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12.540 Principles of the Global Positioning SystemLecture 04: 

12.540 Principles of the Global Positioning System Lecture 04 Prof. Thomas Herring Room 54-611; 253-5941 tah@mit.edu http://geoweb.mit.edu/~tah/12.540

Review: 

Review So far we have looked at measuring coordinates with conventional methods and using gravity field Today lecture: Examine definitions of coordinates Relationships between geometric coordinates Time systems Start looking at satellite orbits

Coordinate types: 

Coordinate types Potential field based coordinates: Astronomical latitude and longitude Orthometric heights (heights measured about an equipotential surface, nominally mean-sea-level (MSL) Geometric coordinate systems Cartesian XYZ Geodetic latitude, longitude and height

Astronomical coordinates: 

Astronomical coordinates Astronomical coordinates give the direction of the normal to the equipotential surface Measurements: Latitude: Elevation angle to North Pole (center of star rotation field) Longitude: Time difference between event at Greenwich and locally

Astronomical Latitude: 

Astronomical Latitude Normal to equipotential defined by local gravity vector Direction to North pole defined by position of rotation axis. However rotation axis moves with respect to crust of Earth! Motion monitored by International Earth Rotation Service IERS http://www.iers.org/

Astronomical Latitude: 

Astronomical Latitude

Astronomical Latitude: 

Astronomical Latitude By measuring the zenith distance when star is at minimum, yields latitude Problems: Rotation axis moves in space, precession nutation. Given by International Astronomical Union (IAU) precession nutation theory Rotation moves relative to crust

Rotation axis movement: 

Rotation axis movement Precession Nutation computed from Fourier Series of motions Largest term 9” with 18.6 year period Over 900 terms in series currently (see http://geoweb.mit.edu/~tah/mhb2000/JB000165_online.pdf) Declinations of stars given in catalogs Some almanacs give positions of “date” meaning precession accounted for

Rotation axis movement: 

Rotation axis movement Movement with respect crust called “polar motion”. Largest terms are Chandler wobble (natural resonance period of ellipsoidal body) and annual term due to weather Non-predictable: Must be measured and monitored

Evolution (IERS C01): 

Evolution (IERS C01)

Evolution of uncertainty: 

Evolution of uncertainty

Behavior 2000-2006 (meters at pole): 

Behavior 2000-2006 (meters at pole)

Astronomical Longitude: 

Astronomical Longitude Based on time difference between event in Greenwich and local occurrence Greenwich sidereal time (GST) gives time relative to fixed stars

Universal Time: 

Universal Time UT1: Time given by rotation of Earth. Noon is “mean” sun crossing meridian at Greenwich UTC: UT Coordinated. Atomic time but with leap seconds to keep aligned with UT1 UT1-UTC must be measured

Length of day (LOD): 

Length of day (LOD)

Recent LOD: 

Recent LOD

LOD compared to Atmospheric Angular Momentum: 

LOD compared to Atmospheric Angular Momentum

LOD to UT1: 

LOD to UT1 Integral of LOD is UT1 (or visa-versa) If average LOD is 2 ms, then 1 second difference between UT1 and atomic time develops in 500 days Leap second added to UTC at those times.

UT1-UTC: 

UT1-UTC Jumps are leap seconds, last one before 2006 was 1999. Historically had occurred at 12-18 month intervals

Transformation from Inertial Space to Terrestrial Frame: 

Transformation from Inertial Space to Terrestrial Frame To account for the variations in Earth rotation parameters, as standard matrix rotation is made

Geodetic coordinates: 

Geodetic coordinates Easiest global system is Cartesian XYZ but not common outside scientific use Conversion to geodetic Lat, Long and Height

Geodetic coordinates: 

Geodetic coordinates WGS84 Ellipsoid: a=6378137 m, b=6356752.314 m f=1/298.2572221 (=[a-b]/a) The inverse problem is usually solved iteratively, checking the convergence of the height with each iteration. (See Chapters 3 &10, Hofmann-Wellenhof)

Heights: 

Heights Conventionally heights are measured above an equipotential surface corresponding approximately to mean sea level (MSL) called the geoid Ellipsoidal heights (from GPS XYZ) are measured above the ellipsoid The difference is called the geoid height

Geiod Heights: 

Geiod Heights National geodetic survey maintains a web site that allows geiod heights to be computed (based on US grid) http://www.ngs.noaa.gov/cgi-bin/GEOID_STUFF/geoid99_prompt1.prl New Boston geiod height is -27.688 m

NGS Geoid model: 

NGS Geoid model NGS Geoid 99 http://www.ngs.noaa.gov/GEOID/GEOID99/

Spherical Trigonometry: 

Spherical Trigonometry Computations on a sphere are done with spherical trigonometry. Only two rules are really needed: Sine and cosine rules. Lots of web pages on this topic (plus software) http://mathworld.wolfram.com/SphericalTrigonometry.html is a good explanatory site

Basic Formulas: 

Basic Formulas

Basic applications: 

Basic applications If b and c are co-latitudes, A is longitude difference, a is arc length between points (multiply angle in radians by radius to get distance), B and C are azimuths (bearings) If b is co-latitude and c is co-latitude of vector to satellite, then a is zenith distance (90-elevation of satellite) and B is azimuth to satellite (Colatitudes and longitudes computed from DXYZ by simple trigonometry)

Summary of Coordinates: 

Summary of Coordinates While strictly these days we could realize coordinates by center of mass and moments of inertia, systems are realized by alignment with previous systems Both center of mass (1-2cm) and moments of inertia (10 m) change relative to figure Center of mass is used based on satellite systems When comparing to previous systems be cautious of potential field, frame origin and orientation, and ellipsoid being used.