introduction to mapping science: lecture #3 (modeling the earth in gis) models of the earth the...
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Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Models of the Earth
The earth can be modeled as a
– sphere,
– oblate ellipsoid
– geoid
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Earth Shape: Sphere and Ellipsoid
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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.
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Earth as Ellipsoid
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Earth Models and Datums
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Definition: Datum
A mathematical model that describes the shape of the ellipsoidCan be described as a reference mapping surfaceDefines 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 measurementsDatums are the basis for coordinate systemsLarge diversity of datums due to high precision of GPSAssigning the wrong datum to a coordinate system may result in errors of hundreds of meters
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Commonly used datums
Datum Spheroid Region of use
NAD 27 Clark 1866Canada, US,
Atlantic/Pacific Islands, Central America
NAD 83 GRS 1980Canada, US, Central
America
WGS 84 WGS 84 Worldwide
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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.
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Types of Scale
Graphical
Verbal
Representative Fraction
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Graphical Scale
10 0 5 10 miles
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Verbal Scale
One inch equals one mile, or;
One inch to one mile
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Representative Fraction
Ratio between map distance and ground distance for equivalent point
Unit free. Ratio is true regardless of units
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Representative Fraction
Expressed as fraction
1/100,000
Or Ratio
1:100,000
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Location Reference Systems
Relative
Absolute
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Relative Location Systems
Real world descriptions
Corner of 34th and Fifth
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Absolute Location Systems
Absolute description
Uses mathematical coordinates to define the position of grid intersections with respect to a defined (accepted) origin
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Absolute Location Systems
Common Absolute Location Systems
Global Coordinate System
Cartesian Coordinate Systems
Universal Transverse Mercator (UTM) Coordinate System
State Plate Coordinate System
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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.
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Spherical coordinate system
Unprojected
Expressed in terms of two angles
latitude
longitude
Latitude and longitude are traditionally measured in degrees, minutes, and seconds (DMS).
The Global Coordinate System
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Latitudepositive in northern hemisphere
negative in southern hemisphere
Longitudepositive east of Prime Meridian
negative west of Prime Meridian
Origin for the Global Coordinate System
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Geographic Coordinates as Data
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
The Global Coordinate System
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Computationally, it is much simpler to work with Cartesian coordinates than with spherical coordinates
x,y coordinatesreferred to as “eastings” & “northings”defined units, e.g. meters, feet
Cartesian Coordinate Systems
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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.
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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.
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Projection
Method of representing data located on a curved surface onto a flat planeAll projections involve some degree of distortion of:
DistanceDirectionScaleAreaShape
Determine which parameter is importantProjections can be used with different datums
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Cylindrical Projections
Used for entire world
Parallels and meridians form straight lines
Tangency: only one point touches surface
Secancy: projection surface cuts through globe, this reduces distortion of larger land areas
Example Cylindrical Projections
Shapes and angles within small areas are true (7.5’ Quad)
Distances only true along equator
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Conic Projections
Can only represent one hemisphere
Often used to represent areas with east-west extent (US)
Secant at 2 standard parallels
Distorts scale and distance, except along standard parallels
Areas are proportional
Directions are true in limited areas
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
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
Orthographic:
Used for perspective views of hemispheres
Area and shape are distorted
Distances true along equator and parallels
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Mathematical Relationships
ConformalityScale is the same in every directionParallels and meridians intersect at right anglesShapes and angles are preservedUseful for large scale mappingExamples: Mercator, Lambert Conformal Conic
EquivalenceMap area proportional to area on the earthShapes are distortedIdeal for showing regional distribution of geographic phenomena (population density, per capita income)Examples: Albers Conic Equal Area, Lambert Azimuthal Equal Area, Peters, Mollweide
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Mathematical Relationships
EquidistanceScale is preserved Parallels are equidistantly placedUsed for measuring bearings and distances and for representing small areas without scale distortionLittle angular distortionGood compromise between conformality and equivalenceUsed in atlases as base for reference maps of countriesExamples: Equidistant Conic, Azimuthal Equidistant
CompromiseCompromise between conformality, equivalence and equidistanceExample: Robinson
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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.
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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.
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
USA In The UTM Zones
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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)
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Transverse Mercator Projection applied to each 6o zone to minimize distortion
UTM Zone Projection
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Advantage of UTM Settings
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Map Projections for State Plane Coordinate System
N-S zones: Transverse Mercator Projection
E-W zones: Lambert conformal conic projection
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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.
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Determining datum or projection for existing data
MetadataData about data
May be missing
SoftwareSome allow it, some don’t
ComparisonOverlay 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
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
Selecting Datums and Projections
Consider the following:Extent: world, continent, regionLocation: polar, equatorialAxis: N-S, E-W
Select Lambert Conformal Conic for conformal accuracy and Albers Equal Area for areal accuracy for E-W axis in temperate zonesSelect UTM for conformal accuracy for N-S axisSelect Lambert Azimuthal for areal accuracy for areas with equal extent in all directions Often the base layer determines your projections
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
GIS Capability
A GIS package should be able to move between – map projections, – coordinate systems, – datums, and – ellipsoids.
Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS)
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.