页面提取自-星载SAR用到的坐标系关系_Thesis

AppendixA.CoordinateFramesandTransformationsSeveraldifferentcoordinateframesareusedinthiswork.Eachofthemhassomeadvantagesconcerningthecalculationsthataredoneonitsbasis.MostoftheframesareCartesian,becausetheyallowrathersimplevectoralgebra.Andbeyondit,asetofsimplematricescanbeusedtoperformvectorrotationaroundeachaxis[AppendixA.2].ThefinalresultsaremostlygiveninellipsoidalWGS84coordinatestorelatethemtothesurfaceoftherotatingearth[AppendixA.1.2].A.1DefinitionoftheCoordinateFramesA.1.1EarthCenteredInertialSystem(IN)vsatPsatFigA.1:EarthCenteredInertialSystem98AppendixTheEarthCenteredInertialSystemisarighthandedCartesiancoordinateframe.Allvectorsrelatedtoanorbitalplatform(e.g.position,velocity)canbecalculatedindependentlyfromtheearth’srotationinthissystem.Definition:Thepointoforiginisthegeocenter.Thez-axisisparalleltothespinaxisoftheearth,thex-axisliesintheequatorialplaneandpointstowardsthevernalequinox(γ).Orthogonaltothex-z-plane,they-axiscompletestherighthandedcoordinatesystem.A.1.2EarthCenteredEarthFixedSystem(EF)LiketheEarthCenteredInertialSystem,theEarthCenteredEarthFixedSystemisrighthandedandCartesian.Inthiscoordinateframeallvectorscanbeexpressedrelatedtotherotatingearth.Fig.A.2:EarthCenteredEarthFixedSystemwithellipsoidalcoordinatesDefinition:Againthepointoforiginisthegeocenter.Thez-axisisalignedwiththez-axisoftheEarthCenteredInertialSystem.Thex-axisliesintheequatorialplaneandpointstowardstheintersectionofequatorandzeromeridian.They-axiscompletestherighthandedsystem.RelativetotheEarthCenteredInertialSystem,theearthfixedsystemrotatesaroundthez-axiswiththeangularvelocityofω.Allvectorsintheearthfixedsystemcanalsobeexpressedbyellipsoidalcoordinates.ThesearerelatedtotheWGS84ellipsoid.Theparametersare:aearth6378137msemimajoraxisoftheWGS84ellipsoidAppendix3EF+ya−bxeearth0.081819190eccentricityoftheWGS84ellipsoidω0.72921151467E-04rad/sangularvelocityTab.A.1:ParametersoftheWGS84EllipsoidTheellipsoidalcoordinatesare:ΦlatitudeΛlongitudehheightabovetheWGS84ellipsoidTab.A.2:EllipsoidalCoordinatesNotethatthepointwhereallellipsoidalcoordinates(Φ,Λ,h)arezeroisnotthecenteroftheEFframebutthepointofintersectionbetweenzeromeridianandequatoronthesurfaceoftheWGS84ellipsoid.TheearthfixedCartesiancoordinatescanbederivedfromtheellipsoidalcoordinatesbyusingtheHelmertprojection[NI06]:xEF=(RN+h)⋅cos(Φ)⋅cos(Λ)yEF=(RN+h)⋅cos(Φ)⋅sin(Λ)(A.1.1)z=R⋅(1−e2)+h⋅sin(Φ)where:EFNearthRN=1−e2aearth·sin2(Φ)(A.1.2)earthTheellipsoidalcoordinatesareobtainedfromtheearthfixedCartesiancoordinatesbyinverseHelmertprojection:yΛ=atanEFxEF22h=xEF+yEFcos(Φ)−1−e2aearth·sin(Φ)Φ=atanearthz+e′2⋅b⋅sin3(θ)2223where:xEF+yEF⋅−eearth⋅aearth⋅cosθ(A.1.3)θ=atanaearthzEFb⋅22EFEFb=a⋅1−e2earthearth22eearth=aearth−baearthe′=22earth=aearth⋅eearth=eearthba⋅1−e21−e2earthearthearth100AppendixrnormoftheCartesianvectortothepointPλgeographiclongitudeδgeocentriclatitudeForcalculationsconcerningthegravitationmodelanditsgradient,theearthfixedCartesiancoordinatesmustbeexpressedinsphericalcoordinates.Inliterature,usuallythepolarangleϑisused,whichistheanglebetweentheCartesianz-axisandthevectortothepointPwhichshallbeexpressedinsphericalcoordinates.Here,thegeocentriclatitudeisused,whichistheanglebetweenthevectortothepointPandtheCartesianx-y-plane.Themodifiedsphericalcoordinatesare:Fig.A.3:ModifiedSphericalCoordinatesTheCartesiancoordinatesexpressedinmodifiedsphericalcoordinatesare:xEF=r⋅cos(δ)⋅cos(λ)yEF=r⋅cos(δ)⋅sin(λ)zEF=r⋅sin(δ)(A.1.4)Thetransformationmatrixisgivenbytherelationbetweentheunitvectors.Fromgeometryweobtain:er=cos(δ)⋅cos(λ)⋅ex,EF+cos(δ)⋅sin(λ)⋅ey,EF+sin(δ)⋅ez,EFeλ=−sin(λ)⋅ex,EF+cos(λ)⋅ey.EFeδ=−sin(δ)⋅cos(λ)⋅ex,EF−sin(δ)⋅sin(λ)⋅ey,EF+cos(δ)⋅ez,EF(A.1.5)with:λ,δofthevectorthemodifiedsphericalcoordinateframeisrelatedtoAppendix5=v=vThevectorsinmodifiedsphericalcoordinatesandinearthfixedcoordinatesare:vr,MSvx,EFvMSλ,MS;vδ,MSvEFy,EFvz,EF(A.1.6)Nowthecoordinatetransformationisdonebywritingequation(A.1.5)asmatrixequation:v=TEF⋅vMSMSEFvr,MScos(δ)⋅cos(λ)cos(δ)⋅sin(λ)sin(δ)vx,EF(A.1.7)≙vλ,MS=−sin(λ)cos(λ)0⋅vy,EFvδ,MS−sin(δ)⋅cos(λ)−sin(δ)⋅sin(λ)cos(δ)vz,EFItisimportanttonotethattheunitvectorsofthemodifiedsphericalcoordinateframedependonaCartesianreferencevectorthatcorrespondstothelocationofthepointoforiginofthemodifiedsphericalcoordinatesintheCartesianframe.A.1.3OrbitalPlaneSystem(OS)Fig.A.3:OrbitalPlaneSystemStrictlyspeaking,anorbitalplanecanonlybedefinediftheorbitisanidealKeplerellipse.Becauseofgravitationalinfluences,thisisnotthecaseinreality.NeverthelesstheOrbitalPlaneSystemisusedtocalculateaKeplerorbitbyasimplesetofequations.Definition:Thepointoforiginisthegeocenter.Thez-axisisorthogonaltotheorbitalplaneandthez-directionischosensothatthesatelliteismovingrighthandedaroundthisaxis.Thex-axis102AppendixzxTSyzxTSTSpointstowardstheperigeeandthey-axiscompletestherighthandedCartesianframe.IftheorbitisidealKepler,allz-valuesarezero.The(timedependent)positionvectorsofthesate