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The Earth’s Size•It was Posidonius who used the stars to determine the earth's circumference. “He observed that a given star could be seen just on the horizon at Rhodes. He then measured the star's elevation at Alexandria, Egypt, and calculated the angle of difference to be 7.5 degrees or 1/48th of a circle. Multiplying 48 by what he believed to be the correct distance from Rhodes to Alexandria (805 kilometers or 500 miles), Posidonius calculated the earth's circumference to be 38,647 kilometers (24,000 miles)--an error of only three percent.” More info -source: ESRI
And it’s also a….•Because it’s so close to a sphere, the earth is often referred to as a spheroid: that is a type of ellipsoid that is really, really close to being a sphere
•These are two common spheroids used today: the difference between its major axis and its minor axis is less than 0.34%.
Spheroids•The International 1924 and the Bessel 1841 spheroids are used in Europe while in North America the GRS80, and decreasingly, the Clarke 1866 Spheroid, are used•In Russia and China the Krasovsky spheroid is used and in India the Everest spheroid
Spheroids•One more thing about spheroids: If your mapping scales are smaller than 1:5,000,000 (small scale maps), you can use an authalic sphere to define the earth's shape to make things more simple
•For maps at larger scale (most of the maps we work with in GIS), you generally need to employ a spheroid to ensure accuracy and avoid positional errors
Geoid•While the spheroid represents an idealized model of the earth’s shape, the geoid represents the “true,” highly complex shape of the earth, which, although “spheroid-like,” is actually very irregular at a fine scale of detail, and can’t be modeled with a formula (the DOD tried and gave up after building a model of 32,000 coefficients)
•It is the 3 dimensional surface of the earth along which the pull of gravity is a given constant; ie. a standard mass weighs an identical amount at all points on its surface
•The gravitational pull varies from place to place because of differences in density, which causes the geoid to bulge or dip below or above the ellipsoid
Spheroids and Geoids•We have several different estimates of spheroids because of irregularities in the earth: there are slight deviations and irregularities in different regions
•Before remote satellite observation, had to use a different spheroid for different regions to account for irregularities (see Geoid, ahead) to avoid positional errors
•That is, continental surveys were isolated from each other, so ellipsoidal parameters were fit on each continent to create a spheroid that minimized error in that region, and many stuck with those for years
The Geographic Graticule/Grid•Once you have a spheroid, you also define the location of poles (axis points of revolution) and equator (midway circle between poles, spanning the widest dimension of the spheroid), you have enough information to create a coordinate grid or “graticule” for referencing the position of features on the spheroid.
Datums•Three dimensional surface from which latitude, longitude and elevation are calculated
•Allows us to figure out where things actually are on the graticule since the graticule only gives us a framework for measuring, and not the actual locations
•Frame of reference for placing specific locations at specific points on the spheroid
•Defines the origin and orientation of latitude and longitude lines.
Surface-Based Datums•Prior to satellites, datums were realized by connected series of ground-measured survey monuments
•A central location was chosen where the spheroid meets the earth: this point was intensively measured using pendulums, magnetometers, sextants, etc. to try to determine its precise location.
•Originally, the “datum” referred to that “ultimate reference point.”
•Eventually the whole system of linked reference and subrefence points came to be known as the datum.
Surface Based Datums•Starting points need to be very central relative to landmass being measured•In NAD27 center point was Mead’s Ranch, KS (why?)•NAD 27 resulted in lat/long coordinates for about 26,000 survey points in the US and Canada.
•Limitation: requires line of sight, so many survey points required
Surface Based Datums•These were largely done without having to measure distances. How?
•Using high-quality celestial observations and distance measurements for the first two observations, could then use trigonometry to determine distances.
ab
A
With b and c and A known, we can determine a’s location through solving for B and C by the law of sines
Satellite Based Datums•With satellite measurements the center of the spheroid can be matched with the center of the earth.
•Satellites started collecting geodetic information in 1962 as part of National Geodetic Survey
•This gives a spheroid that when used as a datum correctly maps the earth such that all Latitude/Longitude measurements from all maps created with that datum agree.
•Rather than linking points through surface measures to initial surface point, are measurements are linked to reference point in outer space
Common Datums•Previously, the most common spheroid was Clarke 1866; the North American Datum of 1927 (NAD27) is based on that spheroid, and has its center in Kansas.
•NAD83 is the new North American datum (for Canada/Mexico too) based on the GRS80 geocentric spheroid. It is the official datum of the USA, Canada and Central America
•World Geodetic System 1984 (WGS84) is newer spheroid/datum, created by the US DOD; it is more or less identical to Geodetic Reference System 1980 (GRS80). The GPS system uses WGS84.
Lat/Long and Datums•These pre-satellite datums are surface based.
•A given datum has the spheroid meet the earth in a specified location somewhere.
•Datum is most accurate near the touching point, less accurate as move away (remember, this is different from a projection surface because the ellipsoid is 3D)
•Different surface datums can result in different lat/long values for the same location on the earth.
Lat/Long and Datums•Lat/long coordinates calculated with one datum are valid only with reference to that datum.
•This means those coordinates calculated with NAD 27 are in reference to a NAD 27 earth surface, not a NAD 83 earth surface.
•Example: the DMS control point in Redlands, CA is -117º 12’ 57.75961”, 34 º 01’ 43.77884” in NAD 83 and -117 º 12’ 54.61539” 34 º 01’ 43.72995” in NAD 27
•Click here for a chart of the different coordinates for the Capital Dome center under different datums (Peter Dana)
Map Projection•This is the method by which we transform the earth’s spheroid (real world) to a flat surface (abstraction), either on paper or digitally
•Because we can’t take our globe everywhere with us!
•Remember: most GIS layers are 2-D3D
2D
Think about projecting a see-through globe onto a wall
Map Projection•The earliest and simplest map projection is the plane chart, or plate carrée, invented around the first century; it treated the graticule as a grid of equal squares, forcing meridians and parallels to meet at right angles
•If applied to the world as mapped now, it would look like:
Map Projection-distortion•Shape: projection can distort the shape of a feature. Conformal maps preserve the shape of smaller, local geographic features, while general shapes of larger features are distorted. That is, they preserve local angles; angle on map will be same as angle on globe. Conformal maps also preserve constant scale locally
Map Projection-distortion•Area:projection can distort the property of equal area (or equivalent), meaning that features have the correct area relative to one another. Map projections that maintain this property are often called equal area map projections.
•For instance, if S America is 8x larger than Greenland on the globe will be 8x larger on map
•No map projection can have conformality and equal area; sacrifice shape to preserve area and vice versa.
Map Projection-distortion•Distance: Projection can distort measures of true distance. Accurate distance is maintained for only certain parallels or meridians unless the map is very localized. Maps are said to be equidistant if distance from the map projection's center to all points is accurate. We’ll go into this more later.
Map Projection-distortion•Direction:Projection can distort true directions between geographic locations; that is, it can mess up the angle, or azimuth between two features; projections of this kind maintain true directions with respect to the map projection's center. Some azimuthal map projections maintain the correct azimuth between any two points. In a map of this kind, the angle of a line drawn between any two locations on the projection gives the correct direction with respect to true north.
Map Projection-distortion•Hence, when choosing a projection, one must take into account what it is that matters in your analysis and what properties you need to preserve
•Conformal and equal area properties are mutually exclusive but some map projections can have more than one preserved property. For instance a map can be conformal and azimuthal
•Conformal and equal area properties are global (apply to whole map) while equidistant and azimuthal properties are local and may be true only from or to the center of map
•Mercator (left)•World Cylindrical Equal Area (above)•The distortion in shape above is necessary to get Greenland to have the correct area; •The Mercator map looks good but Greenland is many times too big
•Mercator maintains shape and direction, but sacrifices area accuracy
•The Sinusoidal and Equal-Area Cylindrical projections both maintain area, but look quite different from each other. The latter distorts shape
•The Robinson projection does not enforce any specific properties but is widely used because it makes the earth’s surface and its features look somewhat accurate
Map Projection-General Types•Cylindrical projection: created by wrapping a cylinder around a globe and, in theory, projecting light out of that globe; the meridians in cylindrical projections are equally spaced, while the spacing between parallel lines of latitude increases toward the poles; meridians never converge so poles can’t be shown
Cylindrical Map Types1. Tangent to great circle: in the simplest case, the
cylinder is North-South, so it is tangent (touching) at the equator; this is called the standard parallel and represents where the projection is most accurate
2. If the cylinder is smaller than the circumference of the earth, then it intersects as a secant in two places
Cylindrical Map TypesSecant projections are more accurate because projection is
more accurate the closer the projection surface is to the globe and a when the projection surface touches twice, that means it is on average closer to the globe
The distance from map surface to projection surface is described by a scale factor, which is 1 where they touch
Cylindrical map distortion• A north-south cylindrical Projections cause major
distortions in higher latitudes because those points on the cylinder are further away from from the corresponding point on the globe
• Scale is constant in north-south direction and in east west direction along the equator for an equatorial projection but non constant in east-west direction as move up in latitude
• Requires alternating Scale Bar based on latitude
Cylindrical map distortion•If such a map has a scale bar (see map in 104 Aiken), know that it is only good for those places and directions in which scale is constant—the equator and the meridians•Hence, the measured distance between Nairobi and the mouth of the Amazon might be correct, but the measured distance between Toronto and Vancouver would be off; the measured distance between Alaska and Iceland would be even further off
Cylindrical map distortion•Why is this? Because meridians are all the same length, but parallels are not.•This sort of projection forces parallels to be same length so it distorts them •As move to higher latitudes, east-west scale increases (2 x equatorial scale at 60° N or S latitude) until reaches infinity at the poles; N-S scale is constant
Conic Projection•Is most accurate where globe and cone meet—at the standard parallel
•Distortion generally increases north or south of it, so poles are often not included
•Conic projections are typically used for mid-latitude zones with east-to-west orientation. They are normally applied only to portions of a hemisphere (e.g. North America)
Map Projection-Specific Types•Mercator: This is specific type of cylindrical projection
•Invented by Gerardus Mercator during the 16th Century
•It was invented for navigation because it preserves azimuthal accuracy—that is, if you draw a straight line between two points on a map created with Mercator projection, the angle of that line represents the actual bearing you need to sail to travel between the two points
Map Projection-Specific Types•Transverse Mercator: Invented by Johann Lambert in 1772, this projection is cylindrical, but the axis of the cylinder is rotated 90°, so the tangent line is longitudinal, rather than the equator
•In this case, only the central longitudinal meridian and the equator are straight lines All other lines are
represented by complex curves: that is they can’t be represented by single section of a circle
•Transverse Mercator projection is not used on a global scale but is applied to regions that have a general north-south orientation, while Mercator tends to be used more for geographic features with east-west axis.
•It is used in commonly in the US with the State Plane Coordinate system, with north-south features
Map Projection-Specific Types•Lambert Conformal Conic:invented in 1772, this is a form of a conic projection
•Latitude lines are unequally spaced arcs that are portions of concentric circles. Longitude lines are actually radii of the same circles that define the latitude lines.
Map Projection-Specific Types•The Lambert Conformal Conic projection is a slightly more complex form of conic projection because it intersects the globe along two lines, called secants, rather than along one, which would be called a tangent
Map Projection-Specific Types•Albers Equal Area Conic projection: Again, this is a conic projection, using secants as standard parallels but while Lambert preserves shape Albers preserves area
•It also differs in that poles are not represented as points, but as arcs, meaning that meridians don’t converge
•Latitude lines are unequally spaced concentric circles, whose spacing decreases toward the poles.
•Developed by Heinrich Christian Albers in the early nineteenth century for European maps
Map Projection-Specific Types•Albers Equal Area Conic: It preserves area by making the scale factor of a meridian at any given point the reciprocal of that along the parallel.
•Scale factor is the ratio of local scale a point on the projection to the reference scale of the globe; 1 means the two are touching and greater than 1 means the projection surface is at a distance
Other Selected Projections• More Cylindrical equal area: (have straight meridians and parallels, the meridians are equally spaced, the parallels unequally spaced)
• Behrmann cyclindrical equal-area: single standard parallel at 30 ° north
•Gall’s stereographic: secant intersecting at 45° north and 45 ° south
•Peter’s: de-emphasizes area exaggerations in high latitudes; standard parallels at 45 or 47 °
Thanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder for links
Other Selected Projections• Azimuthal projections:
•Azimuthal equidistant: preserves distance property; used to show air route distances
•Lambert Azimuthal equal area: Often used for polar regions; central meridian is straight, others are curved
•Oblique Aspect Orthographic
•North Polar StereographicThanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder for links
Other Selected Projections• More conic projections
•Equidistant Conic: used for showing areas near to, but on one side of the equator, preserves only distance property
•Polyconic: used for most of the early USGS quads; based on on an infinite number of cones tangent to an infinite number of parallels; central meridian straight but other lines are complex curves
Thanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder for links
Other Selected Projections• Pseudo-cylindrical projections: resemble cylindrical projections, with straight, parallel parallels and equally spaced meridians, but all meridians but the reference meridian are curves
•Mollweide: used for world maps; is equal-area; 90th meridians are semi-circles
• Robinson:based on tables of coordinates, not mathematical formulas; distorts shape, area, scale, and distance in an attempt to make a balanced map
Thanks to Peter Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder for links
Coordinate Systems•Map projections, like we discussed in last lecture provide the means for viewing small-scale maps, such as maps of the world or a continent or country (1:1,000,000 or smaller)
•Plane coordinate systems are typically used for much larger-scale mapping (1:100,000 or bigger)
Coordinate Systems•Projections are designed to minimize distortions of the four properties we talked about, because as scale decreases, error increases
•Coordinate systems are more about accurate positioning (relative and absolute positioning)
•To maintain their accuracy, coordinate systems are generally divided into zones where each zone is based on a separate map projection
Reason for PCSs•Remember from before that projections are most accurate where the projection surface is close to the earth surface. The further away it gets, the more distorted it gets
•Hence a global or even continental projection is bad for accuracy because it’s only touching along one (tangent) or two (secant) lines and gets increasingly distorted
Reason for PCSs•Plane coordinate systems get around this by breaking the earth up into zones where each zone has its own projection center and projection.
•The more zones there are and the smaller each zone, the more accurate the resulting projections
•This serves to minimize the scale factor, or distance between projection surface and earth surface to an acceptable level
•There is a false origin (zero point) in each zone
•In the transverse Mercator projection, the “cylinder” touches at two secants, so there is a slight bulge in the middle, at the central meridian. This bulge is very very slight, so the scale factor is only .9996
•The standard meridians, where the cylinder touches