02/18/03 12.540 Lec 04 1 12.540 Principles of the Global Positioning System Lecture 04 Prof. Thomas Herring 02/18/03 12.540 Lec 04 2 Review – Examine definitions of coordinates – – Time systems – Start looking at satellite orbits ? So far we have looked at measuring coordinates with conventional methods and using gravity field ? Today lecture: Relationships between geometric coordinates 1 02/18/03 12.540 Lec 04 3 Coordinate types – – Orthometric heights (heights measured about an (MSL) – Cartesian XYZ – Geodetic latitude, longitude and height ? Potential field based coordinates: Astronomical latitude and longitude equipotential surface, nominally mean-sea-level ? Geometric coordinate systems 02/18/03 12.540 Lec 04 4 Astronomical coordinates – Latitude: Elevation angle to North Pole (center of star rotation field) – Longitude: Time difference between event at Greenwich and locally ? Astronomical coordinates give the direction of the normal to the equipotential surface ? Measurements: 2 02/18/03 12.540 Lec 04 5 Astronomical Latitude http://www.iers.org/ ? 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 02/18/03 12.540 Lec 04 6 Astronomical Latitude d f a z d To Celestial body Rotation Axis f a = Z d -d declination Zenith distance= 90-elevation Geiod 3 02/18/03 12.540 Lec 04 7 Astronomical Latitude – Rotation axis moves in space, precession nutation. Given by International Astronomical Union (IAU) precession nutation theory – Rotation moves relative to crust 02/18/03 12.540 Lec 04 8 Rotation axis movement http://bowie.mit.edu/~tah/mhb2000/JB000165_online.pdf) ? By measuring the zenith distance when star is at minimum, yields latitude ? Problems: ? Precession Nutation computed from Fourier Series of motions ? Largest term 9” with 18.6 year period ? Over 900 terms in series currently (see ? Declinations of stars given in catalogs ? Some almanacs give positions of “date” meaning precession accounted for 4 02/18/03 12.540 Lec 04 9 Rotation axis movement monitored ? 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 02/18/03 12.540 Lec 04 10 Evolution (IERS C01) -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 1900.0 1920.0 1940.0 1960.0 1980.0 2000.0 X Pole (") Pole Position (") (0.5"=15m) Year Y Pole (") CIO 1900-1905 5 6 02/18/03 12.540 Lec 04 11 Evolution of uncertainty 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 1900.0 1920.0 1940.0 1960.0 1980.0 2000.0 s X Pole (") s Y Pole (") s Pole Position (") (0.02"=0.6m) Year CIO 1900-1905 02/18/03 12.540 Lec 04 12 Behavior 1993-2001 -10.0 -5.0 0.0 5.0 10.0 0.05.010.015.020.0 Pole Position X Pole (m) Y Pole (m) 1993 2001 ? 02/18/03 12.540 Lec 04 13 Astronomical Longitude GST =1.0027379093UT1+ J 0 GMST { y e Precession 34 J 0 = 24110.54841+ 8640184.812866 T Julian Centuries { + 0.093104T 2 - 6.2 ￥10 -6 T 3 ? Based on time difference between event in Greenwich and local occurrence ? Greenwich sidereal time (GST) gives time relative to fixed stars + D cos 1 2 4 02/18/03 12.540 Lec 04 14 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 7 8 02/18/03 12.540 Lec 04 15 Length of day (LOD) -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 1800.0 1850.0 1900.0 1950.0 2000.0 LOD = Difference of day from 86400. seconds LOD (ms) LOD (ms) LOD (ms) Year 02/18/03 12.540 Lec 04 16 Recent LOD -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1992.0 1994.0 1996.0 1998.0 2000.0 2002.0 LOD = Difference of day from 86400. seconds LOD (ms) LOD (ms) LOD (ms) Year 9 02/18/03 12.540 Lec 04 17 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. 02/18/03 12.540 Lec 04 18 UT1-UTC -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1993.0 1994.0 1995.0 1996.0 1997.0 1998.0 1999.0 2000.0 2001.0 UT1-UTC (discontinuities are leap-seconds) UT1-UTC (s) UT1-UTC (s) Year ? ? 02/18/03 12.540 Lec 04 19 Transformation from Inertial Space to Terrestrial Frame made x i Inertial { = P Precession { N Nutation { S Spin { W Polar Motion { x t Terrestrial { ? To account for the variations in Earth rotation parameters, as standard matrix rotation is 02/18/03 12.540 Lec 04 20 Geodetic coordinates X = (N + h) f l Y = (N + h) f sinl Z = ( b 2 a 2 N + h)sinf N = a 2 a 2 2 f +b 2 sin 2 f ? Easiest global system is Cartesian XYZ but not common outside scientific use ? Conversion to geodetic Lat, Long and Height cos cos cos cos 10 02/18/03 12.540 Lec 04 21 Geodetic coordinates – a=6378137 m, b=6356752.314 m – f=1/298.2572221 (=[a-b]/a) ) 02/18/03 12.540 Lec 04 22 Heights ? WGS84 Ellipsoid: ? The inverse problem is usually solved iteratively, checking the convergence of the height with each iteration. ? (See Chapters 3 &10, Hofmann-Wellenhof ? 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 11 NGS Geoid model 02/18/03 12.540 Lec 04 23 ? http://www.ngs.noaa.gov/cgi-bin/GEOID_STUFF/geoid99_prompt1.prl -27.688 m Geiod Heights ? National geodetic survey maintains a web site that allows geiod heights to be computed (based on US grid) ? New Boston geiod height is 02/18/03 12.540 Lec 04 24 NGS Geoid 99 http://www.ngs.noaa.gov/GEOID/GEOID99/ 12 ? ? ? 02/18/03 12.540 Lec 04 25 Spherical Trigonometry ? http://mathworld.wolfram.com/SphericalTrigonometry.html is a ? 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) good explanatory site 02/18/03 12.540 Lec 04 26 A B C a c b A B C are angles a b c are sides (all quanties are angles) Sine Rule sin a sin A = sin b sin B = sin c sin C Cosine Rule sides Cosine Rule angles a = b c + sin b sin c A b = c c + sin csin a B c = b a + sin asin b C cos A = -cos B cosC + sin B sin C cos a cos B = -cos A cosC + sin A sin C cosb cosC = -cos A cos B + sin A sin B cos c Basic Formulas cos cos cos cos cos cos cos cos cos cos cos cos 13 02/18/03 12.540 Lec 04 27 Basic applications satellite D 02/18/03 12.540 Lec 04 28 Summary of Coordinates ? While strictly these days we could realize coordinates by center of mass and moments of inertia, 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 ? potential field, frame origin and orientation, and ellipsoid being used. ? 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 ? (Colatitudes and longitudes computed from XYZ by simple trigonometry) are realized by alignment with previous systems When comparing to previous systems be cautious of 14