lecture 2 map projections and gis coordinate...
TRANSCRIPT
Lecture 2
Map Projections and GIS
Coordinate Systems
Tomislav Sapic
GIS Technologist
Faculty of Natural Resources Management
Lakehead University
• The two main reasons why map projections are needed:
Maps and GIS data are most often displayed on flat surfaces, be it a paper sheet
or a computer screen.
Spatial calculations, such as distance, can be much easier performed within a
two-dimensional coordinate system of a flat surface than a three-dimensional
coordinate system of an oval (ellipsoid) surface.
Map Projections
•Map projections are mathematical formulas used
to transfer shapes from an oval surface (the Earth
surface – more accurately, a model of it) onto a flat
surface.
• (Map) Projection: a systematic presentation of
intersecting coordinate lines on a flat surface upon
which features from the curved surface of the earth
or the celestial sphere may be mapped (Merriam
Webster Dictionary).
y
x
y
x
y
x
• Although GIS features are a model of real-world features existing on an oval surface, their
positional references, i.e., x, y, coordinates, are stored and displayed in a GIS software within a
plane coordinate system.
• The employed plane coordinate system is either projected or geographic.
• Projected coordinate systems are derived from map projections and geographic coordinate
systems from latitude and longitude degrees.
• Coordinate systems are referenced to the physical earth surface through the use of datums.
The Shape of the Earth and the Model Representing It
• The Earth is an irregular body.
• The shape of the Earth is given the form of a surface that is everywhere perpendicular to the
direction of gravity – so called equipotential surface.
• The force and direction of gravity are affected by irregularities in the density of the Earth’s
crust and mantle, therefore the Earth’s form is somewhat irregular.
• The true shape of the Earth is described as a geoid, defined as that equipotential surface that
most closely corresponds to mean sea level.
• The geoid, with its local irregularities, remains a difficult surface for spatial calculations,
which is why a more simple model was chosen to represent the Earth – the ellipsoid.
• The ellipsoid is an ellipse that has been rotated about its shortest (the minor) axis – the reason
why an ellipsoid is used as the model for the Earth is that the Earth is flattened at the poles, not
by much though, 22 km over 6378 km.
6378 km 6356 k
m
Datum
• When creating an ellipsoid model, the goal
is to get a model that best fits the Earth geoid.
• The mechanism through which an ellipsoid
model, i.e., its coordinate system, is related to
the Earth geoid is called geodetic datum, or
just datum.
• A datum is realized by establishing physical
monuments on the ground, which have x,y,z
coordinates assigned to them, forming a
geodetic coordinate reference system.
• Historically, the practice was to develop an
ellipsoid model and a datum that make a good
local fit to the Earth geoid; meaning that this
level of fit would be lost in other areas on the
Earth.
• With the advent of satellites, a global
ellipsoid has been developed (as part of GRS
1980) that has been internationally accepted
as the best fit globally and by the majority of
countries as the model used locally as well.
Local Ellipsoid
Global Ellipsoid
Source: Ilifee and
Lott (2008).
• Positions on the ellipsoid are expressed through a coordinate system.
Ellipsoidal Coordinate
System
Latitude: the angle north or
south from the equatorial plane.
Longitude: the angle east or west
from an identified meridian.
.
Source: Ilifee and
Lott (2008).
Geocentric Cartesian
Coordinate System
• The formulae involved in computations based on ellipsoidal coordinates
are complex and inappropriate when considering observation made to or
from satellites.
• More appropriately, a set of Cartesian axes is defined with its origin
at the centre of the ellipsoid.
Source: Ilifee and
Lott (2008).
• The center of the Geocentric
Cartesian Coordinate System
is at the centre of the model of
the Earth, which may not
coincide with the centre of the
Earth.
http://itrf.ensg.ign.fr/
GIS/index.php
• International Terrestrial Reference System
(http://www.iers.org/IERS/EN/DataProducts/ITRS/itrs.html) is realized through a
geocentric Cartesian coordinate reference set known as the International Terrestrial
Reference Frame.
• It doesn’t require an ellipsoid, the centre of the ITRS is at the mass centre of
gravity of the Earth.
• Because tectonic plates are constantly moving, the physical stations with ITRF
coordinates are constantly moving as well and their coordinates get updated for the
velocity of the plate (of the location on the plate).
• ITRF is used to improve the accuracy of NAD83.
Datums Commonly Used in Resource Management GIS in Canada
WGS84 (World Geodetic System of 1984)
- datum developed and used by GPS.
-WGS84 gets aligned with the ITRF, meaning that at high levels of accuracy it
departs from the datums tied to a particular plate.
- no physical monuments.
NAD83 (North American Datum of 1983)
-developed based on GRS 1980 – the ellipsoid defined through the use of
satellites.
- tied to the North American tectonic plate, meaning that over time it diverges
from WGS84 – currently by ~1.5 - 2 m.
NAD27 (North American Datum of 1927)
- developed based on the Clarke Ellipsoid of 1866.
- discontinued from use but there are still quite a few GIS datasets with it.
- its coordinate difference from NAD83 depends on the location – ~20 m
around Thunder Bay.
To be measured in a different datum, the x,y position on the Earth needs to be
transformed between the two datums. The datum transformation is especially important
between the datums that significantly differ (for particular practical purposes); it can be
very specific and needs to be paid attention to.
Map Projections - Distortions
• “Projection” refers to the notion of shining light through the earth surface and projecting latitudes,
longitudes and geographic features onto a developable surface.
• In the process of projection the earth is represented by a model – an ellipsoid. A datum, then, links the
ellipsoid to the actual surface of the earth and by doing so, links the map projection/coordinate system to the
surface of the earth.
• Because of the transfer from an oval to a flat surface a degree of distortion is inevitable.
• A developable surface is such a surface that can be unravelled without increasing distortion – a cylinder,
cone, plane.
• However, a developable surface is not a necessary intermediary step in designing map projections – some
projections are defined in pure mathematical terms.
Source: Ilifee and Lott
(2008).
Cylindrical
Conic
Plane
• Distortions do not have to happen across all spatial aspects, they can be eliminated or
drastically reduced in one aspect while recognizing their existence in other aspects.
Preserving distances along the meridians
Preserving areas Preserving shape
Source: Ilifee and
Lott (2008).
Map Projection
Category
Maintained Distorted
Equivalent (Equal-area) areal scale angle, shape, distance
Equidistant distance (over some
portions)
direction, area
Azimuthal angular relationships from a
central point
shape, distance, area
Conformal small shapes and because of
that angles as well
large area
Map Projection Categories and Resulting Distortions
• Secant projections, compared to tangent projections, result in increased low and decreased high
distortion.
Conical tangent and
secant projection
Cylindrical tangent and
secant projection
• The points (lines) where the
ellipsoid and the developable
surface are in common
become part of the map
projection’s parameters.
Source: Ilifee and
Lott (2008).
• Projected features are placed within a plane, Cartesian coordinate system grid.
The graticule represents meridians and parallels.
UTM (Universal Transverse Mercator) Projection
• Cylindrical, secant, conformal projection
• UTM projection is derived by positioning the cylinder east-west. “Transverse”
indicates that the cylinder is perpendicular to the one in the standard Mercator
projection, where it is positioned north-south .
• “Universal” points to the fact that the projection is world-wide
• UTM is made of 60 zones, 6 degrees of longitude wide (360/6 = 60), around the
world, positioned north-south.
• Small shapes and
local angles accurate.
• Area minimally
distorted within each
zone.
• 6 degrees of longitude wide ~ 672 km on the Equator, becoming narrower towards north and south.
• x and y coordinates are expressed in meters.
• The central meridian has a false easting of 500000 m to avoid negative values within a zone.
• x coordinates are expressed relative to the central meridian x coordinate; e.g., x = 400,000 in a zone means that the point is 100 km west from the zone’s central meridian, and x = 600,000 means that the point is 100 km east from the zone’s central meridian.
• The equator has a false northing of 10,000,000 m to avoid negative values in the southern hemisphere.
• Y coordinates in the northern hemisphere represent the distance from the Equator; in the southern hemisphere they represent 10,000,000 – distance in meters from the equator.
• Stretches from 84°N to 80°S.
• Scale factor = 1 along secant meridians and 0.99960 along the central meridian.
• Secant longitude 180 km from the central meridian on the Equator.
Secant
meridians
UTM Zone and Its Coordinate System
Canada stretches
across 16 zones, 7
- 22
Ontario stretches
across 4 zones, 15 –
18.
UTM in Canada
Thunder Bay is in UTM zone 16, which has meridian 87° for its central
meridian; the central meridian for UTM zone 15 is at 93° (six degrees
apart), and so on.
The central meridian
for each of the UTM
zones lies on one of
geographic meridians,
six degrees apart from
the central meridians
on each side.
UTM Projection (cont’d)
• UTM is suitable as a map projections for areas whose width is similar to the width of one UTM zone. Because many administrative areas cross UTM zone boundaries but are stored as features in a same dataset, a decision is then made in which UTM zone should the dataset be projected. • Features covering an area that is wider than two UTM zones should be projected in another, more suitable, map projection, such as Canada Lambert Conformal Conic, in Canada.
• MNR used to apply false northing (y shift) of -4 000,000 m to GIS datasets. These
datasets would also have NAD27 for their datum.
Lambert Conformal Conic Projection
• Conformal, often secant projection, with two added parallels.
• Small shapes and local angles are accurate.
• Area minimally distorted near the standard parallels.
• Correct scale along the standard parallels.
Parameters:
• Units
• 1st standard parallel
• 2nd standard parallel
• Central meridian
• Latitude of projection’s
origin
• Suitable for areas or countries extending east-west (Canada, USA), large and medium scales
(widths of two or more UTM zones).
•Extremely widely used – LCC and Transverse Mercator between them account for 90% of base
map projections world wide.
• ESRI’s Canada Lambert Conformal Conic is a customized map projection package that includes
the datum NAD 1983 and is designed with parameters that suit the geographic extent of Canada.
Web Mercator Projection
• Started by Google in 2005 and has become the
standard Web map projection, used in Google Maps,
Bing Maps, OpenStreet Map, Web map services.
• A mathematical formula that is a variant of the
Mercator, cylindrical, projection.
• Ellipsoid coordinates are transferred onto a
developable surface using the formulas of the
spherical Mercator (Stefanakis 2015).
• Advantages: north-up orientation, quicker
computation, easier tiling.
• Disadvantage: large deviations away from the
Equator - ~40 km at 70° latitude (Stefanakis 2015).
• Should be used for visualizations only (correct
spatial calculations on the Web are done on servers
by transferring the coordinates back to the ellipsoid
coordinates).
Mercator projection of the world between 82°S and 82°N
(https://en.wikipedia.org/wiki/Mercator_projection)
Google Maps
Source: (ArcGIS 10 Help 2012)
Meridians (Longitudes) and Parallels (Latitudes)
Parallels – lines of equal
latitude, running parallel to
the equator on the surface
of the ellipsoid.
Meridians – lines of equal
longitude, running pole to
pole on the surface of the
ellipsoid
Geographic Coordinate System
• Not a map projection, it is a Cartesian coordinate system composed of longitudes (meridians) and latitudes
(parallels), and expressed in decimal degrees.
• Longitudes range 180 degrees, start from Greenwich, positive to the east and negative to the west
(International Date Line is on the opposite side of Greenwich).
• Latitudes range 90 degrees, start from the Equator and end on the poles, positive to the north and negative
to the south.
• Decimal degrees (DD) are converted from degrees according to DD = deg + min/60 + sec/3600 .
0 – 180 0 – (-180)
0 –
90
0
– (-9
0)
As a standard point of geographic reference, GCS is often used in GIS datasets stored in broadly
shared databases; however, GCS should never be used for displaying or for spatial calculations.
UTM Zone 16, NAD83 GCS, WGS84
• Contrary to datums, map projection positions (GIS data) can be straightforward
converted (projected) between map projections (and back) without a loss in accuracy.
This assumes that the datum stays the same.
• If the datum stays the same, no specification of transformation is required.
• E.g., a shapefile in UTM, Zone 16, NAD83, can be projected to a new shapefile, in
the Canada Lambert Conformal Conic projection, and back, and features’ x,y
coordinates should stay the same.
• As well, speaking of projecting data, GIS data geographically laying in UTM zone 16,
for example, can be projected to any other UTM zone.
GIS Datasets and Map Projections in ArcGIS
The description of the map projection of a shapefile is stored in its .prj file.
(a) Dataset’s projection
is properly defined.
(b) Dataset’s projection
is undefined.
(c) Dataset’s projection is
improperly defined.
Y = 5363900
X = 333800
thunder_bay.shp
thunder_bay.prj
UTM, Zone 16, NAD83
Y = 5363900
X = 333800
thunder_bay.shp
?
Y = 5363900
X = 333800
thunder_bay.shp
thunder_bay.prj
UTM, Zone 15, NAD83
Fix: - Define the shapefile
with the proper map
projection.
Delete in Win. Expl. the
.prj and the .xml files
and define the shapefile
with the proper map
projection
GIS datasets can come in several different states with respect to map projections:
Important tips in dealing with coordinate systems in GIS:
- A dataset needs to have a defined coordinate system (map projection) in order to be
projected into a different map projection.
- A dataset which does not have a defined map projection (it lacks the projection, .prj,
file) has to have its map projection defined to the map projection in which its features’
coordinates are.
- A very common and grave mistake is to attempt to project a GIS file with an undefined
map projection into a new map projection by assigning through the Define function the
new map projection to the undefined file. A GIS file with an undefined map projection
first needs to be properly defined (with the map projection in which the file`s features
are), and then projected into a new map projection – i.e., into a new file that now has
features in the new map projection.
- If a mistake is made in defining the projection for a shapefile, erase the projection file
(.prj extension) and the metadata file (.xml extension) in the Microsoft file manager.
- X and y values shown in ArcCatalog can be used to get a general idea about the
projection if the definition is missing: y shift issue, UTM versus Lambert, decimal
degrees versus distance units (e.g., metres), etc.
- Try to create a habit of having all datasets with a defined projection.
Sources:
ArcGIS 10 Help. 2012. About Map Projections.
http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#//003r000
0000q000000.htm (January 22, 2012)
Ilifee J. and R. Lott. 2008. Datums and Map Projections: For Remote Sensing,
GIS and Surveying. Whittles Publishing.
Furuti C. A. 2011. Cartographical Map Projections.
http://www.progonos.com/furuti/MapProj/Normal/TOC/cartTOC.html
(January 20, 2011)
Stefanakis, E. 2015. Web Mercator: the de facto standard, the controversy, and
the opportunity. Gogeomatics Magazine.
http://www.gogeomatics.ca/magazine/web- mercator-the-de-facto-
standard-the-controversy-and-the-opportunity.htm# .