Principles of the Global Positioning System - Coordinate Systems

Principles of the Global Positioning System 1 Coordinate Systems • Today we cover: – Definition of coordinates – Conventional “realization” of coordinates – Modern realizations using spaced based geodetic systems (such as GPS). 2 Coordinate system definition • To define a coordinate system you need to define: – Its origin (3 component) – Its orientation (3 components, usually the direction cosines of one axis and one component of another axes, and definition of handed-ness) – Its scale (units) 3 Coordinate system definition • In all 7 quantities are needed to uniquely specify the frame. • In practice these quantities are determined as the relationship between two different frames • How do we measure coordinates • How do we define the frames 4 Measuring coordinates • Direct measurement (OK for graph paper) • Triangulation: Snell 1600s: Measure angles of triangles and one-distance in base triangle • Distance measured with calibrated “chain” or steel band (about 100 meters long) • “Baseline” was about 1 km long • Triangles can build from small to larges ones. • Technique used until 1950s. 5 Measuring coordinates • Small errors in the initial length measurement, would scale the whole network • Because of the Earth is “nearly” flat, measuring angles in horizontal plane only allows “horizontal coordinates” to be determined. • Another technique is needed for heights. 6 Measuring coordinates • In 1950s, electronic distance measurement (EDM) became available (out growth of radar) • Used light travel times to measure distance (strictly, travel times of modulation on either radio, light or nearinfrared signals) 7 Measuring coordinates • Advent of EDM allowed direct measurements of sides of triangles • Since all distances measured less prone to scale errors. • However, still only good for horizontal coordinates 8 Accuracies • Angles can be measured to about 1 arc second (5x10-6 radians) • EDM measures distances to 1x10-6 (1 part-per-million, ppm) • Atmospheric refraction 300 ppm • Atmospheric bending can be 60” (more effect on vertical angles) 9 Height coordinates • Two major techniques: – Measurement of vertical angles (atmospheric refraction) – “Leveling” measurement of height differences over short distances (<50 meters). – Level lines were used to transfer height information from one location to another. 10 Other methods • Maps were made with “plotting tables” (small telescope and angular distance measurements-angle subtended by a known distance • Aerial photogrammetry coordinates inferred from positions in photographs. Method used for most maps 11 Other methods • What is latitude and longitude • Based on spherical model what quantities might be measured • How does the rotation of the Earth appear when you look at the stars? • Concept of astronomical coordinates 12 Geodetic coordinates: Latitude North P Geoid Earth's surface Local equipotenital surface gravity direction Normal to ellipsoid φa φg Equator 13 Longitude Rotation of Earth x λ Longitude measured by time difference of astronomical events 14 Astronomical coordinates • Return to later but on the global scale these provide another method of determining coordinates • They also involve the Earth’s gravity field • Enters intrinsically in triangualtion and trilateration through the planes angles are meaured in 15 Web sites about geodetic measurements • http://www.geography.wisc.edu/sco/geo detic/horizontal.html Geodetic control for Wisconsin • http://www.ngs.noaa.gov/ is web page of National Geodetic Survey which coordinates national coordinate systems 16 Earth’s Gravity field • All gravity fields satisfy Laplace’s equation in free space or material of density ρ. If V is the graviational potential then ∇ V =0 2 ∇ V = 4 πGρ 2 17 Solution to gravity potential • The homogeneous form of this equation is a “classic” partial differential equation. • In spherical coordinates solved by separation of variables, r=radius, λ=longitude and θ=co-latitude V(r,θ, λ)=R(r)g(θ)h(λ) 18