# Models of the Earth sphere, oblate ellipsoid geoid

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Models of the Earth sphere, oblate ellipsoid geoid
The earth can be modeled as a sphere, oblate ellipsoid geoid

Earth Shape: Sphere and Ellipsoid

Definitions: Ellipsoid
Also referred to as Spheroid, although Earth is not a sphere but is bulging at the equator and flattened at the poles Flattening is about 21.5 km difference between polar radius and equatorial radius Ellipsoid model necessary for accurate range and bearing calculation over long distances  GPS navigation Best models represent shape of the earth over a smoothed surface to within 100 meters

The Spheroid and Ellipsoid
The sphere is about 40 million meters in circumference. An ellipsoid is an ellipse rotated in three dimensions about its shorter axis. The earth's ellipsoid is only 1/297 off from a sphere. Many ellipsoids have been measured, and maps based on each. Examples are WGS84 and GRS80.

Earth as Ellipsoid

WGS 84 Geoid defines geoid heights for the entire earth
Geoid: the true 3-D shape of the earth considered as a mean sea level extended continuously through the continents Approximates mean sea level WGS 84 Geoid defines geoid heights for the entire earth

Earth Models and Datums

Definition: Datum A mathematical model that describes the shape of the ellipsoid Can be described as a reference mapping surface Defines the size and shape of the earth and the origin and orientation of the coordinate system used. There are datums for different parts of the earth based on different measurements Datums are the basis for coordinate systems Large diversity of datums due to high precision of GPS Assigning the wrong datum to a coordinate system may result in errors of hundreds of meters

Commonly used datums Datum Spheroid Region of use NAD 27 Clark 1866
Canada, US, Atlantic/Pacific Islands, Central America NAD 83 GRS 1980 Canada, US, Central America WGS 84 Worldwide

The Datum An ellipsoid gives the base elevation for mapping, called a datum. Examples are NAD27 and NAD83. The geoid is a figure that adjusts the best ellipsoid and the variation of gravity locally. It is the most accurate, and is used more in geodesy than GIS and cartography.

Geoid

Map Scale The ratio of the distance between two points on the map and the real world (earth) distance between the same two points Map measurement / Real world measurement

Map Scale Map scale is based on the representative fraction, the ratio of a distance on the map to the same distance on the ground. Most maps in GIS fall between 1:1 million and 1:1000. A GIS is scaleless because maps can be enlarged and reduced and plotted at many scales other than that of the original data. To compare or edge-match maps in a GIS, both maps MUST be at the same scale and have the same extent. The metric system is far easier to use for GIS work.

Types of Scale Graphical Verbal Representative Fraction

Graphical Scale 10 5 10 miles

Verbal Scale One inch equals one mile, or; One inch to one mile

Representative Fraction
Ratio between map distance and ground distance for equivalent point Unit free. Ratio is true regardless of units

Representative Fraction
Expressed as fraction 1/100,000 Or Ratio 1:100,000

Large Scale vs. Small Scale
Small scale = Large area Small scale = Large Denominator 1:500,000 Large scale = Small area Large scale = Small denominator 1:50,000

Location Reference Systems
Relative Absolute

Relative Location Systems
Real world descriptions Corner of 34th and Fifth

Absolute Location Systems
Absolute description Uses mathematical coordinates to define the position of grid intersections with respect to a defined (accepted) origin

Absolute Location Systems
Common Absolute Location Systems Global Coordinate System Cartesian Coordinate Systems Universal Transverse Mercator (UTM) Coordinate System State Plate Coordinate System

Geographic Coordinates
Geographic coordinates are the earth's latitude and longitude system, ranging from 90 degrees south to 90 degrees north in latitude and 180 degrees west to 180 degrees east in longitude. A line with a constant latitude running east to west is called a parallel. A line with constant longitude running from the north pole to the south pole is called a meridian. The zero-longitude meridian is called the prime meridian and passes through Greenwich, England. A grid of parallels and meridians shown as lines on a map is called a graticule.

The Global Coordinate System
Spherical coordinate system Unprojected Expressed in terms of two angles latitude longitude Latitude and longitude are traditionally measured in degrees, minutes, and seconds (DMS).

Origin for the Global Coordinate System
Latitude positive in northern hemisphere negative in southern hemisphere Longitude positive east of Prime Meridian negative west of Prime Meridian

Geographic Coordinates as Data

The Global Coordinate System
Longitude Angle formed by a line going from the intersection of the prime meridian and the equator to the center of the earth, and a second line from the center of the earth to the point in question

The Global Coordinate System
Latitude Angle formed by a line from the equator toward the center of the earth, and a second line perpendicular to the reference ellipsoid at the point in question

Cartesian Coordinate Systems
Computationally, it is much simpler to work with Cartesian coordinates than with spherical coordinates x,y coordinates referred to as “eastings” & “northings” defined units, e.g. meters, feet

Map Projections A transformation of the spherical or ellipsoidal earth onto a flat map is called a map projection. The map projection can be onto a flat surface or a surface that can be made flat by cutting, such as a cylinder or a cone. If the globe, after scaling, cuts the surface, the projection is called secant. Lines where the cuts take place or where the surface touches the globe have no projection distortion.

Map Projections (cont.)
Projections can be based on axes parallel to the earth's rotation axis (equatorial), at 90 degrees to it (transverse), or at any other angle (oblique). A projection that preserves the shape of features across the map is called conformal. A projection that preserves the area of a feature across the map is called equal area or equivalent. No flat map can be both equivalent and conformal. Most fall between the two as compromises. To compare or edge-match maps in a GIS, both maps MUST be in the same projection.

Projection Method of representing data located on a curved surface onto a flat plane All projections involve some degree of distortion of: Distance Direction Scale Area Shape Determine which parameter is important Projections can be used with different datums

Projections The earth is “projected” from an imaginary light source in its center onto a surface, typically a plate, cone, or cylinder. Planar or azimuthal Conic Cylindrical

Cylindrical Projections
Secancy: projection surface cuts through globe, this reduces distortion of larger land areas Tangency: only one point touches surface Used for entire world Parallels and meridians form straight lines

Example Cylindrical Projections
Shapes and angles within small areas are true (7.5’ Quad) Distances only true along equator

Conic Projections Can only represent one hemisphere
Often used to represent areas with east-west extent (US)

Lambert is used in State Plane Coordinate System
Albers is used by USGS for state maps and all US maps of 1:2,500,000 or smaller Lambert is used in State Plane Coordinate System Secant at 2 standard parallels Distorts scale and distance, except along standard parallels Areas are proportional Directions are true in limited areas

Azimuthal Projections
Often used to show air route distances Distances measured from center are true Distortion of other properties increases away from the center point

Orthographic: Used for perspective views of hemispheres Area and shape are distorted Distances true along equator and parallels Lambert: Specific purpose of maintaining equal area Useful for areas extending equally in all directions from center (Asia, Atlantic Ocean) Areas are in true proportion Direction true only from center point Scale decreases from center point

Other Projections Pseudocylindrical
Unprojected or Geographic projection: Latitude/Longitude There are over 250 different projections!

Pseudocylindrical: Used for world maps Straight and parallel latitude lines, equally spaced meridians Other meridians are curves Scale only true along standard parallel of 40:44 N and 40:44 S Robinson is compromise between conformality, equivalence and equidistance

Mathematical Relationships
Conformality Scale is the same in every direction Parallels and meridians intersect at right angles Shapes and angles are preserved Useful for large scale mapping Examples: Mercator, Lambert Conformal Conic Equivalence Map area proportional to area on the earth Shapes are distorted Ideal for showing regional distribution of geographic phenomena (population density, per capita income) Examples: Albers Conic Equal Area, Lambert Azimuthal Equal Area, Peters, Mollweide

Mathematical Relationships
Equidistance Scale is preserved Parallels are equidistantly placed Used for measuring bearings and distances and for representing small areas without scale distortion Little angular distortion Good compromise between conformality and equivalence Used in atlases as base for reference maps of countries Examples: Equidistant Conic, Azimuthal Equidistant Compromise Compromise between conformality, equivalence and equidistance Example: Robinson

Local Coordinate Systems
A coordinate system is a standardized method for assigning codes to locations so that locations can be found using the codes alone. Standardized coordinate systems use absolute locations. A map captured in the units of the paper sheet on which it is printed is based on relative locations or map millimeters. In a coordinate system, the x-direction value is the easting and the y-direction value is the northing. Most systems make both values positive.

Coordinate Systems for the US
Some standard coordinate systems used in the United States are geographic coordinates universal transverse Mercator system military grid state plane To compare or edge-match maps in a GIS, both maps MUST be in the same coordinate system.

USA In The UTM Zones

The Universal Transverse Mercator Coordinate System
60 zones, each 6° longitude wide Zones run from 80° S to 84° N Poles covered by Universal Polar System (UPS)

UTM Zone Projection Transverse Mercator Projection
applied to each 6o zone to minimize distortion

UTM Coordinate Parameters
Unit meters Zones: 6o longititue N and S zones separate coord X-origin 500,000 m east of central meridian Y-origin equator

Zone central meridian Eastings = 500,000 meters North pole Northings = 10,000,000 meters allows overlap between zones form mapping purposes give all eastings positive numbers tell if we are to the east or west of the central meridian provide relationship between true north and grid north

State Plane Coordinate System
Each state has one or more zones Zones are either N-S or E-W oriented (except Alaska) Each zone has separate coordinate system and appropriate projection Unit: feet No negative numbers

Map Projections for State Plane Coordinate System
E-W zones: Lambert conformal conic projection N-S zones: Transverse Mercator Projection

Pros and Cons of SPCS Advantages:
The system is used primarily for engineering applications e.g. utility companies, local governments to do accurate surveying of facilities network (sewers, power lines) More accurate than UTM. feet vs. meters SPCS deals with smaller area Disadvantages: Lack of universality cause problems for mapping over large areas such as across zones and states

Projections and Datums
Projections and datums are linked The datum forms the reference for the projection, so... Maps in the same projection but different datums will not overlay correctly Tens to hundreds of meters Maps in the same datum but different projections will not overlay correctly Hundreds to thousands of meters.

Determining datum or projection for existing data
Metadata Data about data May be missing Software Some allow it, some don’t Comparison Overlay may show discrepancies If locations are approx. 200 m apart N-S and slightly E-W, southern data is in NAD27 and northern in NAD83

Selecting Datums and Projections
Consider the following: Extent: world, continent, region Location: polar, equatorial Axis: N-S, E-W Select Lambert Conformal Conic for conformal accuracy and Albers Equal Area for areal accuracy for E-W axis in temperate zones Select UTM for conformal accuracy for N-S axis Select Lambert Azimuthal for areal accuracy for areas with equal extent in all directions Often the base layer determines your projections

GIS Capability A GIS package should be able to move between
map projections, coordinate systems, datums, and ellipsoids.

Summary There are very significant differences between datums, coordinate systems and projections, The correct datum, coordinate system and projection is especially crucial when matching one spatial dataset with another spatial dataset.