Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS) Models of the Earth The...

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


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

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 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


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


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.


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 0 5 10 miles


Verbal Scale

One inch equals one mile, or;

One inch to one mile



Ratio between map distance and ground distance for equivalent point

Unit free. Ratio is true regardless of units



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.


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


Latitudepositive in northern hemisphere

negative in southern hemisphere

Longitudepositive east of Prime Meridian

negative west of Prime Meridian

Origin for the Global Coordinate System


Geographic Coordinates as Data


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



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



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


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 planeAll projections involve some degree of distortion of:

DistanceDirectionScaleAreaShape

Determine which parameter is importantProjections 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

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


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


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


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

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


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


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)


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

UTM Zone Projection


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


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


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

N-S zones: Transverse Mercator Projection

E-W zones: Lambert conformal conic 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

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


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


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.

Introduction to Mapping Science: Lecture #3 (Modeling the Earth in GIS) Models of the Earth The...

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