Copyright Peter Dana, Geography Department, University of Texas
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POSC Specifications Version 2.2 |
Epicentre Usage Guide Coordinate Systems - A Deeper Understanding |
[Coordinate System Table of Contents]
Earth based coordinate systems forms an extensive field of study. The following is an introductory tutorial to some of the basics in the field, with information about how these basic ideas are implemented in Epicentre. It is useful for those who have only a brief introduction to the concepts, and wish to gain a further understanding. It is also useful for those who need to translate their understanding to Epicentre terms.
An illustration of the importance of the coordinate system is shown in the following table taken from the University of Texas Coordinate System web site.
Copyright Peter Dana, Geography Department, University of Texas
Geographic coordinates of positions are always expressed as a latitude and longitude. The latitude and longitude are related to a particular earth figure, which may be a sphere for most atlas maps and imprecise work, but for rigorous purposes is a mathematical figure which much more closely approximates the shape of the real earth. The earth's shape is assumed to be an ellipsoid of revolution, (the figure defined by an ellipse rotated round its minor axis), sometimes (though not in Epicentre) termed a spheroid. For historical reasons there are several such ellipsoids in use for mapping different countries of the world, each defined by the length of its semi-major axis, - about 6378000m, - and either its semi-minor axis or, more usually, by the reciprocal of its flattening, - a figure about 298. Each of these ellipsoids is generally matched to one or several countries. An ellipsoid appropriate for one region of the earth is not always appropriate to other regions. Recent work based on satellite data has seen the development of ellipsoids that are used worldwide - such as GRS80.
Another property is needed to uniquely specify geographical positions. This is the position and orientation of the ellipsoid relative to the earth. The term used to describe this fitting of an ellipsoid to the earth is the geodetic datum. Many geodetic datums exist throughout the world, each usually associated with the national survey of a particular country or continent. More recently several new geodetic datums have been successively derived, from steadily accumulating satellite and other data, to provide for a best world wide fit.
Each geodetic datum is defined by a set of numerical elements which define where the imaginary geographical graticule of lines of latitude and longitude lies on the earth's surface, i.e. the direction of the poles and the rotated position of the graticule about this direction. It is normally fixed by attributing coordinates to a starting point, fundamental point or datum, and defining azimuth direction for the national survey control framework. The latitude and longitude and an azimuth at the datum point was originally established using the best means possible at the time and selecting an ellipsoid which best fitted the earth's surveyed surface for the country or continent in question. The geodetic datum may then have taken the name of the fundamental surveyed point, e.g. Kalianpur for India, Pulkovo for Russia. In the case of the newer earth centered and satellite derived geodetic datums, where the datum is effectively the center of the earth ellipsoid, a different type of label with world or international connotations is used, e.g. WGS 84.
Thus all precisely surveyed and mapped points and features on the earth's surface will be uniquely defined in position by stating their coordinates, (grid or geographical), the projection if grid, and the geodetic datum, including its earth ellipsoid. Therefore in order to be unambiguous and precise in the definition of position it is essential that as well as quoting latitude, longitude and height it is also necessary to specify the geodetic datum, including ellipsoid, and the level to which the height coordinate values are related. Geographic coordinates may be transformed to grid coordinates, or projected coordinates of Epicentre, or may be transformed to other geographic coordinates on a different datum. See Section 3.4, "Projections and Projected Coordinate System formulas" and Section 3.5, "Geographic Coordinate System Transformations".
While geographic coordinate systems and projected coordinate systems in Epicentre deal with two-dimensional locations, there is the third dimension that must be considered when giving a location. The third dimension is the height or elevation.
Definition of heights or levels requires consideration of further parameters and reference levels or surfaces, and are relevant to some of the vertical coordinate systems of Epicentre. Additionally they become relevant when considering three dimensional cartesian coordinates1 which feature in the datum shift problem represented by Bursa-Wolf and Molodensky in Epicentre's coordinate transformation methods.
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Note 1: Another type of coordinate system allowd for in Epicentre is the geocentric coordinate system. This coordinate system is a cartesian (x,y,z) coordinate system with origin near the center of the earth, and z-axis approximately parallel to the earth's axis of rotation. But locations are rarely given in these coordinates. These coordinate systems are used mainly as intermediates in specifying and applying datum shifts such as Bursa-Wolf and Molodensky transformations. Usage of these coordinate systems, therefore, is not covered in this article, since earth surface locations are rarely given in these coordinates. |
One surface of importance in the description of elevations is the ellipsoidal surface. Heights may be referenced from the ellipsoid surface. A second is the earth's topographical surface. A third, and widely used, surface relevant to defining a height value for a point or object is the Geoid. The Geoid is an equipotential surface in the earth's gravity field and is the surface with which a spirit level aligns itself. The shape of this surface is governed by the distribution of the earth's mass and consequent gravity variations. It is mathematically complex and difficult to completely define to high accuracy but it approximates to a tideless mean sea level - and is often called "Mean Sea Level" - and might be imagined to extend continuously through the continental land masses. It is important because it is the surface to which surveying levelling data relates.
The surface of a sphere or ellipsoid cannot be flattened out without tearing, stretching, or otherwise distorting it. This incompatibility between these figures and a plane lead to distortions in projected coordinate systems.
Since all map projections attempt to represent the curved surface of the earth on a flat sheet of paper there are inevitable distortions in the result, - in relative areas, in scale, in shape, or in all three. Although a great number of different map projections are in general use for atlas and sheet maps and there are many others which have been devised to theoretically provide better representations of the earth's surface for showing particular countries or features, only a limited number have been adopted for national topographic mapping purposes. In general, those used for large and medium scale topographic mapping have the particular property that they preserve shape, i.e. the shape of the land and sea areas, and thus of topographic, geological, and other features. Hence they are also deemed most suitable for large and medium scale (1:250,000 or larger) hydrocarbon exploration/production mapping. Such maps are said to be orthomorphic (shape preserving) or conformal (angle preserving). The latter term is used in the Epicentre documentation.
In order to preserve shape, other distortions have to be accepted. In particular, these map projections do not preserve a constant scale. In order that areas and scale do not suffer extreme distortion towards the extremities of the mapped areas, the projections are usually restricted to cover limited areas within which the scale distortion is not evident or unbecoming. Indeed these projections are generally arranged to minimize scale distortion such that often several elements of the same projection, usually arranged in regular bands of longitude or latitude, may be needed to achieve complete coverage of a particular country.
All projections have mathematical formulas which define the relationship between the latitude/longitude graticule on the earth and its representation on the map sheet, or the relationship between the geographic coordinates (latitude and longitude) of points and their projected coordinates (grid or rectangular coordinates) on the projection or map. In Epicentre terms, each set of formulas define a coordinate transformation method, and the use of the formulas by specifying a set of parameter values defines a coordinate transformation. See Section 3.4, "Projections and Projected Coordinate System formulas". Formulas for the projection methods included in Epicentre reference data are also documented in that section.
In addition to transformations between a geographic coordinate system and a projected coordinate system, there are transformations between geographic coordinate systems. Since an ellipsoid is "positioned and oriented" to the earth, these transformations tend to be a repositioning (shift the center of the ellipsoid) and reorientation (rotation of the axes) of the ellipsoid. There may also be a rescaling, since the ellipsoids may differ in size. These types of coordinate transformations are called datum shifts. See Section 3.5, "Geographic Coordinate System Transformations".
As previously indicated, quoted positions on the earth's surface must always be qualified by the geodetic datum (including the ellipsoid) to which they are related. Before satellite surveying methods arrived, such positions would generally automatically be related to the national survey framework in a country, because they would usually have been connected to one or more national control points. They would thus automatically assume a relationship with the national geodetic datum. If such a connection was not possible, the survey may well have been based on an astronomically surveyed point which itself became the datum for that particular survey and any others related to it.
For the last twenty years, with much surveying being performed using satellite positioning methods, the coordinate data delivered by the GPS or earlier Transit satellite instrumentation is referred to a satellite system datum, which is invariably not in sympathy with the country's historical national survey control and mapping datum. Satellite datums use earth-centered or "Geocentric" ellipsoids whereas the position and orientation of the ellipsoids used for much national mapping was arranged to best fit the data available for the national area at the time, often assuming coincidence of the ellipsoid and geoid surface at the selected datum point. These ellipsoids were generally neither earth-centered nor with their Z-axial directions consistent with the earth's polar axis direction.
So in order to express coordinates derived in one datum system in another, such as, for example, deriving national geodetic system latitude and longitude values from GPS surveyed latitude and longitude values, it is necessary to apply a numerical transformation to the coordinates, using either the Molodensky or Bursa Wolf coordinate transformations of Epicentre. An explanation of these operations, other geodetic transformations, and the relevant formulas appear Section 3.5, "Geographic Coordinate System Transformations".