EAS-591T – Space Geodetic Measurements of Active Crustal
Motions
LAB 2
Coordinates can be expressed using various datums, all related to each other through geometrical transformations. The objective of this lab is to review some of them and to write small programs to convert coordinates between them.
U Geodetic datum = size and shape of the Earth (= coordinate system) + origin and orientation of the coordinate system (and their time derivatives…). A minimum of 5 parameters are necessary to define a datum:
- The translation of its origin with respect to the Earth’s center of mass (t_{X}, t_{Y}, t_{Z})
- The size and shape of the Earth, usually approximated as an ellipsoid: semi-major axis a and flattening f.
U There are many different datums in use to express coordinates. The shape of the Earth is usually considered to be an ellipsoid, geodetic coordinates are therefore often refered to as “ellipsoidal” coordinates. In the particular case where the semi-major and semi-minor axes are equal, the ellipsoid becomes a sphere and coordinates are called “spheric” coordinates.
U Datums are derived from measurements, from classical triangulation surveys to space-based techniques. Therefore, they evolve as the precision and capabilities of the observation techniques improve.
U Using an incorrect datum to express coordinates can result in position errors of hundreds of meters. Different countries, agencies, and applications use different datums. Great care must be used when manipulating coordinates to ensure that they are associated with a well-defined datum.
U The simplest datum is an Earth Centered, Earth Fixed cartesian datum (ECEF). Its origin is the center of mass of the Earth’s. It is define by three right-handed orthogonal axis X, Y, Z. Units are meters. The Z axis coicincides with the Earth’s rotation axis. The (X,Y) plane coincides with the equatorial plane. The (X,Z) plane contains the Earth’s rotation axis and the prime meridian.
U Coordinates are usually expressed as latitude, longitude, and height on a given ellipsoid = ellipsoidal coordinates. The Prime Meridian is the origin for longitudes. The Equator is the origin for latitudes.
- The geodetic latitude of a point is the angle from the equatorial plane to the vertical direction of a line normal to the reference ellipsoid.
- The geodetic longitude of a point is the angle between a reference plane and a plane passing through the point, both planes being perpendicular to the equatorial plane.
- The geodetic height at a point is the distance from the reference ellipsoid to the point in a direction normal to the ellipsoid.
U Hundreds of geodetic datums are in use around the world. The Global Positioning system is based on the World Geodetic System 1984 (WGS-84). Coordinate values resulting from interpreting latitude, longitude, and height values based on one datum as though they were based in another datum can cause position errors in three dimensions of up to one kilometer
U Corodinates can also be refering to a local topocentric datum. The origin of this datum is any point you choose on the surface of the Earth. It has 3 right-handed orthogonal axis: u (for “up”) is vertical (= perpendicular to the local equipotential surface) and points upwards, n (for “north”) is in the local horizontal plane and points to the geographic north, e (for “east”) is in the local horizontal plane and points to the geographic east. Units are meters.
U Datum conversions.
- Geodetic to ECEF:
- ECEF to geodetic:
- ECEF to local topocentric:
The conversion is a combination of 3 rotations needed to align the ECEF axis with the NEU axis:
_{}
where [X,Y,Z] is the vector to be transformed (in meters), and l and f the latitude and longitude of the reference point, respectively.
U See also:
http://www.colorado.edu/geography/gcraft/notes/datum/datum.html
http://www.hdic.jmu.edu/sic/content/std_datum.html
http://www.nima.mil/GandG/tm83581/toc.htm
http://www.lsgi.polyu.edu.hk/cyber-class/geodesy/syllabus.htm
http://www.uz.ac.zw/engineering/GeoInformatics/Notes/GEODESY/Geodesy1.htm
Assigmnent: Write matlab functions to:
Convert XYZ ECEF coordinates to ellipsoidal [xyz2wgs]
Convert ellipsoidal coordinates to XYZ ECEF [wgs2xyz]
Convert ellipsoidal coordinates to ellipsoidal [wgs2wgs]
Convert a vector in XYZ
ECEF coordinates to local topocentric coordinates [xyz2neu]
These functions should have as arguments the input coordinates plus any additional information neededEx.: [lat,lon,ele] = xyz2wgs(x,y,z,a,f)