SPATIAL REFERENCE SYSTEMS


 Duane Warren
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1 SPATIAL REFERENCE SYSTEMS We will begin today with the first of two classes on aspects of cartography. Cartography is both an art and a science, but we will focus on the scientific aspects. Geographical Information Systems use spatially referenced data. Features located on the Earth's surface, which is curved and 3dimensional, are generally depicted on a flat dimensional plane (i.e. computer monitor screen, or printed page). This is achieved by projecting a model of the earth onto either a flat surface or a surface which can be flattened out, but this by necessity results in a distortion of some of the properties of the data. Using different map projections, the distortion with regard to some properties can be reduced, but only at the expense of increased distortion with regard to other properties. It is therefore important, when using a GIS, to be aware of the strengths and weaknesses of different projections, and if necessary to transform the data to a more suitable projection. We will look at map projections in more detail next day. But, before we do that, we need to consider the principal methods used to reference spatial location. First, however, it is useful to consider what is meant by map scale. MAP SCALE To show a portion of the earth's surface on a map, it is necessary to reduce the size of the original. The extent of the reduction is known as the scale. Scale can be expressed in different ways. For example, the following all refer to the same scale: inch = mile inch = 63,360 inches :63,630 The last notation, referred to as the representative fraction (RF), is preferable because it avoids the use of specific units. Older Irish Ordnance Survey maps were drawn at scales of /4 inch to the mile, / inch to the mile, 6 inches to the mile, etc. but today they are more likely to be :50,000, :5,000 or :0,000 (which are their near equivalents). There is also a popular :50,000 series. Only five :50,000 sheets are required to cover the whole of Ireland. The :0,000 sheets are more likely to be used by people who require a lot more degree of detail about smaller areas. Scales with a large RF (e.g. :0,000) are referred to as large scale, whereas those with a small representative fraction (e.g. :50,000) are small scale. Large scale maps have a high resolution (i.e. show more detail), but cover a smaller extent than small scale maps of a similar size. (N.B. Many people, including geographers who should know better, erroneously refer to small scale maps as 'large scale' because they cover a large area. However, the correct terminology becomes evident if you think in terms of fractions: e.g. /0,000 is a bigger fraction than /50,000). One might assume that scale is irrelevant in GIS because the scale changes each time one zooms in or out. However, scale in traditional paper maps is closely related to accuracy  large scale maps depict more detail than small scale maps, which tend to be much more generalised. Although the scale in a GIS may get larger as you zoom in, the accuracy of the features depicted does not change. Thus, although digital data can easily be displayed at a variety of scales, its accuracy is fixed and is the equivalent of a paper map drawn at a specific scale. A lot of digital data were originally digitised from paper maps in such instances it is important to know the scale of the map from which the data were digitised. This information should be included as part of the metadata. THE SHAPE OF THE EARTH When attempting to specify a location on the Earth s surface, it is necessary to identify its location relative to a specific model of the Earth. Different models can be used, depending upon the degree of accuracy required and also (in some cases) the region in which the place of interest is located. The Earth can be assumed to be a sphere at small scales (i.e. < :5,000,000), but at larger scales (> :,000,000) it should be treated as a spheroid (or ellipsoid). When treated as a sphere, the radius can be taken as 6,37 km (or 6,370,997 metres).  
2 At larger scales the Earth is modelled as an oblate spheroid  i.e. its semiminor axis b, around which it rotates, is shorter than its semimajor axis a. The degree of flattening (or ellipticity) f of an ellipsoid is defined as: f = (ab)/a. A value of 0.0 would indicate a sphere, whereas.0 would indicate a flat disc. The ellipticity of the Earth is estimated in the World Geodetic System 984 (WGS84) as /98.57 or indicating that the ellipsoid is very similar to a sphere (hence the term spheroid). However, at large scales the differences between modelling the Earth as a spheroid or a sphere are large enough to be significant. The Earth is not a perfect spheroid, so the best estimates of its parameters (i.e. semimajor and semiminor axes) vary depending upon the local data used to estimate them. A number of different estimates, defining different spheroids, are widely used. Details of the spheroid used form part of the datum (see below) which must be provided for the full specification of a reference system. The Geoid Locational data calculated relative to a spheroid are generally sufficiently accurate for most GIS purposes, even at large scale. (Indeed measurements based upon a sphere may be sufficient at scales less than :50,000, and are almost certainly sufficient below :,000,000). However, for very detailed measurements, especially if you need to measure elevation, even a spheroid does not provide the accuracy required. Under such circumstances it is necessary to make measurements relative to the geoid. The geoid takes account of local variations in the shape of the Earth (i.e. the local departures from the shape defined by a spheroid, which may be up to 00 metres in extent). However, this is not the same thing as the surface formed by the local terrain (i.e. land surface). Mountains and oceans can be thought of as localised departures from the geoid. The geoid is defined as 'the equipotential surface that most closely corresponds to mean sea level'. All points on an equipotential surface experience the same degree of gravitational pull. However, equipotential surfaces are irregular, reflecting gravity anomalies caused by variations in the density of the Earth's crust and mantle. The gravitational pull decreases as one moves out from the centre of the earth, so different equipotential surfaces occur at different heights. The geoid is the equipotential surface which most closely corresponds to mean sea level. However, mean sea level also varies from place to place  it is a couple of metres higher, for example, at one end of the Panama Canal than it is at the other end. Mapping agencies in different countries have consequently adopted slightly different estimates of mean sea level and therefore define the geoid slightly differently.  
3 As illustrated in the diagram, the height of a peak will vary depending on whether it is measured from the geoid (orthometric height H), an ellipsoid (ellipsoidal height h), or a sphere (spheroidal height h). COORDINATE SYSTEMS Spherical Coordinates Treating the Earth as a sphere, location on the Earth's surface can be expressed in terms of angles measured from the centre of the Earth. Lines of latitude (or parallels) run from 'east to west' and measure the angle (  phi) north or south of the Equator. Lines of longitude (or meridians) run from pole to pole and define the angle (  lambda) east or west of a line running from pole to pole through Greenwich in London (the prime meridian). Treating the earth as a sphere, one degree of longitude at the Equator equals. km whereas at the poles it vanishes into nothing. One degree of latitude, however, measures approximately the same distance everywhere. Angles are expressed in degrees, minutes and seconds (DMS). There are 60 minutes in a degree, and 60 seconds in a minute. There are 360 degrees in a circle. Latitude is usually expressed as a positive value if it is in the northern hemisphere, and as a negative value if it is in the southern hemisphere. Longitude is likewise expressed as a positive value if it is east of Greenwich and negative if it is west of Greenwich. Computers are not really comfortable handling minutes and seconds, so decimal degrees (DD) are more frequently used in GIS. These express minutes and seconds as portions of whole degrees (i.e. minute = degrees, second = degrees). Spheroidal Coordinates A somewhat similar system of latitude and longitude can be used to specify location relative to a spheroid. One notable difference is that latitude is measured as the angle between the equatorial plane and a line perpendicular to the surface, known as the spheroidal normal, rather than the centre of the Earth. Coordinates defined in this way are known as geodetic coordinates, whereas those measured from the centre of a sphere are known as geographic coordinates. However, the difference in practice is extremely small and many texts use the term 'geographic coordinates' as a generic term for both systems. The term Global Reference System is also sometimes used as a generic term for latitude and longitude. The height h of the Earth s surface at a point (i.e. altitude) is measured relative to the spheroid (reference ellipsoid). One degree of latitude does not exactly correspond to the same distance at all places in reality because the Earth is a distorted spheroid rather than a sphere. The south pole, for example, is slightly closer to the Equator than the north pole, so each degree in the southern hemisphere is slightly smaller that a degree in the northern hemisphere
4 It should be noted that the geodetic coordinates of any given place will vary depending upon the parameters of the spheroid used to model the Earth. As noted previously, a full specification of the coordinate system therefore requires details of the spheroid to be included in the datum. Cartesian Coordinates The formulae used for calculations based upon geodetic coordinates are complex, and are not very suitable for observations made from satellites. A Cartesian system is therefore sometimes used to express the location of points on the surface of the spheroid in 3dimensional space as measured along three axes with an origin located at the centre of the spheroid. The Z axis is aligned with the minor (i.e. polar) axis of the spheroid, the X axis lies in the equatorial plane and is aligned with the prime (i.e. Greenwich) meridian, whilst the Y axis is also in the equatorial plane but at right angles to the X axis. Formulae are available in text books for converting between Cartesian and geodetic coordinate systems. Planar Coordinates Maps are created by projecting features on the Earth's surface onto a plane (i.e. flat surface). We will discuss the properties of different types of map projections next day, but for present it should be noted that large scale maps generally use a different coordinate system (e.g. Irish Grid coordinates) to any of those discussed above. Small scale maps usually locate features using coordinates expressed in degrees latitude and longitude. Large scale maps use a dimensional Cartesian system, with X and Y axes. The X axis measures the distance from a defined origin in an eastwest direction, and the Y axis measures the distance in a northsouth direction. The X and Y axes always intersect at right angles, so the term rectangular coordinates is sometimes used. Places to the east of the origin are identified by positive coordinates, and places to the west by negative coordinates. Likewise, places north of the origin are identified by positive values, and those south of the origin by negative values. It would be awkward in practice to use negative coordinates, so the origin of the map is artificially displaced to the south and to the west to create a false origin, so that all points on the map have positive X and Y coordinates. Needless to say, the locations of the origin and the false origin need to be included in the datum
5 Each projection uses a different coordinate system. Converting grid references used by one planar coordinate system to those used by another planar system (sometimes referred to as secondary projection) normally involves converting the coordinates from the first system into geographic coordinates (e.g. decimal degrees) before converting these into the coordinates used by the second system (although this is usually hidden by the software used). Digitiser Coordinates Digitiser tables use yet another Cartesian coordinate system. The origin of this system is typically in the centre of the table, and the location of points to the right or left is measured along the X axis in the preferred units of the digitiser (e.g. centimetres or inches). The vertical location is measured along the Y axis. Positive values are used for points to the right and above the origin; negative values for points to the left and below the origin. Digitiser tables normally provide an option of digitising using either digitiser coordinates or realworld (i.e. projected) coordinate system used by the map which is being digitised. If the map is digitised using digitiser coordinates, it will be necessary to convert the coordinates to either geographic coordinates or a realworld system if the data are to be used with data from other sources because the digitiser coordinates will have an arbitrary origin depending upon where the map happens to be taped onto the table. This conversion will also need to take account of the fact that the grid on the map will probably be at an angle to coordinate system defined by the digitiser. It should be noted that a map cannot be digitised directly into geographic coordinates because the Global Reference System is not a planar coordinate system. MEASURING DISTANCES The distance between two points in a planar coordinate system can be easily calculated using Pythagoras s theorem, i.e. d = ( ) + ( y y ) x x However, the distance calculated between two points on map projected using a planar coordinate system will normally provide an inaccurate estimate of the actual distance between the points on the Earth s surface, due to the fact that all planar maps necessarily distort the curvature of the Earth. The distance between two points can be calculated from their geographic coordinates. Assuming a spherical Earth, the distance between two points along the arc forming part of the great circle through the points is given as: where R is the radius of the Earth. d = Rcos [ sin sin + cos cos cos( )] The calculations are more complicated if the Earth is treated as a spheroid, but similar principles apply using geodetic coordinates. When measuring distances using a GIS, one should establish whether the software can calculate distances from geographic coordinates, or whether it simply applies Pythagoras formula to planar coordinates. Likewise, one should also establish whether the software takes account of the projection used if using a planar coordinate system. Similar considerations apply when drawing circles. A circle having a fixed radius in the real world will appear distorted when projected onto a plane. The converse is also true. The radius of a circle drawn on a map will generally vary by direction if measured on the Earth s surface. Thus when defining a circular buffer in a GIS, what appears to be a circle on the map will not necessarily have a consistent radius in the real world. DATUMS If you wish to convert locational information depicted on a map to a different projection, you will need to know not only what projection was used to draw the map (and its associated parameters), but also information about the  5 
6 reference system used to express these locations as geodetic coordinates. This information is referred to as the datum. The datum must include information about the size and shape of the spheroid, the position of its centre, as well as information about the origin used by the local or national mapping agency. The spheroid information may take the form of the name of its originator and its date of origin (e.g. Clarke 866), or it may specify the values of the semimajor axis a and semiminor axis b (e.g. for Clarke 866, a=6,378,06.4m, b=6,356,538.8m) or it may specify the length of the semimajor axis a and the ellipticity f (from which the length of the semiminor axis can be easily calculated). The origin is a point selected by the mapping agency. This is usually a point where the selected spheroid is assumed to coincide with the geoid. If the spheroid is not assumed to coincide with the geoid at the origin, then the geoidspheroid separation must also be specified. Different mapping agencies using the same spheroid consequently tend to define the centre of the spheroid at slightly different locations in 3dimensional space, reflecting the local perturbations of the geoid at the selected origin. However, provided the datum is specified, it is possible, although mathematically complicated, to convert between reference systems. In the diagram above, a local mapping agency is using the same spheroid (i.e. same dimensions) as the one defined in the World Geodetic System (WGS84), but because the local datum is defined so that the spheroid coincides locally with the geoid, the centre of the spheroid in the local datum is displaced relative to the centre of the spheroid in WGS84. North American maps traditionally used the spheroid estimated by Clarke in 866. This was adopted as part of the North American Datum, 97 (or NAD7). Following the availability of more accurate data from satellites, a new datum was introduced for North America in 983 (NAD83) based on the world Geodetic Reference System (GRS80) spheroid. The US military uses a slightly modified version of this called the World Geodetic System (WGS84). The spheroid specified by WGS84 has the parameters a=6,378,37m, f=/ WGS84 is used by GPS in North America, but a modified version called European Terrestrial Reference Frame (ETRF89), also based on GRS80, was established by the International Agency for Geodesy (IAG) and adopted by European mapping agencies for GPS in 989. The Irish realisation of ETRF89, known as IRENET95, was adopted by the IAG in 996. The Irish Grid uses a datum called Ireland 965, based on a spheroid originally defined by Airy in 830 (but modified in 849). OSGB uses a datum called OSGB36, based on an unmodified Airy spheroid. OSI introduced a new planar coordinate system in 00 which is more compatible with GPS called Irish Transverse Mercator. This is based on the GRS80 spheroid
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