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Advances in Space Research 42 (2008) 727–736

Using the Global Navigation Satellite System and satellite altimetryfor combined Global Ionosphere Maps

S. Todorova *, T. Hobiger 1, H. Schuh

Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstr. 27-29, 1040 Vienna, Austria

Received 31 October 2006; received in revised form 25 July 2007; accepted 14 August 2007

Abstract

For deriving global maps of the Total Electron Content (TEC) from space geodetic techniques usually observations from the GlobalNavigation Satellite System (GNSS) are taken. However, the GNSS stations are inhomogeneously distributed, with large gaps particu-larly over the sea surface.

Within this study we create Global Ionosphere Maps (GIM) from GNSS data and additionally introduce satellite altimetry observa-tions, which help to compensate the insufficient GNSS coverage of the oceans. The obtained global maps are in 2 h intervals and dailyvalues of Differential Code Biases (DCB) for all the GNSS satellites and receivers are also estimated. The combination of the data fromaround 160 GNSS stations and two satellite altimetry missions – Jason-1 and TOPEX/Poseidon – is performed on the normal equationlevel. The comparison between the integrated ionosphere models and the GNSS-only maps shows a higher accuracy of the combinedGIM over the seas. The study aims at improved combined global TEC maps, which should make best use of the advantages of eachparticular type of data and have higher accuracy and reliability than the results derived by the two methods if treated individually.� 2007 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Global Ionosphere Maps; DCB; Global Navigation Satellite System; Satellite altimetry; Combination

1. Introduction

1.1. The ionosphere

The ionosphere can be defined as that part of the upperatmosphere where the density of free electrons and ions ishigh enough to influence the propagation of electromag-netic radio frequency waves (Hargreaves, 1992). The ioni-sation process is primarily driven by the Sun and itsactivity varies strongly with time, as well as with geograph-ical location. When electromagnetic waves travel throughthe ionosphere, the integration between the electromag-netic field and the free electrons influences both the speed

0273-1177/$34.00 � 2007 COSPAR. Published by Elsevier Ltd. All rights rese

doi:10.1016/j.asr.2007.08.024

* Corresponding author.E-mail address: stodo@mars.hg.tuwien.ac.at (S. Todorova).

1 Present address: Space-Time Standards Group, Kashima SpaceResearch Center, National Institute of Information and CommunicationsTechnology, 893-1 Hirai, Kashima, 314-0012 Ibaraki, Japan.

and the propagation direction of the signals. This effect isknown as ionospheric refraction (Hartmann and Leitinger,1984) and has to be considered in the determination of thepropagation velocity of the signals of all space geodetictechniques operating with electromagnetic waves. Theionospheric refraction can be determined in terms of STEC(Slant Total Electron Content), which is the integral of theelectron density along the signal path S (Eqs. (1) and (2)).STEC is measured in Total Electron Content Units(TECU), with 1 TECU equivalent to 1016 electrons/m2.Eqs. (1) and (2) express the effect in meters of the ionisedmedium on phase and group signal propagation:

Iphase ¼Z

S1� 40:28N e

f 2

� �� 1

� �dS

¼ � 40:28

f 2

ZS

N edS ¼ � 40:28� 1016

f 2STEC; ð1Þ

rved.

728 S. Todorova et al. / Advances in Space Research 42 (2008) 727–736

Igroup ¼Z

S1þ 40:28N e

f 2

� �� 1

� �dS ¼ 40:28

f 2

ZS

N edS

¼ 40:28� 1016

f 2STEC; ð2Þ

where f is the carrier frequency in Hz and Ne the free elec-tron density in the medium.

Usually, observing at two different radio frequenciesallows the elimination of the ionospheric influence and –on the other hand – provides information about the iono-sphere parameters. If the behaviour of the ionosphere isknown, the ionospheric refraction can be computed viaEqs. (1) and (2) and used to correct single-frequencymeasurements.

1.2. GNSS as a tool for investigating the ionosphere

The Global Navigation Satellite System (GNSS), pres-ently consisting on GPS (Global Positioning System) andGLONASS (GLObal NAvigation Satellites System), pro-vides information about the ionospheric refraction,enabling high resolution ionosphere imaging in longitude,latitude and time (e.g. Brunini, 1997; Schaer, 1999 and ref-erences therein). Both observables of the system – carrierphase and code measurements – are affected by the iono-sphere. According to Eqs. (1) and (2), this effect dependson the signal frequency f and on the STEC between thesatellite and the receiver. Thus, forming the so-called geom-etry-free linear combination by subtracting simultaneousobservations at the two different frequencies L1 and L2,and in this way removing all frequency-independent effects(such as clock errors, troposphere delay etc.), leads to anobservable, which contains only the ionospheric refractionI and the inter-frequency hardware biases Dbk and Dbi

(usually in nanoseconds), associated with the satellite k

and the receiver i. In the case of carrier phase observationsthe ionospheric observable reads as:

Uki;4 ¼ Uk

i;1 � Uki;2 ¼ �n4aIk

i þ cðDbk � DbiÞ; ð3Þ

where Uki;1 and Uk

i;2 are the carrier phase observations at thetwo frequencies, corrected by the carrier phase ambiguities(see Section 2.1 – dual-frequency carrier-phase smoothedcode observations), Ik

i is the ionospheric refraction betweenthe satellite and the receiver related to L1 (in meters),n4 ¼ 1� f 2

1 =f 22 is a factor for relating the ionospheric

refraction on L4 to L1 and a is a constant used to convertmeters into TECU. As it emerges from Eq. (3), the geom-etry-free linear combination is very appropriate for extract-ing information about the ionosphere. It has to be noted,that the derived ionospheric parameters are affected bythe inter-frequency hardware biases (e.g. Mannucci et al.,1998), also called Differential Code Biases (DCB), so whenmodelling the ionosphere it is necessary to estimate them asadditional unknowns.

In 1998 a special Ionosphere Working Group of theInternational GNSS Service (IGS) was initiated for

developing global ionospheric TEC gird (Feltens andSchaer, 1998; Hernandez-Pajares, 2004). Up to now, fourAnalysis Centres (AC) – Centre for Orbit Determinationin Europe (CODE) (Schaer, 1999), European SpaceAgency (ESA) (Feltens, 1998), Jet Propulsion Laboratory(JPL) (Mannucci et al., 1998), and Universidad Politec-nica de Cataluna (UPC) (Hernandez-Pajares et al.,1999), deliver daily global maps of vertical TEC andDCB values in the IONospheric EXchange (IONEX) for-mat (Schaer et al., 1998) by using different estimationmethods. Since the end of 2005 a combined IGS solutionis also available.

1.3. Ionosphere parameters from satellite altimetry data

Satellite altimetry missions with double-frequency radaraltimeter on-board, such as TOPEX/Poseidon (T/P) andJason-1, also provide information about the ionosphere.The T/P mission was launched in August 1992 for observ-ing the ocean circulation and was operational till October2005. Jason-1, launched in December 2001, is the follow-on to T/P and has inherited its main features – orbit,instruments, measurement accuracy, etc. The orbit altitudeof the two missions is 1336 km.

The primary sensor of both T/P and Jason-1 is theNASA Radar Altimeter operating at 13.6 GHz (Ku-band)and 5.3 GHz (C-band), simultaneously. The two widelyseparated frequencies allow TEC to be detected directlyfrom the nadir altimetry sampling data along the satellitetrack (Imel, 1994).

1.4. TEC estimation from GNSS and satellite altimetry

In order to asses the precision of the TEC estimatesfrom GNSS and from satellite altimetry measurements,the results of the two methods have often been compared(Brunini et al., 2005 and references therein). Althoughthere is a good general agreement between GNSS andaltimetry derived TEC, there are still some open ques-tions. One of them is related to the better understandingof the frequency-dependent systematic errors in thealtimetry measurements, which would bias both thesea-level height and the TEC estimates (Chelton et al.,2001). Moreover, several studies have shown that T/Pand Jason-1 overestimate the vertical TEC by about 3–4 TECU compared to the values delivered by GNSS(i.e. Brunini et al., 2005). This is a contradiction becauseopposite to GNSS, the altimetry derived TEC is not sen-sitive to the plasmaspheric contribution (TEC between�1300 and �20,000 km height above the Earth surface)due to the lower orbit altitude of the altimetry satellites.On the other hand, most of the ionosphere models fromGNSS data are based on the Single Layer Model(described in Section 2.1), which does not account wellfor the ionospheric contribution above the altitude ofthe altimetry missions (Brunini et al., 2005). The differ-ences between GNSS and altimetry derived TEC as well

S. Todorova et al. / Advances in Space Research 42 (2008) 727–736 729

as the systematic errors of the Jason-1 and T/P satellitesare treated in more detail in Sections 3.2, 3.3 and 3.4.

It has to be pointed out, that when using a single layermodel, the STEC values derived from GNSS measurementshave to be converted into vertical TEC (VTEC), while thealtimetry missions deliver directly the vertical values. Themapping function used for this conversion is also a poten-tial error source for the GNSS TEC estimates. Finally, forcomparing with altimetry TEC, the values derived fromGNSS have to be interpolated for regions far from theobserving stations, i.e. such comparisons are performedin the worst scenario for GNSS.

2. Methodology

2.1. Global models of the ionosphere in 3D

The global maps created in this study represent the ion-osphere in longitude, latitude and time and are based onthe Single Layer Model (SLM). SLM assumes that all freeelectrons are concentrated in an infinitesimally thin layerabove the Earth surface. The height H of this thin shell isusually set between 300 and 450 km. A signal transmittedfrom the satellite to the receiver crosses the ionosphericshell in the so-called ionospheric pierce point. The zenithangle at that point is z 0 and the signal arrives at the groundstation with zenith distance z. The relation between themeasured STEC along the ray path and the vertical valueat the pierce point is given by a mapping function (Eq.(4)).The mapping function for the transformation betweenSTEC and VTEC reads as:

F ðzÞ ¼ STEC

VTEC¼ 1

cos z0¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� Re

ReþH sinðazÞ� �2

r ; ð4Þ

Fig. 1. Data used for the IGG GIM: (a) IGS GNSS stations and (b) fo

In this study the Modified Single Layer Model (MSLM)mapping function approximating the JPL Extended SlabModel (ESM) mapping function was adopted. The bestfit with the ESM mapping function is achieved at H

= 506.7 km and a = 0.9782 (as given in http://aiuws.unibe.ch/ionosphere/mslm.pdf). The GNSS-derived STEC valuesare extracted from the geometry-free linear combinationapplied on dual-frequency carrier-phase smoothed codeobservations. The phase-smoothed pseudorange observa-tions are adapted from statistic comparison of continuoustime series of dual-frequency code and phase measure-ments. This method allows the approximate determinationof the ambiguities of the L1 and L2 carriers and leads tosignificant reduction of the noise of the original codemeasurements (Schaer, 1999). Data from around 180stations of the IGS is used with sampling rate of 30 s. Inthe case of satellite altimetry, the original ionosphericcorrection from T/P and Jason-1 is adopted and convertedinto VTEC by a factor depending on the operationalfrequency of the altimeter.

In this work for the global representation of VTEC aspherical harmonic expansion up to degree and order 15was chosen (Schaer, 1999):

EVðb;sÞ¼Xnmax

n¼0

Xn

m¼0

eP nmðsinbÞðanm cosðmsÞþbnm sinðmsÞÞ; ð5Þ

where EV is the Vertical Total Electron Content, b is thegeomagnetic latitude of the ionospheric pierce point,s = kG + UT � p (with kG – geographical longitude) isthe sun-fixed longitude of the ionospheric pierce point,eP nm is the normalized Legendre function of degree n

and order m, and anm and bnm are the unknown coeffi-cients of the spherical expansion. A Matlab-based soft-ware was developed for computation of 12 two-hourlyGIM per day and of the corresponding RMS (Root

otprints of T/P and Jason-1 in two-hourly intervals, day 022 2005.

730 S. Todorova et al. / Advances in Space Research 42 (2008) 727–736

Mean Square) maps, and daily values of the DCB for allthe GNSS satellites and receivers. The final outputs arein the IONEX format.

2.2. Combination

For the combination of GNSS and altimetry data aleast-squares adjustment is applied on each set of observa-tions and then the normal equations are combined. This isdone by adding the relevant normal matrices obtainedfrom the two types of observations:

NCOMB ¼ N GNSS þ N ALT

¼ ATGNSS � pGNSS � AGNSS þ AT

ALT � pALT � AALT ð6Þ

Fig. 2. (a) VTEC and (b) RMS map, IGG GN

where N and A are the corresponding normal and designmatrices and p denotes a matrix with the weights on itsmain diagonal.

At this stage of our work we adopt equal weights(pGNSS = 12) for all GNSS observations in both theGNSS-only and the combined solution. Later on a moreelaborate weighting of the GNSS data will be performed,accounting for the difference between the time in whichevery single observation has been made and the nominaltime of the corresponding ionosphere map. As for the rela-tive weighting of the altimetry data, different strategies arepossible. On the one hand, due to the much higher numberof GNSS measurements compared to satellite altimetry, theT/P and Jason-1 data should be overweighted, in order toincrease its impact on the combined GIM. When adopting

SS-only model, day 022 2005, 17:00 UT.

S. Todorova et al. / Advances in Space Research 42 (2008) 727–736 731

the a priori variance of ralt0 ¼ 0:25 TECU (pALT = 42) for

the altimetry measurements, the differences between thecombined and the GNSS-only GIM can reach up to±10 TECU and the RMS of the combined maps decreaseswith up to 1 TECU over the areas with altimetry observa-tions. In the case of overweighting, however, it becomescrucial to assess the bias between GNSS and altimetryTEC, discussed in Section 1.4. On the other hand, if wetake into account the higher noise of the altimetry measure-ments compared to the carrier-phase smoothed code obser-vations from GNSS, a lower weight should be applied onall T/P and Jason-1 derived observations. For the com-bined GIM presented in Sections 3.2 and 3.3 the adoptedweight is pALT ¼ 1

4

2, which corresponds to a priori vari-

ance ralt0 ¼ 4 TECU. It has to be pointed out, that the rela-

tive weighting acts like a scaling factor for the contributionof the altimetry data in the combined GIM. It is a verycomplex issue, depending on the different spatial and tem-poral distribution of the observations and on their specificsystematic errors. Therefore, the relative weighting of thetwo types of measurements needs to be optimised and isa matter of further investigation. As a next step, a variancecomponent estimation can be carried out in order to assessthe optimal relative weighting of the two data sets.

Fig. 3. Satellite DCB for GPS and GLONASS, IGG GNSS-only model for dCODE.

Nevertheless, it can be anticipated that combining GNSSand altimetry data will improve the general robustnessand reliability of the GIM and their quality particularlyover the oceans. As it can be seen in Fig. 1 the spatial dis-tribution of the altimetry observations helps to balance thegaps between the GNSS stations. Moreover, the combina-tion procedure allows the independent estimation of tech-nique-specific constant time delays and can thus be usedto indicate and model the technique-specific systematics.

3. Results

Both the GNSS-only and the combined solutions devel-oped within this work are referred to as IGG (Institute ofGeodesy and Geophysics) GNSS-only and combined mod-els. The current results of our study are presented below,considering as example the outcomes for day 022 in 2005.

3.1. GNSS-only model

Fig. 2 shows the IGG GNSS-only VTEC and RMSmaps for 17:00 UT. As expected, the precision of the mapsis lower in areas where no GNSS sites are located, which ismainly above the sea surface and in the southern polar

ay 022 2005 vs. the monthly DCB provided by the IGS Analysis Center

Fig. 4. (a) VTEC and (b) RMS map, IGG combined model, day 022 2005, 17:00 UT.

732 S. Todorova et al. / Advances in Space Research 42 (2008) 727–736

region. The a posteriori sigma of the GNSS-only solution isrgnss

ap ¼ 4:74 TECU. The mean bias between the estimatedVTEC maps and the GIM provided by the IGS AnalysisCenter CODE2 is 0.4 TECU with a standard deviation of±0.5 TECU.

In parallel to the ionosphere parameters, DifferentialCode Biases for all GNSS satellites and ground stationsare computed daily as constant values, with a zero-meancondition imposed on the satellite DCB. The estimatedvalues are routinely compared with the monthly DCBprovided by the IGS Analysis Center CODE. Such a

2 Global Ionosphere Maps Produced by CODE, http://www.aiub.unibe.ch/content/research/gnss/code_research/igs/global_ionosphere_maps_produced_by_code/index_eng.html.

comparison for the satellite DCB can be seen onFig. 3. Taking into account, that the daily values esti-mated here are compared with the more robust monthlyaverage of the DCB provided by CODE, it can be statedthat the agreement is fairly good.

3.2. Combined solution

The integrated GIM for 17:00 UT on the same day022 2005, is shown on Fig. 4. A closer look at the differ-ence between the combined maps and the GNSS-onlysolution (Fig. 5) shows a general lowering of the RMSof the combined model of up to 0.13 TECU especiallyover the areas coinciding with the footprints of Jason-1(Fig. 5b). The a posteriori sigma of the combined solu-

S. Todorova et al. / Advances in Space Research 42 (2008) 727–736 733

tion is also slightly lower: rcombap ¼ 4:71 TECU. As for the

VTEC (Fig. 5a), there is a slight increase of the VTECvalues (up to 0.8 TECU) particularly over the oceansand in the low southern latitudes, where nearly no GNSSobservations are available. The decrease of VTEC in theequatorial area, coinciding with the region of highestionospheric activity for this map, can be interpreted asthe signature of the plasmaspheric component. Asalready mentioned, the altimetry measurements do notaccount for the topside ionosphere and therefore, despiteof the discussed TEC overestimation, the integration ofaltimetry data in the GNSS GIM leads to a decreaseof the obtained TEC over the area where the plasma-spheric contribution reaches its maximum.

Fig. 5. (a) VTEC and (b) RMS map, IGG combined min

3.3. Validation of the results by using observations from

GNSS stations not included in the model

One basic problem we are facing in this study is the val-idation of the combined GIM. Since the measurementsfrom T/P and Jason-1 are included in the models, this datacannot be used for assessment of the quality of the com-bined solutions. Therefore, we adopted a procedure, which– slightly modified – is used for self-consistency testing ofthe IGS GIM. As reference for this test we use the STECmeasured from GNSS stations, which are not included inour GNSS-only and combined models. Moreover, the cho-sen reference stations are located in the ocean or at thecoast, far from other GNSS receivers. These conditions

us IGG GNSS-only model, day 022 2005, 17:00 UT.

734 S. Todorova et al. / Advances in Space Research 42 (2008) 727–736

assure that the STEC measured at the external stations canbe regarded as independent of the data used in the adjust-ment. Moreover, due to their location, the observations atthese sites should coincide better with the prediction of thecombined model, in which data sampled above the sea isincluded.

The observed values at the reference station to a givensatellite are obtained by building the differences betweenthe geometry-free linear combination L4 obtained at thehighest elevation and the rest of the observations to thesame satellite. In this way the receiver and satellite hard-ware delay bi

k is eliminated and the remaining DSTEC val-ues can be used as a reference observation:

L4elevj

ref � L4elevmaxref ¼ STEC

elevj

ref þ bik � STECelevmax

ref � bik

¼ STECelevj

ref � STECelevmaxref

¼ DSTECref ð7Þ

j = 1, 2, . . . ,n, where n is the number of elevations.The model DSTEC values are obtained by interpolating

the VTEC corresponding to the reference station from theIGG combined and GNSS-only GIM and mapping it backto STEC using the same elevations. Then DSTEC is com-puted via Eq. (7) for both the GNSS-only and the com-bined model. Finally, the standard deviations of thedifference DSTECref � DSTECmodel are computed andcompared:

stdcomb ¼ stdðDSTECref � DSTECcombÞ ð8Þstdgnss-only ¼ stdðDSTECref � DSTECgnss-onlyÞ ð9Þ

A negative difference between the values of stdcomb andstdgnss-only for a given elevation indicates that the TEC pre-dicted by the combined model reproduces the observed ref-erence value better than the GNSS-only GIM.

Fig. 6. Comparison of the predictions from the combined and the GNSS-only GIM for day 022 2005 with observed values obtained from GNSSstations not included in the models; negative values of Dstd indicate betterperformance of the combined model for the given elevation.

For testing the IGG combined and GNSS-only modelsthree reference stations were chosen: ASPA (u = �14.3�,k = �170.7�, island of Tutuila, American Samoa), kour(u = 5.2�, k = �52.8�, Atlantic coast, French Guiana)and PDEL (u = 37.7�, k = �25.7�, Sao Miguel Island,Azores). The results shown in Fig. 6 indicate, that theIGG combined model performs better than the GNSS-onlyGIM for more than 50% of the elevations for all three sta-tions. It can be expected, that the optimisation of the rela-tive weighting and proper treatment of the bias betweenGNSS and altimetry TEC will lead to further improvementof the performance of the combined GIM.

3.4. Comparison with raw Jason-1 data

The rapid and final Global Ionosphere Maps producedby the IGS AC and described in Section 1.2, are routinelyvalidated with T/P and Jason-1 TEC (Orus et al., 2003). Inorder to assess the self-consistency of our approach and theaccuracy of the GIM created within this work, the IGGmodels were also included in this validation procedure, inwhich the VTEC delivered by Jason-1 along its track iscompared with the corresponding values from the globalmaps from GNSS data. The comparison was performedfor eight consecutive days in 2005 including both, theIGG GNSS-only and the combined GIM with integratedT/P and Jason-1 data. It has to be noted, that the weightof the altimetry data in the IGG combined models includedin this test is pALT = 42. Furthermore, the IGG GNSS-onlysolution used for this comparison was obtained frompseudorange observations, whereas the GNSS data incor-porated in the IGG combined model was obtained fromphase-smoothed code measurements.

Fig. 7 shows the mean bias and percentage error of the finalCODE, ESA, JPL, UPS, and the combined IGS GIM as wellas the two IGG solutions compared to Jason-1. The compar-ison is performed in time (Fig. 7a) and in latitude (Fig. 7b).The two IGG models agree well with the results of the IGSAC and particularly with the CODE solution, which is alsoobtained by using spherical harmonics. The Jason-1 dataintegrated in the combined IGG model causes a lowering ofthe difference compared to the GNSS-only models.

In low and mid latitudes (±40�) nearly all global modelscoincide with the measurements from Jason-1 within1 TECU (Fig. 7b). The differences increase at higher lati-tudes, and, on the contrary, in the equatorial area theybecome even negative. This behaviour of the differencesbetween GNSS and altimetry derived TEC, which can beobserved for nearly all AC’s GIM, can be regarded as avisualisation of the insufficient performance of the altime-try measurements in low latitudes, caused by the contribu-tion of the topside ionosphere.

4. Conclusions and outlook

It was shown that the combined GIM from GNSSand altimetry data have the potential to contribute to

Fig. 7. Jason-1 minus IGS final GIM and IGG GNSS-only and combined solutions, mean bias and percentage error in (a) time and (b) latitude over eightdays (days 022–029 2005).

S. Todorova et al. / Advances in Space Research 42 (2008) 727–736 735

the accuracy of the global ionosphere modelling as wellas to the better understanding of the ionosphere as awhole. Still, the combined GIM must be further opti-mised, with main focus on the weighting of the individ-ual results from the different techniques. An importanttopic is to consider and properly model the technique-specific error sources. Therefore, the next step in ourstudy is the estimation of an altimetry bias in parallelto the parameters of the ionosphere. Taking into accountthat Jason-1 and T/P seem to overestimate the TECcompared to GNSS, it can be assumed that the altimetrymeasurements are biased by a constant instrumental off-set. Using the combination procedure, realised by stack-ing of the normal equations, it is possible to developcombined ionosphere models with additional estimationof daily biases for the altimetry satellites, similar to theGNSS DCB. If computed as a single unknown, this biaswill include the plasmaspheric component, additionally tothe actual instrumental delay. In a next step the behav-iour of the differences between the GNSS-only solutionand the altimetry TEC (discussed in Sections 3.2 and3.4) can be implemented for the development of a func-tion, which accounts for both the instrumental bias andthe plasmaspheric contribution, instead of a constantbias for the altimetry satellites.

Finally, in order to achieve a global coverage and higheraccuracy and reliability of the ionosphere models, the com-bination method can be adopted also for ionospheric datafrom other space geodetic techniques, such as VLBI andDORIS.

Acknowledgements

Project P19564-N10 is funded by the Austrian Sci-ence Fund (FWF). Thanks to the International GNSSService (IGS) and to ADSCentral, GeoForschungsZen-trum Potsdam (GFZ), for the free supply with GNSSand altimetry data. We are also grateful for the freeavailability of the Generic Mapping Tools (GMT) andthe GPS Toolkit (GPStk) software. Many thanks toDr. Manuel Hernandez-Pajares for his assistance andto two anonymous reviewers for very helpful commentsand suggestions.

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