CEE 8813 · GIS Lecture 2 · Datums

Datum Lab

A latitude and a longitude name a place only after you say which datum they belong to. A datum is an ellipsoid of a chosen size, placed so that it fits the Earth's real shape. Change the datum and the same numbers land somewhere else, about 17 m away at Georgia Tech, which is enough to move a crash off the road it happened on.

1 · Shape of the Earth · slides 2–4

Ellipsoid Sphere, R = 6,371 km Geoid (schematic bumps)

Which ellipsoid?

Exaggerate the flattening

×40
×1 · true shape×100

Exaggerate the geoid undulations

×10,000
×1 · invisible×20,000

Latitude

33.75°N
Exact values · WGS 84The numbers are exact. Only the drawing is exaggerated.
Semi-major axis a
Semi-minor axis b
Flattening 1/f
1° of latitude
1° of longitude
Latitude is measured on the normal

h = H + N. GPS gives h, the height above the ellipsoid. Survey benchmarks and map elevations give H, the orthometric height above the geoid. N is the geoid height, the geoid's distance above the ellipsoid; it is negative where the geoid dips below. Worldwide N runs from about −106 m (Indian Ocean) to about +85 m (near New Guinea).

2 · Fit a datum · slides 5–6

Local fit

Schematic
RMS gap · North America
RMS gap · South America
RMS gap · whole Earth
0 m 0 m 0 m

Drag the ellipse, or use the sliders. Everything here is schematic. The geoid is a made-up smooth curve with realistic heights (about −106 m to +85 m), and every gap is exaggerated thousands of times relative to the size of the Earth (the tag on the drawing gives the factor). The gaps and RMS values are computed from that curve.

3 · Same numbers, different place · slides 7–8

Printed coordinates

Read as NAD 27
In lane widths
Read as NAD 83≈ 1–2 mtoday, NAD83(2011) vs WGS 84 (G2139)

Georgia, drawn in Georgia Statewide Lambert. Each arrow points from the WGS 84 reading to the NAD 27 reading; arrow lengths are exaggerated. Click a place to pick it.

In the lab. The crash CSV (Collisions Dataset.csv) is in GCS WGS 1984. The curve and road layers are NAD 1983 based. Mixing WGS 84 and NAD 83 costs about 1–2 m, which is inside the accuracy of most crash locations. Mixing in NAD 27 data costs about 15–26 m in Georgia, enough to put a crash on the wrong road. The NAD 27 shift shown here is the standard 3-parameter CONUS shift (EPSG:1173, ArcGIS name NAD_1927_To_WGS_1984_4); the NADCON grids that ArcGIS Pro can also use can differ from it by a few metres.

4 · Your lab data speaks four coordinate systems

Lab layerStored as (.prj)DatumWhat the numbers areX as storedY as stored
Collisions Dataset.csvDisplay XY DataGCS_WGS_1984WGS 1984Longitude and latitude in decimal degrees. West longitudes are negative.
D7_Curve_Inventory_GT.shpcurves, polylinesNAD_1983_StatePlane_Georgia_West_FIPS_1002_FeetNAD 1983Transverse Mercator in US survey feet. Central meridian −84.1667°, scale 0.9999, false easting 2,296,583.333 ft, latitude of origin 30°.
GeorgiaRoadSegments_SR.shpstate routes, polylinesNAD_1983_Georgia_Statewide_LambertNAD 1983Lambert Conformal Conic in US survey feet. Standard parallels 31°25′N and 34°17′N, central meridian −83.5°, latitude of origin 0°.
GA_Counties.shpFEMA risk index, polygonsWGS_1984_Web_Mercator_Auxiliary_SphereWGS 1984Web Mercator in metres, the system of online basemaps.
For comparisonnot a lab layerNAD_1983_UTM_Zone_16NNAD 1983UTM zone 16N in metres (EPSG:26916). Central meridian −87°, false easting 500,000 m.

Define Projection vs Project

Pick a mistake and see where the data lands. Define Projection only writes the label (the .prj file) and never moves a coordinate. Project computes new coordinates in another system. ArcGIS Pro's on-the-fly projection draws layers with different but correct .prj files on top of each other, so a layer that does not line up is almost always mislabelled.

Try this in class

  1. In Shape of the Earth, drag the flattening slider down to ×1. The ellipse becomes a circle to the eye, because a − b is only 21.4 km, 0.34 % of the radius. Drag it back to ×100 to see why the lecture still calls the Earth an ellipsoid.
  2. Pick Clarke 1866 and tick “All three ellipsoids”. Even at ×100 the three outlines sit on top of each other. Ask the class why NAD 27 and NAD 83 still disagree by 17 m in Atlanta, then answer it with Fit a datum.
  3. In Fit a datum, press the local preset and read the three RMS gaps, then press the geocentric preset. Then drag the ellipse yourself and try to beat both. You cannot, because the local preset is already the best possible fit for North America, and the geocentric preset is the best possible fit for the whole Earth.
  4. Slide the latitude from 0° to 80° and watch 1° of longitude shrink from 111 km toward zero while 1° of latitude barely changes. At Atlanta a buffer “of 0.001°” would reach 111 m north and south but only 93 m east and west, so on the ground it is an oval.
  5. In Same numbers, different place, compare Savannah with Brasstown Bald. Count lane widths on the arrow and decide whether the crash would still sit on the same road.
  6. Step through the four mistakes in the playground. For each one, say out loud which tool fixes it: Define Projection, Project, or redoing Display XY Data.

Pitfalls students hit

  • Define Projection is not Project. Define Projection only writes the .prj label and never moves coordinates; use it only to correct a missing or wrong label. Project computes new coordinates in another system. Projecting data that was labelled wrong bakes the error into the coordinates.
  • Display XY Data needs the right GCS. The crash CSV is in GCS WGS 1984, so pick that coordinate system in the dialog (XY Table To Point in ArcGIS Pro). Put Longitude in the X field and Latitude in the Y field.
  • Check the datum transformation. When layers sit on different datums, ArcGIS Pro applies a geographic transformation, listed under Map Properties > Transformation, and Project has a Geographic Transformation box. NAD_1983_To_WGS_1984_1 is a zero shift that treats the two as identical; for NAD 27 data pick a NADCON or other NAD 1927 transformation on purpose.
  • Undocumented data. If a file has no .prj and you must guess, the lecture's advice is to try NAD 1983 first (slide 8), then check it against a layer you trust.
  • Degrees are not distances. Never buffer or measure in a GCS. Project to a coordinate system in feet or metres first, such as State Plane Georgia West for the Atlanta curves.

Back to Lecture 2: from the Earth's shape to a datum

Slides 2 to 8 build a datum in steps: angles on a round Earth, an ellipsoid, the geoid, fitting one to the other, and the datums North America actually uses.

Geographic coordinate systemSlide 2

A GCS locates places with angles in degrees on a three-dimensional surface. Latitude is the angle up from the equator; on an ellipsoid it is measured along the normal, the line perpendicular to the horizon. Longitude is the angle east or west of the Prime Meridian.

Georgia: at 33.75°N one degree of longitude is about 92.7 km and one of latitude about 110.9 km, so a degree is not a fixed distance.

Spheroid (ellipsoid)Slide 3

The Earth bulges at the equator. An ellipsoid is set by its semi-major axis a (equatorial radius) and semi-minor axis b (polar radius). For GRS 80 and WGS 84, a = 6,378,137 m and b = 6,356,752.314 m.

Why exaggerate: a − b is 21.4 km, so at true scale the ellipsoid looks like a circle.

GeoidSlide 4

The geoid is the surface of equal gravity potential that matches mean sea level. It is lumpy because mass inside the Earth is uneven. The NASA picture on the slide exaggerates the lumps, and so does this page.

Heights: GPS heights are ellipsoidal (h); survey and map elevations are orthometric (H, above the geoid); h = H + N.

Datum: local vs geocentricSlides 5–6

A datum shifts an ellipsoid so it fits the geoid. A local datum fits one region best and fits elsewhere worse (best for North America, worse for South America). A geocentric datum puts the ellipsoid's center at the Earth's center of mass and fits the whole Earth.

Real numbers: the NAD 27 ellipsoid sits about 238 m from the Earth's center of mass.

The same flagpole, two datumsSlide 7

The slide's flagpole has one position on the ground but two pairs of coordinates, one in GCS NAD 1983 and one in GCS NAD 1927. Read one pair in the other datum and you are pointing at a different spot.

Georgia: 17.4 m at Tech Tower, 26.1 m in Savannah, 15.9 m on Brasstown Bald.

Datums used in North AmericaSlide 8

NAD 27 uses the Clarke 1866 spheroid and is still found in old data sets. NAD 83 uses GRS 80 and is the usual datum for mapping; NAD 83 HARN refines it with a high-accuracy network. WGS 84 is geocentric and is what most GPS units report.

Lab: crashes are WGS 1984, curves and roads are NAD 1983, counties are WGS 1984 Web Mercator.

A datum answers “where is the ellipsoid, and how big is it”. A projection, the next part of Lecture 2, answers “how do we flatten it onto paper”. Every projected coordinate system in the lab (State Plane, Georgia Lambert, Web Mercator, UTM) sits on top of a datum, which is why the .prj of a projected layer still names NAD 1983 or WGS 1984.

Next: Projection Light Lab (slides 9–24)