Map Projections and Coordinate Systems

Surveying 101 for GIS Professionals

2013 Kentucky GIS Conference Jeremy Gould – Kentucky Transportation Cabinet

Agenda

• Geographic Coordinate Systems • Ellipsoids • Geoid • Horizontal Datums • Projected Coordinate Systems • Project Datum Factors • Vertical Datums

Geographic Coordinate Systems

• Geographic Coordinates Systems use radial coordinates to locate a point on a specifically defined sphere (ellipse). These are called spherical coordinates.

Cartesian point P can also be represented in spherical coordinates ( Φ= +/-degrees latitude λ,φ, γ) where: λ= +/- degrees longitude γ= + radial distance from center

Geographic Coordinate Systems

° ° ° ° Equator 0 ° latitude Prime Meridian 0 ° longitude

Ellipsoids

• Ellipsoids are flattened spheroids that when referenced to the earth can be rotated and/or shifted to best fit the earth (geoid) either in part or in whole

Geoid

• The

geoid

is an equipotential gravimetric surface resulting in an irregular and non-mathematical approximation of the earth’s size and shape relative to a base of reference that best fits global mean sea level in a least squares sense • The

geoid

is a 3 dimensional surface along which the pull of gravity is a specified constant • The geoid is a measured and interpolated surface and not a mathematically defined surface • Differences in the density of the Earth cause variation in the strength of the gravitational pull, in turn causing regions to dip or bulge above or below the reference ellipsoid

Geoid

Gravity Recovery And Climate Experiment (GRACE) Gravimeters

Ellipsoids

There are Global Ellipsoids and Regional (local) Ellipsoids Two Global ellipsoids are GRS80 and WGS84

Horizontal Datums

• A datum is a reference surface • A geodetic datum consists of two major components – Ellipsoid with a spherical coordinate system and origin – Set of points and lines that have been surveyed • A geodetic

datum

is a three dimensional Euclidian reference frame defined relative to an associated

ellipsoid

oriented to achieve a best fit statistical approximation of the geoid either in part or in whole.

• The

North American Datum (NAD)

has been defined by two different ellipsoids, the Clarke ellipsoid of 1866, which was oriented to best fit the North American continent and is the basis of NAD27, and the Global Reference System ellipsoid of 1980 (GRS80) which is a globally defined ellipsoid and the basis of NAD83.

Ellipsoid, Geoid, and Datum

CLARKE 1866 Ellipsoid (NAD’27) GRS80 Ellipsoid (NAD’83) GEOID

Earth Mass Center Approximately 236 meters

Horizontal Datums

• Lat and Long locations of given benchmarks in the NAD27 datum will likely be different from the lat and long of that same benchmark in the NAD83 or WGS84 datum's.

– The monumented points do not move – This is described as a

datum shift

• Shift in coordinate locations from WGS84 to NAD83 is often less than 1 meter • Datum shifts between NAD27 and NAD83 are often 100’s of meters

Horizontal Datums

• Geographic Position (Lat-Long) (variations between datums for same position) DATUM 1

z w



1 P

Example: Datum 1 may have a long-lat of (-85.31 °, 37.55 °) Datum 2 may have a long-lat of (-85.30 °, 37.54 °) The same point has different coordinates because of the shift/rotation of the ellipsoid 

1 u v



2

DATUM 2 

2 y x

Projected Coordinate Systems

• A mapping projection is a geometric tool that allows a portion of a spherical surface to be represented on a two dimensional surface such as a flat sheet of paper or computer screen in a spatially consistent manner.

• A State Plane Coordinate System is a specialized mapping projection that allows direct conversion between spherical geographic coordinates of latitude (  ) and longitude (  ), and rectangular Cartesian coordinates of northing ( (

x

).

y)

and easting

Projected Coordinate Systems

• So how do we get from our Geographic Coordinates to a Projected Coordinate System?

Projected Coordinate Systems

Cylindrical Conical Planar

Projected Coordinate Systems

• Transverse Mercator Projection SF

<

1 Practical Limit of Projection (SF 

k

0 ) SF

>

1 Axis of Cylinder Intersection of Ellipsoid and Projection Cylinder (SF = 1) Polar Axis Grid Origin Central Meridian (SF =

k

0 )

Universal Transverse Mercator Coordinate System

Projected Coordinate Systems

KENTUCKY PROJECTIONS UTM Zones 16 & 17 Transverse Mercator (Secant Cylinder)

UTM Zone 16 UTM Zone 17

Projected Coordinate Systems

• Lambert Conic Projection (Northern Hemisphere)

Central Meridian Polar Axis North Standard Parallel (SF = 1) South Standard Parallel (SF = 1) Parallel of Grid Origin (Base Parallel)

State Plane Coordinate Systems

• State Plane zones are sometimes identified by the Federal Information Processing System (FIPS) Codes as shown below

Projected Coordinate Systems

KENTUCKY PROJECTIONS North and South State Plane Lambert Conformal Conic (Secant Cone)

State Plane North Zone State Plane South Zone

Projected Coordinate Systems

KENTUCKY SINGLE ZONE PROJECTION

Single Zone (all counties) Single Zone coverage area (entire state).

Projected Coordinate Systems

KENTUCKY SPCS – NORTH AND SOUTH ZONES

NORTH ZONE SOUTH ZONE

Projected Coordinate Systems

Kentucky Projections NAD83 State Plane Coordinate System (Lambert Conformal Conic) Parameter Central Meridian North Std Parallel South Std Parallel Base Parallel False Northing False Easting Single Zone 85 ° 45’ W 38 ° 40’ N 37 ° 05’ N 36 ° 20’ N 1,000,000 m 1,500,000 m North Zone 84 ° 15’ W 38 37 37 ° 58’ N ° 58’ N ° 30’ N 0 m 500,000 m South Zone 85 ° 45’ W 37 36 36 ° 56’ N ° 44’ N ° 20’ N 500,000 m 500,000 m Linear unit of measure for all zones is the U.S. Survey Foot (USFt) (1 USFt = .3048006096012 meters)

Projected Coordinate Systems

COORDINATE SPACE COMPARISON 1,500,000 m 4921245 ft 1,250,000 m 4101038 ft 1,000,000 m 328083 ft 750,000 m 2460623 ft 500,000 m 1640415 ft 250,000 m 820207 ft 0 m 0 ft NAD'83 SINGLE ZONE NAD'83 SOUTH ZONE NAD'83 NORTH ZONE NAD'27 NORTH ZONE NAD'27 SOUTH ZONE EASTING

Projected Coordinate Systems

S1 G1 E1 SF>1 E1 < G1 < S1 SF=1 G2 < E2 < S2 S2 E2 G2 SF<1

E = Distance on ellipsoid G = Distance on grid S = Distance on surface SF = Grid Scale Factor  = Geodetic latitude Ellipsoid Topographic Surface (Ground) Projection Grid

SF=1 SF>1 E3 E3 < S3 < G3 S3 G3

Equatorial Plane

Project Datum Factor

• A Project Datum Factor (PDF) converts grid distances (state plane coordinates) to ground/surface distances.

• If you were to use a total station to measure distance between two points on the ground and then used GPS to measure the location of the same two points and calculate the distance between those two points on the state plane grid, the two distances would be close but not exactly the same. This is due to the curvature of the earth combined with the elevation above sea level of the project location. The grid (state plane projection) is trying to represent the elevated, curved surface of the earth on a flat plane at sea level.

• The PDF was more prevalent before GPS became popular because total stations were the primary tools used for surveying.

– Projects were designed using the PDF. This allowed surveyors in the field to measure directly from the designed plans, without having to apply the PDF on the fly in the field.

Project Datum Factor Example

Project Datum Factor Example

0’s Inverse of PDF 1/1.000059148

Project Datum Factor Example

Conversion Among Coordinate Systems

• Exact or approximate mathematical formulas have been developed to convert to and from geographic coordinates (lat and long) to all commonly used coordinate projections • Care must be taken when converting among projections that use different datums – A datum transformation must be used to convert from one geographic coordinate system to another

Conversion Among Coordinate Systems

Inverse of PDF Inverse of PDF

Vertical Datums

• Many Vertical Datums • GPS provides Elipsoid height

Summary

• Geographic Coordinate Systems • Ellipsoids • Geoid • Horizontal Datums • Projected Coordinate Systems • Project Datum Factors

References

• • http://kartoweb.itc.nl/geometrics/index.html

http://resources.arcgis.com/en/help/main/10.1/index.html#//003r0000000 1000000 • http://training.esri.com/gateway/index.cfm?fa=catalog.webCourseDetail& courseid=24 • http://transportation.ky.gov/Highway-Design/Pages/Survey Coordination.aspx

• http://www.agc.army.mil/Missions/Corpscon.aspx

• • Basic GIS Coordinates, Second Edition: Jan Van Sickle http://www.esri.com/news/arcuser/0703/geoid1of3.html

Questions?

Hopefully after this talk your project won’t look like this.