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ClaudioBrunini
UniversidadNacionaldeLaPlata,
ConsejoNacionaldeInvestigaciones
CientíficasyTécnicas
LaPlata
Argentina
SIRGASPresident
LauraSánchez
DeutschesGeodätisches
Forschungsinstitut
München
Germany
SIRGASVice-President
HermannDrewes
InternationalAssociation
ofGeodesy
Munich
Germany
SIRGASScientificCouncil
SchoolonReferenceSystems,Crustal
DeformationandIonosphereMonitoring
Contents
1.Coordinatesystems,typesofcoordinates
2.Geodeticreferencesystemsandframes
3.CoordinatesdeterminationfromGNSS
4.Verticalreferencesystems
5.ReferencesystemandframefortheAmericas(SIRGAS)
6.Crustaldeformation,observationandmodelling
7.Ionospheremodellingandanalysis
1.Coordinatesystems,typesofcoordinates
Acoordinatesystemprovidesthebasisforuniquedeterminationof
thepositionofpointsinlines,surfacesorspaces(1D,2D,3D).
Todefineacoordinatesystemonehastospecify
0.thetypeofcoordinates(rectilinear,curved,plane,spatial);
1.thelocationoftheorigin;
2.theorientationoftheaxes;
3.theunitofmeasure.
Coordinatescannotbedetermineddirectlybutonlyw.r.t.asystem!
Weshalldiscussthefollowingtypesofcoordinatesystems:
•Globalthree-dimensionalCartesiancoordinates[X,Y,Z]
•,h]
•LocaltopocentricCartesiancoordinates[x,y,z],[n,e,u]
•Planecoordinates(Mercator,Lambert,azimuthal)[N,E]
•Coordinatesreferringtoirregularsurfaces[H](seechapter4)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-1
1.1GlobalCartesian3Dcoordinates
•Cartesiansystemshaverectilinear
orthogonalaxes.
•Wedistinguishright-handand
left-handsystems.Ingeodesywe
usetheright-handsystem.
Greenwich
Pole
•Inglobalsystems,theoriginisthe
centreofmassoftheEarth;
thedirectionoftheZ-axisis
towardstheconventionalpoleof
rotation,thedirectionoftheX-axis
isclosetotheGreenwichmeridian;
theunitofmeasureisthemetre.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-2
•Theseparametersarecalleddatum.
TransformationofCartesiancoordinates(1)
•Acoordinatetransformationisthechangeofcoordinatesfrom
onecoordinatesystemtoanotheronewithadifferentdatum.
•Acoordinatetransformationappliesparameterswhichcanbe
derivedempiricallyfromasetofidenticalpointsinbothsystems.
•Forundeformedcoordinatesetsinthree-dimensionalCartesian
systemsitiscommontousethe7-parameterssimilarity
transformation(afterHelmert1893):
)()(12
13
23
3
2
1
)()(
**
0
0
0
PPPD
Z
Y
X
M
Z
Y
X
RR
RR
RR
T
T
T
Z
Y
X
Z
Y
X
)(12
13
23
3
2
1
*
'1
'1
'1
P
Z
Y
X
MRR
RMR
RRM
T
T
TT1,T2,T3:TranslationsinX,Y,Z
Ri:RotationsaroundX-,Y-,Z-axes
M:Factorofunitmeasures(scale)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-3
TransformationofCartesiancoordinates(2)
•TheHelmerttransformation
producescorrelationsofT,R,M
parametersifthecentreofpoints
(i.e.theaverageofcoordinates)
differsfromtheorigin.
•Toavoidthiscorrelation,the
transformationofMolodensky-
Badekasisused:
Z
Y
X
ZZ
YY
XX
RR
RR
RR
M
Z
Y
X
p
p
p
XY
XZ
YZ
''
''
''
1
1
1
•X,Y,Z:origincoordinates
oftheinitialinthenewsystem.
•RX,RY,RZ:Rotationsofaxes.
•X'P,Y'P,Z'P:Averageofpoint
coordinatesintheinitialsystem.
•M:Factorofmeasures(scale).
X',Y',Z'=initialsystem
X,Y,Z=newsystem
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-4
1.2Globalsphericalcoordinates
•Aglobalsphericalsystemis
preferred(andsufficient)formany
representations,e.g.thegravityfield,
globalgeography,kinematicsof
tectonicplates,etc.
•rP:radialgeocentricdistance,
P:latitude,
P:polardistance(co-latitude90-P),
P:geocentriclongitude.
p
p
P
X
Y
Z
P
Globalsphericalcoordinatescan
easilybeconvertedtoCartesian
coordinatesX,Y,Z.sin
sincos
coscos
cos
sinsin
cossin
rr
Z
Y
X
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-5
Ellipsoidalcoordinates(1)
•ThefigureoftheEarthisbetter
approximatedbyanellipsoidthan
byasphere,becausetheEarth’s
rotationproducesaflatteningwith
semi-minoraxis(b)about21km
shorterthansemi-mayoraxis(a).
•Ingeodesyweuseanellipsoidthat
optimallyfitsthegeoidaccording
totheGauss-Listingdefinition,i.e.
coincidingwiththemeansealevel
instaticequilibrium.
ap
p
X
Z
b
Y
a
ba
f
a
ba
e
22
Polarflattening:
Eccentricity:
•Theshapeoftheellipsoidisgiven
bysemi-mayoraxis(a)andpolar
flattening(f)oreccentricity(e).
Q
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-6
Ellipsoidalcoordinates(2)
•Theellipsoidalsurfacesystemis
extendedintospacebyinclusionof
the(ellipsoidal,geometric)heighth
ofpointPabovetheellipsoid,
measuredalongthenormalvector.
TheprojectionofPuponthe
ellipsoidcorrespondstopointQ.
•Thecoordinatestriad(,,h)is
calledcurvilinearcoordinates.
p
p
X
Z
Y
Yp
Zp
P
Q
sin))1((
sincos)(
coscos)(
2
hNe
hN
hN
Z
Y
X
r
•Theellipsoidalcoordinatescanbe
convertedtoCartesiancoordinates
X,Y,ZusingthemeridianradiusN:
N=a/(1–e2sin20.5
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-7
ConversionofCartesiantoellipsoidalcoordinates
p
p
X
Z
Y
Yp
Zp
P
Q
•Aconversionisthechangebetween
twotypesofcoordinatesreferringto
thesamedatum.
•ConversionofCartesiancoordinates
toellipsoidalcoordinatesisdoneby
aniterationprocess:
atan(Z+e22+Y2)0.5
atanY/X
–N
atan3q)/(p–e2acos3q)]
2/(1–e2)
b=a(1–f)
p=(X2+Y2)0.5
q=atan[(Za)/(pb)]
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-8
Transformationofellipsoidalcoordinates
Thechangeofellipsoidalcoordinatesfromasourcesystemtoa
transformedsystemcandirectlybehandledlikeadisplacement
(offset)usingtheformulaeofMolodensky:
t=s+d;t=s+d,ht=hs+dh
with:
d"=(–dXsinscoss–dYsinssins+dZcoss+(adf+fda)sin2
s)/(rssin1")
d"=(–dXsins+dYcoss)/(nscosssin1")
dh=dXcosscoss+dYcosssins+dZsins+(adf+fda)sin2
s–da
where:
dX,dY,dZ=parametersofgeocentrictranslation.
da=differenceofsemi-mayoraxesofthetransformedandthesourceellipsoid.
df=differenceoftheflatteningoftheellipsoids.
da=at–as,df=ft–fs=1/(1/ft)–1/(1/fs).
rsandns=curvatureradiusofthemeridiansectionsandthefirstvertical,respectively,
inagivenlatitudesontheinitialellipsoid:
rs=as(1–es
2)/(1–es
2sin2
s)3/2,ns=as/(1–es
2sin2
s)1/2
Theformulaefordanddprovidethechangesinandinsecondsofarc.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-9
1.3Local(topocentric)coordinates(1)
•Localtopocentriccoordinatesrefer
toalocalreferencesystemrelatedto
theEarth’sgravityfield:Theorigin
istheobserver(“topocentre”P),the
orientationisgivenwithrespectto
thelocalvertical(zenith,plumbline):
•Thez-axispointstothezenith,
rectangulartotheplanex,y;
•Thex-axispointstothenorthofthe
meridian;
•They-axispointstotheeast,thus
formingaleft-handsystem.
•x,y,zareequivalenttodenotations
north(n),east(e)andup(u).
p
p
P
X
Z
Y
Yp
Zp
x(n)
z(u)
y(e)
Topocentriccoordinatesystems
aretypicallyusedinastronomy
(horizontalcoordinates,heights
andazimuths).
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-10
Local(topocentric)coordinates(2)
•PointPiisdescribedw.r.t.theoriginP
•
theangleinthehorizonplanebetween
themeridianplaneofPandtheplane
formedbythenormalsinPandPi.
•
verticalplanebetweentheellipsoidal
verticalandtheconnectinglineP-Pi,
countedpositivefromthezenith.
•Thedistancesisonlyusedinlocal
geodeticsystems(notinastronomic
systems).
P
ne
u
Pi
Thetopocentricsystemsare
fundamentalforconnecting
measurementsofdifferent
techniques(localties).
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-11
Conversionoftopocentrictogeocentriccoordinates
Ecuador
=0
X
Y
Z
MeridianofGreenwich
=0
Topocentricorigin
=O
,=O
,h=hO
X=Xo,Y=Yo,Z=Zo
U=V=W=0
Geocentricorigin
=0,=0,h=-a
X=Y=Z=0
0
0
0
1
*
Z
Y
X
U
N
E
R
Z
Y
X
00
00000
00000
1
sincos0
sincossinsincos
coscoscossinsin
T
RR
with:
X=XO–EsinO–NsinOcosO+UcosOcosO
Y=YO+EcosO–NsinOsinO+UcosOsinO
Z=ZO+NcosO+UsinO
V(north)
U(east)
W(height)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-12
1.4Planecoordinates
•Planecoordinatesystemsenablerepresentingtheellipsoidal(or
spherical)surfaceinaplanethroughmathematicalorgeometrical
specifications,e.g.byprojections.
•Astherepresentationofacurvedsurfaceinaplaneisnotpossible
withoutdistortion,onehastodecideonarepresentationdistorting
lesstheangles,thedistancesortheareas,respectively.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-13
Theorientationofthesurfacescanbe
•normal(axisofthesurfaceparalleltothe
Earth’srotationaxis),
•transversal(axisparalleltotheequator),
•oblique(axisinanarbitrarydirection).
Inprincipletherearethreetypesofprojections:
•Projectionsontoacone,
•Projectionsontoacylinder,
•Projectionsontoaplane.
Projections
ObliqueNormal
Cylindrical
Transversal
AzimuthalConical
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-14
Specialtypesofprojectionsanddistortions
SecantTangential
Cylindrical
Themetricdistortions
maybegroupedinto:
•Conformity:doesnot
present(diferencial)
angulardistortion.
•Equivalence:
presentscoextensive
(equal)areas.
•Equidistance:(some)
meridian(s)and
parallel(s)represent
thetrue(relationsof)
length.
AzimuthalConical
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-15
Gauss-Krügerplaneconformcoordinatesystem
•TheGauss-Krügersystemisbasedon
atransversalcylindricalprojection.It
isnotageometricalprojectionbuta
mathematicalconformtransformation
withoutdifferentialangulardistortion.
4500000
3500000
2500000
2600000
5500000
3400000
3600000
4400000
4600000
5400000
•Itisregionallyappliedanddividedin
zonesof3°longitudeextension.
•Thecentralmeridianandtheequator
arestraightlines.Theothermeridians
andparallelsarecomplexcurves.
•Thescaleistruealongthecentral
meridianandconstantalongthelines
paralleltothemeridian.Itincreases
withthedistancefromthemeridian.
Alemania
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-16
UniversalTransversalMercator(UTM)System
•TheUTMsystemissimilartoGauss-Krüger.
•Itisgloballyappliedanddividedinzonesof
6°longitudeextension.
•Toreducethedistortionatthezonelimits,a
scalefactorof0,9996isappliedinthecentral
meridian,sothatthelinesin1°37’eastand
westdistancepresentthetruescale(1,0000).
•TheEarthfrom84ºNto80ºSisdividedinto20
stripesof60zonesof6ºlongitudeextension.
•Thebordermeridiansaredivisibleby6and
thezonesarenumberedfrom1to60.
•Thestripesof8°(12°inthenorthernmost)are
designatedfromsouthtonorthbyletters(Cto
XwithoutIandO)beginningwithCin80ºS.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-17
ConicalconformLambertsystem
•Themeridiansarestraight
linesconvergenttothepole.
•Theanglesbetweenthe
meridiansaretrue.
•Theparallelsareunequally
separated;theyareconcentric
circlesaroundthepoles.
•Twostandardparallelsare
length-preserving(truescale).
•Thereissymmetryaround
eachstandardparallel.
•Thepoleclosesttoastandard
parallelisapoint,theother
poleisnotshown.
Thedistortionisconstantalong
eachparallel.Thereisnodistortion
alongthestandardparallels.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-18
Standardparallelsin20°N
and60°N,centralmeridian
90ºW
Polarstereographicsystem
•Themeridiansarestraight
linesconvergenttothepole.
•Theanglesbetweenthe
meridiansaretrue.
•Theparallelsareunequally
separated;theyareconcentric
circlesaroundthepoles.
•Onestandardparallelis
length-preserving(truescale).
•Thereissymmetryaround
eachstandardparallel.
•Thepoleclosesttoastandard
parallelisapoint,theother
poleisnotshown.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20131-19
Thedistortionisconstantalong
eachparallel.Thereisnodistortion
alongthestandardparallel.
ReferenceSystem:Definitionofstandards,parameters,models,etc.
servingasthebasisfortherepresentationofthegeometryofthe
Earthsurfaceanditstemporalvariation(e.g.,speedoflightc0,
standardgravitationalparameterGM,modelsofspecialandgeneral
relativity,modelsoftheatmosphere(ionosphereandtroposphere),
three-dimensionalorthogonalCartesiancoordinatesystemwithits
temporalvariationconsistentwiththeEarth’srotation).
ReferenceFrame:Realisation(materialisation)ofareference
systembyasetofphysicalandmathematicalquantities(e.g.,a
numberofphysicallymarkedpointsattheEarthsurfacewithgiven
three-dimensionalCartesiancoordinatesX,Y,Zforafixedepoch
anditslinearvariationswithtime(dX/dt,dY/dt,dZ/dt),i.e.constant
velocities(vX,vY,vZ).
2Geodeticreferencesystemsandframes
2.1Definitionofreferencesystemsandframes
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-1
Geodeticdatum
Geodeticdatum:Parametersfixingtheorigin,orientationandscale
ofacoordinatesystemw.r.t.theEarth(e.g.,inaglobal3DCartesian
system:theoriginisrealisedinthegeo-centre,theorientationis
realisedbythepositionoftheEarthrotationpoleandthedirectionof
oneconventionalreferencelongitudeatadefinedepoch,thescaleis
realisedbythemetreunitbasedonthespeedoflightinvacuum).
Important:
1.Referencesystemscannotbedeterminedbymeasurements,but
theyaredefinedconventionally;i.e.,thegeodeticcoordinatesand
directionsarenotestimablebutneedadefinedcoordinatesystem.
2.Referenceframesmustrealisethereferencesystemstrictly
accordingtoitsdefinition(e.g.geocentricandnotcrustfixed).
3.Thegeodeticdatummustbegivenunambiguously.3D-systems
require7parameters.Theremustnotbefixedmorecoordinates.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-2
Hierarchyofreferencesystems
DenominationPrincipalvectorsApplicationexample
———————————————————————————
Observationline-of-sightmeasurementof
system(local)gravitydirections&distances
Horizonsystemgravityterrestrialnetworks
(regional)
EquatorsystemEarthsatellitegeodesy
(global)celestialorigin
Celestialsystemcelestialoriginradioastronomy
(extragalactic)celestialreferencepole
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-3
Examplesofglobalreferencesystems
Terrestrialreferencesystems(TRS)
WorldGeodeticSystem1984(WGS84):Globalterrestrial
referencesystem,originallyestablishedfororbitdeterminationof
TRANSITDopplersatellites(WGS72),lateradoptedfortheorbit
determinationofNAVSTARGPSsatellites(WGS84).
WGS84adoptedtheITRF(seenextslide)foritsrealisationin2002.
InternationalTerrestrialReferenceSystem(ITRS):Reference
systemoftheInternationalEarthRotationandReferenceSystems’
Service(IERS)establishedfordeterminingthecelestial(ICRS)and
terrestrial(ITRS)referencesystemsandtheirinterrelation,i.e.the
orientationandrotationoftheEarthinspace(EOP,ERP).
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-4
Globalandregionalreferenceframes
InternationalTerrestrialReferenceFrame(ITRF):
MaterialisationoftheITRSbystationsattheEarthsurfaceandgiven
coordinatesforafixedepochanditsvariationswithtime(velocities).
ITRF2008:morethan900pointsinmorethan500sites.
TheITRFservesalsoforthepreciseorbitdeterminationoftheGPS
satellitesbytheInternationalGNSSService(IGS).
SistemadeReferenciaGeocéntricoparalasAméricas(SIRGAS):
DensificationoftheITRF,initiallyestablishedforSouthAmericaby
aGPScampaignwith58stationsin1995andextendedtothe
Caribbean,CentralandNorthAmericain2000with184stations.At
presenttherearemorethan250permanentstations.
Examplesofreferenceframes
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-5
PreliminarySouthAmericanDatum1956(PSAD56):
EstablishedbyastronomicallyobservedcoordinatesinLaCanoa,
Venezuela,referringtotheinternationalellipsoid1924(Hayford).
Deviationfromgeo-centre:X=-288m,Y=175m,Z=-376m.
SouthAmericanDatum1969(SAD69):
EstablishedinChúa,Brazil,referringtotheGRS67ellipsoid.
Deviationfromgeo-centre:X=-57m,Y=1m,Z=-41m.
NorthAmericanDatum1983(NAD83):
Adjustmentofastronomical,DopplerandVLBImeasurements
referringtotheGRS80ellipsoid.
Deviationfromgeo-centre:X=1,0m,-
InternationalTerrestrialReferenceFrame(ITRF2008):
UseofEarthgravityfieldparametersC11=S11=C10=0incomputing
theorbitsofLasersatellitesforstationpositioning;orientation
accordingtoadefinedmeridian;metricscaleunit.
Examplesofgeodeticdatums
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-6
Traditionalandmodernreferencesystems
Whydoweneednewreferencesystems?Whycan’twecontinue
usingthetraditionalsystems(PSAD56,SAD69,NAD83)?
Becauseweareusingsatellitetechniques.Wehavetoguarantee
identicalreferencesystemsforterrestrialpointsandsatellites.
Modernreferencesystemsmustbedefinedbyaglobaldatum:
TheoriginistheEarth’scentreofmass(geo-centreX0=Y0=Z0=0),
becausesatellitesareorbitingaroundthegeo-centre(Kepler’slaw).
Theorientationisgivenbytheearthrotationaxis(Z-axis)andthe
conventionofazero-longitude(X-axisintheGreenwichmeridian).
Themetricscaleisgivenbythespeedoflightand(incaseofuseof
satellitetechniques)bythestandardgravitationalparameterGM.
Modernreferencesystemsarealwaysthree-dimensional:
X,Y,Z,ortransformedtoNorth,East,Heightw.r.t.anellipsoid.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-7
Traditionalreferencesystems:
-regionaldefinition
-nationalrealisation
-localdatum
Manysystemsaroundtheglobe
Traditionalandmodernreferencesystems
Modernreferencesystems:
-globaldefinition
-globalandregionalrealisation
-geocentricdatum
Onesystemallovertheworld
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-8
Organisationofmodernreferenceframes
Themodernreferencesystemsareinstalledandmaintainedbythe
servicesoftheInternationalAssociationofGeodesy(IAG).
Thereisoneserviceforeachofthegeodeticobservationtechniques:
-InternationalGNSS(GlobalNavigationSatelliteSystems)Service
(IGS),
-InternationalLaserRangingService(ILRS),
-InternationalVLBI(VeryLongBaselineInterferometry)Service
forGeodesyandAstrometry(IVS),
-InternationalDORIS(DopplerOrbitographyandRadio-positioning
IntegratedbySatellite)Service(IDS).
TheInternationalEarthRotationandReferenceSystems’Service
(IERS)coordinatestheactivitiesforestablishingandmaintaining
theInternationalTerrestrialReferenceFrame(ITRF).
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-9
InternationalEarthRotation
andReferenceSystemsService
IERSDirectingBoard(http://www.iers.org)
CentralBureau
DataCentre
TechniqueCentres
(ExternalServices)
AnalysisCoordinator
ResearchCentres
CelestialReference
EarthOrientation
TerrestrialReference
RapidService/Predicts
Conventions
GeophysicalFluids
ProductCenters
ILRS
IVS
IDS
IGS
Electr.InterfacesDocuments
Users
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-10
GeneralstructureofIAGServices
-PresidentandelectedDirecting(orGoverning)Board.
-CentralBureau(CoordinatingOffice).
-Stationnetworkforobservationsmaintainedbyindividual
institutionsattheirowncostswithOperationCentre(s).
-Satellitesoperatedbyspaceagenciesattheirowncosts(incaseof
satelliteobservationservices).
-CorrelationCentresoftheobservationdata(incaseofVLBI).
-DataCentrescollectingtheobservationdataandtheresultsofthe
analysisandcombination.
-AnalysisCentresforprocessingthedataandgeneratingproducts
forscientificandpracticaluse.
-CombinationCentresforevaluatingandcombiningtheproducts
generatedbythedifferentanalysiscentres.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-11
InternationalGNSSService(IGS)
IGSTrackingNetwork(http://igs.org)
427stationsintotal,360inoperation(2013-09-15)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-12
Int.SatelliteLaserRangingService(ILRS)
ILRSTrackingNetwork(http://ilrs.gsfc.nasa.gov)
50stationsinoperation(2013-01-09)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-13
InternationalVLBIService(IVS)
IVSTrackingNetwork(http://ivscc.gsfc.nasa.gov)
31stationsinoperation(2013-08-16)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-14
InternationalDORISService(IDS)
IDSTrackingNetwork(http://ids.cls.fr)
75stationsintotal,59inoperation(2010-07-21)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-15SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-1-16
(www.dgfi.badw.de)
InternationalCelestial
ReferenceFrame(ICRF)
Internat.Terrestrial
Refer.Frame(ITRF)
realisedbythecoordinatesof
markedstationsattheEarth’s
crustwith3-Dcoordinates
realisedbythecoordinatesof
QuasiStellarRadioSources
(Quasars)observedbyVLBI
2.2Celestial(inertial)referencesystemandframe
Celestialreferencesystem=Conventionalinertialsystem
-Inertialsystemsarenon-acceleratedsystemswithoutexternalforce.
-Theoriginallawsofphysics(Newton’slaws,celestialmechanics,
relativitytheory,...)refertoinertialsystems.
Earthorientationparameters(EOP)
-Rotationoftheterrestrialwithrespecttothecelestialsystem
Terrestrialreferencesystem=Conventionalglobalsystem
-TerrestrialreferencesystemsareassociatedwiththesolidEarth.
-Theyparticipateinallthemotionsofthesolidearthandallthe
deformationsoftheEarth’scrust.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-1
Celestial(inertial)referencesystem
Eclipticand
equatorialsystems
Coordinatesofan
objectinspace
(star,radiosource):
-equatorialsystem:
rightascension
declination
-eclipticsystem:
eclipticlongitude
eclipticlatitude
(hardlyeverused)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-2
Realisationofthecelestialreferencesystembythe
InternationalCelestialReferenceFrame(ICRF)
TheInternationalCelestialReferenceSystem(ICRS)isrealisedby
theastronomiccoordinates(rightascensionanddeclination)of
quasi-stellarastronomicradiosources(Quasars)observedbythe
VeryLongBaselineInterferometry(VLBI).Thisrealisationis
called“InternationalCelestialReferenceFrame”(ICRF).
ThecurrentICRF(ICRF2of2009)containsthecoordinatesof
extragalacticradiosourcesforepoch2000.0inthreecategories:
-295datumdefiningsources,
-3080additionalglobalsources,
-39instablesourceswithspecialtreatment,
-3414sourcesintotal.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-3
RealisationoftheICRFbyVLBI
VeryLongBaseline
Interferometry(VLBI)
Twotelescopesreceivesignalsof
thesameradiosource(Quasar)at
differenttimes.
StationO‘Higgins,Antarctica,
ofBKG/DLR,Germany
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-4
PrincipleofVLBIanalysis(1)
1.Observation
Simultaneousrecordingofthe
signalsfromthesameQuasars.
2.Correlation
Calculationofthemaximum
correlationbydelayingthe
signalofonereceiver:t.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-5
PrincipleofVLBIanalysis(2)
3.CorrectionsofVLBI
observations
-Ionosphericrefraction
-Troposphericrefraction(dry
andwet)
-Time(clock)correction
-Telescopecalibration
(deformations,eccentricities,
signaltraveltimewithinthe
telescopetoelectroniccentre)
Thecorrectionsaresimilarto
GPSbecausethefrequencies
aresimilar.
Correlators(IGGBonn,Germany)
Onecorrelatorisneededforeach
baseline.Todaywehavebroadband
datatransmission(e-VLBI)andnear
real-timecomputercorrelation.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-6
PrincipleofVLBIanalysis(3)
4.Analysis
Computationoftheterrestrialbaseline
lengthandthedirectiontotheQuasar.
5.Adjustment
ComputationoftheQuasar
coordinates()relativeto
theterrestrialreferenceframe
bysimultaneousadjustment
ofallobservations.ICRF2
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-7
ExampleofICRF2coordinates
DesignationSource0()0()C-meanfirstlastNexpNobs
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-8
InterrelationofCRFandTRF
Thecelestialbodies(stars,radiosources)andartificialsatellites
usedforspacemeasurementsintheTerrestrialReferenceSystem
(TRS)donotparticipateintheEarthrotation.Thereforewehave
toknowtheEarthOrientationParameters(EOP)withrespectto
theCelestialReferenceSystem(CRS)atanyinstantaneoustime
ofmeasurementsattheEarthsurface.
TheEOPprovidetheconnectionbetweentheCRFandtheTRF.
TheyarederivedfromthecontinuousobservationoftheEarth’s
rotation.
WedistinguishbetweentheorientationoftheEarthinspace
(precessionandnutation),theorientationwithrespecttothebody
oftheEarth(polarmotion),andthevariationoftherotational
velocity(variationofUniversalTimeUTorLengthofDayLOD,
respectively).
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-9
ConnectionbetweenCRFandTRFviaEOP
Thereferencesystemsmustbeconsistent:CRF-EOP-TRF
Ecliptic
Equator
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-10
Variationoforientationinspace
ThecauseofthevariationofEarthorientationinspace(precession
andnutation)istheasymmetriceffectofthelunisolargravitational
forcesontheequatorialbulge.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-11
Precessionandnutation
Thelunisolarforces
produceawobbleof
therotationaxisofthe
Earthinspace.
Traditionallyitwas
distinguishedbetween
thefrequenciesof
precession(lower
frequency)andnutation
(higherfrequencies).
Thisdifferentiationwas
terminatedwiththe
IAU2000andIUGG
2003resolutions.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-12
Precessionandnutationintheinertialspace
Thesystem2000oftheInternationalAstronomicalUnion(IAU2000)
fixestheoriginofrightascensionatepoch2000.0(“nonrotating”).
Thenewprecession-nutationmodel(,)includesalltheeffects.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-13
P=R3(-z)R2()R3(-)N=R1(--)R3(-)R1()
Reductionofprecessionandnutationin
geodeticandastronomicalobservations
ConventionalInertial
System(CIS)
RealisedasICRFby
Quasarcoordinates
Celestial
IntermediateSystem
Equatorialsystemat
afixedepoch
(e.g.t0=2000.0)
Precession-NutationInstantaneous(true)
equatorialsystem
Z-axis=celestial
poleatafixedepoch
(e.g.t0=2000.0)
Z-axis=Celestial
IntermediatePole
(CIP)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-2-14
2.3RotationandtidesoftheEarth
Symmetryaxis
1.TheEarthrotationaxisdoesnot
coincidewiththesymmetryaxis
(principalaxisofinertia)
2.Theangularmomentum
variesduetodisplacements
andmotionsofmasses.
VariationofEarthrotationrelativetotheEarth’sbody
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-1
Polarmotion(XP,YP)andvariationofthe
rotationalvelocityUT1-UTC
CTP=Conventional
TerrestrialPole
GAST=GreenwichAppa-
rentSiderealTime
ZCT
YCTXCT
zT
x
T
yT
CTP
Greenwich
meantime
meridian
Conventional
equator
Instantaneous
(true)pole
MGAST
xp
-yP
UT1=UniversalTime
UTC=UniversalTime
Coordinated(TAI)
DUT=UT1-UTC
=VariationofGAST
dDUT=VariationofLOD
(LengthofDay)
EarthRotationParameters
(ERP:XP,YP,DUT)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-2
ReductionofERPvariations
Celestial
intermediatesystem
Conventional
TerrestrialSystem
(CTS)
Instantaneous(true)
equatorialsystem
Polarmotion(XP,YP),
andvariationof
angularvelocity
(DUT=UT1–UTC)
Globalsystemata
fixedepoch
(e.g.t=2000.0)
Z-axis=Celestial
IntermediatePole
(CIP)
Z-axis=axisofthe
coordinatesystem
forthepointsinthe
conventionalsystem
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-3
Thesolutionisacircularmotionwiththeradius
pR1²+2²=0,2”andperiod1/=305days(Euler)
RotationoftheEarthasarigidbody
Inaninertialcelestialsystem(Eulerequation):
dH/dt=L(L=lunisolartorqueprecession,nutation)
Inaterrestrialsystem(polarmotion):
dH/dt+×H=0(ifaxisHaxisthereisnomotion!)
WithprincipalmomentsofinertiaofthebodyoftheEarthA,B,C:
A1
H=I·=B·2
C3
Lawofconservationoftheangularmomentuminaninertialsystem:
dH/dt=0(H=angularmomentum)
H=I·(I=momentofinertia,=rotationalvelocity)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-4
RotationoftheEarthasanonrigidbody
I(t)=I+I(t)timevariablemomentofinertiaduetomass
displacementsintheterrestrialsystem
H(t)=I(t)·timevariableangularmomentum
dH/dt+×H=LLiouvilleequation(insteadofEulerequation)
Inadditiontothevariationofthemomentofinertiabythedisplaced
massesthereisaneffectbythemotion(velocityv)ofthemasses:
h(t)=(x×v)dMangularmomentumduetothemotionofmasses
H(t)=I(t)·+h(t)“forced”angularmomentum
ThesolutionoftheLiouvilleequationresults(inthesamewayasthe
Eulerequation)intheperiodandtheamplitudeofthecircularmotion
oftheEarthrotationpole.ThisisforanelasticEarth:
1/=435days(Chandlerperiod)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-5
Angularmomentumcausedbymovingmasses
Thethermodynamicenergygeneratesan
accelerationofmassesintheatmosphericand
oceanicpressurefield
d²x/dt²=dv/dt=-1/dp/dx
p+dp
x
p
ThemotionofmassesinterfereswiththeEarth
rotationandcausestheCoríolisacceleration
d²x/dt²=dv/dt=-2×v
v
StrongestmassmovementsintheEarthsystem
arethoseoftheatmosphereandtheoceans.
Temperaturevariations(dT)changethemass
density.Thevolume(dv)variesopposingtothe
pressure(p)andproducesthermodynamicenergy
dQ=c·dT+p·dv
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-6
VariationofEarthrotationbymovingmasses
Incaseofequilibriumbetweentheaccelerationscausedbypressure
andbytheCoriolisforcethereisa“geostrophic”motion(Greek:
strephein=toturn)
1/dp/dx-2×v=0
Thehighpressureoftheatmosphereisontherightofcurrentsin
thenorthernhemisphereandontheleftinthesouthernhemisphere.
dv/dt
CH
p
p+dp
CH
p
p+dp
vg
dv/dtvg
Thismotionhasnoaccelerationandresultsdifferentinthenorthern
andsouthernhemispheres:
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-7
Circulationoftheatmosphere(principalwinds)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-8
Circulationoftheoceans(principalcurrents)
Thedeviationsofon-goingatmosphericandoceaniccurrentsfrom
thegeostrophiccurrentsgeneratethevariationsofEarthrotation.
Humboldt
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-9
ObservationofvariationsofEarthrotationby
spacegeodetictechniques(VLBI,SLR,GPS)
Polarmotion[]=[arcsec]
131minlatitudeandequatoriallongitude
<~18m>
1ms46cmlinearmotionattheequator
Variationofrotationalvelocity(LOD[sec])
-0,002
-0,001
0,000
0,001
0,002
0,003
0,004
198219841986198819901992199419961998200020022004200620082010
[sec]IERSC04
LODElNiñoeffect
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-10
Partitioningofpolarmotion
MotionwithChandlerperiod(435days)[]
Motionwithannualperiod(365,25days)[]
-0,3
-0,2
-0,1
0,0
0,1
0,2
0,3
0,4
0,5
0,6
198219841986198819901992199419961998200020022004200620082010
xChandler
yChandler
-0,3
-0,2
-0,1
0,0
0,1
0,2
0,3
0,4
0,5
0,6
198219841986198819901992199419961998200020022004200620082010
xannual
yannual
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-11
CausesofvariationsofEarthrotation
YPcomponentandatmosphericangularmomentum[“]
LODandangularmomentumofatmosphericmassmotions[s]
0,0
0,1
0,2
0,3
0,4
0,5
198219841986198819901992199419961998200020022004200620082010
yannual
yAAMmass
-0,001
0,000
0,001
0,002
0,003
0,004
198219841986198819901992199419961998200020022004200620082010
LOD
AAMmotionz
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-12
ImportanceofvariationsofEarthrotationfor
referencesystems
ThevariationsofEarthrotationwithrespecttotheinertialspace
(precessionandnutation)affectthecoordinatesofcelestialbodies
(rightascencionanddeclination).Theymustbereducedfromthe
positionsinthecelestialreferencesystem(Quasars,satelliteorbits).
ThevariationsofEarthrotationwithrespecttotheEarth’sbody
(polarmotionandUT1/LOD)affecttheterrestrialcoordinates
(latitudeandlongitude).Theymustbereducedfromthepositionsin
theterrestrialreferencesystem(referenceframe).
IfthevariationsofEarthrotationarenotreduced,theyproduce
errorsintheprecisepositioningusingspacetechniques(e.g.GPS)
cominguptoseveralmetres.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-13
acagacagacag
Lunisolartidalacceleration(1)
Centrifugalacceleration
ac=2
*(r*-Rcos)
Gravitationalacceleration
ag=GM*/(R2+r2
*-2Rr*cos)
Totalacceleration:(Doodson’sconstantD*=0,75GM*R2/r3
*)
-radial:ar=2/RD*(cos2*+1/3)-tangential:a=2/RD*sin2
Celestialbody*M*:MEarthR:r*D*
Moon1:81,31:602,628m²/s²
Sun332946:11:234811,208m²/s²
EffectsofMoonandSun
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-14
Lunisolartidalacceleration(2)
Mostimportantpartialtides(Torge1989)
SymbolPeriod[h]Amplitude[ms-2]Origin(Lunar/Solar)
Longperiodcomponents
M0102,9Lconstantflattening
S047,7Sconstantflattening
Diurnalperiods
K123,93436,9L,Sinclinationoftheecliptic
O125,82310,6Lprincipalwave
P124,07144,6Sprincipalwave
Semidiurnalcomponents
M212,42375,6Lprincipalwave
S212,00174,8Sprincipalwave
N212,6671,9Leclipticorbit
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-15
Lunisolartidalacceleration(3)
Variationofamplitudesasafunctionoflatitude
Acceleration[=10nms-2]Inclination[10-3"]
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-16
DeformationcausedbyEarthtides(1)
DeformationoftheelasticEarthgeneratedbythepotentialVm:
rel=h·rm=h·Vm/g(h=Lovenumber)
Deformationofthelevelsurface(W+Vm)generatedbythemass
displacements:
Vd=k·Vm(k=Lovenumber)
W=gravitationalpotential
Vm=tidalpotential
Vd=deformationpotential
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-17
Horizontalsurfacedeformation(l=Shidanumber):
xel=l(1/g)(V/)yel=l(1/g)(V/cos)
Gravimetricamplitudefactor:Inclinationamplitudefactor:
=1-1,5k+h=1+k-h
h(f),k(f),l(f),(f),(f)arefrequencydependentparametersreflecting
theelasticbehaviourofthebodyoftheEarth.
DeformationcausedbyEarthtides(2)
MagnitudesLunareffectSolareffect
Equipotentialsurfacedeformation~36cm~16cm
h0,64(topographydeformation)~23cm~10cm
k0,32(subsequentleveldeformation)~12cm~5cm
l0,16(horizontaldeformation)~10-8=1m/100m
1,164(gravimetricamplitude)-1,1...+0,5--2
0,674(inclinationamplitude)0,017"0,008"
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-18
ObservationofEarthtides
SuperconductingGravimeterHorizontalpendulum
Verbaandert-Melchior
Zöllner
principle
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-19
Observedangle
Plumbline
Rotationaxis
Observationofoceantides
Principleof
atidegauge
BM
well
zero
mark
ruleheight
rulezero
Principleofsatellitealtimetry
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-20
Oceantidemodel
ExampleM2-model:Amplitude(dashedlines[cm])andphase([h])
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-21
ImportanceofEarthandoceantidesfor
terrestrialreferencesystems
TidesaffecttheEarthsurfaceintwoways:
•DeformationcausedbygravitationalforcesofMoonandSun;
•Deformationcausedbyvariableoceanandatmosphereloading.
Theeffectsontheterrestrialreferenceframemustbereduced(e.g.
bytheterrestrialmodelFES2004andtheoceanmodelScherneck).
Inthe(geometric)terrestrialreferencesystemwereducetheentire
effectincludingthedeformationcausedbythepermanentpotential
ofMoonandSun.Theresultisasystemcompletelyfreeoftidal
effects(“tidefreesystem”).
Inthemeasurementofterrestrialgravitytheeffectofthepermanent
potentialisnotreducedbutonlythetemporaleffects.Theresultisa
systemoftheaveragetides(“meantidesystem”).
Asaconsequencepositionsandgravityvaluesarenotconsistent.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-3-22
2.4Terrestrialreferencesystemandframe
Celestial
Intermediate
System
Z
Y
X
Z‘
Y‘
X‘
Satellite(XS,YS,ZS)
Station
(XP‘,YP‘,ZP‘)
3
1
2
)'(
i
PiSiXXS
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-1
Requiredconsistencyofreferencesystems
Z‘
Y‘
X‘
Thedifferencebetweenthereferencesystems(originandorientation)
enterscompletelyintothecoordinates(e.g.polarmotionupto18m)
Consequencesofnon-
consistentsystems
1.Single(precise)point
positioning(PPP)
Z
Y
X
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-2
Z‘
Y‘
X‘
Ascalefactorof2...3·10-7permetredifferenceinthereference
systemsentersintothebaselines(18mpolarmotion:5mm/km)
Z
Y
X
Requiredconsistencyofreferencesystems
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-3
Consequencesofnon-
consistentsystems
2.Relative(differential)
positioningbyGPS
ConsistencyofCRSandTRSbytransformation
ConventionalInertialSystem
(CIS)
Coordinates(angles)
oftheQuasars(ICRF)
CelestialIntermediatePole
(CIP)
Precession-NutationN(,)
XPole,YPole,UT1-UTCPolarmotion,
DUTvariation
ConventionalTerrestrialSystem
(CTS)
Pointcoordinates(3-D)
attheEarthsurface
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-4
InternationalTerrestrialReferenceSystem(ITRS)
Theterrestrialreferencesystemmustbeconsistentwiththecelestial
referencesystemusedforsatelliteorbits.ITRSisa3-DCartesiansystem.
•TheoriginoftheITRSisdefinedbytheEarth’scentreofmass
(includingatmosphereandoceans),becausesatellitesareorbiting
aroundthegeo-centreaccordingtoKepler’slaws.
•Theorientationoftheaxesisdefinedconventionallybyfixingthe
Z-axisclosetotheEarthrotationaxisandtheX-axisclosetothe
Greenwichmeridianinitspositionsatagivenepoch.Thetime
evolutionmustbegivenbystationmotionparameters(velocities)
notgeneratingaglobalrotationwithrespecttotheEarthrotation.
•Thescaleunitismetricasdefinedbythespeedoflightinvacuum
accordingtotheInternationalSystemofUnits(SI)andbythe
standardgravitationalparameterGM(forsatellitetechniques).
•StandardsandmodelsaredefinedbytheIERSConventions.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-5
RealisationofthedatumoftheITRS
•VLBIprovidestheorientationbytheEOP(,,XP,YP,UT1);
•SLRandVLBIprovidethescaleunitbymeasuringdistancesbased
onthemetredefinition(speedoflightreducedtotheatmosphere).
(GPSdoesnotmeasuredistancesbutdifferencesofdistances)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-6
Thesphericalharmonicsof
theEarthgravityfieldare:
C11=Xdm/aM
S11=Ydm/aM
C10=Zdm/aM
Thecentreofmassistheintegral
overallthemassesoftheEarth:
X0=Xdm/M
Y0=Ydm/M
Z0=Zdm/M
Thedatumisrealisedbycombinationofgeodeticspacetechniques:
•SLRprovidestherelationtothegeo-centrebydeterminingtheorbit
intheEarthgravityfieldwithcorrespondingparameters:
AgravitymodelwithC11=S11=C10providesgeocentriccoordinates!
PrincipleofSatelliteLaserRanging(SLR)
Measurement
ofthedistance
Earth-satellite
-Earth
•Laser-rangingisalwaysglobal.
•Thedistancesfromthetelescopes
tosatellitesaremeasuredbythe
signal’straveltimeasroundtrip.
•Thereisonlyoneclock,therefore
thereisnoneedofestimating
clockcorrections(likeinGPS).
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-7
Realisationoftheorigininthegeo-centrebySLR
Kepler
orbit
dobs
Rsta
Rsat
Rsat
geoc–dobs=Rstageocentric
Laserrangingprocessingincludes
thecomputationofglobalorbits.
EmployinganEarthgravityfield
modelwithsphericalharmonic
coefficientsC11=S11=C10=0
providesephemeridesreferringto
thecentreofmass(geo-centre).
Subtractingthemeasureddistance
fromthegeocentricradiusvector
providesgeocentriccoordinatesof
thetelescope.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-8
ThisisdifferentinGPS,wherewe
getonlybaselinesbecauseclock
errorshavetobeestimated.
True
orbit
Realisationoftheorientationbyconvention
Onecouldrealisetheorientationalsobythedegree-twospherical
harmonicsofthegravityfieldwhichprovidethesymmetryaxes
(axesofmaximummomentofinertia)ofthemassesoftheEarth:
C21=dm/a2M,S21dm/a2M
S22dm/2a2M
However,thesecoefficientscannotbedeterminedwiththerequired
precision.Thereforetheorientationisrealisedbyconvention,fixing
theZ-axisclosetotheEarthrotationaxisandtheX-axisclosetothe
GreenwichmeridianasgivenbytheBureauInternationaldel’Heure
(BIH,apredecessoroftheIERS)fortheepoch1984,0.
Theseparametershavetobeextrapolatedintimeinawaythatno
globalrotationoftheEarthcrust(realisedbytheobservation
stations)remainsinthecoordinatestimeseries(seelaterslides).
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-9
Realisationofthemetricscale
•Thescaleunitoftheterrestrialreferencesystemisthemetreas
definedintheInternacionalSystemofUnits(SI)bythespeedof
lightinvacuum.Toreducethespeedforatmosphericeffects,an
atmospheremodelisintroducedanditsparametersarecorrectedby
estimationinthecommondataadjustment.
•Theestimatedparametersaredependedonthesignalfrequencyand
correlatedwithinstrumentalproperties(e.g.electronicdelayofthe
signal,phasecentres,deformations),stationheightsandothers.
Thereforewegetdifferent“metric”scalesfordifferenttechniques.
•Inordertoavoiddistorsionsbetweenthetechniques,thescaleis
definedbyoneortwotechniques(SLRandVLBIinITRF2008)
andthescalesoftheothersareestimatedw.r.t.these.
(Thebestwouldbetoeliminatealltheinstrumentaleffectsbyan
externalcalibrationresultinginidenticalscalesforalltechniques.)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-10
EvolutionoftheITRSintime(1)
Geo-centre
Z
Y‘
X‘
Satellite(XS‘,YS‘,ZS‘)
transformed
fromCIStoCTS
Station
(XP‘,YP‘,ZP‘)
Theterrestrialstationsaremovingbecauseofcrustaldeformations.
Satellitesdonotparticipateinthismovement.
Z‘
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-11
EvolutionoftheITRSintime(2)
•BesidesthevaryingEarthrotationwehavetocorrectthereference
framecoordinatesbyitsmotionsw.r.t.theCTSinordertoconserve
theconsistencywiththeCIS,i.e.thesatelliteephemerides.
•Principalterrestrialstationmotionsarecausedbyplatetectonics
whichisalong-timeprocess.Thereforetheyaremodelledbylinear
coordinatechanges,i.e.byconstantvelocitiesdX/dt,dY/dt,dZ/dt.
•Thismodellingrequiresakinematicreferencesystemtowhichthe
velocitiesreferandwhichisconsistentwiththeEarthrotation,i.e.
itmustnotgenerateaglobalrotationoftheEarthcrust.
•Thekinematicreferencesystemisdefinedbygeologic/geophysical
modelsoftectonicplatemotiongeneratingnonetrotations(NNR);
-untilITRF91:AM0-2(MinsterandJordan1974,1978);
-untilITRF94:NNRNUVEL-1(ArgusandGordon1991);
-untilpresent:NNRNUVEL-1A(DeMetsetal.1994).
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-12
12mayorplateswithgivenmotionbyrotationvectors(seechapter6)
Geologic-geophysicalplatemodelNNRNUVEL-1A
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-13
INDI
ANTA
NOAM
CARB
SOAM
ARAB
AUST
NAZC
PCFC
EURA
COCOAFRC
Geodeticstationmotions(velocities)shallrefertothismodel.
EURA
•TheInternationalTerrestrialReferenceSystem(ITRS)isrealisedby
theInternationalTerrestrialReferenceFrame(ITRF).
•AccordingtoresolutionsoftheInternationalUnionofGeodesyand
Geophysics(IUGG)theITRFistheonlyterrestrialreferenceframe
tobeusedinscienceandpractice.(WGS84adoptedITRFin2002.)
RealisationoftheITRSbytheITRF
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-14
0,0
5,0
10,0
15,0
20,0
25,0
30,0
35,0
19881990199219941996199820002002200420062008
[cm]
WGS84
ITRFyyyy
Deviationsfromthegeo-centre)
TheITRFisrealisedbycombiningindividualsolutionsofpositions
(andvelocities)providedbyanalysiscentresofdifferenttechniques.
until2000:internationalcallforindividualsolutionsbyIGNParis;
since2005:combinationofsolutionsofIAGservices(IGN,DGFI).
RealisationsoftheITRF
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-15
No.ofstationssol‘s:VLBISLRGPSDORISTotal
ITRF8812056--11
ITRF8911368--14
ITRF9012047--11
ITRF91131571-13
ITRF92155566-17
ITRF93160645-15
ITRF94209615315
ITRF96290427316
ITRF97309456318
ITRF2000477396+8*321+8*
ITRF200533811114
ITRF200857811114
regionaldensifications
•TheIERSProductCentrefortheITRF(IGNParis)releasesacallto
theIAGServices(IVS,ILRS,IGS,IDS)toprovideweekly(incase
ofVLBIsession-wise)solutionsoftheiranalysiscentresincluding
3Dstationpositioncoordinates(X,Y,Z)anddailyEarthOrientation
Parameters(EOP)combinedbytheircombinationcentrestoone
techniquesolutionpreferablyintermsofdatum-free(alternatively
±1m)normalequations.
OrganisationoftheITRFsince2001
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-16
•The(presentlytwo)ITRFcombinationcentres(IGN,DGFIMunich)
-analysetheweeklytimeseriestoidentifythedegreeoffreedomof
thedatum(e.g.eventualconstraintsbyintroducingconditionson
stationcoordinates),grosserrors,jumpsincoordinates,etc.;
-combinetheweeklysolutionsofeachobservationtechniqueinto
onesolutionofcoordinatesforadefinedreferenceepochandits
intra-techniquecombination.
Earthquake
Arequipa,
Peru2001
Examplesofweeklycoordinatesanalysis
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-17
Jumpincoordinatesanddifferentvelocitiesrequired
Antenna
change
Høfn2001
Iceland82,55
82,60
82,65
82,70
82,75
82,80
1043109511471200125213041356140814601513
Height
200420032008200120002002200520062007
Tobeconsideredasanewstationafterantennachange
Seasonal
variation,
Brasilia,
Brasil1105,99
1106,00
1106,01
1106,02
1106,03
1106,04
1043109511471200125213041356140814601513
[m]
Week
Height
200420032008200120002002200520062007
Aminimumoftwoyearsrequiredforvelocityestimation
200520062007200420032008200120002002
Analysisofthenormalequations(DGFI)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-18
•Thenormalequationsmustnothaveanyconstraintsonthedatum
(“datumfree”normalequations)i.e.thedatumdefectmustbe7.
•Afixeddatumindifferentweeksrefersthecoordinatestodifferent
origins,orientationsandscales,whichthencannotbecombined.
•Alltheinputdatawereanalysedforthedatumdefect.Theresultis:
-GPS(IGS):nodefectatall(allparametersarefixed),
-SLR(ILRS):defect3(3translationsandscalearefixed),
-VLBI(IVS)defect6(scalefixed),
-DORIS(IDS)nodefectatall(allparametersarefixed).
•Ifthedefectisnotappropriate,onehasto“liberate”thedatum,i.e.
onehastointroducecolumnsandrowsintothenormalequations
forthosedatumparameterswhichshouldbefree.
(IGNdidnotusenormalequationsbutsolutions+covariancematrices)
InputdataforITRF2008:X,Y,Z,XP,YP,UT1
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-19
Technique
Service
AnalysisCentre
Data:epochs,
weeklyseries
Interval
GPS
IGSAC
NRCOttawa
Weeklysolutions
(LOD)
1997-2008
SLR
ILRSCC
ASIMatera
Weeklysolutions
(LOD)
1983-2008
VLBI
IVSCC
GIUBBonn
24hsessions,free
normalequations
1980-2008
DORIS
IDSCC
CLSToulouse
Weeklysolutions
(LOD)
1993-2008
Total
~1500occupations
~920points
578stations
~4500solutions
withdailyEOPs
(UT1onlybyVLBI)
1980-2008
Combinationprocedure(DGFI)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-20
GPS
n.eq.week1
n.eq.week2
n.eq.weekn
...
Multiannual
X,v,EOP
DORISVLBI
n.eq.week1
n.eq.week2
n.eq.weekn
...
Multiannual
X,v,EOP
n.eq.week1
n.eq.week2
n.eq.weekn
...
SLR
n.eq.week1
n.eq.week2
n.eq.weekn
...
Multiannual
X,v,EOP
Multiannual
X,v,EOP
Accumulationofnormalequations,inputoflocalties,datumfixing
ITRF2008:positions,velocitiesandEOP
(IGNdidnotusenormalequationsbutcombinedbyHelmerttransformation)
localties
Accumulationofnormalequationsofthetimeseries
Localtiesatco-locationstations
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-21
•Thereliabilityofthelocalmeasurementsmaybeevaluatedby
-comparingthemwiththeestimated3-Dcoordinatedifferences,
-comparingthevelocitiesobtainedfromthedifferenttechniques,
-comparingEOPsobtainedfromthedifferenttechniques(DGFIonly)
•Tocombinethenormalequationsofthedifferenttechniquesweneed
connections(coordinatedifferences)betweentheirreferencepoints.
•Thedefinitionandrealisationofthereferencepoints(phasecentres)
iscomplicated
X
andthetiesmustbetransformedtotheglobalframe.
Co-locationsitesofspacetechniques
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-22
No.ofco-locations
4techniques:7
3techniques:22
2techniques:60
no.ties[mm]limit[mm]no.select
GPS-VLBI970.5...1.03233
GPS-SLR1170.5...1.03030
GPS-DORIS1370.5...1.03034
VLBI/SLR-DORIS930.5...1.03028
VLBI-SLR
Localtiesselected
no.vel.no.select
5424
5123
4214
306
1711
Localtiesandidentical
velocitiesselected(DGFI)
Introductionofthedatumparameters
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-23
•DGFI:Theinputdataare(madeto)datumfreenormalequations.
•IGN:TheindividualsolutionsarestackedbyHelmert-transformation
•Thedatumparametersareintroducedinthefinalcombination:
•TheITRF2008datumreferstotheepoch2005.0by
-3translationsgivenbySLR:coordinates’origin=geo-centre,
-3rotationsbytheITRF2005:orientationaccordingtoBIH1984,
-1scalebySLRandVLBI:speedoflightcorrectedbyatmosphere,
-3velocitiesoftranslationsbySLR:velocityorigin=geo-centre,
-3velocitiesofrotation(withEOP):byNNRcondition:
IGN:geophysicalmodelNNRNUVEL-1A;
DGFI:geodeticmodelAPKIM(seechapter6);
-1driftofthescalebySLRandVLBI(onlyIGN).
StationvelocitiesoftheITRF2008
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-24
Differentvelocitiesinasiterefertodifferentperiodseitherdueto
seismiceventsorinstrumentalchanges(e.g.antennachange).
PrecisionoftheITRF2008
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-25
StatisticsofformalerrorsinIGNandDGFIcomputations
T(X)T(Y)T(Z)R(X)R(Y)R(Z)ScaleRMS
SLR
VLBI
GPS
DORIS
-0,1/-0,2
-1,8/0,4
-1,1/0,1
1,3/-0,1
0,0/-0,5
1,3/0,4
0,1/-0,1
0,1/0,4
-0,3/0,1
-0,9/-0,1
-4,9/0,0
-3,0/0,8
0,5/0,3
0,1/0,0
0,4/0,0
0,0/0,0
-1,0/0,4
-1,3/0,0
-1,3/0,1
-2,7/0,0
1,8/0,4
5,3/-0,1
0,1/0,0
-3,3/0,0
-2,0/0,1
2,1/-0,1
2,9/0,0
3,2/-0,1
2,0/0,8
0,4/0,1
1,3/0,2
3,2/1,0
ComparisonofITRF2008byIGNandDGFI([mm]/[mm/a])
[mm][mm/a]
14
0
926
359
145
5443
4
697
456
246
7656
189
41
[mm]
2412
231283
190
7643
531
DGFIcoordinates(t0=2005,0)
[mm/a]
307
498
294
181
64
2088
DGFIvelocities(dX/dt)
IGNcoordinates(t0=2005,0)IGNvelocities(dX/dt)
WGS84inITRF2008
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-4-26
WGS84wasdesignedforsatelliteorbitdeterminationandsinglepoint
positioning(PPP).11stationsarenotsufficientforareferenceframe.
2.5Regionalandnationalreferenceframes
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-1
TheglobalITRFisdensifiedbyregionalreferenceframesobserved
byGPSinconnectionwithglobalIGSstationsincludedintheITRF.
ThefirstnetworkofthistypewasobservedinEuropein1988.
EUNAV-88(pre-campaign)
EUREF-89(densific.)/Turkey
EUREF-NW(Iceld./Greenld.)
EUREF-EAST/Hun/Slov/Czech
EUREF-Poland/Baltic/Bulgaria
EUREF-Cyprus/Iceland/D/NL
EUREF-Lux/Slov/Croa/Roman.
EUREF-Ukraine/Slovenia
EUREF-FYROM/Bosnia
EUREF-Serbia/Monte./Albania
EUREF-Moldavia1999
1988
1989
1990
1991
1992
1993
1994
1995
1996
1998
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-2
SistemadeReferencia
Geocéntricopara
AméricadelSur
(SIRGAS)
58stationsobservedfor
10daysinMay1995
(seechapter5)
SouthAmerican
ReferenceFrame
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-3
184stationsobserved
for10daysinMay2000
extendingthereference
frametoallAmericas.
SistemadeReferencia
Geocéntricoparalas
Américas(SIRGAS)
TheAmericas
ReferenceFrame
Thecontinentalframes
arefurtherdensifiedby
nationalreferenceframes
Peru
Colombia
NationaldensificationsbyGPScampaigns
Argentina
Venezuela
19981994
Ecuador
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-4
Disadvantagesofreferenceframesbycampaigns
•Thereferenceframecoordinates(positionsatadefinedreference
epoch)serveasreliablevaluesforconnectingthenewpoints,but:
ThepointsaremovingwiththeEarth’scrust
-continuouslywiththetectonicplates,
-irregularlyindeformationzones,
-sporadicallybyearthquakes.
•Thecoordinateshavetobetransformedfromthedefinitionepoch
totheobservationepochofnew(connected)stations.Thisis
problematicifthesemotionsarenotknown(observed).
•Itismuchbettertoestablishreferenceframesbycontinuously
observingstations.Thisisdoneinallcontinents.
•Whenusingthereferenceframeforpracticalpositioningonehasto
occupysimultaneouslythereferencestationsinadditiontothenew
stationsbyotherreceivers(moreGPSreceiversrequired).
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-5
EuropeanPermanentTrackingNetwork(EPN)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-6
AfricanReferenceFrame(AFREF)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-7
Asia-PacificReferenceFrame
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-8
NorthAmericanReferenceFrame(NAREF)
(6differentreferenceframesofdifferentinstitutions)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-9
SIRGASnetworkof
continuouslyobserving
stations(SIRGAS-CON)
(>300stations)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-10
CoveringtheareaofMexico,
CentralAmerica,TheCaribbean,
SouthAmericaandsomepartsof
Antarctica(seechapter5)
Nationaldensificationsbycontinuously
observingstations(examples)
Brazil(RBMC)Colombia(MAGNA-ECO)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-11
Realisationoftheregionalreferenceframes(1)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-12
TheregionalreferenceframesaredensificationsoftheITRF.Asthe
ITRFcontainsalsoSLR,VLBIandDORISstations,andtofacilitate
theeasyaccess,theIGSprovidesanITRFextractforGNSSusers.
TheIGS08fullnetworkconsists
of232stationsandisthebasis
fortheregionaldensifications.
TheIGS08corenetworkconsists
of91stationsandformsthebasis
fortheorbitdeterminations.
Realisationoftheregionalreferenceframes(2)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-13
TheIGSupdatesthereferenceframesbyreprocessingandconsidering
changesinthestationoccupations.ThesenewframesarecalledIGb.
Theanalysiscentresforthecontinentalnetworkscomputeweekly
solutionsofstationcoordinatesreferringtotheactuallyvalidIGb
frame.Usersshouldapplytheseframes(SIRGAS)forpositioning.
Realisationoftheregionalreferenceframes(3)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-14
AllIGSRegionalNetwork
AssociateAnalysisCentres
(RNAAC)deliverweekly
solutionstotheGlobal
NetworkAssociateAnalysis
Centres(GNAAC),where
theyarecombinedtothe
globalpolyhedron(1008
stationsat(2013-09-18).
TheRNAACalsocomputemulti-annualsolutionsincludingprecise
stationcoordinatesatadefinedepoch(t0)anditslinearvariationsin
time(constantvelocitiesv).Theseshallbeusedtoextrapolatethe
coordinatesfromthedefinitionepochtotheobservationepochti:
X(t)=X(t0)+v(ti–t0).
NAREFvelocitiesSIRGASvelocities
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-16
Preciseuseoftheterrestrialreferenceframe
Definition
ofthereference
system
GPSobservation
atepochtiinthe
referencesystemtk
Archive
ofcoordinatesX
forepocht0
Transformationt0ti
X(ti)=X(t0)+T0k+R0k·X(t0)
+dX/dt·(ti–t0)
Calculationof
coordinates
Y(new)
Re-transformationtit0
Y(t0)=Y(ti)–T0k–R0k·Y(tk)
–dY/dt·(ti–t0)
Consistentvelocities:from
thearchiveorfromamodel
Rem.:Adifferenceof1mbetweenreferenceframescausesrelative
errorsof~2…3·10-7
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-17
VelocitymodelforSouthAmericaandCaribbean
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-18
VEMOS09
(SIRGAS09)
VEMOS09
(relative
SOAM)
HowtousetheITRF/IGSstationsindetail?
-ForGNSSpositioningoneshouldusetheIGScoordinatesorits
densifications(e.g.SIRGAS)whicharegiveninSINEXformat.
-Thepointshaveadetailedidentification:SCPCDOOCSDSN
SC=sitecode(4characters),e.g.AZUE
PC=pointcodewithinthesite,e.g.A
DO=monumentidentification(domesno.):iiinsrjjj(e.g.41301M001)
iii=countryno.,e.g.Panama=413,ns=numberofsite,
r=reference(Monument,interSection),jjj=no.ofoccupation
OC=observationtechnique:D=DORIS,L=SLR,P=GPS,R=VLBI
SD=stationdescription(21characters),e.g.CiudaddeChitre,Panama
SN=numberofcoordinatesolutionforthispoint,e.g.0001,
OccupationperiodsaredefinedinanothersectionoftheSINEXfile.
-Coordinates(STAX,STAY,STAZ)andvelocities(VELX,VELY,
VELZ)aregivenwiththisidentification.Observecarefullytheperiod!
-IftheSIRGASweeklysolutionsareused,onehastorealizethat
theyrefertotheepochoftheparticularweek!
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20132-5-19
3.GNSSPositioning
BeganonOctober4,1957withthelaunchofthe
SovietsatelliteSputnik.
Soonafter(1958)WilliamGuierandGeorge
Weiffenbach,attheAppliedPhysicsLaboftheJohns
HopkinsUniversity,computedthesatelliteorbitusing
theDopplershiftmeasuredonthesignalfromapointof
knownposition.
Quicklywasrecognizedthattheproblemcouldbereversed:calculatetheposition
ofthemeasuringpointfromasatelliteofknownorbit.
Thatinitiatedthe‘NavyNavigationSatelliteSystem’(NNSS)programoftheUS
Army,whichspawnedthefirstGNSSknownasTRNASIT.
TheTRANSITconstellationwascomposedby5satellitesandoperatedfrom1960
to1996,beingopenedtocivilusersin1967.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
TheTRANSITsuccessor
From1964to1966higherrequirementsfora
newGNSSwereestablishbytheUSAAF:
determinetheinstantaneouspositionofapoint
atrestormoving,atanytimeandplace,by
simultaneouslymeasuringtherangesto4
satellitesofknownpositions.
ThatrequirementwasfulfilledbytheGPSwhita24-satellitesconstellation.
Satelliteshavebeenlaunchedfrom1978.
Openedtocivilusersin1983.
InitialOperationalCapabilityachievedinDecember1993.
Complete24-satelliteconstellationinMarch1994.
Atotalof64satelliteshavebeenlaunchedtodate.
Presentlythereare31operationalsatellites.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Allthesatellitescarryredundantatomicclocks.
BroadcasttwocarriersnamedL1andL2.
Carriersaremodulatedwithdifferentcivilandmilitarycodes.
Militarycodesareencrypted.
7BlockII-RMsatellitesmodulatenewcivil(CAonL2)andmilitary(M)codes.
4BlockII-Fsatellitesbroadcastanewcarrier(L5)toprovidesafety-of-lifeservices.
Anewgenerationofsatellites(BlockIII-A)willbelaunchedfrom2014.
PresentandplanedGPSstatus
BlockII-R
BlockII-RM
BlockIII-A
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Presently,onlytheUSAGPSandtheRussianGLONASSarefullyoperationalat
globalscale.
ChinaisexpandingitsregionalBeidouintotheglobalCompass.
TheEUisdeployingitsGalileo(presently,only4satellites).
PresentandplanedGNSSstatus
IndiaandJapanare
developingregional
navigationsystems.
Therearemany
augmentationsystemsfor
improvingnavigationintegrity
andaccuracyatlocalor
regionalscale(e.g.:theUSA
WAASortheEUEGNOS).
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
GPSconstelation
6orbitalplanes,55°inclination,60°inlongitude;
31satellite,4+satellitesperplane;
~circularorbits,~26.000kmradii(~20.000km
abovetheEarthsurface);
12hsidereal(11h58mUT)revolutionperiod,
orbitalspeed:3.9km/s.
4+satellitesoverthehorizonatany
placeandtime.
Samegroundtrackeveryday~4mafter.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Navigationprinciple
Aworld-wideinfrastructure
composedbytrackingstations,
terrestrialandground-to-satellites
datalinksandoperational
centres,monitorandcontrolthe
constellation.
Basedonrecentmeasurements,satellitesorbitsare
computedandextrapolatedtothenearfuture.
Extrapolatedorbitsarebroadcastedtousers(the
‘broadcastedephemerides’)bythesatellites.
Measuringsimultaneouslytherangesto4satellites
thereceivercorrectsitsclocksandcomputeits
positioningusingthetrilaterationprinciple.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Theequationofobservation
Satelliteandreceivercoordinatesmustbe
inthesamereferencesystem.
222
R
SSS
RRRR
SS
XYZXYZ
Thereceiverposition(unknown)iscomputedbasedonthesatelliteposition
(known)andthesatellite-to-receiverrange(measured).
Theequationofobservationresults
fromthePythagoras'sTheorem.
wherestandsforalltheunaccountederrors.
S
R
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
3.1mathematicalfoundationThesystemofequationsofobservation
Theequationofobservation
contains3unknowns:
222
R
SSS
RRRR
SS
XYZXYZ
1
2
1
2
3
222
222
222
111
222
3333
RRR
RR
R
R
R
R
RRR
R
R
R
XYZ
XY
XYZ
XYZ
XYZ
Z
XYZ
Solvingtheproblemrequires
simultaneousmeasurementsto,
atleast,3satellites:
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
ReceiverapproximatepositionDirectioncosineofthemeasuredsatellite
Linearizationoftheequationofobservation
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Linearizationoftheequationofobservation
,0,0,,0
22
,,0,
2
coscoscos
SSS
SSSS
RRXRY
RR
S
R
S
RR
S
RR
RZRR
XYZXYZ
XYaaZ
,0,0,0,0
,0
,0
222
,
,0
,0
cos
SSSS
RRRR
S
RS
RXS
R
RRR
XXYYZZ
XX
XXX
approximatereceiver-satelliterange
directioncosinesofthe
measuredsatellite
correctionstothereceiver
approximatecoordinates.
receiverapproximatecoordinates
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Geocentriccoordinatescanbe
transformedtolocalcoordinates
bymeansof:
translationoftheoriginfromthe
geocentertotheapproximatereceiver
position;
rotationoftheaxesbythelatitudeand
longitudeoftheneworigin.
Fromthegeocentrictothelocalsystem
,,XYZ
,,enu
00,
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
0
0
P
X
Z
Y
Y0
Z0
x(n)
z(u)
y(e)
Theequationofobservationinthelocalsystem
Afterapplyingthetransformationtheequationofobservationreads:
,0,,0,0
,0,,0,0
,0,,0
cossincos
coscoscos
cossin
SSS
ReRR
SSS
RnRR
SS
RvR
aEA
EA
aE
,0,0,,0,,0,coscoscosSSS
ReR
S
RnRRRRR
S
v
S
Raaenv
E=satelliteelevation
A=satelliteazimuth
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Thesystemofequationsofobservation
1
,0
2
,0
,0
111
,0,,0,,0,
222
,0,,0,,0,
,0,,0,,0
1
2
,
1
2
coscoscos
coscoscos
coscoscos
ReRnRv
ReRnRv
n
RRR
nn
ReRnR
RRR
R
R
R
n
R
R
RR
R
R
R
nn
vRR
aa
aa
aa
env
env
env
1nL
Usingmatrixnotation
111
,0,,0,,0,
222
,0,,0,,0
1
2
,
,0,,0,
1
,0
,0
2
,0
,
,
1
0
2
coscoscos
coscoscos
coscoscos
ReRnRv
ReRnRv
n
R
R
R
R
n
nn
Re
R
R
R
R
n
RnRv
n
R
R
RR
aa
a
e
n
v
a
aa
3nA1nv31x
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
LeastSquaressolution
1
andminˆTTT
xxAAvvAALvL
aretheunknownestimates;
aretheerrorestimates;
istheunityofweightstandarddeviation;
Thevariance-covarianceoftheunknowns
ˆˆˆˆ,,
T
RRRnevx
supra-indexTstands
fortransposedmatrix
ˆˆLAvx
1
2
2
2
22ˆˆˆ
nnnenv
T
eneeev
vnvevv
qqa
qqq
qqq
xCAA
2
1
ˆ
1
ˆ
n
i
i
n
measurementerrors
cofactormatrixoftheunknowns
geometricalconfigurationofthe
measuredsatellites
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Dilutionoftheprecisionfactors
Thereceiverpositionanditserrorcanbecomputedfrom:
,0
,0
,0
ˆˆ
ˆˆ
ˆˆ
R
R
RR
RR
RRR
nn
ee
vv
n
e
v
Theqfactors(co-factors)dependsonlyonthe
geometricalconfigurationofthemeasuredsatellites
anddefinedtheDilutionofthePrecision(DOP)factors:
222
22
ˆˆˆ
ˆˆˆ
ˆˆˆ
nneevv
nneeH
v
P
Vv
qqqPDOP
qqHDOP
qVDOP
3-D
horizontal
vertical
ˆˆ
ˆˆ
ˆˆ
n
e
n
vv
n
ee
v
q
q
q
good(PDOP<3) Poor(PDOP>6)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Errorsources
Measurementerrors:willbediscussedlateron.
Errorsinthesatelliteposition:
Broadcastedephemerids:±1m
RTIGSorbits:±5cm
FinalIGSorbits:±2.5cm
Approximatereceiverposition(canbeimprovediteratively).
Theseerrorsapplyontheapproximatereceiver-satelliterangeand,withmuchless
importance,onthedirectioncosines:
,0,0,0,0
,0,,0,0
,0,,0,0
,0,,0
222
coscoscos
cossincos
cossin
SSSS
RRRR
SSS
RnRR
SSS
ReRR
SS
RvR
XXYYZZ
EA
aEA
aE
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
3.2pseudo-rangemeasurement
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
GPScarriers
c
f
a=amplitude
f=frequency
=phase
()sin(2)staft
GPSsignalsresultfromthesuperpositionofdifferentmodulationsondifferentcarriers.
Carriersarepuresinusoidalwavesmathematicallydescribedby3parameters:
GPSsatellitesbroadcast2carriers:L1andL2(BlockII-FalsoL5),whose
frequenciesandwavelengthare:
L1:f1=f0x154=1575.42MHz;1=19.05cm
L2:f2=f0x120=1227.60MHz;2=24.45cm
wavelength
speedoflightinvacuum
299792,458km/s
f0=10.23MHz
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
3.2pseudo-rangemeasurement
GPScodes
GPScarriersaremodulatedwithciviland
militarycodesandwiththebroadcasted
ephemerides.
Codesareapseudo-randombinarysequence
(onlytwovalues:0or1)modulatedonthe
phaseofthecarrier(codetransitionschange
180°thephaseofthecarrier).
CA(coarseacquisition)P(precise)
Chiplength~300m~30m
Repetitioninterval1ms1week
ModulateonL1(andL2intheII-RM)L1andL2
Encryptednoantispoofing(partially
usedbyGPSreceivers)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Measurementofthesatellitetoreceiverrange
AssumingsatelliteandreceiverclockssynchronizedtoGPStime:
ccc
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Measurementofthesatellitetoreceiverpseudo-range
1ns30cm!
'tT
''
P
cctTcctT
T=errorofthesatelliteclock
w.r.t.GPStime;
t=errorofthereceiverclock
w.r.t.GPStime.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
correction:thesatelliteclockoffset
Equationofobservationforpseudo-range
unknown:thereceiverclockoffset
,0,0,,0,,0,coscoscosSSSSS
RReRnRv
S
RRRRR
S
RTaaenvctPc
Thesystemofequationofobservationsfornsatellitesreads
111
1,0,,0,,0,
2222
,0,,0,,0,
,0
,0,,0
1
,0
1
2
,,
22
,
0
0
,
1coscoscos
coscoscos
coscoscos
ReRnRv
ReRnRv
nn
nnnR
R
R
R
R
R
ReRn
R
R
R
n
R
v
R
R
R
R
n
aa
T
aaT
T
ec
cn
v
ta
P
a
c
c
P
P
1nL4nA1nv41x
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Errorsources
ErrorsourceCAPBehaviour
Measurement±1m±30cmrandom,increasesatlowelevation.
Multipath±3m±1m
quasi-random,12hsiderealtime
repetitionpattern;increasesatlow
elevation.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Errorsources
Errorsinthesatelliteclockcorrection:
Broadcastephemerid:±5ns(±1.5m)
RTIGSorbits:±150ps(±4.5cm)
FinalIGSorbits:±7.5ps(±2.2cm)
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Carrierphasemeasurement
'ctT
l
NctT
anintegernumber
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Equationsofobservationforcarrierphasemeasurements
11111
,0,0,,0,,0,
22222
,0,0,,0
1
2
,,0,
,0,0
11
,
2
0
2
,,
coscoscos
coscoscos
coscosc
RReRnRv
RRe
R
R
n
R
RRRRR
RRRRR
RR
R
RnRv
nnnn
RReRn
R
envtN
envt
ccTaa
Taa
Ta
l
lc
en
c
c
N
l,0,osn
Rv
n
R
n
RRRcvtNa
LAxvInmatrixform:
111
,0,,0,,0,
222
,0,,0,,0,
,0,,0,,0,
00
00
00
coscoscos
coscoscos
coscoscos
ReRnRv
ReRnRv
nnn
ReRnRv
c
c
c
A
RnReRvRt1
RN2
RNn
RN
nmeasurements
3+1+nunknowns
ambiguities
anintegernumber
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Solutionoftheequationofobservationsystem
Accuratesolutionsareobtainedwiththe‘geometrical’method:
accumulatemeasurementsuntilthenumberofequationsgetsgreaterthan
thenumberofunknowns;and
theconditionnumberofthematrixAgetssmallerenoughtoensureagood
de-correlationofthedifferentunknowns.
directioncosineschangeslowly
1.5hours
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
ambiguity
ErrorsourceCAPL1orL2Behaviour
Measurement±1m±30cm±2mmcarrierphasemeasurements
areultra-precisebut
ambiguousMultipath±3m±1m±10mm
cycleslipsareproducedwhenthereceiver
lostandre-trackasatellitesignal
Errorsources
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
3.3modelstoreduceerrorsources
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
ionizedatmosphere
(~50-1000km)
neutralatmosphere
(~0-50km)
continentaldrift
crustaldeformation
seismicandvolcanoactivity
Earthtides
Earthorientationandirregular
rotation
post-glacialrebound
ocean,hydrologicaland
atmosphericloading
ldif
Effectsthatmustbe
accountedforprecise
positioning
orbit
clock
relativity
antennaphasecenters
windup
instrumentalbiases
multipath
clock
antennaphasecenter
instrumentalbiases
multipath
measurementerrors
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
3.3modelstoreduceerrorsources
PosicionamientoprecisoErrorbudget(ifnotcorrected)
Note:analgorithmisneededto
computetheGPSemissiontime
fromthereceiverreceptiontime.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Satelliteclockismovingwithrespecttothereceiverclock;inaddition,satelliteand
receiverclocksareunderdifferentgravitypotentials.
Relativisticcorrection
2
1
2
SRr
R
ttV
tc2
SRRS
R
ttUU
tc
2
sinRELEeaE
c
Themajorpartoftheseerrorsiscorrectedby
changingthefrequencyofthesatelliteoscillator
from10.23to10.2299999954MHz(~38s/day).
Thevariablepartoftheseerrorscanbe
computedas:
E=eccentricanomaly
a=semi-majoraxis
e=eccentricity
E=geocentricgravitationalconstant
Vr=relativevelocity
US=satellitepotential;UR=receiverpotential
tS=satellitetime;tR=receivertime
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Antennaphasecenters
Themeasuredpseudo-rangeis
betweenantennaphasecenters.
Satellitecoordinatescorrespondto
thecenterofmass
mustbetranslatedtoantenna
phasecenterwithaneccentricity
vector.
Geodeticcoordinatesmustbe
referredtoageodeticmark
theantennaphasecenterofthe
receivermustberelatedtothe
geodeticmarkwithaneccentricity
vector.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
RECCRPCRPCRPCV
Antennaphasecenterofthereceiver
Variablepart(PhaseCenterVariation)
PCVinmm(afterSteigenbergeretal.,2006).
ValuesaredifferentforL1andL2andchangeifaradomeisontheantenna.
Constantparts
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Antennaphasecenterofthesatellites
SSS
ECCPCMPCV
constantpart
variablepart(PhaseCenterVariation)
Theorientationofthex-y-zsystemchangesw.r.t.
theobserveraslongassatellitemoves
PCVinmm(afterSteigenbergeretal.,2006).
ValuesaredifferentforL1andL2.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Carrierphasewind-up
'
'arccos
'
''''
sign
dd
dd
dx
dx
ItisduetothecircularpolarizationoftheGPSsignals.
ValuesaredifferentforL1andL2.
Dependsontherelativeorientationofsatelliteand
receiverantennas.
Changesintherelativeorientationcauseaphase
variationthatthereceivermisunderstandsasarange
variation.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Instrumentalbiases
Thesatelliteandthereceiverelectronicsproducedelays(afew
nanoseconds)inthetimeofemissionandreceptionofthesignals.
Ifnotcorrected,thesedelaysaremisinterpretedasanincreaseof
thepropagationtimeandhenceofthemeasuredpseudo-range(a
fewmeters).
Delaysaredifferentforcodeandcarriermeasurements.
Delaysarefrequencydependent.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Speedoflightintheatmosphere,v,differsfromthevacuum,c:
Errorscausedbytheatmosphere
00
1dr
ndr
vc
c
n
v
refractionindex;n=1invacuum.
Thepropagationintervalsintheatmosphereandinvacuumare:
0
1
'1ndr
c
(neglectingthebending)
andthedifference:
0
1
'1dr
c
whichisequivalenttoarangeerror:
6
00
110cndrNdr
refractivity
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Theionosphereextendsfrom~50kmabovetheErathsurfaceandischaracterizedbythe
presenceoffreeelectrons(electronsdissociatedfromatomsandmolecules).
Itsmassislowerthan0.1%ofthetotalatmosphericmass,butfreeelectronsinteractwith
electromagneticwaveschangingitsspeedofpropagation.
Ionosphericerror
6
2
40.310
INED
f
f=frequency(Hz);
ED=electrondensity(electrons/m3)
-forcarrier/+forcode
16
22
0
40.340.310
IEDdrTEC
ff
TECTotalElectronContent
(TECu1TECu=1016electrons/m3)
frequencyerror(1TECu)
L1:1,57542E+090,162m
L2:1,22760E+090,267m
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
TECvarieswithlocaltime,
season,solaractivity,latitudeand
geomagneticperturbation.
Ionosphericdelaymayreach40
mforalowelevationsatellite.
TECmapscomputedbySIRGAS
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Ionospherefreecombinationofcarrierphasemeasurements
Eliminatestheionosphericerror:
22
12
3122222
1212
2.54571.5457
ff
lll
ffff
2
1
1,1111
40.3
I
TEC
f
lN
2
2
2,2222
40.3
I
TEC
f
lN
311222.54571.5457bNN
~3timesgreaterthantheerrorsinL1orL2measurements
theambiguityisnotanintegeranymore
2222
312312122.54571.54572.54571.545733
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
TDWNNN
hDandhW=scaleheight(m)
p=atmosphericpressure(mb)
T=temperature(K)
e=partialwatervaporpressure(mb)
allreducedtoh=0.
drycomponent
hD~40km
wetcomponent
hW=11km
Troposphericerror
Theneutralatmosphere(troposphere+stratosphere)extends~50km
abovetheearthsurface;accordingtotheHopfieldmodel:
Dry(~90%)+Wet(~20%)components
,0
,0
4
heightvariability
4
5
2
heightvariability
()77.64
()12.963.71810
Dh
Wh
S
D
D
N
H
W
W
N
hhp
Nh
Th
hhee
Nh
TTh
6
6
,0,0
0
6
6
,0,0
0
10
10()
5
10
10()
5
D
W
h
DzDDhD
h
WzWWhW
NhdhNh
NhdhNh
40136148.72273.16DhT
11000Wh
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
/
/
/
/
/
/
/
1
1
1
()
cos()
cos()
cos()
DW
DW
DW
DW
DW
DW
DW
a
b
c
mz
a
z
b
z
zc
,0,0()()()TDzDWzWzmzmz
mD/W(z)=Dry/Wetmappingfunctions(e.g.:‘Vienna’mappingfunction);
aD/W,bD/W,cD/W=empiricallydeterminedfunctionsdependentonlatitude,dayof
yearandheight.
Thetotaltroposphericdelay(dry+wet)forazenithdistancez,isgivenby:
Troposphericcorrection
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Estimationofthetropospheirccorrection
Modelsarenotaccurateenoughtocorrectthetroposphericerror(speciallythewet
component.
Errorsaffectsmostlytheestimationoftheheight.
Anempiricalcorrectionisaddedtoaccountfortheunmodeledcorrection:
,0,0,0TzTzTz
Anempiricalcorrectionisestimatedtogetherwiththestationcoordinates,thereceiver
clockandtheambiguities.
Typicallyonecorrectionperhourisestimated.
ZenithdelayEmpiricalcorrection
Modeledvalue
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Equationofobservationforpointpositioning
,0,0,,0,,0
3
,3
3
coscoscosSSSSS
RRReRnRv
S
WR
RRRR
S
R
R
TR
S
R
S
enaa
m
lvt
bz
c
Ionosphere-freecombinationof
carrierphasemeasurements
Corrections:
satelliteclock
relativity
satelliteandreceiver
antennasphasecentersand
wind-up
troposphere(dry+wet)
coordinatevariations(solid
tides,oceanloading,EOP,etc.)
Receiver-satelliterangecomputed
fromsatelliteephemeridesand
approximatereceivercoordinates
Correctionstotheapproximate
receivercoordinates
Receiver-satellite
directioncosines
Correctionto
receiverclock
Ionosphere-freebias
(includingsatelliteand
receiverinstrumental
biases.
Unmodeled
tropospheric
correction
Measurement+multipatherrors
ontheionospherefree
combination
Tropospherewet
mappingfunction
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
3.4pointanddifferentialpositioningReceiverclockandtroposphericcorrections
isthestandarddeviationforthe
absolutevalue
isthestandarddeviationforthe
temporalvariation
r
t
a
time
dependence
unknownupdating
interval
typeof
constraint
constantreceivercoordinates(assumingthat
geophysicalvariationsaremodeled)
nonenone
Ambiguities(assumingthatcycleslips
havenotoccurred)
Nonenone
varyingreceiverclockoneperepochrelative
troposphericcorrectiononeperhourabs.andrel.
a
r
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Equationofobservationfordifferentialpositioning
Singledifferences
111
1,212lll222
1,212lll
receiver1
receiver2
satellite1
satellite2
Doubledifference
222
1,2
1,2
1,1
111
1,22221lllllll
Errorsinthesatelliteclockscancelout;
Measurementserrorsincreasebya
factor.2
Errorsinthereceiverclockscancelout;
Measurementserrorsincreasebya
factor.5
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Effectonreceivercoordinates
receiver1
receiver2
satellite1
satellite211
1
22
1
1,2
1,222lllll
1
1cosa
1
2cosa
2
1cosa
2
2cosa
1X2X
122
11
1coscosXaXa
122
22
1coscosXaXa
2
1
2
2
1
1
2
1
1
2
cocos
os
s
scoc
Xa
aX
a
a
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Effectonbiasandtroposphere
receiver1
receiver2
satellite1
satellite211
1
22
1
1,2
1,222lllll
11
111;bm
11
12bb
22
12bb
11
1
22
12
1,
1,22
2
bbbbb
11
222;bm
22
111;bm
22
222;bm
1
1
12
1
2mm
1
2
12
2
2mm
22
12
1
1212
1
mmmm
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Correlationsbetweenbaselines
Let‘sassume3receiversand2
satelliteswhichleadsto6single
differencedcombinationspossible
0/
1111
1,2121
2222
1,2121
1111
2,3232
2222
2,3232
1111
3,1313
2222
3,1313
101000
010100
001010
000101
100010
010001
S
llll
llll
llll
llll
llll
llll
A
Thevariance-covariancematrixissingular
receiver1
receiver2
receiver3
0/0/
201010
020101
102010
010201
101020
010102
T
SDSSCAA
red+green=-blue
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Correlationbetweensingleanddoubledifferences
Let’sassumethattheredandgreenindependentbaselinesarechosen(more
measurements,shortlengths,othercriteria):
/
1
1,2
1,2122
1,21,21,21,2
1,2121
2,32,32,32,3
2
2,3
1100
0011
SD
l
llll
llll
lA
0/
1
1
1112
1,2121
2221
1,2122
1112
2,3232
2221
2,3233
2
3
101000
010100
001010
000101
S
l
llll
llll
llll
llll
lA
0/
11
11
22
11
1,211
1,222
/0/1,222
2,322
11
33
22
33
111100
001111
D
SDS
ll
ll
lll
lll
ll
ll
A
AA0/0/
5040
0504
4050
0405
T
DDDDCAA
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Justanexample
Observations
Receivers50
Samplingrate1/15s
Averagesatellitesperepoch6
Observationperiod24h(86400s)
Equationsofobservation50x1/15x86400x6=1728000
unknowns
Receiverscoordinates3x50=150
Receiverclocks50x1/15x86400=288000
Troposphericcorrections50x24=1200
Biases50x31x2=3100
Total292450
Oneperhour;constrained.
Assuming2continuous
satellitesarcsperday(no
cycleslips).
Oneperepoch;constrained.
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
3.5networkcomputation
Thesystemofequationsofobservation
L=Ax+vL
=variance-covariancematrixofthemeasurements
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Normalsystem
Nx=b1T
LN=1T
LbA
1
x=Nb
1
XN
Thesolutionprovidestheunknowns,,theresiduals,,
thestandarddeviationoftheunityofweight,,andthevariance-
covariancematrixoftheunknowns,.
vLAx
1
0
t
L
nm
vv
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
Combinationofsolutions
TheSIRGASprocessingcenterscomputeweekly‘looselyconstrained’normal
equations(thereferenceframeisdefinedbyonlythesatellitecoordinates).
TheSIRGAScombinationcenterscomputethefinalsolutionbystackingthe
individualnormalequationsandimposingthedatumtothenetwork.
FixedsolutionWeightedsolution
Nonetrotation&no
nettranslationsolution
AfterL.Sánchez
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October2013
4.Verticalreferencesystems
Ifthereferencesurfaceandheight
typedependontheEarth'sgravity
field,wetalkaboutaphysical
heightsystem(e.g.orthometric
heightsandgeoid,ornormal
heightsandquasi-geoid)ifnot,it
isageometricalheightsystem
(e.g.ellipsoidalheightsand
referencelevelellipsoid).
Anyverticalreferencesystemisbasicallycomposedby
1)areferencesurface,i.e.thezero-heightlevel(verticaldatum)
2)averticalcoordinate,i.e.atypeofheight
Itsrealisationisgivenbyaverticalnetwork,i.e.asetofpoints,whose
heightsareofthesametypeconsideredin(2)andrefertothedatum
specifiedin(1).
1SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
4.1Geometricalheightsystem
Verticalcoordinate:Ellipsoidalheight(h):length,alongtheellipsoid
normal,fromtheellipsoidtothepointP(X).
-today,hisderivedfromgeocentriccoordinates[X,Y,Z]
-earlier(fortheadjustmentoftriangulationnetworks),h=H+N
Referencesurface:alevel(orequipotential)ellipsoid.
U0canbeunivocallydeterminedasafunction
oftheellipsoidparameters:
.0constUUX
1
2
22
22
220
3
1
12
'
11
3
1
arctan
n
n
m
n
e
b
GM
a
b
ba
ba
GM
U
GMgeocentricgravitationalconstant
asemi-majoraxis
bsemi-minoraxis
(nominalmeanEarth’s)angular
velocity
GM
ba
m
b
ba
e
22
2
22
2
,'
2SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Geometricalheightsystem
Therealisationofageometricalreferencesystemdependson
-theorientationandpositionoftheellipsoidwhitrespecttotheITRS
(referencesystemfor[X,Y,Z]).
-theellipsoidparameters:ifanyofthosechanges,referencesurface
andverticalcoordinatechange.
-theprimarygeocentriccoordinates[X,Y,Z]:iftheychange,the
verticalcoordinatechanges.
Dependenceofhonthe
orientationandposition
oftheellipsoid
Dependenceofhon
theellipsoid
parameters(form
andsize)
3SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Geometricalheightsystem
Dependenceofhonthechangesof[X,Y,Z]
BOGA:Bogotá,Colombia
Subsidence
CONZ:Concepción,Chile
Earthquake
NAUS:Manaus,Brazil
Hydrologicalloading
UYTA:Tacuarembo,Uruguay
IntroductionofITRF2008
Source:www.sirgas.org
4SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Componentsofageometricalverticalreferencesystem
Verticalcoordinate:Ellipsoidalheighthreferredtoacertainepochtand
itsvariationswithtime:
Referencesurface:stationaryintimeandspaceforabettermodelling
ofheightchanges:
Forthat,onehastobefamiliarizedwithstandards,conventions,and
proceduresappliedfortherealisationoftheITRS,i.e.theInternational
TerrestrialReferenceFrame(ITRF)anditsregionaldensificationslike
SIRGAS(LatinAmericaandCaribbean),EPN:EUREFPermanentNetwork
(Europe),etc.
5
.0constUUX
dt
dh
th
X
X;,
SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Remarksonthegeometricalheightsystem
-Ellipsoidalheightsarenotmeasurabledirectly;theymustbederived
fromothergeodeticparameters.
-Theprimarycoordinates[X,Y,Z]mustbecomputedinthesame
referenceframeinwhichtheGNSSsatelliteorbitsaregiven,i.e.ITRFor
itsregionaldensificationsmustbeusedasreferenceframeinGNSS
processing.
-Forglobalcompatibility,thesameellipsoidmustbeusedforthe
conversion[X,Y,Z][,,h],atpresenttheGRS80.
-Ellipsoidalheightsinlocalgeodeticdatums(e.g.PSDA56,NAD27,etc.)
canonlybeobtainedbymeansoftransformations.
-TheverticalpositioninGNSSis2…3timeslessaccuratethanthe
horizontalposition,becausemostoftheerrorsourcesinGNSSactin
radialdirection.
-Accurateellipsoidalheights(atmm-level)requirelongGNSSpositioning
(atleastthreedays)andpost-processingfollowingtheIERSconventions.
Otherwise,accuracyincm-todm-level.
6SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
4.2Physicalheightsystems
(Primary)verticalcoordinate:Leveldifferencebetweenareference
surfaceW0andtheequipotentialsurfacepassingthroughP(X)i.e.WP
Referencesurface:equipotentialsurfacedefinedbyapotentialvalue
(calledW0).Therealisation(geometricalrepresentation)ofthis
equipotentialsurfacewithrespecttoareferenceellipsoidistheso-
calledgeoidcomputation.
Sincetheprimaryobservableareleveldifferences,theW0valueis
usuallyselectedarbitrarily.Inaddition,theW0valuepersedoesnot
provideanyinformationaboutthegeometryofthereferencesurface.
PPPWWCW0
7SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Physicalheightsystems
Cpiscalledgeopotentialnumberanditisgivenin[m2s-2].Tofacilitate
itsuseinpractice,Cpisconvertedinadistance(givenin[m])bydividing
itbyagravityvalue.
representsthemeangravity
valuebetweenthereference
surfaceW0andtheequipotential
surfaceWP.
ThepotentialdifferenceW0-WPisconstant(becausetheyare
equipotentialsurfaces)andtherefore,thevalueofHPdependsonthe
valueof.Asaconsequence,wedistinguishdynamic,orthometricand
normalheights.
g
WW
g
C
HPP
P
ˆˆ
0
8SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Physicalheights:someglossary
•Thegeoidistheequipotential
surfacethatwouldcoincidewith
themeanseasurfaceifthe
oceansandatmospherewerein
equilibrium;undertheinfluence
ofEarth'sgravitationandrotation
alone;withoutotherinfluences
suchaswinds,tidesor
gravitationaleffectsofother
bodies.
•Theco-geoidistheestimated
geoid,i.e.therealgeoidcannot
beknown,becausetheactual
massdistributionandactual
gravityverticalgradientarenot
knownprecisely.Theassumption
ofhypothesesaboutthat
produces“somethingsimilar”to
thegeoid(theco-geoid),butno
thetruegeoid.
•
•
9SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Physicalheights:someglossary
•Thetelluroidisthesurface
definedbythosepointsQ,whose
normalpotentialUQisidentical
withtheactualpotentialWPof
thepointsPontheEarth’s
surface,i.e.UQ=WP
•Theheightanomalyisthe
distance,alongthenormalplumb
line,betweentheEarth’ssurface
andthetelluroid.Whenplotted
abovetheellipsoidtheresulting
surfaceiscalledthequasi-geoid.
•Telluroid,quasi-geoidandco-
geoidarenotequipotential
surfaces(thegravityvectorisnot
perpendiculartothem).
•Geoidandquasi-geoidare
identicalinoceanareas.
•
•
•
•••••••••••••••••••••••••••••••••••••••••••
10SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Physicalheights:someglossary
•Ifisthemeanreal
gravityvalue,alongthe
plumbline,between
Earth’ssurfaceandgeoid,
wegetorthometric
heights.
•Ifisthemeannormal
gravityvaluebetween
telluroidandellipsoid(or
betweenEarth’ssurface
andthequasi-geoid),we
getnormalheights.
•Ifisaconstantnormal
gravityvalue,weget
dynamicheights.
g
WW
g
C
HPP
P
ˆˆ
0
11SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Geopotentialnumbersinpractice
dngdWg
dn
dW
;
IntegratingdnbetweentwopointsAandB,wehave:
i.e.theleveldifferencednbetweentwopoints(A,B),locatedontheEarth‘ssurface
andontwodifferentequipotentialsurfaces,correspondsto:
isthemeangravityvaluebetweenthetwoequipotentialsurfacesWAandWB
B
A
BAAB
B
A
WWHHgngdWˆ
g
WW
HHdnBA
ABBA
ˆ
Thederivativeofthegravitypotentialina
givendirectionisequaltothecomponent
ofthegravityalongthisdirection.Along
theplumbline(perpendiculartothelevel
surfaces),itis:
12SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
BB
BBdngngWWC
00
0
Ifthestartingpointisonthegeoid(W0),
wehavethegeopotentialnumberofB
g
C
g
WW
HBB
B
ˆˆ
0
gistheaverageofthegravityalongthe
levellinglineconnecting0andB.
TheheightvalueforBisthen
Note:gandaredifferentvalues!
Geopotentialnumbersinpractice
Spiritlevellingmeasuresthegeometricheight
differencebetweentwopoints,without
takingcareoftheeffectofthegravityonthe
levelsurfaces(non-parallelism),i.e.ina
closedlevellingloop
whichalsomeans
0dn
B
BHdn
0
Levelledheightdifferencesarenotthesameasphysicalheightdifferences!!
13SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Thelevelledheightdifferencesmustbeconvertedintopotentialdifferencesorinto
physicalheightdifferencestosatisfytheloopmisclosureconditionofzero,i.e.
Afterwardstheyareadjustedbythemethodofconditionequationsorbythemethod
ofparametervariation.kisknownasthegravitycorrection(orreduction)tolevelling.
Inpractice,thepotentialdifferencefromgeometric(spirit)levellingbetweentwo
benchmarksIandIIisgivenby:
Geopotentialnumbersinpractice
0dW0kdn
DIIDIIBCBCABABIAIAIIIdngdngdngdngW,
2
BA
AB
gg
g
14SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Geopotentialnumbersinpractice
Requiredinputdata:
1)Levelleddifferences(dn)withsystematicerrorsreduced(i.e.
atmosphericrefraction,rodgraduationerrors,temperatureeffecton
theinstrumentandrods,etc.).Randomerrorsarecompensatedby
meansoftheadjustment.
Accuracystandardsinlevelling:
Standarddeviationofaleveldifference
measuredintwodirections(back-sight
andfore-sigth):
Standarddeviationforaleveldifference
comparedwithknownheightdifferences
(junctionwithexistingverticalnetworks):
First-order
Second-order:
Third-order:
Fourth-order:mmRs
mmRsmmRs
mmRsmmRs
mmRsmmRs
km
kmkm
kmkm
kmkm
62
525
323
222
R:lengthin[km]ofthelevellinglineassociatedwiththemeasuredheightdifference.
15SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Geopotentialnumbersinpractice
Requiredinputdata:
2)Lengthofthelevellingsegments:thestandarddeviationoflevelling
increasesinverselyproportionallywiththedistance.Thelevelling
linesshallbeadjustedinoneblockandweightedwiththeinverseof
thisstandarddeviation.ForexampleinSouthAmerica:
kmdn
dnkmdn
Rs
pRmms
16/11
4
16SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Geopotentialnumbersinpractice
Requiredinputdata:
3)Measurementepochs:totakeintoaccountcrustalvertical
movements
A
A
B
B
C
C
C
D
D
D
E
E
Levellingsegmentsmeasuredindifferentepochs
Changesinthelevelledheight
differencesarecausedby
verticalmovments?Orare
theyobservationerrors?
17SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Geopotentialnumbersinpractice
Requiredinputdata:
4)(Real)gravityvalues
(observedorinterpolated)at
levelledpointstocompute
thegeopotentialnumbers.
A
B
CD
E
Usualavailabilityofgravityforlevelling
Height[m]
mgforR=1km
[10-5ms-2]
mgforR=2km
[10-5ms-2]
10
20
30
40
50
400
200
133
100
80
566
283
189
141
113
70
100
200
500
57
40
20
8
81
57
28
11
1000
2000
4000
4
2
1
8
4
2
Minimumaccuracyof
gravityvaluesforthe
determinationof
geopotentialnumbers
18SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Fromgeopotentialnumberstophysicalheights
o
DYNC
H
DYN
ABAB
DYN
AB
knH
dn
g
n
g
k
B
Ao
o
B
Ao
oDYN
AB45
45
45
45
Dynamicheights:
Frompotentialdifferences:Usingthedynamiccorrection:
Normalgravityforthesurfaceofthelevelellipsoidatcertainlatitude,normally45°.
Normalheights:
Frompotentialdifferences:Usingthenormalcorrection:
Meannormalgravityalongthenormalplumblinebetweentelluroidandellipsoid(analytically
estimable,iterativ)
o
N
H
N
Nm
m
N
dH
H
C
H
0
1
;
N
ABAB
N
AB
knH
N
B
o
o
B
mN
A
o
o
A
m
B
Ao
oN
AB
HHn
g
k45
45
45
45
45
45
][sin211
!2
1
2
12
2
2
22
2
2
ms
a
H
a
H
fmfH
H
H
H
NN
o
N
o
N
o
om
asemi-majoraxis,fflattening,latitudeofthepoint,
GM
ba
m
22
m
19
-900
-450
0
Correccióndinámica(45°)[cm]Correccióndinámica(0°)[cm]
0
1000
2000
3000
Perfiltopográfico[m]
PUERTADEHIERRO
LAMATABARRANCABERMEJA
MEDELLIN
LAMATAVALLEDELRÍOMAGDALENA
CORDILLERACENTRAL
SABANASDECÓRDOBAYSUCRE
Dynamiccorrectionvs.topography(levellinglooplength:1364km)
-1200
-600
0
Correccióndinámica(45°)[cm]Correccióndinámica(0°)[cm]
0
2000
4000
Perfiltopográfico[m]
CUCUTA
BOGOTA
BARRANCABERMEJA
MEDELLIN
LAMATA
BOGOTA
CORDILLERA
ORIENTAL
CORDILLERACENTRAL
VALLEDELRÍO
MAGDALENAVALLEDERRÍOMAGDALENA
VALLEDELRÍO
CATATUMBO
ALTODESANTURBAN
CORDILLERA
ORIENTAL
Dynamiccorrectionvs.topography(levellinglooplength1820km)
Fromgeopotentialnumberstophysicalheights
DynamiccorrectionintwoColombianlevellinglines
Courtesy
20SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Fromgeopotentialnumberstophysicalheights
NormalcorrectionintwoColombianlevellinglines
0
50
Molodenki,Vignal,Hirvonen[cm]
0
2000
4000
Perfiltopográfico[m]
BOGOTA
IBAGUE
MEDELLIN
CALI
MEDELL
VALLEDELRÍOCAUCA
LADERADELACORDILLERA
CENTRAL
ALTODELASDELICIAS
CORDILLERACENTRAL
VALLEDELRÍOMAGDALENA
CORDILLERAORIENTAL
VALLEDELRÍO
MAGDALENA
CORDILLERACENTRAL
Normalcorrectionvs.topography(levellinglooplength1640km)
-15
0
15
Molodenki,Vignal,Hirvonen[cm]l
0
750
1500
2250
Perfiltopográfico[m]
QUIBDO
MEDELLIN
BUENAVENTURA
(Mareógrafo)
CALI
BUENAVENTURA
(Mareógrafo)
CORDILLERA
OCCIDENTAL
VALLEDELRÍO
CAUCA
RÍOATRATO/RÍOSANJUAN
CORDILLERACENTRAL
CORDILLERA
OCCIDENTAL
Normalcorrectionvs.topography(levellinglooplength1143km)
Courtesy
21SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Fromgeopotentialnumberstophysicalheights
Orthometricheights:
Frompotentialdifferences:Usingtheorthometriccorrection:
MeanrealgravityalongtheplumblinebetweenEarth’ssurfaceandgeoid.Itcanonlybe
estimatedbymeansofhypothesesaboutthe(unknown)Earth’sinternalmassdistributionand
the(unknown)verticalgravitygradient.Eachdifferenthypothesisproducesadifferenttypeof
orthometricheight.
Someexamplesoforthometrichypotheses:
O
H
O
Om
m
O
dHg
H
g
g
C
H
0
1
;
O
ABAB
O
ABknH
O
B
o
o
B
mO
A
o
o
A
m
B
Ao
oO
ABH
g
H
g
n
g
k45
45
45
45
45
45
mg
Helmert:
2
1083818,0086,3
2
16
02
O
p
pppHm
H
ggggg
Units:gp,gm,gH/2,g0[ms-2];p[10-3kgm-3];HO[m]
FirstmethodofRamsayer:;
Ledersteger:
O
p
O
AP
pm
H
H
ggggg
ˆ
2
1
000
n
i
O
i
O
H
n
H
1
1ˆ
2
10086,3
2
1
10086,3
16
1
6
O
p
n
i
O
iim
H
xHxg
n
g
22SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Fromgeopotentialnumberstophysicalheights
OrhometriccorrectionsintwoColombianlevellinglines
0
2000
4000
Perfiltopográfico[m]
BOGOTA
IBAGUE
MEDELLIN
CALI
MEDELLIN
VALLEDELRÍOCAUCA
LADERADELACORDILLERA
CENTRAL
ALTODELASDELICIAS
CORDILLERACENTRAL
VALLEDELRÍOMAGDALENA
CORDILLERAORIENTAL
VALLEDELRÍO
MAGDALENA
CORDILLERACENTRAL
-50
0
50
100
Helmert[cm]Ramsayer(I)[cm]Ramsayer(II)[cm]Ramsayer(III)[cm]Baranov[cm]Ledersteger[cm]
Orthometriccorrectionsvs.topography(levellinglooplength1640km)
0
750
1500
2250
Perfiltopográfico[m]
QUIBDO
MEDELLIN
BUENAVENTURA
(Mareógrafo)
CALI
BUENAVENTURA
(Mareógrafo)
CORDILLERA
OCCIDENTAL
VALLEDELRÍO
CAUCA
RÍOATRATO/RÍOSANJUAN
CORDILLERACENTRAL
CORDILLERA
OCCIDENTAL
-20
0
20
Helmert[cm]Ramsayer(I)[cm]Ramsayer(II)[cm]Ramsayer(III)[cm]Baranov[cm]Ledersteger[cm]
Orthometriccorrectionsvs.topography(levellinglooplength1143km)
Courtesy
23SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Fromgeopotentialnumberstophysicalheights
Comparisonofnormal(Molodenskii)andorhometriccorrections
24SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Physicalheightssummary
DynamicheightsOrthometricheightsNomalheights
Definitionof
o:constantnormalgravityvalueatanarbitrary
latitude(usually=45°).
gm:Meanrealgravityvaluealongtheplumbline
betweenthegeoidandP.
m:Meannormalgravityvaluealongthenormal
plumblinebetweentheellipsoidandthetelluroid(or
betweenthequasi-geoidandP).
DescriptionSimpleconversiontoheightunits(scaled
geopotentialnumbers)
o
DYNC
H
Distance,alongtheplumbline,betweenthesurface
pointPandthegeoid.
O
H
O
Om
m
O
dHg
H
g
g
C
H
0
1
;
Distance,alongthenormalplumbline,betweenthe
ellipsoidandthetelluroid(orbetweenthequasi-
geoidandP)
N
H
N
Nm
m
N
dH
H
C
H
0
1
;
Correction
(forlevelling)
Magnitude:<20m
DYN
ABAB
DYN
ABknH
dn
g
n
g
k
B
Ao
o
B
Ao
oDYN
AB45
45
45
45
Magnitude:mm...dm
O
ABAB
O
ABknH
O
B
o
o
B
mO
A
o
o
A
m
B
Ao
oO
ABH
g
H
g
n
g
k45
45
45
45
45
45
Magnitude:mm...dm
N
ABAB
N
ABknH
N
B
o
o
B
mN
A
o
o
A
m
B
Ao
oN
AB
HHn
g
k45
45
45
45
45
45
RemarksNogeometricalmeaning
Pointsonthesamelevelsurfacehavethe
sameheightvalue
Hypothesesarenotrequired
Referencesurface:thegeoid
NhHO
h:ellipsoidalheight,N:geoidundulation
Heightsofpointsonthesamelevelsurfacediffer
inthesamemannerasthegmgravityvalues
Hypothesesaboutmassdensityanddistribution
aswellasaboutthegravityverticalgradient
Hgarenecessary.ThevalueofHO
dependsontheadoptedhypotheses.
gmcannotbeestimatedunivocally,only
approximately.
Referencesurface:thequasi-geoid(closetothe
geoidbutnotalevelsurface)
hHN
h:ellipsoidalheight,:heightanomaly
Pointsonthesamelevelsurfaceandatthe
samelatitudehavethesamenormalheights.In
othercases,heightsdifferinthesamemanner
asmvarieswiththelatitude.
Hypothesesarenotrequired
misestimableunivocally.
25SchoolonReferenceSystems,CrustalDeformationandIonosphereMonitoring,PanamaCity,21-23October20134-
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
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Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes
Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes

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Sirgas 2013 school_on_reference_systems_deformation_ionosphere_notes