Introduction to GPS: Theory and Applications

1. GS 608 Introduction to GPS: Theory and Applications Undergraduate and Graduate, 3 credit hours AU 2001 Department of Civil and Environmental Engineering and Geodetic Science

2. Part I SPATIAL REFRENCE SYSTEMS AND FRAMES GS608 References: http://www.geocities.com/CapeCanaveral/1224/theory/theory.html (and referenced links) http://www.colorado.Edu/geography/gcraft/notes/datum/datum_f.html

3. Now, where in the world am I?

5. Local and Global Ellipsoid

7. Maps are 2-dimensional abstractions of reality • As such, they are not located in their “real” geographic setting • They are removed from real coordinates Thus: • We must have a system for locating objects once they are depicted on a map, in geographic space • These systems are called Spatial Reference Systems

9. Reference systems • Abstract (Cartesian) • Geographical (geometric) • Earth is projected into 2-dimensional space • Or, can be viewed in 3-dimensions

10. Grid can vary from map to map and make comparison between maps difficult

15. Sphere as an approximation of the Earth surface Geographical coordinates Latitude, longitude and height above the sphere (measured along the normal to the sphere)

21. Point is located on the surface of the reference sphere • r is a radius of the reference sphere approximating the shape of the Earth • XYZ triad is placed at the center of the sphere For points with elevation h above the reference sphere:

22. If the triad XYZ is ECEF (Earth-centered and Earth-fixed), centered at the center of mass of the reference sphere with radius r • And the plane XZ is located in the reference (zero) meridian plane • And the plane XY is located in the equatorial plane • The polar coordinates  and  can be referred to as geographical (spherical) coordinates, latitude and longitude.

24. Geographical coordinate systems • Based on spherical shape of the Earth • Geodetic coordinate systems • Based on ellipsoidal shape of the Earth • Varying systems use different reference ellipsoids (spheroids) • Ellipsoid is an approximation of the shape of the Earth: the Earth is an oblate ellipsoid, nearly spherical, but bulging at the equator.

27. Terminology confusion with geographic coordinates • 3 types of co-ordinates define different perpendiculars: • astronomical coordinates • physically defined perpendicular, based on the gravity • geodetic coordinates • mathematically defined perpendicular, based on the reference surface, specifically ellipsoid, used for large and medium scale mapping, and in geodesy. • geographic coordinates • mathematically defined perpendicular, based on the reference surface, typically spheres, used for small scale mapping

30. Spherical Coordinates Based on Ellipsoid of Revolution • They are consistent from map to map, but making measurements necessary to use them may be difficult as • degrees of longitude vary in distance from about 69 miles at the equator to 0 mile at the poles • Geodetic coordinates are not directly measurable in the field, they can be observed by astronomical methods and reduced to the ellipsoid

31. WGS84

33. Coordinate Frame Geometry

36. Coordinate Conversion: Cartesian X,Y,Z to Geodetic Lat, Lon, h • Precise (accurate) conversion can be performed • Iteratively • direct computation of longitude • iterate for latitude and height • Closed formulas (Borkowski, 1989; see the handout, IERS Conventions, 1996, p.12)

37. Short Definitions 1/2 Map Projection A Map Projection defines the mapping from geographic coordinates on a sphere or the geodetic coordinates on a spheroid to a plane. Reference System Shape, size, position and orientation of a (mathematical) Reference Surface, (e.g. Sphere or Spheroid). It normally is defined in a superior, geocentric three-dimensional coordinate system (for example WGS84, ITRF).

38. Short Definitions 2/2 Reference Surface Mathematically (e.g. Sphere or Spheroid) or physically (Geoid) defined surface to approximate the shape of the earth for referencing the horizontal and/or vertical position. Reference Frame A set of control points to realize a Reference System. Geodetic Datum Traditional Term for Reference System.

39. Reference Systems: SUMMARY 1/2 • A coordinate system is most commonly referred to as three mutually perpendicular axes, scale and a specifically defined origin • An access to the coordinate system is provided by coordinates of a set of well defined reference points • Coordinate system and an ellipsoid create a datum; ellipsoid must be defined by two parameters (a and f or a and e); ellipsoid must be oriented in space • Modern systems, especially these derived from GPS observations are Earth-centered, Earth-fixed (ECEF)

41. Reference Systems: SUMMARY 2/2 • Geodetic datum defines the size and shape of the earth and the origin and orientation of the coordinate systems used to map the earth. • Numerous different datums have been created and used so far, evolving from those describing a spherical earth to ellipsoidal models derived by modern techniques, such as satellite observations • Modern geodetic datums range from flat-earth models used for plane surveying to complex, global models, which completely describe the size, shape, orientation, gravity field, and angular velocity of the earth. • Potential problems: • Referencing geodetic coordinates to the wrong datum can result in significant position errors • The diversity of datums in use today and the technological advancements that have made possible global positioning measurements with sub-decimeter accuracies require careful datum selection and careful conversion between coordinates in different datums.

44. Datums are created by geodesists, while cartography, surveying, navigation, and astronomy are the end users • National Imagery and Mapping Agency (NIMA), former Defense Mapping Agency created WGS84 – World Geodetic Datum 84 • National Geodetic Survey (NGS) created NAD83 – North American Datum 83 • International Earth Rotation Service (IERS)createdITRFxx, where xx stands for the reference year at which the frame was (re)established or (re)computed • ITRF stands for International Terrestrial Reference Frame • ITRF coordinates can be expressed in WGS84 at 10 cm level • Newest ITRF refers to epoch 2000 (ITRF2000)

45. ITRF200 Reference Frame

46. What is ITRF ? • The International Earth Rotation Service (IERS) has been established in 1988 jointly by the International Astronomical Union (IAU) and the International Union of Geodesy and Geophysics (IUGG). The IERS mission is to provide to the worldwide scientific and technical community reference values for Earth orientation parameters and reference realizations of internationally accepted celestial and terrestrial reference systems • In the geodetic terminology, a reference frame is a set of points with their coordinates (in the broad sense) which realize an ideal reference system • The frames produced by IERS as realizations of ITRS are named International Terrestrial Reference Frames (ITRF). • Such frames are all (or a part of) the tracking stations and the related monuments which constitute the IERS Network, together with coordinates and their time variations.

47. ITRF97 • The reference frame definition (origin, scale, orientation and time evolution) is achieved in such a way that ITRF97 is in the same system as the ITRF96 • Station velocities are constrained to be the same for all points within each site; • ITRF97 positions were estimated at epoch 1997.0; • Transformation parameters (at epoch 1997.0) and their rates from ITRF97 to each individual solution were also estimated. • Transformation between ITRF at epoch 1997.0 and other frames: • Ri represent rotations, D scale change and Ti stands for translation; i=1,2,3

49. TRANSFORMATION PARAMETERS AND THEIR RATES FROM ITRF94 TO OTHER FRAMES ---------------------------------------------------------------------------------------------- SOLUTION T1 T2 T3 D R1 R2 R3 EPOCH Ref. cm cm cm 10-8 .001" .001" .001" IERS Tech. . . . . . . . Note #, page RATES T1 T2 T3 D R1 R2 R3 cm/y cm/y cm/y 10-8/y .001"/y .001"/y .001"/y ---------------------------------------------------------------------------------------------- ITRF93 0.6 -0.5 -1.5 0.04 -0.39 0.80 -0.96 88.0 RATES -0.29 0.04 0.08 0.00 -0.11 -0.19 0.05 18 82 ITRF92 0.8 0.2 -0.8 -0.08 0.0 0.0 0.0 88.0 18 80 ITRF91 2.0 1.6 -1.4 0.06 0.0 0.0 0.0 88.0 15 44 ITRF90 1.8 1.2 -3.0 0.09 0.0 0.0 0.0 88.0 12 32 ITRF89 2.3 3.6 -6.8 0.43 0.0 0.0 0.0 88.0 9 29 ITRF88 1.8 0.0 -9.2 0.74 0.1 0.0 0.0 88.0 6 34

50. X,Y,Z (Lat, Lon, h) based on the definition of WGS84 ellipsoid