3.0 Introduction The
surface of the Earth is anything but uniform. The oceans,
can be treated as reasonably uniform, but the surface or topography of
the land masses exhibits large vertical variations between mountains and valleys. These variations make it impossible to approximate the shape of the Earth with any
reasonably simple mathematical model. Consequently, two main reference surfaces have been established to approximate the shape of the Earth. One reference surface is called the . These are illustrated in the figure below.ellipsoid
3.1 The Geoid and the vertical datum We can simplify matters by imagining that the entire Earth’s surface is covered by water. If we ignore tidal and current effects on this ‘global ocean’, the resultant water surface is affected only by gravity.
This has an effect on the shape of this surface because the
direction of gravity - more commonly known as plumb line - is dependent
on the mass distribution inside the Earth. Due to irregularities or mass
anomalies in this distribution the 'global ocean' results in an undulated
surface. This surface is called the
Where a mass deficiency exists, the
Geoid will dip below the mean ellipsoid. Conversely, where a mass surplus
exists, the Geoid will rise above the mean ellipsoid. These influences
cause the Geoid to deviate from a mean ellipsoidal shape by up to +/-
100 meters. The deviation between the Geoid and an ellipsoid
is called the . The biggest presently known undulations are
the minimum in the Indian Ocean with N = -100 meters and the maximum in
the northern part of the Atlantic Ocean with N = +70 meters (figure below). geoid undulationDeviations (undulations) between the Geoid and the WGS84 ellipsoid.
Tool for the calculation of a geoid undulation (compared to WGS84) at a point whose latitude and longitude is specified (external link):
The Geoid is used to describe heights. In order to establish the Geoid as reference for heights, the ocean’s water level is registered at coastal places over several years using tide gauges (mareographs). Averaging the registrations largely eliminates variations of the sea level with time. The resulting water level represents an approximation to the Geoid and is called the Every nation or groups of nations have
established those mean sea level points, which are normally located close
to the area of concern. For the Netherlands and Germany, the local mean sea level is realized through the Amsterdam tide-gauge (zero height). We can determine the height of a point in the Netherlands or Germany with respect to the Amsterdam tide gauge using a technique known as geodetic levelling (figure (b) below). The result of this process will be the height above local mean sea level. The height determined with respect to a tide-gauge station is known as the Obviously, there are several realizations of local mean sea levels (also called Care must be taken when using heights from another local vertical datum. This might be the case in the border area of adjacent nations. An example, the tide gauge (zero height) of the Netherlands differs -2.34 metres from the tide gauge (zero height) of the neighbouring country Belgium (figure below). Even within a country, heights may differ depending on to which tide gauge, mean sea level point, they are related. An example, the mean sea level from the Atlantic to the Pacific coast of the USA increases by 0.6 to 0.7m.
The local vertical datum (or height datum) is implemented through a levelling network (figure (a) below). A levelling network consists of benchmarks, whose height above mean sea level has been determined through geodetic levelling. The implementation of the datum enables easy user access. The surveyors do not need to start from scratch (i.e. from the Amsterdam tide-gauge) every time they need to determine the height of a new point. They can use the benchmark of the levelling network that is closest to the point of interest (figure (b) below).
The use of satellite-based positioning equipment (e.g. GPS) to determine heights with respect to a reference ellipsoid (e.g. WGS84) is becoming more in use. These heights are known as the
As a result of satellite gravity missions, it is currently possible to determine the orthometric height (height H above the Geoid) with centimetre level accuracy. It is foreseeable that a global vertical datum may become ubiquitous in the next 10-15 years. If all published maps are also using this global vertical datum by that time, heights will become globally comparable, effectively making local vertical datums redundant for GIS users.
3.2 The ellipsoid Above, we have defined a physical surface, the Geoid, as a reference surface for heights. We also need a reference surface for the description of the horizontal coordinates (i.e. geographic coordinates) of points of interest. Since we will later project these horizontal coordinates onto a mapping plane, the reference surface for horizontal coordinates requires a mathematical definition and description. The most convenient geometric reference is the may be used. An ellipsoid is formed when an ellipse is rotated about its minor axis. This ellipse which defines an ellipsoid or spheroid is called a meridian ellipse (notice that ellipsoid and spheroid are used here as equivalent and interchangeable words).sphere
The shape of an ellipsoid may be defined
in a number of ways, but in geodetic practice the definition is usually
by its semi-major axis The ellipsoid may also be defined by its semi-major axis a and eccentricity e, which is given by: Given one axis and any one of the other three parameters, the other two can be derived. Typical values of the parameters for an ellipsoid are:
3.3 The sphere As can be seen from the dimensions of the
Earth ellipsoid, the semi-major axis
The consequence is that instead of using the ellipsoid, the sphere might be sufficient for certain mapping tasks. In practice, maps at scale 1:5,000,000 or smaller can use the mathematically simpler sphere without the risk of large distortions. At larger scales, the more complicated mathematics of ellipsoids are needed to prevent these distortions in the map. The mathematically simpler sphere may be used as reference surface for maps at small-scale.
3.4 Local and global ellipsoids Many different ellipsoids have been defined in the world. The Geoid, a globally best fitting ellipsoid for it, and a regionally best fitting ellipsoid for it, for a chosen region.
With increasing demands for global surveying, work is underway to develop In 1924, the general assembly of the IUG in Madrid introduced the ellipsoid determined by Hayford in 1909 as the At the general assembly 1967 of the IUGG in Luzern, the 1924 reference system was replaced by the For some time, the Geodetic Reference System 1967 was used in the planning of new geodetic surveys. For example, the Australian Datum 1966 and the South American datum 1969 are based upon this ellipsoid. However, at its general assembly 1979 in Canberra the IUGG recognized that the Geodetic Reference System 1967 no longer represented the size and shape of the Earth to an adequate accuracy. Consequently, it was replaced by the
3.5 The local horizontal datum Ellipsoids have varying position and orientations. An ellipsoid is positioned and oriented with respect to the local mean sea level (or Geoid) by adopting a latitude (f ) and longitude (l) and ellipsoidal height ( Examples of local horizontal datums with their underlying ellipsoid and difference in position (datum shift) with respect to WGS84.
Several hundred local horizontal datums exist in the world. The reason is obvious: Different local ellipsoids with varying position and orientation had to be adopted to best fit the local mean sea level in different countries or regions. An example is the Potsdam Datum, the local horizontal datum used in Germany. The fundamental point is in Rauenberg and the underlying ellipsoid is the Bessel ellipsoid ( The local horizontal datum is realized through a so-called Within this framework, users do not need to start from scratch (i.e. from the fundamental point) in order to determine the geographic coordinates of a new point. They can use the monument of the triangulation network that is closest to the new point. The extension and re-measurement of the network is nowadays done through satellite measurements.
Computations of a triangulation network for a local horizontal datum were usually made with optical survey instruments such as a theodolite or total station. These instruments are levelled by means of spirit bubbles. The bubbles follow the influence of the Earth's gravity which means that the computations
have to be corrected with respect to the ellipsoid. Without these corrections the computations may be distorted by some centimeters or even decimeters because of the local difference between the direction of the plumb line (the normal to the Geoid) and the vertical direction on the ellipsoid (the normal to the ellipsoid). This difference in direction is known as the
3.6 The global horizontal datum Local horizontal datums have been established to fit the Geoid well over the area of local interest, which in the past was never larger than a continent. With increasing demands for global surveying activities are underway to establish global reference surfaces. The motivation is to make geodetic results mutually comparable and to provide coherent results also to other disciplines like astronomy and geophysics. The most important global (or geocentric) spatial reference system for the GIS community is the X,Y,Z). The Z-axis points towards a mean Earth north pole. The X-axis is oriented towards a mean Greenwich meridian and is orthogonal to the Z-axis. The Y-axis completes the righthanded reference coordinate system (figure (a) below).
The ITRS is realized through the The ITRF96 datum was established at the 1st of January, 1997. This means that the measurements use data up to 1996 to fix the geocentric coordinates ( Global horizontal datums, such as the ITRF2000 or WGS84, are also called To implement the ITRF in a region, a densification of control stations is needed to ensure that there are enough coordinated reference points available in the region. These control stations are equipped with permanently operating satellite positioning equipment (i.e. GPS receivers and auxiliary equipment) and communication links. Examples for (networks consisting of) such permanent tracking stations are the AGRS in the Netherlands and the SAPOS in Germany. We can easily transform ITRF coordinates (
Hundreds of existing local horizontal and vertical datums are still relevant because they form the basis of map products all over the world. For the next few years, we will be required to deal with both local and global datums until the former are eventually phased out. During the transition period, we will require tools to transform coordinates from local horizontal datums to a global horizontal datum and vice versa (section 5 on coordinate transformations). The organizations that usually develop transformation tools and make them available to the user community are provincial or National Mapping Organizations (NMOs) and cadastral authorities.
de By, R.A. (editor), , Kraak, M.J., Sun, Y., Weir, M.J.C. and van Westen, C.J. Knippers, R.A.Principles of geographic information systems (Chapter 4.2 on spatial referencing), 2nd edition, ITC Educational Textbook, ITC, Enschede, 2001.
Geometric Fundamentals of Mapping. Non-published
educational notes, ITC, Enschede.
Geometric Aspects of Mapping. Non-published
educational notes, ITC, Enschede, 1999.
Georeferencing and Coordinate
Transformations. Non-published educational notes. ITC, Enschede, 1996. |