Geodesy and Map Projections

Geodesy and Map Projections

Geodesy, Map Projections and Coordinate Systems Geodesy - the shape of the earth and definition of earth datums Map Projection - the transformation of a curved earth to a flat map Coordinate systems - (x,y,z) coordinate systems for map data Readings: Introduction Revolution in Earth Measurement

Some images and slides from Michael Dennis, National Geodetic Survey and Lewis Lapine, South Carolina Geodetic Survey Traditional Surveying uses benchmarks as reference points Global Positioning uses fixed GPS receivers as reference points (Continuously Operating Reference System, CORS)

Differential GPS Differential GPS uses the time sequence of observed errors at fixed locations to adjust simultaneous measurements at mobile receivers A location measurement accurate to 1 cm horizontally and 2cm vertically is now possible in 3 minutes with a mobile receiver More accurate measurements if the instrument is left in place

longer This has to take Tectonic Motions into account Tectonic Motions From Sella et al., Types of Coordinate Systems (1) Global Cartesian coordinates (x,y,z) for the whole earth (2) Geographic coordinates (f, l, ll, z) (3) Projected coordinates (x, y, z) on a local

area of the earths surface The z-coordinate in (1) and (3) is defined geometrically; in (2) the z-coordinate is defined gravitationally Geographic Coordinates (f, l, ll, z) Latitude (f, l) and Longitude (l) defined using an ellipsoid, an ellipse rotated about an axis Elevation (z) defined using geoid, a surface of constant gravitational potential Earth datums define standard values of the ellipsoid and geoid

Shape of the Earth We think of the earth as a sphere It is actually a spheroid, slightly larger in radius at the equator than at the poles Ellipse An ellipse is defined by: Focal length = Distance (F1, P, F2) is

constant for all points on ellipse When = 0, ellipse = circle For the earth: Major axis, a = 6378 km Minor axis, b = 6357 km Flattening ratio, f = (a-b)/a ~ 1/300 Z b

O F1 P a X F2

Ellipsoid or Spheroid Rotate an ellipse around an axis Z b a O a X Rotational axis Y Standard Ellipsoids Ellipsoid

Major Minor Flattening axis, a (m) axis, b (m) ratio, f Clarke (1866) 6,378,206 6,356,584 1/294.98 GRS80 6,378,137 6,356,752 1/298.57

Ref: Snyder, Map Projections, A working manual, USGS Professional Paper 1395, p.12 Geodetic Datums World Geodetic System (WGS) is a global system for defining latitude and longitude on earth independently of tectonic movement (military) North American Datum (NAD) is a system defined for locating fixed objects on the earths surface and includes tectonic movement (civilian)

Horizontal Earth Datums An earth datum is defined by an ellipse and an axis of rotation NAD27 (North American Datum of 1927) uses the Clarke (1866) ellipsoid on a non geocentric axis of rotation NAD83 (NAD,1983) uses the GRS80 ellipsoid on a geocentric axis of rotation WGS84 (World Geodetic System of 1984) uses GRS80, almost the same as NAD83 Adjustments of the NAD83 Datum

Slightly different (f, l, l) for benchmarks Continuously Operating Reference System Canadian Spatial Reference System National Spatial Reference System High Accuracy Reference Network Representations of the Earth Mean Sea Level is a surface of constant gravitational potential called the Geoid Sea surface

Ellipsoid Earth surface Geoid THE GEOID AND TWO ELLIPSOIDS l CLARKE 1866 (NAD27)

GRS80-WGS84 (NAD83) Earth Mass Center Approximately 236 meters GEOID WGS 84 and NAD 83 North American

Datum of 1983 (NAD 83) (Civilian Datum of US) l International Terrestrial Reference Frame (ITRF) includes updates to WGS84 (~ 2 cm) World Geodetic System of 1984 (WGS 84) is

reference frame for Global Positioning Systems Earth Mass Center 2.2 m (3-D) dX,dY,dZ GEOID Definition of Latitude, f, l

m O q f, l S p n r (1) Take a point S on the surface of the ellipsoid and define

there the tangent plane, mn (2) Define the line pq through S and normal to the tangent plane (3) Angle pqr which this line makes with the equatorial plane is the latitude f, l, of point S Cutting Plane of a Meridian P Prime Meridian Equator

Meridian p la n e Definition of Longitude, l l = the angle between a cutting plane on the prime meridian and the cutting plane on the meridian through the point, P -150 180E, W 150

-120 120 90W (-90 ) 90E (+90 ) P l -60

-30 -60 30 0E, W Three systems for measuring elevation Orthometric heights (land surveys, geoid)

Ellipsoidal heights (lidar, GPS) Tidal heights (Sea water level) Conversion among these height systems has some uncertainty Trends in Tide Levels (coastal flood risk is changing) Charleston, SC + 1.08 ft/century

1900 2000 Galveston, TX + 2.13 ft/century - 4.16 ft/century 1900 Juneau, AK 2000 1900

2000 Geoid and Ellipsoid Earth surface Ellipsoid Ocean Geoid Gravity Anomaly

Gravity anomaly is the elevation difference between a standard shape of the earth (ellipsoid) and a surface of constant gravitational potential (geoid) Definition of Elevation Elevation Z P z = zp z = 0 Land Surface

Mean Sea level = Geoid Elevation is measured from the Geoid Gravity Recovery and Climate Experiment (GRACE)

NASA Mission launched in 2002 Designed to measure gravity anomaly of the earth Two satellites, 220 km apart, one leading, one trailing Distance between them measured by microwave to 2m High gravity force pulls satellites together Lower gravity force, lets them fly apart more Gravity anomaly = difference from average Gravity Recovery and Climate Experiment (GRACE) Force of gravity responds to changes in water volume Water is really heavy! Gravity is varying in time and space. Gravity Anomaly of Texas, 2002 2012

Normal In 2011, we lost 100 Km3 of water or 3 Lake Meads GRACE and Texas Reservoir Water Storage Surface water reservoir storage is closely correlated with the GRACE data Grace Satellites Normal In 2011 we lost 100 Km3 of water overall Surface Water Reservoirs Normal

In 2011 we lost 9 Km3 of water from reservoirs Vertical Earth Datums A vertical datum defines elevation, z NGVD29 (National Geodetic Vertical Datum of 1929) NAVD88 (North American Vertical Datum of 1988) takes into account a map of gravity anomalies between the ellipsoid and the geoid Converting Vertical Datums

Corps program Corpscon (not in ArcInfo) Point file attributed with the elevation difference between NGVD 29 and NAVD 88 NGVD 29 terrain + adjustment = NAVD 88 terrain elevation Importance of geodetic datums NAVD88 NGVD29 (cm) NGVD29 higher

in East More than 1 meter difference NAVD88 higher in West Orthometric datum height shifts are significant relative to BFE accuracy, so standardization on NAVD88 is justified Geodesy and Map Projections Geodesy - the shape of the earth and definition of earth datums Map Projection - the transformation of a

curved earth to a flat map Coordinate systems - (x,y) coordinate systems for map data Earth to Globe to Map Map Scale: Map Projection: Scale Factor Representative Fraction = Globe distance

Earth distance (e.g. 1:24,000) = Map distance Globe distance (e.g. 0.9996) Types of Projections Conic (Albers Equal Area, Lambert Conformal Conic) - good for East-West land areas

Cylindrical (Transverse Mercator) - good for North-South land areas Azimuthal (Lambert Azimuthal Equal Area) - good for global views Conic Projections (Albers, Lambert) Cylindrical Projections (Mercator) Transverse

Oblique Azimuthal (Lambert) Web Mercator Projection (used for ESRI Basemaps) Web Mercator is one of the most popular coordinate systems used in web applications because it fits the entire globe into a square area that can be covered by 256 by

256 pixel tiles. The spatial reference for the ArcGIS Online / Google Maps / Bing Maps tiling scheme is WGS 1984 Web Mercator (Auxiliary Sphere). Coordinate Systems Universal Transverse Mercator (UTM) - a global system developed by the US Military Services State Plane Coordinate System - civilian

system for defining legal boundaries Texas Centric Mapping System - a statewide coordinate system for Texas Coordinate System A planar coordinate system is defined by a pair of orthogonal (x,y) axes drawn through an origin Y X Origin (xo,yo)

(f, lo,lo) Universal Transverse Mercator Uses the Transverse Mercator projection Each zone has a Central Meridian (lo), zones are 6 wide, and go from pole to pole 60 zones cover the earth from East to West Reference Latitude (f, lo), is the equator (Xshift, Yshift) = (xo,yo) = (500000, 0) in the Northern Hemisphere, units are meters UTM Zone 14

-99 -102 -96 6 Origin -120 -90 Equator -60

State Plane Coordinate System Defined for each State in the United States East-West States (e.g. Texas) use Lambert Conformal Conic, North-South States (e.g. California) use Transverse Mercator Texas has five zones (North, North Central, Central, South Central, South) to give accurate representation Greatest accuracy for local measurements ArcGIS Spatial Reference Frames Defined for a feature dataset in ArcCatalog

XY Coordinate System Projected Geographic Z Coordinate system Domain, resolution and tolerance

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