree at the 1.5 mm level WRMS in the height and the 0.8 mmlevel WRMS in the horizontal. Further studies are needed tounderstand the sources of the remaining inconsistencies.

Acknowledgments This work was partly funded by the CNESthrough a TOSCA grant. All the plots, except Fig. 6, have been madewith the General Mapping Tool (GMT) software (Wessel and Smith1991).

References

Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007)ITRF2005: a new release of the International Terrestrial ReferenceFrame based on time series of station positions and Earth Ori-entation Parameters. J Geophys Res 112(B09401). doi:10.1029/2007JB004949

Altamimi, Z and Collilieux, X and Métivier, L (2011) ITRF2008:an improved solution of the International Terrestrial ReferenceFrame. J Geodesy. doi:10.1007/s00190-011-0444-4

Blewitt G (2003) Self-consistency in reference frames, geocenter defi-nition, and surface loading of the solid Earth. J Geophys Res 108.doi:10.1029/2002JB002082

Blewitt G, Lavallée D (2000) Effect of annually repeating signalson geodetic velocity estimates. Tenth General Assembly of theWEGENER Project (WEGENER 2000), San Fernando, Spain,September 18–20

Collilieux X, Altamimi Z, Coulot D, Ray J, Sillard P (2007) Com-parison of very long baseline interferometry, GPS, and satellitelaser ranging height residuals from ITRF2005 using spectral andcorrelation methods. J Geophys Res 112(B12403). doi:10.1029/2007JB004933

Collilieux X, Altamimi Z, Ray J, van Dam T, Wu X (2009) Effectof the satellite laser ranging network distribution on geocentermotion estimation. J Geophys Res 114(B04402). doi:10.1029/2008JB005727

Collilieux X, Altamimi Z, Coulot D, van Dam T, Ray J (2010) Impactof loading effects on determination of the International TerrestrialReference Frame. Adv Space Res 45:144–154. doi:10.1016/j.asr.2009.08.024

Collilieux X, Métivier L, Altamimi Z, van Dam T, Ray J (2010b) Qual-ity assessment of GPS reprocessed Terrestrial Reference Frame.GPS Solut. doi:10.1007/s10291-010-0184-6

Farrell WE (1972) Deformation of the earth by surface loads. Rev Geo-phys Space Phys 10(3):761–797

Fritsche M, Dietrich R, Rülke A, Rothacher M, Steigenber-ger P (2010) Low-degree earth deformation from reprocessedGPS observations. GPS Solut 14(2):165–175. doi:10.1007/s10291-009-0130-7

Guo J, Ou J, Wang H (2010) Robust estimation for correlated obser-vations: two local sensitivity-based downweighting strategies. JGeodesy 84(4):243–250. doi:10.1007/s00190-009-0361-y

Han D, Wahr J (1995) The viscoelastic relaxation of a realistically strat-ified earth, and a further analysis of postglacial rebound. GeophysJ Int 120:287–311. doi:10.1111/j.1365-246X.1995.tb01819.x

Herring T (2004) GLOBK: Global Kalman filter VLBI and GPS anal-ysis program version 4.1. Tech. rep., Massachusetts Institute ofTechnology, Cambridge

Lavallée DA, van Dam TM, Blewitt G, Clarke PJ (2006) Geocentermotions from GPS: A unified observation model. J Geophys Res111:5405. doi:10.1029/2005JB003784

Mangiarotti S, Cazenave A, Soudarin L, Crétaux JF (2001) Annual ver-tical crustal motions predicted from surface mass redistribution

123

http://dx.doi.org/10.1029/2007JB004949http://dx.doi.org/10.1029/2007JB004949http://dx.doi.org/10.1007/s00190-011-0444-4http://dx.doi.org/10.1029/2002JB002082http://dx.doi.org/10.1029/2007JB004933http://dx.doi.org/10.1029/2007JB004933http://dx.doi.org/10.1029/2008JB005727http://dx.doi.org/10.1029/2008JB005727http://dx.doi.org/10.1016/j.asr.2009.08.024http://dx.doi.org/10.1016/j.asr.2009.08.024http://dx.doi.org/10.1007/s10291-010-0184-6http://dx.doi.org/10.1007/s10291-009-0130-7http://dx.doi.org/10.1007/s10291-009-0130-7http://dx.doi.org/10.1007/s00190-009-0361-yhttp://dx.doi.org/10.1111/j.1365-246X.1995.tb01819.xhttp://dx.doi.org/10.1029/2005JB003784

14 X. Collilieux et al.

and observed by space geodesy. J Geophys Res 106:4277–4292.doi:10.1029/2000JB900347

McCarthy D, Petit G (2004) IERS Technical Note 32-IERS Conventions(2003). Tech. rep., Verlag des Bundesamts fur Kartographie undGeodasie, Frankfurt am Main, Germany. http://maia.usno.navy.mil/conv2003.html

Petrov L, Boy J (2004) Study of the atmospheric pressure loading signalin very long baseline interferometry observations. J Geophys Res109(B18). doi:10.1029/2003JB002500

Ray J, Altamimi Z, Collilieux X, Dam TMvan (2008) Anomalous har-monics in the spectra of GPS position estimates. GPS solution12(1):55–64. doi:10.1007/s10291-007-0067-7

Rodell M, Houser PR, Jambor U, Gottschalck J, Mitchell K, MengCJ, Arsenault K, Cosgrove B, Radakovich J, Bosilovich M, EntinJK, Walker JP, Lohmann D, Toll D (2004) The Global Land DataAssimilation System. Bull Amer Meteor Soc 85(3):381–394.doi:10.1175/BAMS-85-3-381

Stammer D, Wunsch C, Fukumori I, Marshall J (2002) State esti-mation improves prospects for ocean research. Eos Trans AGU83(27):289–295. doi:10.1029/2002EO000207

Tregoning P, van Dam TM (2005) Effects of atmospheric pressure load-ing and seven-parameter transformations on estimates of geocen-ter motion and station heights from space geodetic observations. JGeophys Res 110:3408. doi:10.1029/2004JB003334

van Dam TM, Herring TA (1994) Detection of atmospheric pressureloading using very long baseline interferometry measurements. JGeophys Res 99:4505–5417. doi:10.1029/93JB02758

van Dam TM, Blewitt G, Heflin MB (1994) Atmospheric pressure load-ing effects on Global Positioning System coordinate determina-tions. J Geophys Res 99:23939–23950. doi:10.1029/94JB02122

van Dam TM, Wahr J, Milly PCD, Shmakin AB, Blewitt G, LavalléeD, Larson KM (2001) Crustal displacements due to continentalwater loading. Geophys Res Lett 28:651–654. doi:10.1029/2000GL012120

Wessel P, Smith WHF (1991) Free software helps map and display data.EOS Trans 72:441. doi:10.1029/90EO00319

Wu X, Argus DF, Heflin MB, Ivins ER, Webb FH (2002) Site distribu-tion and aliasing effects in the inversion for load coefficients andgeocenter motion from GPS data. Geophys Res Lett 29(24):2210.doi:10.1029/2002GL016324

Wu X, Heflin MB, Ivins ER, Argus DF, Webb FH (2003) Large-scaleglobal surface mass variations inferred from GPS measurements ofload-induced deformation. Geophys Res Lett 30(14):1742. doi:10.1029/2003GL017546

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http://dx.doi.org/10.1029/2000JB900347http://maia.usno.navy.mil/conv2003.htmlhttp://maia.usno.navy.mil/conv2003.htmlhttp://dx.doi.org/10.1029/2003JB002500http://dx.doi.org/10.1007/s10291-007-0067-7http://dx.doi.org/10.1175/BAMS-85-3-381http://dx.doi.org/10.1029/2002EO000207http://dx.doi.org/10.1029/2004JB003334http://dx.doi.org/10.1029/93JB02758http://dx.doi.org/10.1029/94JB02122http://dx.doi.org/10.1029/2000GL012120http://dx.doi.org/10.1029/2000GL012120http://dx.doi.org/10.1029/90EO00319http://dx.doi.org/10.1029/2002GL016324http://dx.doi.org/10.1029/2003GL017546http://dx.doi.org/10.1029/2003GL017546

Strategies to mitigate aliasing of loading signals while estimating GPS frame parametersAbstract1 Introduction2 Strategy2.1 Method to compute position time series2.2 Synthetic data and tests2.3 Description of tests2.4 Evaluation of the aliasing error

3 Evaluation of the methods3.1 Scale3.2 Downweighting height3.3 Using a sub-network3.4 Estimating the deformation field3.5 NNR-condition

4 Application to real data5 Discussion6 Conclusion and recommendationsAcknowledgmentsReferences