1 Lecture 2 Map Projections and GIS Coordinate Systems Tomislav Sapic GIS Technologist Faculty of Natural Resources Management Lakehead University
2 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). 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.
3 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. x y y x y x
4 The Shape of the Earth and the Model Representing It 6356 km 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 km
5 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).
6 Positions on the ellipsoid are expressed through a coordinate system. Ellipsoidal Coordinate System Source: Ilifee and Lott (2008). Latitude: the angle north or south from the equatorial plane. Longitude: the angle east or west from an identified meridian..
7 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. 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. Geocentric Cartesian Coordinate System Source: Ilifee and Lott (2008).
8 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). GIS/index.php
9 Datums Commonly Used in Resource Management GIS in Canada WGS84 (World Geodetic System of 1984) - geodetic coordinate reference system (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 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.
10 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. 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. Cylindrical Plane Conic Source: Ilifee and Lott (2008).
11 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 shape Preserving areas Source: Ilifee and Lott (2008).
12 Map Projection Categories and Resulting Distortions Map Projection Category Maintained Distorted Equivalent (Equal-area) areal scale angle, shape, distance Equidistant Azimuthal Conformal distance (over some portions) angular relationships from a central point small shapes and because of that angles as well direction, area shape, distance, area large area
13 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.
14 Projected features are placed within a plane, Cartesian coordinate system grid. The graticule represents meridians and parallels. Source: Ilifee and Lott (2008).
15 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.
16 UTM Zone and Its Coordinate System 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 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 along the central meridian. Secant longitude 180 km from the central meridian on the Equator. Secant meridians
17 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 in Canada Canada stretches across 16 zones, 7-22 Ontario stretches across 4 zones, 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.
18 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 ,000 m to GIS datasets. These datasets would also have NAD27 for their datum.
19 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 1 st standard parallel 2 nd 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.
20 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. Google Maps 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)
21 Meridians (Longitudes) and Parallels (Latitudes) Meridians lines of equal longitude, running pole to pole on the surface of the ellipsoid Parallels lines of equal latitude, running parallel to the equator on the surface of the ellipsoid. Source: (ArcGIS 10 Help 2012)
22 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/ (-180) (-90) 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 display or for spatial calculations.
23 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.
24 GIS Datasets and Map Projections in ArcGIS The description of the map projection of a shapefile is stored in its.prj file. GIS datasets can come in several different states with respect to map projections: (a) Dataset s projection is properly defined. (b) Dataset s projection is undefined. (c) Dataset s projection is improperly defined. thunder_bay.shp thunder_bay.shp thunder_bay.shp Y = X = thunder_bay.prj UTM, Zone 16, NAD83 Fix: - Y = X = ? Define the shapefile with the proper map projection. Y = X = thunder_bay.prj UTM, Zone 15, NAD83 Delete in Win. Expl. the.prj and the.xml files and define the shapefile with the proper map projection
25 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 a 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 x and y values, etc. - Try to create a habit of having all datasets with a defined projection.
26 Sources: ArcGIS 10 Help About Map Projections q htm (January 22, 2012) Ilifee J. and R. Lott Datums and Map Projections: For Remote Sensing, GIS and Surveying. Whittles Publishing. Furuti C. A Cartographical Map Projections. (January 20, 2011) Stefanakis, E Web Mercator: the de facto standard, the controversy, and the opportunity. Gogeomatics Magazine. mercator-the-de-factostandard-the-controversy-and-the-opportunity.htm#.
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