principles of the global positioning system, lecture 2
TRANSCRIPT
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12.540 Principles of the Global Positioning System
Lecture 02 Prof. Thomas Herring
http://geoweb.mit.edu/~tah/12.540
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Coordinate Systems
• Today we cover: – Definition of coordinates – Conventional “realization” of coordinates – Modern realizations using spaced based
geodetic systems (such as GPS).
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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)
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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
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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.
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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.
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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)
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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
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Accuracies
• Angles can be measured to about 1 arc second (5x10-6 radians)
• EDM measures distances to 1x10-6 (1part-per-million, ppm)
• Atmospheric refraction 300 ppm • Atmospheric bending can be 60” (more
effect on vertical angles)
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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.
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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
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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
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Geodetic coordinates: Latitude North
Equator
Geoid
gravity direction
Normal to ellipsoid
ga
Local equipotenital surface Earth's surface
P
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Longitude
Longitude measured by time difference of astronomical events 02/13/12 12.540 Lec 02 14
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Astronomical coordinates
• Return to later but on the global scale these provide another method ofdetermining coordinates
• They also involve the Earth’s gravityfield
• Enters intrinsically in triangulation and trilateration through the planes anglesare measured in
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Height determination
• Height measurements historically are very labor intensive
• The figure on the next page shows how the technique called leveling is used to determine heights.
• In a country there is a primary leveling network, and other heights are determinedrelative to this network.
• The primary needs to have a monument spacing of about 50 km.
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Leveling• The process of leveling is to measure
height differences and to sum these to get the heights of other points.
Orthometric height of hill is h1+h2+h3
N is Geoid Height. Line at bottom is ellipsoid
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Leveling
• Using the instrument called a level, the heights on the staffs are read and the difference in the values is the height differences.
• The height differences are summed to get the height of the final point.
• For the primary control network: the separation of the staffs is between 25-50 meters.
• This type of chain of measurements must be stepped across the whole country (i.e., move across thecountry in 50 meter steps: Takes decades and wasdone).
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Leveling problems
• Because heights are determined by summing differences, system very prone to systematic errors; small biases in the height differences due to atmospheric bending, shadows on the graduations and many other types of problem
• Instrument accuracy is very good for first-order leveling: Height differences can be measured to tens of microns.
• Accuracy is thought to about 1 mm-per-square-root-km for first order leveling.
• Changes in the shapes of the equipotential surface with height above MSL also cause problems.
• The difference between ellipsoidal height and Orthometric height is the Geoid height
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Trigonometric Leveling
• When trying to go the tops of mountains, standard leveling does not work well. (Image trying to do this to the summit of Mt. Everest).
• For high peaks: A triangulation method is used call trigonometric leveling.
• Schematic is shown on the next slide • This is not as accurate as spirit leveling
because of atmospheric bending.
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Trigonometric Leveling schematic
• Method for trigonometric leveling. Method requires that distance D in known and the elevation angles are measured. Trigonometry is used to compute h
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Trigonometric Leveling
• In ideal cases, elevation angles at both ends are measured at the same time. This helps cancel atmospheric refraction errors.
• The distance D can be many tens of kilometers. In the case of Mt. Everest, D was over 100 km (the survey team was not even in the same country; they were in India and mountain is in Nepal).
• D is determined either by triangulation or after 1950 by electronic distance measurement (EDM) discussed later
• The heights of the instruments, called theodolites, above the ground point must be measured. Note: this instrument height measurement was not needed for leveling.
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Web sites about geodetic measurements
• http://sco.wisc.edu/surveying/networks.php Geodetic control for Wisconsin
• Try search “trilateration network” search. Finding maps of networks is now difficult (replaced with GPS networks)
• http://www.ngs.noaa.gov/ is web page of National Geodetic Survey which coordinates national coordinate systems
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Earth’s Gravity field
• All gravity fields satisfy Laplace’s equation in free space or material of density . If V is the gravitational potential then
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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, =longitude and =co-latitude
V (r,,) R(r)g( )h() 02/13/12 12.540 Lec 02 25
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Summary
• Examined conventional methods of measuring coordinates
• Triangulation, trilateration and leveling
• Astronomical positioning uses external bodies and the direction of gravity field
• Continue with the use of the gravity field.
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12.540 Principles of the Global Positioning SystemSpring 2012
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