02/11/02 12.540 Lec 02 1 12.540 Principles of the Global Positioning System Lecture 02 Prof. Thomas Herring 02/11/02 12.540 Lec 02 2 Coordinate Systems ? Today we cover: – Definition of coordinates – Conventional “realization” of coordinates – Modern realizations using spaced based geodetic systems (such as GPS). 02/11/02 12.540 Lec 02 3 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) 02/11/02 12.540 Lec 02 4 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 02/11/02 12.540 Lec 02 5 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. 02/11/02 12.540 Lec 02 6 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. 02/11/02 12.540 Lec 02 7 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 near- infrared signals) 02/11/02 12.540 Lec 02 8 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 02/11/02 12.540 Lec 02 9 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) 02/11/02 12.540 Lec 02 10 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. 02/11/02 12.540 Lec 02 11 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 02/11/02 12.540 Lec 02 12 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 02/11/02 12.540 Lec 02 13 Geodetic coordinates: Latitude North Equator Geoid gravity direction Normal to ellipsoid φ g φ a Local equipotenital surface Earth's surface P 02/11/02 12.540 Lec 02 14 Longitude x Rotation of Earth λ Longitude measured by time difference of astronomical events 02/11/02 12.540 Lec 02 15 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 02/11/02 12.540 Lec 02 16 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 02/11/02 12.540 Lec 02 17 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 ? 2 V = 0 ? 2 V = 4πGρ 02/11/02 12.540 Lec 02 18 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)