Datums and Spheroids

The Earth’s Shape and Size • It is only comparatively recently that we’ve been able to say what both are • Estimates of shape by the ancients have ranged from a flat disk, to a cube to a cylinder to an oyster. • Pythagoras was the first to postulate it was a sphere • By the fifth century BCE, this was firmly established. • But how big was it? ©2008 Austin Troy

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.” • -source: ESRI ©2008 Austin Troy

So, what shape IS the earth? • Earth is not a sphere, but an ellipsoid, because the centrifugal force of the earth’s rotation “flattens it out”. Source: ESRI • This was finally proven by the French in 1753 • The earth rotates about its shortest axis, or minor axis, and is therefore described as an oblate ellipsoid ©2008 Austin Troy

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 Source: ESRI • These are two common spheroids used today: the difference between its major axis and its minor axis is less than 0.34%.... ©2008 Austin Troy

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 ©2008 Austin Troy

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 ©2008 Austin Troy

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 • Overall these differences are small~100 meters ©2008 Austin Troy

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 ©2008 Austin Troy

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. ©2008 Austin Troy

The Geographic Graticule/Grid • This is a location reference system for the earth’s surface, consisting of: • Meridians: lines of longitude and • Parallels: lines of latitude • Prime meridian is at Greenwich, England (that is 0º longitude) • Equator is at 0º latitude Source: ESRI ©2008 Austin Troy

The Geographic Graticule/Grid • This is like a planar coordinate system, with an origin at the point where the equator meets the prime meridian • The difference is that it is not a Grid because grid lines must meet at right angles; this is why it’s called a graticule instead • Each degree of latitude represents about 110 km, although that varies slightly because the earth is not a perfect sphere ©2008 Austin Troy

The Geographic Grid/Graticule • Latitude and longitude can be measured in degrees, minutes, seconds (e.g. 56° 34’ 30”); minutes and seconds are base-60, like on a clock • Can also use decimal degrees (more common in GIS), where minutes and seconds are converted to a decimal • Example: 45° 52’ 30” = 45.875 ° ©2008 Austin Troy

The Geographic Grid/Graticule • Latitude lines form parallel circles of different sizes, while longitude lines are half-circles that meet at the poles • Latitude goes from 0 to 90º N or S and longitude to 180º E or W of meridian; the 180º line is the date line ©2008 Austin Troy Source: ESRI

Horizontal Datums • Definition: a 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 • Hence, a datum provides frame of reference for placing specific locations at specific points on the spheroid • Defines the origin and orientation of latitude and longitude lines. ©2008 Austin Troy

Horizontal Datums • A datum is essentially the model that is used to translate a spheroid into locations on the earth • A spheroid only gives you a shape—a datum gives you locations of specific places on that shape. • Hence, a different datum is generally used for each spheroid • Two things are needed for datum: spheroid and set of surveyed and measured points ©2008 Austin Troy

Geoid and the vertical datum • There is also a vertical datum, based on the geoid • The geoid serves as the earth’s reference elevation when “sea level” is inadequate, which it is in many cases because the sea is not everywhere and because the sea can be affected by wind or weather • The geoid provides the reference elevation from which vertical measurements can be taken. ©2008 Austin Troy

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. ©2008 Austin Troy

Surface Based Datums • Starting points need to be very central relative to landmass being measured • In NAD27 center point was Mead’s Ranch, KS • NAD27 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 were required • Problem: errors compound with distance from the initial reference. This is why central location needed for first point ©2008 Austin Troy

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. Mead’s Ranch With b and c and A known, we can determine a’s location through solving for B and C by the law of sines B=A(sin(b))/(sin(a)) D c B A E a b C Secondary Measured point ©2008 Austin Troy

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, measurements are linked to reference point in outer space ©2008 Austin Troy

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 a 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. ©2008 Austin Troy

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. • So, just giving lat and long is not enough!!! ©2008 Austin Troy

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 NAD27 are in reference to a NAD27 earth surface, not a NAD83 earth surface. • Example: the DMS control point in Redlands, CA is -117º 12’ 57.75961”, 34 º 01’ 43.77884” in NAD83 and -117 º 12’ 54.61539” 34 º 01’ 43.72995” in NAD27 ©2008 Austin Troy

Datum Shift • NAD83 is superior to NAD27 because: • NAD83 is more accurate and NAD27 can result in a significant horizontal shift • When we go from a surface-oriented datum to a spheroid-based datum, the estimated position of survey benchmarks improves; this is called datum shift • That shift varies with location: 10 to 100 m in the cont. US, 400 m in Hawaii, 35 m in Vermont • Click here for an example from Peter Dana ©2008 Austin Troy

Datums and Spheroids