Comparative analysis of synthetic GNSS time series - Bias ...

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Comparative analysis of synthetic GNSS time series -Bias and precision of velocity estimations

Stephane Mazzotti, Aline Déprez, Eric Henrion, Christine Masson, FrédéricMasson, Jean-Luc Menut, Marianne Métois, Jean Matthieu Nocquet, Lucie

Rolland, Pierre Sakic, et al.

To cite this version:Stephane Mazzotti, Aline Déprez, Eric Henrion, Christine Masson, Frédéric Masson, et al.. Compar-ative analysis of synthetic GNSS time series - Bias and precision of velocity estimations. [ResearchReport] RESIF. 2020. �hal-02460380�

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RENAG–09/2019–SyntheticGNSStimeseriescomparisonsandvelocityestimations 1

RENAG-RESIFReportComparativeanalysisofsyntheticGNSStimeseries-Biasandprecisionofvelocityestimations



ComparativeanalysisofsyntheticGNSStimeseries-BiasandprecisionofvelocityestimationsStephaneMazzotti1,AlineDeprez2,EricHenrion3,ChristineMasson1,FrédéricMasson3,Jean-LucMenut4,MarianneMétois5,Jean-MatthieuNocquet4,6,LucieRolland4,PierreSakic7,AnneSocquet2,AlvaroSantamaría-Gómez8,PierreValty9,MathildeVergnole4,PhilippeVernant1

1.GéosciencesMontpellier(GM),UMR5243,UniversitédeMontpellier,CNRS.PlaceE.Bataillon,34095Montpelliercedex,France.

2.InstitutdesSciencesdelaTerre(ISTerre),UGA,CNRS,USMB,IRD,IFSTTAR.38058GrenobleCedex9,France.

3.InstitutdePhysiqueduGlobe,UMR7516,UniversitédeStrasbourg/EOST,CNRS.5rueRenéDescartes,F-67084StrasbourgCedex,France.

4.Géoazur,UniversitéCôted'Azur,CNRS,ObservatoiredelaCôted'Azur,IRD.250rueAlbertEinstein,SophiaAntipolis,06560Valbonne,France.

5.LaboratoiredeGéologiedeLyon(LGTPE),UniversitéClaudeBernardLyon1,bat.Géode,69100Villeurbane,France.

6.InstitutdePhysiqueduGlobeParis.1rueJussieu,75238Pariscedex05,France.

7.Helmholtz-Zentrum,GeoForschungsZentrum.Telegrafenberg(GFZ),D-14473Potsdam,Germany.

8.GET,UniversitédeToulouse,CNES,CNRS,IRD,UPS.Toulouse,France.

9.Servicedegéodésieetnivellement,IGN.73,Av.deParis,94160SaintMande,France.

ThisreportsummarizestheanalysescarriedoutbetweenFebruary2018andJune2019withintheRENAGconsortium(RéseauNationalGNSS,http://renag.resif.fr),partoftheRESIFresearchinfrastructure (FrenchSeismologicandGeodeticNetwork,https://www.resif.fr).TheprojectwasimplementedasaninternalexercisedesignedtoassessthevariabilityinGNSStimeseriesanalysesandderivedvelocitiesfortheRENAG-RESIFstations.

Synthetictimeseriescreations,statisticalanalysesofsolutions,andfiguresweredoneusingtheRsoftware(RCoreTeam,2016).


Summary105synthetictimeseriesreplicatingGNSS3DpositionseriesareanalyzedindependentlybyninedifferentgroupswithintheRENAGconsortiuminordertocharacterizethevariabilityin estimations of long-term velocities. The main objective is not a detailed study of theparametersandsourcescontrollingvelocityvariations,butsimply toestablish first-orderconclusions regarding theuncertaintiesonGNSSvelocityestimationsasa functionof thedifferentanalysismethodsandsoftware.Becausethetruevelocitiesareknown,ourresultsarepresentedintermsofvelocitybiases(i.e.deviationsoftheestimatedvelocitiesrelativeto the expected values). Statistics on these biases can then be used as indicators of thepotentialprecisionofactualGNSSvelocities.

Tofirstorder,theninemethodsandsoftwareoftimeseriesanalysisprovidehorizontal(resp.vertical)velocityestimationsatprecisionsbetterthan1.0mm/a(resp.2.0mm/a).Noneofthetestedmethodsorsoftwareclearlystandsoutassignificantlybetterorworsethantheothers.However, a group of four solutions (including the unweighted average of all ninesolutions) provides systematically better results than the others. They are based on astandardtimeseriesanalysisusingaleast-squareinversionofaparametricmodel(velocity,seasonalterms,offsets)witheitherautomaticandmanualoffsetdetectionmethods.

FortimeserieswithnoiseanddurationcharacteristicscorrespondingtoclassicalGNSSdata(e.g.,RENAG-RESIFstations),thevelocitybiases(andthuspotentialGNSSvelocityprecision)arecharacterizedbythefollowingstatistics:

• Medians ca. 0.1 mm/a (horizontal components) and 0.1–0.3 mm/a (verticalcomponent).

• 95th percentiles ca. 0.2–0.7 mm/a (horizontal components) and 0.5–2.0 mm/a(verticalcomponent).

• RMS (root-mean-square) ca. 0.1–0.3 mm/a (horizontal components) and 0.3–0.9mm/a(verticalcomponent).

Inadditiontothevariabilityofvelocityestimationsasafunctionoftheanalysismethods,first order information can be derived regarding the solution combination and velocityuncertainties:

• Theunweightedaverageofallnineanalysesyieldsresultssystematicallyintheuppertierofallindividualsolutions.

• Formal velocityuncertainties (standard errors) calculatedon thebasis of colored-noisemodelsarestaticallyrepresentativeofthevelocitybiases.

• In contrast, formal velocity uncertainties (standard errors) calculated using othermethods (whitenoiseorstatisticalvariance)arenot representativeof thevelocitybiases(resp.significantlytoolowortoohigh).


1)SyntheticdatadescriptionSyntheticpositiontimeseriesaregeneratedforasetof35stationsgroupedin5categoriesrepresenting the spectrum of actual GNSS series characteristics (noise level, number ofoffsets,completeness).Fromthehighesttothelowestquality,thefivegroupsare:“SC”(SuperClean, stations SC01–SC07); “RC” (Rather Clean, stations RC01–RC07); “MR” (Mid Range,stationsMR01–MR07);“RU”(RatherUgly,stationsRU01–RU07);“SU”(SuperUgly,stationsSU01–SU07).Thelong-termvelocities(cf.eq.1)rangebetween-2and+2mm/a,withthemajorityca.-0.5–+0.5mm/a,inordertorepresenttherelativevelocitiesbetweenstationsinFranceandWesternEurope(i.e., intraplatevelocitiesinalocalreferenceframesuchasthosemeasuredbyRENAG-RESIFstations).

NB: Stations in the SU group correspond to rare cases of extreme noise levels and are notrepresentative of standard GNSS series used in geodetic studies. They are provided forinformationbutarenotincludedintheresultsandconclusions.

Foreachstation,threedailypositiontimeseriesx(t)aregeneratedtorepresenttheNorth,EastandUpcomponents:

𝑥(𝑡) = 𝑥& + 𝑣𝑡 + 𝐴*𝑠𝑖𝑛(𝜔*𝑡 + 𝜙*) + 𝐴0𝑠𝑖𝑛(𝜔0𝑡 + 𝜙0) + 𝐵2𝐻(𝑡, 𝑇2) + 𝜀(𝑡, 𝜅, 𝐶) (1)

withx0:interceptofthepositionseries;v:linearvelocity;𝜔*and𝜔0:periodsoftheannualandsemi-annualsignals;A1andA2:amplitudesoftheannualandsemi-annualsignals;BiandTi:amplitudeanddateoftheithoffsetexpressedasaHeavisidefunction:

𝐻(𝑡, 𝑇2) = 0, 𝑡 < 𝑇2; 𝐻(𝑡, 𝑇2) = 1, 𝑡 ≥ 𝑇2 (2)

TheoffsetdatesTi are identical for the three componentsbut theamplitudesBi canvarybetween each component. A random daily dispersion 𝜀(𝑡, 𝜅, 𝐶) is added to each seriesfollowing a colored-noise model (Kasdin, 1995), with 𝜅 and C the spectral index andamplitude(expressedasthedailyRMS_Root-Mean-Square)ofthedispersion.Theannualandsemi-annualphases(𝜙*and𝜙0)arerandomlygeneratedforeachseries.TheparametervaluesaregiveninTable1.

Foreachqualitygroup,syntheticseriesareproducedfordurationsof10–12,19and29years(resp.,2or3stations,2or3stations,2stations).ThesedurationsarechosentorepresentstandardGNSSdatausedingeodynamicstudies(10–20years),plusprolongeddataallowingtestingtheimpactonvelocityestimationsofanextra10yearsofdata.

ThemethodtodetectandmanageoffsetsisacriticalelementofGNSStimeseriesanalysisand velocity estimation (Gazeau et al., 2013; Masson et al., 2019). Our objective is not adetailedstudyofthiseffectbutonlytoprovidesetsofresultsrepresentativeofactualGNSSdata. In order to include offset management as a study parameter, each station isaccompaniedbyapseudo“logfile”comprisingdatesof“equipmentchanges”.Thesedatescorrespondpartly toactualoffsets imposed in theseriesandpartly to “falsealarms” (i.e.,dateswithnoimposedoffset).Inaddition,someoftheimposedoffsetsarenotincludedinthelogfiles.


The position time series for the 35 stations are shown in Appendix 1. The full datasetcomprises,foreachstation,textfileswithpositionsanddates,pseudologs,trueparametervalues(velocities,offsetdatesandamplitudes).Itisavailableinthearchiveaccompanyingthisreportanduponrequesttotheauthors.

NB: The true parameter values were not transmitted to the analysis groups during theprocessingphase.

Table1.Timeseriesparametersofthe5stationcategories SuperClean RatherClean MidRange RatherUgly SuperUgly

completeness (% ofavailabledays) 98 95 90 80 70

k -0.30 -0.50 -0.70 -0.90 -1.10

N 1.8 2.0 2.2 2.8 3.9

C(mm) E 2.2 2.6 2.7 3.5 5.0

U 4.0 4.9 5.3 7.0 8.0

N 1.0 1.0 1.0 1.0 1.0

A1(mm) E 1.0 1.0 1.0 1.0 1.0

U 3.0 3.0 3.0 3.0 3.0

N 0.5 0.5 0.5 0.5 0.5

A2(mm) E 0.5 0.5 0.5 0.5 0.5

U 1.0 1.0 1.0 1.0 1.0

number ofoffsets 0-2 1-3 1-4 2-4 3-5


2)TimeseriesanalysismethodsandsoftwareThesynthetictimeseriesareanalyzedindependentlybydifferentlaboratoriesandresearchdepartmentsoftheRENAGconsortiumusingvarioussoftwareandmethods,yieldingnineindependentsolutions:

• S1- MIDAS software (Blewitt et al., 2016). Automatic velocity and uncertaintyestimationsbased statistical analysisofposition couplesoverone-year spread.Nooffsetdetectionorestimation.Dataanalysis:A.Deprez,A.Socquet,ISTerre.1

• S2- homemade software (MATLAB based). Least-square inversion of parametricmodel(eq.1),uncertaintiescomputedusingCATSsoftware(Williams,2008).Offsetsimposedbasedonmanualdetection.Dataanalysis:E.Henrion,EOST.

• S3-homemadesoftware(Rbased).Least-squareinversionofparametricmodel(eq.1),uncertaintiescomputedwithcolored-noiseapproximation.Automaticdetection+manualverificationofoffsets.Dataanalysis:C.Masson,GM.

• S4-softwaredevelopedatINGV,Roma.Least-squareinversionofparametricmodel(eq. 1), uncertainties computed using CATS software (Williams, 2008). Manualdetectionofoffsets.Dataanalysis:M.Métois,LGLTPE.

• S5- Pyacs software (Tran, 2013; Nocquet, 2018). Least-square inversion ofparametricmodel(eq.1),uncertaintiescomputedwithwhite-noiseapproximation.Automaticdetectionandrepairofdiscontinuities(outliersandoffsets)withF-ratiotest for thesignificance (99%confidence levelminimumfor robustdetection)andreprocessing after manual verification of offsets. Data analysis: J.-L. Menut, J.-M.Nocquet,L.Rolland,M.Vergnolle,Geoazur.2

• S6-HECTORsoftware(Bosetal.,2013)andGeodeZYXToolbox(Sakicetal.,2019).MaximumLikelihood estimation of velocities, annual and semi-annual parametersusing white-and power-law noise. Offsets based on logsheet information + visualdetectionwithamanual"pointandclick".Dataanalysis:P.Sakic,GFZ.

• S7-SARIsoftware(Santamaría-Gómez,2019.).Least-squareinversionofparametricmodel (eq. 1), uncertainties computed using white-noise approximation. Manualdetectionofoffsets.Dataanalysis:A.Santamaria,GET.

• S8- GeoTS software (Gazeaux et al., in prep). Joint estimation (least-square) ofvelocity, annual and semi-annual, andoffsetswithwhiteandautoregressivenoise.OffsetsaredetectedusingHotellingstatistics.Offsetdatesandnoisecharacteristicsarecommontothe3components.Dataanalysis:P.Valty,J.Gazeau,IGN.

• S9- TSView software (MIT, http://www-gpsg.mit.edu/~tah/GGMatlab/#_tsview).Least-square inversion of parametricmodel (eq. 1), uncertainties computed usingcolored-noiseapproximation.Manualdetectionofoffsets.Dataanalysis:P.Vernant,GM.

1SolutionS2isusedtocomputevelocityproductsbytheEPOSGNSSprocessingcenterlocatedatISTerre.https://www.epos-ip.org/tcs/gnss-data-and-products/news/epos-gnss-data-and-products-presents-data-analysis-double-difference2Exceptforthelaststep(reprocessingaftermanualvisualization),SolutionS5isusedbytheRENAGDataCentertoproduceoperationalpositiontimeseriesfortheRENAG-RESIFstations.http://renag.resif.fr/ts/charts/stop/


Inaddition,theaverageofallninesolutionsiscomputedwithequalweightingofallsolutions:

• S10-unweightedaverage.

Methodsummary

• S1:statisticsonvelocitiesoverone-yearspread;• S2,S4,S7andS9:least-squareinversionofaparametricmodelwithoffsetsbasedon

logsfilesormanagedmanually;• S3, S5 and S8:least-square inversion of a parametricmodelwith automatic offset

detection(plusmanualverification);• S6:MaximumLikelihoodEstimationwithoffsetdetection(plusmanualverification

basedonlogfiles).

Velocityuncertaintysummary

• S1:statisticsonvelocities;• S2andS4:CATSfullcolored-noiseestimation;• S3,S6andS8:colored-noiseapproximation;• S5andS7:white-noiseapproximation;• S9:colored-noiseapproximationwithautoregressivemodel.


3)ResultsandcomparisonwithtargetvelocitiesHereafter,thedifferencebetweenanestimatedvelocityandthetrue(synthetic)velocityisreferredtoasabias(i.e.,thedeviationfromtheexpectedresult).Inordertoprovidestatisticsforthewholedataset,wecomputethreestatisticalestimatorsofthebiases:

• dV50:median(50thpercentile)ofthedistributionofabsolutevaluesofbiases;• dV95:95thpercentileofthedistributionofabsolutevaluesofbiases;• dVRMS:Root-mean-square(RMS)ofthedistributionofbiases.

Because of the limited number of synthetic data (105 series from 35 stations, 5 qualitygroups,3seriesdurations),ourstudyonlyprovidesfirstorderresultsonthevariabilityofvelocityestimationsfromdifferentmethodsandsoftware.Thissmallsampledoesnotallowdetailedanalysesof specificparametereffectsandsuchconclusions shouldnotbedrawnfromourresults.

A synthesis of each individual solution is provided in a summary figure (Figs. S1–S10 inAppendix 2) showing the estimated velocities versus the true velocities for the threecomponents, as well as the distribution of the velocity biases relative to the associatedvelocityuncertainties.

Overallresultsorganizedpersolutions(S1–S10)andperqualitygroupsareshowninFigures1and2.Inallfigures,acolorcodeisusedtorepresentthefivequalitygroupsandstationcodes:SuperClean/SC=darkgreen;RatherClean/RC=lightgreen;MidRange/MR=lightblue;RatherUgly/RU=orange;SuperUgly/SU=red.

IndividualsolutionstatisticsaregiveninTable2(excludingstationsfromgroupSU).Inordertodiscusstheaveragequalityofeachsolution,weuseasimplemetricthatconsistincountingthenumberoftimesagivensolutionfallsinthebestthree(uppertier)orworstthree(lowertier)foreachcomponent(N,E,U)andstatisticalindicator(dV50,dV95,dVRMS).EachsolutionoverallscoreconsistsintheratioNuppertier/Nlowertier;e.g.,ascoreof2/4indicatesthatthesolutionappearstwiceintheuppertierandfourtimesinthelowertier(overatotalofnineindicators).

Result 1. To first order, all nine solutions are associatedwith horizontal (resp. vertical)velocitybiasessmallerthan1.0mm/a(resp.2.0mm/a)ofthesamemagnitudes.

Result2.Nosinglesolutionstandsoutassignificantlybetterorworsethantheothers(Figs.1and2).However,oursimpleclassificationof thesolutions in theupperand lower tiersindicatesthreemaincategories(Table2):

• SolutionsS3,S4andS10(unweightedaverage)aresignificantlymoreoftenintheuppertierthaninthelowertier;SolutionS7alsofalls inthiscategorybutwithslightlylowerscore;

• SolutionsS1,S2,S5andS8aremoreofteninthelowertierthanintheuppertier;• SolutionsS6andS9areasoftenintheupperandlowertiers.


Result3.VelocitybiasesforstationsfromgroupsSC–RU,representativeofstandardGNSSdata,varybetweenca.0.1and2.0mm/adependingonthecomponent(N,E,U)andindicator(Table2).Specifically:

• Medianbiases(dV50)areca.0.1mm/a(horizontalcomponents)and0.1–0.3mm/a(verticalcomponent).

• dV50=0.04–0.11mm/aNorthcomp.;• dV50=0.04–0.15mm/aEastcomp.;• dV50=0.07–0.36mm/aUpcomp.

• The95thpercentiles(dV95)showalargevariabilitybetweensolutions,ca.0.2–0.7

mm/a(horizontalcomponents)and0.5–2.0mm/a(verticalcomponent).• dV95=0.23–0.53mm/aNorthcomp.;• dV95=0.24–0.71mm/aEastcomp.;• dV95=0.43–2.03mm/aUpcomp.

• RMSofbiases(dVRMS)arepositionedbetweenthemedianand95thstatistics,ca.

0.1–0.3mm/a(horizontalcomponents)and0.3–0.9mm/a(verticalcomponent).• dVRMS=0.14–0.24mm/aNorthcomp.;• dVRMS=0.13–0.28mm/aEastcomp.;• dVRMS=0.28–0.96mm/aUpcomp.

ThedifferencesbetweentheNorthandEastcomponentsinallindicatorsreflectthehighernoiseamplitude(dailydispersion)ofthelatter(cf.Table1).

Table2.Velocitybiasstatisticsfortheninesolutions(S1–S9)andaverage(S10).Statisticsaregiveninmm/aforthe28stationsinqualitygroupsSC,RC,MRandRU(SUstationsarenotincluded,cf.Section1).Solutionsflaggedingreen/redareinupper/lowertierforagivencomponentandindicator(line).Thelastline(Score)givesthetotalresultforagivensolutionasnumberofuppertier(green)/lowertier(red)classifications.

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10

dV50 0.09 0.11 0.04 0.05 0.07 0.05 0.07 0.08 0.06 0.05

North dV95 0.26 0.53 0.26 0.23 0.41 0.33 0.31 0.26 0.27 0.23

dVRMS 0.16 0.24 0.14 0.18 0.23 0.16 0.15 0.16 0.13 0.13

dV50 0.15 0.07 0.05 0.06 0.06 0.04 0.06 0.12 0.11 0.06

East dV95 0.38 0.71 0.35 0.24 0.51 0.42 0.35 0.65 0.52 0.32

dVRMS 0.21 0.28 0.22 0.13 0.23 0.19 0.16 0.28 0.23 0.14

dV50 0.36 0.16 0.10 0.13 0.21 0.11 0.17 0.14 0.07 0.13

Up dV95 2.03 1.54 0.43 0.83 0.98 1.81 0.77 1.07 1.01 1.04

dVRMS 0.96 0.74 0.28 0.44 0.46 0.71 0.46 0.47 0.77 0.42

Score 1/5 0/7 8/0 7/1 1/4 3/2 5/1 1/4 4/4 7/0


Result4.Thevertical(Up)componentisassociatedwithvelocitybiasesca.2–5timeslargerthanthehorizontalcomponentsandwithastrongervariabilitybetweensolutions(Table2,Figs.1and2).Thiseffectislikelyduetothelargernoiseamplitudecoupledwithlargeroffsetamplitudes compared to the horizontal components. As a result, the various offsetmanagementmethodsyieldahighervariabilityinthevelocityestimations.

Inparticular,solutionS1(MIDASmethod)standsoutasslightlyworsethantheothersonthethree indicators for theverticalcomponent(Table2).Thismightpointoutaneedforbettertuningofthemethodforserieswithhighnoise/largeoffsets.Incontrast,solutionS3(automaticoffsetdetection)providesbetterresultsthantheothersforthedV95anddVRMSof the vertical component, suggesting that automatic offset detection may be a betteralternativeinthesecases(comparedtomanualdetection).

Result5.Theunweightedaverageofallninesolutions(S10)yieldsresultssystematicallyintheuppertierofallindividualsolutions(Table2,Figs.1and2).

Figure 1. Velocity biases organized first per solutions and then per quality groups.Absolute value of the velocity bias of each station is represented by a circle for individualsolutionsandfortheaverage.Resultsineachsolutionaregroupedbyqualityindicatedbycolors


(fromSCtoSU=greentored,cf.text).Foreachsolutionthemedian,95thpercentileandRMSofallstationsaregiveninmm/yr(NB:thesestatisticsincludestationsfromgroupSU,incontrastwiththoseinTable2).N,E,U:North,EastandVerticalcomponents.

Figure 2. Velocity biases organized first per quality groups and then per solutions.Absolute value of the velocity bias of each station is represented by a circle for individualsolutions(1.0=S1,1.1=S2…1.8=S9;2.0=S1,2.1=S2…)andadiamondfortheaverage.Solutionsaregroupedbyqualityindicatedbycolors(fromSCtoSU=greentored,cf.text).Foreachqualitygroupthemedian,95thpercentileandRMSofallsolutions(S1–S9)aregiveninmm/a.N,E,U:North,EastandVerticalcomponents.


Result6.Tofirstorder,theeffectofthetimeseriesdurationcanbeestimatedbycomparingtheseriesof10–12,19and29years.ThiseffectisshowninFigure3(excludingseriesfromqualitygroupSU).Asexpected,thelengtheningoftheseriesdurationyieldsadecreaseofthevelocitybiases.Thisdecreaseisnon-linearandstronglysensitivetotheseriesparameters:

• Lengtheningofseriesfrom10–12to19yearsresultsinadecreaseofvelocitybiasesca.50%,varyingbetween30%and80%dependingonthequalitiesandcomponents.

• Lengtheningofseriesfrom19to29yearsresultsinadecreaseofvelocitybiasesca.25%,varyingbetween6%and60%dependingon thequalitiesandcomponents (with one case of increase by 15%, probably an example of badstatisticsfromthesmallsample).


Figure3.Velocitybiasesorganizedper seriesdurations.Absolute values of the velocitybiases(excludingqualitygroupSU)areshownasboxplots(quartilesandoutliers).Themedian,95thpercentileandRMSaregivenforeachduration(10–12,19and29years).

Result7.Theninesolutionsareassociatedwithdifferentmethodstoestimatethevelocityformaluncertainties(standarderrors,cf.Section2).AssumingaGaussiandistributionofthevelocity estimations, these uncertainties should be such that 68.3% (resp. 95.4%) of thevelocitybiasesareinferiorto1(resp.2)timesthestandarderrors.Thecomparisonofthevelocitybiasesrelativeto theirstandarderrors foreachsolution(Appendix2)allowsthefollowingobservations:

• Methodsbasedoncolored-noisemodels(solutionsS2,S3,S4,S6andS8)yieldreasonable estimations of the velocity standard errors (although on averageslightlytoosmall):80–91%ofthevelocitybiasesareinferiortotwicetheirstandarderrors.

• Unsurprisingly,methods based on white-noise models (solutions S5 and S7)yieldstandarderrorssystematicallytoosmall:only20–25%ofthevelocitybiasesareinferiortotwicetheirstandarderrors.AveragedV/sratiosvarybetween6and14,suggestingstandarderrorstoosmallbyafactorof3–7.

• TheMIDASmethod(solutionS1)yieldsstandarderrorssystematicallytoolarge,especiallyforthehorizontalcomponentsforwhich100%ofthevelocityaresmallerthantwicetheirstandarderrors.dV/sratiossuggestthatthestandarderrorsaretoolargebyafactorca.2.

• TheTSView“RealSigma”method(solutionS9)yieldsstandarderrorsslightlytoosmall,withca.60%ofthevelocitybiasesinferiortotwicetheirstandarderrors(averagedV/sratiosca.2).


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Appendix1.Synthetictimeseries

EachfigureshowsthedailypositionseriesofthestationNorth,EastandUpcomponents(inmm)asafunctionofthedate(inyr).Theorangeverticalbarsindicatethedatesofoffsetsimposedintheseries.Cf.textforstationcodes.

StationSC01


StationSC02


StationSC03


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StationSC06


StationSC07


StationRC01


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StationMR01


StationMR02


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StationRU01


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StationSU01


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StationSU07


Appendix2.Solutionsummaryfigures

Eachfigureshowsasummaryfortheindividualsolution(S1–S9)andtheaverage(S10).FortheNorth,EastandVerticalcomponents(N,E,U), theestimatedvelocitiesandtheir95%confidenceintervals(twostandarderrors)areshownrelativetothetruevelocities.Numbersabovethegraphsarethemedianofthevelocitybiasabsolutevalues(dV50)perqualitygroupandforallstations.Thebottom-rightgraphshowstheratiosofvelocitybiases(dV)overtheirstandarderrors(s).Thenumbersaboveindicatethepercentageofratiosinferiorto2andthemeansoftheratios(N,E,andUcomponents).

SolutionS1

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.04 − 0.11 − 0.11 − 0.07 − 0.12 mm/yr All: 0.09 mm/yr

S1 − (N)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.09 − 0.18 − 0.17 − 0.17 − 0.48 mm/yr All: 0.17 mm/yr

S1 − (E)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.13 − 0.5 − 0.81 − 0.42 − 0.68 mm/yr All: 0.38 mm/yr

S1 − (U)

0.0

0.5

1.0

1.5

2.0

2.5

Comp. N, E, U

abs(

dV) /

s

V < 2*sigma: 98.1 % N: 0.4 , E: 0.7 , U: 0.9


SolutionS2

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.01 − 0.12 − 0.12 − 0.25 − 0.98 mm/yr All: 0.12 mm/yr

S2 − (N)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)V

Estim

. (m

m/y

r)

0.02 − 0.03 − 0.09 − 0.22 − 1.28 mm/yr All: 0.13 mm/yr

S2 − (E)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.1 − 0.11 − 0.18 − 0.39 − 1.07 mm/yr All: 0.29 mm/yr

S2 − (U)

01

23

4

Comp. N, E, U

abs(

dV) /

s

V < 2*sigma: 90.5 % N: 0.9 , E: 0.9 , U: 0.8


SolutionS3

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.01 − 0.03 − 0.08 − 0.2 − 0.28 mm/yr All: 0.08 mm/yr

S3 − (N)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)V

Estim

. (m

m/y

r)

0.01 − 0.08 − 0.05 − 0.18 − 0.63 mm/yr All: 0.08 mm/yr

S3 − (E)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.04 − 0.04 − 0.19 − 0.34 − 0.8 mm/yr All: 0.19 mm/yr

S3 − (U)

01

23

45

67

Comp. N, E, U

abs(

dV) /

s

V < 2*sigma: 81.9 % N: 1.1 , E: 1.2 , U: 1.3


SolutionS4

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.01 − 0.06 − 0.08 − 0.15 − 0.17 mm/yr All: 0.08 mm/yr

S4 − (N)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)V

Estim

. (m

m/y

r)

0.02 − 0.11 − 0.11 − 0.14 − 0.53 mm/yr All: 0.1 mm/yr

S4 − (E)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.1 − 0.21 − 0.12 − 0.18 − 0.45 mm/yr All: 0.15 mm/yr

S4 − (U)

020

040

060

080

0

Comp. N, E, U

abs(

dV) /

s

V < 2*sigma: 91.4 % N: 0.8 , E: 29.5 , U: 0.7


SolutionS5

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.04 − 0.19 − 0.07 − 0.13 − 0.3 mm/yr All: 0.11 mm/yr

S5 − (N)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)V

Estim

. (m

m/y

r)

0.02 − 0.21 − 0.14 − 0.14 − 1.08 mm/yr All: 0.14 mm/yr

S5 − (E)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.03 − 0.22 − 0.31 − 0.5 − 0.32 mm/yr All: 0.23 mm/yr

S5 − (U)

020

4060

80

Comp. N, E, U

abs(

dV) /

s

V < 2*sigma: 21.9 % N: 8.5 , E: 13.1 , U: 14.2


SolutionS6

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.01 − 0.06 − 0.04 − 0.12 − 0.08 mm/yr All: 0.05 mm/yr

S6 − (N)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)V

Estim

. (m

m/y

r)

0.01 − 0.04 − 0.05 − 0.14 − 0.28 mm/yr All: 0.09 mm/yr

S6 − (E)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.05 − 0.02 − 0.17 − 0.23 − 0.62 mm/yr All: 0.16 mm/yr

S6 − (U)

01

23

4

Comp. N, E, U

abs(

dV) /

s

V < 2*sigma: 85.7 % N: 0.9 , E: 0.9 , U: 1.4


SolutionS7

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.01 − 0.03 − 0.13 − 0.25 − 0.37 mm/yr All: 0.11 mm/yr

S7 − (N)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)V

Estim

. (m

m/y

r)

0.01 − 0.05 − 0.07 − 0.11 − 0.81 mm/yr All: 0.09 mm/yr

S7 − (E)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.04 − 0.15 − 0.21 − 0.33 − 0.53 mm/yr All: 0.21 mm/yr

S7 − (U)

05

1015

2025

3035

Comp. N, E, U

abs(

dV) /

s

V < 2*sigma: 21.9 % N: 6.6 , E: 6.4 , U: 6.4


SolutionS8

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.01 − 0.1 − 0.04 − 0.18 − 0.41 mm/yr All: 0.1 mm/yr

S8 − (N)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)V

Estim

. (m

m/y

r)

0.03 − 0.09 − 0.12 − 0.23 − 1.48 mm/yr All: 0.19 mm/yr

S8 − (E)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.04 − 0.05 − 0.22 − 0.4 − 1.14 mm/yr All: 0.22 mm/yr

S8 − (U)

02

46

810

Comp. N, E, U

abs(

dV) /

s

V < 2*sigma: 81 % N: 1.4 , E: 1.7 , U: 1.1


SolutionS9

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.01 − 0.1 − 0.09 − 0.11 − 0.13 mm/yr All: 0.07 mm/yr

S9 − (N)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)V

Estim

. (m

m/y

r)

0.01 − 0.08 − 0.17 − 0.21 − 0.48 mm/yr All: 0.15 mm/yr

S9 − (E)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.05 − 0.06 − 0.06 − 0.55 − 0.65 mm/yr All: 0.21 mm/yr

S9 − (U)

02

46

8

Comp. N, E, U

abs(

dV) /

s

V < 2*sigma: 58.1 % N: 1.8 , E: 1.8 , U: 2.2


SolutionS10–unweightedaverage

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.01 − 0.05 − 0.09 − 0.05 − 0.26 mm/yr All: 0.05 mm/yr

mu − (N)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)V

Estim

. (m

m/y

r)

0.02 − 0.05 − 0.07 − 0.14 − 0.58 mm/yr All: 0.09 mm/yr

mu − (E)

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

V True (mm/yr)

V Es

tim. (

mm

/yr)

0.07 − 0.12 − 0.17 − 0.39 − 0.55 mm/yr All: 0.17 mm/yr

mu − (U)

01

23

Comp. N, E, U

abs(

dV) /

s

V < 2*sigma: 92.4 % N: 0.7 , E: 0.9 , U: 0.9

Comparative analysis of synthetic GNSS time series - Bias ...

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