The South African Coordinate Reference System, Study Guides, Projects, Research of Engineering Science and Technology

Download The South African Coordinate Reference System and more Study Guides, Projects, Research Engineering Science and Technology in PDF only on Docsity! Chief Directorate: National Geo-spatial Information Private Bag X 10, Mowbray, 7705; Tel: 021-6584300; Fax: 021-6891351; Van der Sterr Building, Rhodes Avenue, Mowbray, 7705 The South African Coordinate Reference System 1. INTRODUCTION The Chief Directorate: National Geo-spatial Information (CD:NGI) is mandated, in terms of section 3A(1)(d) of the Land Survey Act (Act 8 of 1997) to “establish and maintain a national control survey network”. All cadastral parcels and surveys, as well as most engineering surveys and Geographic Information System (GIS) based projects are referenced to this national control survey network. Numerous map projections and coordinate systems are used in South Africa, especially for mapping purposes. The official “issue“ coordinates of the national control survey network (and hence most surveys) are reported in the Gauss Conform coordinate system referenced to the Hartebeesthoek94 datum. This coordinate system/geodetic datum combination is known as the South African Coordinate Reference System (SACRS). These two components are inseparable in the definition of SACRS and a different datum, for example, would constitute and different coordinate reference system. There is widely held misconception that the coordinate system has changed in 1999, when in fact the geodetic datum has changed, resulting in a new definition of the SACRS. This is perpetuated by the use of the words “Lo” and “WG” for coordinates referenced to Cape Datum and Hartebeesthoek94 respectively. This paper will define the various elements of the SACRS in detail and, in particular, distinguish between a coordinate system (projected) and a geodetic datum. 2. DEFINITIONS* * All definitions from (ISO 19111:2007(E)), unless otherwise stated, 2.1 coordinate reference system Set of mathematical rules for specifying how coordinates are to be assigned to points that are related to the real world by a datum. 2.2 Datum parameter or set of parameters that define the position of the origin, the scale, and the orientation of a coordinate system 2.3 easting (E) Distance in a coordinate system, eastwards (positive) and westwards (negative) from a north-south reference line. 2.4 ellipsoid Surface formed by the rotation of an ellipse about a main axis. 2.5 geodetic coordinate system Coordinate System in which position is specified by geodetic latitude, geodetic longitude and (in the three-dimensional case) ellipsoidal height. 2.6 geodetic datum Datum describing the relationship of a coordinate system to the Earth. A set of constants specifying the coordinate system used for geodetic control. A complete geodetic datum provides, as a minimum, definition for orientation, scale and dimensions for the reference ellipsoid. The concept is generally expanded to include the published coordinates of control stations within the system. (CGCC 1998) 2.7 map projection Coordinate conversion from a geodetic/ellipsoidal coordinate system to a plane. 2.8 northing (N) distance in a coordinate system, northwards (positive) or southwards (negative) from an east-west reference line 2.9 projected coordinate system Two-dimensional coordinate system resulting from a map projection. 2.10 southing (x) Distance in a coordinate system, southwards (positive) and northwards (negative) from an east-west reference line. 2.11 vertical datum Datum describing the relation of gravity-related heights to the earth. In most cases the vertical datum will be related to a defined mean sea level. Ellipsoidal heights are treated as related to a three-dimensional ellipsoidal coordinate system referenced to a geodetic datum. 2.12 westing (y) Distance in a coordinate system, westwards (positive) and eastwards (negative) from a north-south reference line. 3.5 Hartebeesthoek94 and the WGS84 Reference Frame  The World Geodetic System 1984 (WGS 84) is a Conventional Terrestrial Reference System that includes in its definition a reference frame, a reference ellipsoid, a consistent set of fundamental constants, and an Earth Gravitational Model (EGM) with a related global geoid (Malys et. al, 1997)  The global geocentric reference frame and collection of models known as the World Geodetic System 1984 Reference Frame (WGS84RF) has evolved significantly since its creation in the mid-1980s. The WGS84RF has been redefined periodically.  GPS satellite orbits and control segment positions operate in the WGS84RF.  The WGS84RF should not be confused with the WGS84 ellipsoid.  Since 1997, the WGS84RF has been maintained within 10cm, and more recently within 5cm of the current ITRF. The latest realisation of the WGS84RF is G1150 (Merringan el at, 2002).  Hence, the differences between Hartebeesthoek94 and the WGS84RF would be of the same magnitude as Hartebeesthoek94 and the current ITRF realisation (see 3.4 above). 3.6 Connecting/referencing to Hartebeesthoek94  For a point/data to be referenced to Hartebeesthoek94 datum: Direct connection: the position/s must be determined relative to any point in the national control survey network (horizontal), such as the 29000 trigonometrical beacons and 20000 town survey marks. This would constitute direct connection. Indirect connection: can be achieved by determining a position that has already been directly connected.  Note: When data is collected using autonomous GPS (which operates in the WGS84RF, and has a typical accuracy of 5m), it can be deemed to be referenced to Hartebeesthoek94. This is because the uncertainty in position is an order of magnitude larger than the difference in position of a point in the respective datums.  When using real-time TrigNet services (which is referenced to ITRF2005), users will have to occupy points referenced to Hartebeeshhoek94 to establish a localised relationship between the respective datums. 4. THE TRANSVERSE MERCATOR PROJECTION Johann Heinrich Lambert was a German/French mathematician and scientist. His mathematics was considered revolutionary for its time and is still considered important today. In 1772 he released both his Conformal Conic projection and the Transverse Mercator projection. The Transverse Mercator projection is the transverse aspect of the Mercator projection, which is a cylindrical projection, turned about 90 so that the projection is based on meridians and not the parallels. Figure 4.1: Normal aspect of the Cylindrical Projection, eg: Mercator Figure 4.2:Transverse aspect of the Cylindrical Projection, eg: Transverse Mercator The Transverse Mercator projection, in its various forms, is the most widely used projected coordinate system for world topographical and offshore mapping. All versions (e.g. Gauss Conform, Gauss Kruger, and Universal Transverse Mercator) have the same basic characteristics and formulas. The differences which distinguish the different forms of the projection, and which are applied in different countries arise from variations in the choice of the coordinate transformation parameters, namely the latitude of the origin, the longitude of the origin (central meridian), the scale factor at the origin (on the central meridian), and the values of false easting and false northing, which embody the units of measurement, given to the origin. Additionally there are variations in the width of the longitudinal zones for the projections used in different territories. The following table indicates the variations in the projection parameters which distinguish the different forms of the Transverse Mercator projection: Name Areas used Central meridian(s) Latitude of origin CM Scale Factor Zone width False Easting at origin False Northing at origin Transverse Mercator Various, world wide Various Various Various Usually less than 6° Various Various Gauss Conform (Transverse Mercator south oriented) South Africa 2° intervals E of 11°E 0° 1 2° 0m 0m UTM North hemisphere World wide 6° intervals° E & W of 3° E & W Always 0° Always 0.9996 Always 6° 500000m 0m UTM South hemisphere World wide 6° intervals E & W of 3° E & W Always 0° Always 0.9996 Always 6° 500000m 10000000m Gauss-Kruger Former USSR , Germany, S. America Various, according to area of cover Usually 0° Usually 1.000000 Usually less than 6°, often less than 4° Various but often 500000 prefixed by zone number Various Table 4.1: Different forms of the Transverse Mercator Projection 5. THE GAUSS CONFORM COORDINATE SYSTEM The Gauss Conform coordinate system (as used in South Africa) uses the Transverse Mercator map projection formulae modified to produce westings (y) and southings (x) instead of northings (N) and eastings (E). Note: The Gauss Conform projection is used in the southern hemisphere only. This projection is used for the computation of the plane westings (yLo) and southings (xLo) coordinates, commonly (but incorrectly) known as the “Lo coordinate system". 5.1 Coordinate System Conventions 5.1.1 Reference longitude / central meridian (zone/belt)  These 2 longitude wide zones (belts) are centred on every odd meridian, i.e. (15 E, 17 E, …. 35 E as well as 37 E for (Marion and Prince Edward Islands) as central meridian. Example; Longitude 19 E is the central meridian (CM) of the belt between 18 E and 20 E.  The origin of each belt is the intersection of each uneven degree of longitude (longitude of origin = Lo) and the equator.  Each zone is named after the longitude of origin i.e. Lo 17°, Lo 19°, Lo 21° etc. and is independent of geodetic datum Figure 5.1: Gauss Conform zones (continental South Africa*) Note: Marion and Prince Edward Islands on Lo 37°E, 5.1.2 Latitude at natural origin /reference Latitude:  The equator 0°E, is the latitude of reference or origin of the Gauss Conform Coordinate System. 5.1.3 x (southings)  Coordinates are measured southwards from the equator  Increases from the equator (where x = 0m) towards the south pole (with a maximum of ± 3 840 000m for continental South Africa).  Similar to the “northing” coordinates but sign in opposite. 5.1.4 y (westings)  Coordinates are measured from the Central Meridian (Lo) of the respective zone.  Increases from the CM (where y=0) in a westerly direction.  “y” is +ve west of the CM and –ve east of the Central Meridian.  Since the zone width is 2 (1 ) either side of the Central Meridian, the “y” value should range between +105000 m and - +105000 m in South Africa.  Unless specifically intended, a feature with a “y” ordinate exceeding the abovementioned values should be referenced to the adjacent Central Meridian. 5.1.5 False Origin  There is no false origin (y = 0m at Central Meridian and x = 0m at equator) 5.1.6 Order of Coordinates 1 9 E 2 2 E 2 3 E 2 5 E 2 7 E 2 9 E 3 1 E 2 4 E 2 6 E 2 8 E 3 0 E 3 2 E 2 0 E 2 1 E 1 6 E 1 7 E 1 8 E 2 22 2 b b - a 'e 2 = e'2.cos2 ; = tan 10 65536 436598 16384 110256 256 1754 64 452 4 3 e. e. e. e. .e 1 A 10 65536 727658 2048 22056 512 5254 16 152 4 3 e. e. e. e. .e B 10 16384 103958 4096 22056 256 1054 64 15 e. e. e. e. C 10 131072 311858 2048 3156 512 35 e. e. e. D 10 65536 34658 16384 315 e. e. E 10 131072 693 e. F 2 B C D E F 2 4 6 8 10 B a. 1 - e . A. - .sin 2 .sin 4 - .sin .sin8 - .sin10  = - (where is the longitude of the central meridian) 22 sin.e - 1 a N 2 4 3 2 4 2 6 5 2 4 2 2 2 x B .Nsin .cos .N.sin .cos . 5 9 4 - 2 24 .N.sin .cos . 61 - 58 270 - 330 ... 720 ... 58 - 14 18 - 5.cos.N. 120 - 1.cos.N. 6 .N.cos y 222425 5 223 3    Note: The Gauss Conform system is used in the southern hemisphere only. 5.2.3 Conversion of Gauss Conform Co-ordinates (y, x) to Geographical Co-ordinates ( ) a = semi-major axis of the reference ellipsoid in metres. b = semi-minor axis of the reference ellipsoid in metres. Given: y (Gauss Conform ordinate in metres, westing) x (Gauss Conform ordinate in metres, southing) Lo ( ) (Reference Longitude/ central meridian in integer degrees) Find: (Latitude in degrees decimal, positive south) (Longitude in degrees decimal, positive east) Formulae: Convert to radians 2 22 2 a b - a e 2 22 2 b b - a 'e b a b - a n 4 64 12 4 1 n n 1a. n 1x. 6sin.n. 96 151 4sin.n. 16 21 2sin.n 16 9 -n . 2 3 323f f = tan f f 222 f cos.e' 2 3 f 22 2 f sin.e - 1 e - 1a. M f 22 f sin.e - 1 a N 2 f 2 f 4 f 2 f 2 f3 ff f 4 ff f 2 f .9 - 4 - 3 5. N.M . 24 y N.M . 2 y - 4 f 4 f 2 f 2 f 2 f 2 f f 5 f 5 2 f 2 f f 3 f 3 ff 3 - 24 .8 28 6 5. cos.N 1 . 120 y 2 1. cos.N 1 . 6 y - cos.N y  = - (i.e. make the sign of the latitude negative, for the southern hemisphere)  (where is the longitude of the central meridian) 5.2.4 Sample Coordinates Geographical Coordinates Name Latitude Longitude dd mm ss.sssss dd mm ss.sssss Cape Town -33 48 17.26765 18 30 19.23450 Durban -29 45 17.23457 29 58 26.56340 Johannesburg -26 13 25.23450 28 02 33.03451 Gauss Conform Coordinates* Y x Central Meridian of Projection (Lo) Cape Town 45803.274 3742119.361 19 °E Johannesburg -104178.755 2902034.431 27 °E Johannesburg 95682.219 2901968.897 29 °E Durban -94214.530 3293328.957 29 °E Durban 99235.716 3293372.454 31 °E * Note: When the same point is projected onto an adjacent central meridian, both the y and x coordinates will change. The y coordinate will differ in sign and substantially in magnitude. The x coordinate will differ by about 100m due to varying distortion characteristics. 6. THE SOUTH AFRICAN COORDINATE REFERENCE SYSTEM It must be stressed that any position reported in the SACRS must be referenced to both the Hartebeesthoek94 datum and the Gauss Conform Coordinate System as defined above. Any position reported in other projections, (e.g. the standard Transverse Mercator or UTM projection), or another datum (e.g. Cape Datum) would, by definition, not be referenced to the SACRS. To summarise: SACRS = Gauss Conform Coordinate System (south oriented version of standard Transverse Mercator Projection) referenced to the Hartebeesthoek94 Datum. Figure 5.1: Current SACRS definition Figure 5.2: SACRS definition prior to January 1999 6.1 DEFINING THE SOUTH AFRICAN COORDINATE REFERENCE SYSTEM IN SOFTWARE 6.1.1 Defining Hartebeesthoek94 Datum within your GIS/GNSS software  Choose ellipsoid as WGS 84  Assign datum name as Hartebeesthoek94  Define relationship from World Geodetic Reference System 1984 (WGS 84) to Hartebeesthoek94 in terms of Moledensky (3D Cartesian shifts) with Translations dX = 0, dY = 0 and dZ=0 (although this is not strictly true, it is an acceptable in practice)