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MIT 12 215 - Lecture Notes

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112.215 Modern NavigationThomas Herring ([email protected]),MW 10:30-12:00 Room 54-322http://geoweb.mit.edu/~tah/12.21509/21/2009 12.215 Modern Naviation L02 2Todayʼs ClassLatitude and Longitude• Finish discussion of reference systems (end of Lec 1)• Simple spherical definitions• Geodetic definition: For an ellipsoid• Astronomical definition: Based on direction of gravity• Relationships between the types• Coordinate systems to which systems are referred• Temporal variations in systems209/21/2009 12.215 Modern Naviation L02 3Simplest Global Reference Frame• Geometric: Origin at thecenter of mass of the Earth;Orientation defined by a Z-axis near the rotation axis;one “Meridian” (planecontaining the Z-axis) definedby a convenient location suchas Greenwich, England.• Coordinate system would beCartesian XYZ.ZXYCenter of MassGreenwichMeridian09/21/2009 12.215 Modern Naviation L02 4Simple System• The use of this type of simple system is actually arecent development and is the most common systemused in GPS.• Until the advent of modern “space-based geodeticsystems” (mid-1950s), coordinate systems were muchmore complicated and based on the gravity field of theEarth.• Why?309/21/2009 12.215 Modern Naviation L02 5Potential based coordinate systems• The basic reason is “realization”: Until distancemeasurements to earth-orbiting satellites and galactic-based distance measurements, it was not possible toactually implement the simple type measurementsystem.• Conventional (and still today) systems rely on thedirection of the gravity vector• We think in two different systems: A horizontal one(how far away is something) and a vertical one (heightdifferences between points).09/21/2009 12.215 Modern Naviation L02 6Conventional Systems• Conventional coordinate systems are a mix ofgeometric systems (geodetic latitude and longitude)and potential based systems (Orthometric heights).• The origin of conventional systems are also poorlydefined because determining the position of the centerof mass of the Earth was difficult before the first Earth-orbiting artificial satellite. (The moon was possiblebefore but it is far enough away that sensitivity centerof mass of the Earth was too small).409/21/2009 12.215 Modern Naviation L02 7Simple Geocentric Latitude andLongitude• The easiest form of latitude and longitude tounderstand is the spherical system:• Latitude: Angle between the equatorial plane and thepoint. Symbol φc (in this class)• Latitude is also the angle between the normal to thesphere and the equatorial plane• Related term: co-latitude = 90o-latitude. Symbol θc (inthis class). Angle from the Z-axis• Longitude: Angle between the Greenwich meridianand meridian of the location. Symbol λc09/21/2009 12.215 Modern Naviation L02 8Geocentric quantities• GeocentricLatitude andLongitude• Note: Vectorto P is alsonormal to thesphere.ZXYGreenwichMeridianPλφMeridian of point PEquatorHθR509/21/2009 12.215 Modern Naviation L02 9Geocentric relationship to XYZ• One of the advantages of geocentric angles is that therelationship to XYZ is easy. R is taken to be radius ofthe sphere and H the height above this radius€ φc= tan−1(Z / X2+ Y2)λc= tan−1(Y / X)R + Hc= X2+ Y2+ Z2€ X = (R + Hc)cosφccosλcY = (R + Hc)cosφcsinλcZ = (R + Hc)sinφc09/21/2009 12.215 Modern Naviation L02 10Problem with Geocentric• Geocentric measures are easy to work with but theyhave several serious problems• The shape of the Earth is close to an bi-axial ellipsoid(i.e., an ellipse rotated around the Z-axis)• The flattening of the ellipsoid is ~1/300(1/298.257222101 is the defined value for the GPSellipsoid WGS-84).• Flattening is (a-b)/a where a is the semi-major axisand b is the semi-minor axis.• Since a=6378.137 km (WGS-84), a-b=21.384 km609/21/2009 12.215 Modern Naviation L02 11Geocentric quantities• If the radius of the Earth is taken as b (the smallestradius), then Hc for a site at sea-level on the equatorwould be 21km (compare with Mt. Everest28,000feet~8.5km).• Geocentric quantities are never used in any largescale maps and geocentric heights are never used.• We discuss heights in more in next class and whenwe do spherical trigonometry we will use geocentricquantities.09/21/2009 12.215 Modern Naviation L02 12Ellipsoidal quantities• The most common latitude type seen is geodeticlatitude which is defined as the angle between thenormal to the ellipsoid and the equatorial plane. Wedenote with subscript g.• Because the Earth is very close to a biaxial ellipsoid,geodetic longitude is the same as geocentric longitude(the deviation from circular in the equator is only a fewhundred meters: Computed from the gravity field ofthe Earth).709/21/2009 12.215 Modern Naviation L02 13Geodetic LatitudeNorthEquatorGeoidgravity directionNormal to ellipsoidφgφaLocal equipotenital surfaceEarth's surfacePAstronomical Latitude also shown09/21/2009 12.215 Modern Naviation L02 14Relationship between φg and XYZ• This conversion is more complex than for thespherical case.€ X = (N + hg)cos(φg)cos(λg)Y = (N + hg)cos(φg)sin(λg)Z = [(1− e2)N + hg]sin(φg)where e2= 2 f − f2 and N (North - South radius of curvature) isN2= a2/[1− e2sin2(φg)]809/21/2009 12.215 Modern Naviation L02 15Inverse relationship• The inverse relationship between XYZ and geodeticlatitude is more complex (mainly because to computethe radius of curvature, you need to know the latitude).• A common scheme is iterative: € a N'= a / 1− e2sinφ'r'= X2+ Y2[1− e2N' /(N '+h')]φ'= tan−1(Z /r')h'= X2+ Y2/cosφ'−N' or h'= Z/sinφ'−(1− e2)N 'iterate to a until h' change is small09/21/2009 12.215 Modern Naviation L02 16From http://www.colorado.edu/geography/gcraft/notes/datum/gif/xyzllh.gifClosed formexpression forsmall heights909/21/2009 12.215 Modern Naviation L02 17Other items• A discussion of geodetic datum and coordinate systemscan be found at:http://www.colorado.edu/geography/gcraft/notes/datum/datum.html• Geodetic longitude can be computed in that same wayas for geocentric longitude• Any book on geodesy will discuss these quantities inmore detail (also web searching on geodetic latitude willreturn many hits).• The difference between astronomical and geodeticlatitude and longitude is called “deflection of the vertical”09/21/2009 12.215 Modern Naviation L02 18Astronomical latitude and longitude• These have similar definitions to geodetic latitude andlongitude


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