3.1 The figure of the Earth
The earth's surface is anything but uniform. Only a part of it, the oceans, can be treated as reasonably uniform. But the surface or topography of the land masses show large vertical variations between mountains and valleys which make it impossible to approximate the shape of the earth with any reasonably simple mathematical model.
We can simplify matters by the idealization of expanding the oceans below the landmasses and make the assumption that the water can flow freely also there. If we then neglect tidal and current effects on this "global ocean", the resultant water surface is remaining affected only by gravity. This has a certain consequence 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" is forced to be an undulated surface. This surface is called the geoid or the "physical figure of the earth". The plumb line through any surface point is always perpendicular to it.
Perspective view of the Geoid (Geoid undulations 15000:1)
If the earth was of uniform density and the earth's topography didn't exist, the geoid would have the shape of an oblate ellipsoid centered on the earth's center of mass. Unfortunately, the situation is not this simple. 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 (see figure). The deviation between the geoid and an ellipsoid is called the Geoid undulation (N).
Relationships between the earth's surface, the geoid and a reference ellipsoid
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.
Surveying observations are usually made with instruments levelled by means of spirit bubbles. Since these bubbles follow the influence of the earth's gravity the observations are made relative to the geoid.
3.2 The Geoid as reference surface for Heights
In order to establish the geoid as reference for height measurements, the ocean's water level is registered at coastal places over several years using tide gauges (mareographs). Averaging the registrations, as long as they are periodic, largely eliminates variations of the sea level with time. The resulting water level represents an approximation to the geoid and is called the Mean Sea Level (MSL).
Every nation or groups of nations have established those observation points, which are normally located close to the area of concern. For the Netherlands the geodetic tide gauge station is in Amsterdam, for France in Marseille, for Greece in Saloniki, etc. Starting from these stations the heights of points on the earth's surface can be measured using geodetic levelling techniques.
Differential leveling for height measurements (Mean Sea Level is the starting point for the height measurements)
Since all the height values within a country are related to a particular reference point, called levelling datum or vertical datum, care must be taken when using heights from another system. This might be the case in the border area of adjacent nations. Even within the territory of a state, heights may differ depending on to which tide gauge they are related. As an example, the MSL from the Atlantic to the Pacific coast of the USA increases by 0.6 to 0.7 m. The MSL from the Netherlands differs -2.34 meters from the MSL from Belgium. As consequence heights in the border area differ (see figure below; note that the contours end at the border).
Fragment of the Dutch topo map showing the border of Belgium and the Netherlands. The Mean Sea Level of Belgium differ -2.34m from the MSL of The Netherlands. As a result , contour lines are abruptly ending at the border.
The same care must be taken when using GPS measurements. GPS measurements are taken relative to the WGS84 ellipsoid. GPS heights have to be adjusted before they can be compared to heights given on topographic maps, which are related to a MSL point.
3.3 Approximations of the Earth's figure
The curvature of the geoid displays discontinuities at abrupt density variations inside the earth. Consequently, the geoid is not an analytic surface and it is thereby not suitable as a reference surface for the determination of locations. If we are to carry out computations of positions, distances, directions, etc. on the earth's surface, we need to have some mathematical reference frame. The most convenient geometric reference is the oblate ellipsoid as it provides a relatively simple figure which fits the geoid to a first order approximation. For small scale mapping purposes we can also use the sphere which fits the geoid to a second order approximation.
3.3.1 The Ellipsoid
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 (Note that ellipsoid and spheroid are being treated as equivalent and interchangeable words).
A cross section of an ellipsoid, used to represent the Earth's surface, indicating what is major and minor axis radius
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 and flattening. Flattening f is dependent on both the semi-major axis a and the semi-minor axis b.
f = (a - b) / a
The ellipsoid may also be defined by its semi-major axis b 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:
a = 6378135.00m b = 6356750.52m
f = 1/298.26 e = 0.08181881066
3.3.2 The Sphere
As can be seen from the dimensions of the earth ellipsoid, the semi-major axis a and the semi-minor axis b differ only by a bit more than 21 km. A better impression on the earth's dimensions may be achieved if we refer to a more "human scale". Considering a sphere of approximately 6 m in diameter then the ellipsoid is derived by compressing the sphere at each pole by 1 cm only. This compression is rather small compared to the dimension of the semi-major axis a.
The ellipsoid and the sphere, a comparison
The consequence is that instead of using the ellipsoid, the sphere might be sufficient for certain mapping tasks.
The sphere as reference surface for small-scale mapping
In practice 1:5,000,000 is recommended as the largest scale at which the spherical approximation can be made.
3.4 The Ellipsoid as Reference surface for Locations
3.4.1 Local Reference Ellipsoids
It is important to realize that topographic maps are drawn and geodetic positions are defined with respect to a horizontal datum (also referred to as geodetic datum or reference datum). A horizontal (or geodetic) datum is defined by the size and shape of an ellipsoid as well as several known positions on the physical surface at which latitude and longitude measured on that ellipsoid are known to fix the position of the ellipsoid. In the United States we use the North American Datum, in Japan the Tokyo Datum, in some European countries the European Datum, in Germany the Potsdam Datum, etc.
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. As a consequence, the differences between the geoid and the reference ellipsoid may be ignored. This allows accurate maps to be drawn in the vicinity of the datum. The figure below shows that a position on the geoid will have a different set of latitude and longitude coordinates in each reference datum. In this figure the North American datum is extrapolated to Europe. Even though the datum fits the geoid in the North American continent well, it does not fit the European geoid. Conversely, if the European datum is extrapolated to the North American continent, the similar result is found.
The geoid and two best fitting local ellipsoids for a chosen region
Widely in use are the following ellipsoids generally named after their generator:
Horizontal datums are defined by the size and shape of an ellipsoid, as well as its position and orientation. There are a few hundred of these local horizontal datums defined in the world. The table below shows some examples of local datums , which use the same ellipsoid (Clarke 1866 or Hayford), but in different positions (referred as datum shifts).
Examples of reference datums, with its reference ellipsoid and datum shift values
Two reference ellipsoids in different position.
3.4.2 Global Reference Ellipsoids
With increasing demands for global surveying activities are going on to establish also global reference ellipsoids. Especially the International Union for Geodesy and Geophysics (IUGG) is involved in establishing those reference figures. The motivation is to make geodetic results mutually comparable and to provide coherent results also to other disciplines like astronomy and geophysics.
The geoid and a globally best fitting ellipsoid
In 1924 in Madrid, the general assembly of the IUGG introduced the ellipsoid determined by Hayford in 1909 as the International Ellipsoid. In contrary to local reference ellipsoids, which apply only to a region or local area of the earth's surface, global reference systems approximating the geoid as a mean earth ellipsoid. However, according to present knowledge, the values for this earth model only give an insufficient approximation. At the general assembly 1967 of the IUGG in Luzern, the 1924 reference system was replaced by the Geodetic Reference System 1967 (GRS 1967). It represents a good approximation (as of 1967) to the mean earth figure. The geometric ellipsoidal parameters a, b and f are given in the table below. The Geodetic Reference System 1967 has found application especially in the planning of new geodetic surveys.
At its general assembly 1979 in Canberra the IUGG recognized that the Geodetic Reference System 1967 no longer represents the size and shape of the earth to an adequate accuracy. Consequently, it was replaced by the Geodetic Reference System 1980 (GRS 1980 see table). The World Geodetic System 1984 (WGS84) is based on the GRS 1980 and provides the basic reference frame for GPS (Global Positioning System) measurements.
Note Nowadays, geodesists are able to measure the Earth-as-a-whole with the aid of artificial satellites. However, a re-adjustment of all existing local geodetic survey reference systems is not to be expected for the time being due to the great efforts in applying coordinate transformation and changing existing maps. For the time being, local reference systems retain their practical importance for national mapping activities.
3.5 Relationships between reference surfaces
In summary, when speaking of the size and shape of the earth and positions on it, there are three surfaces to be considered:
- The topography - the physical surface of the earth.
- The Geoid - the level surface (also a physical reality).
- The Ellipsoid - the mathematical surface for computations.
Surveying observations are made on the earth surface relative to the geoid. Before using observations in geodetic computations, they must be corrected for locational differences between the geoid and the reference ellipsoid. These corrections are small and may for some purposes be ignored if a reference ellipsoid is chosen so as to closely fit the geoid in the area of concern.
Mean Sea Level (MSL) points, an approximation to the geoid, are used as reference surfaces for height measurements ( orthometric heights).
The earth surface, and two reference surfaces, the geoid and a reference ellipsoid. Orthometric heights are measured from a Mean Sea Level point, an approximation to the geoid.
Ellipsoidal heights have to be adjusted before they can be compared to the orthometric heights given on topographic maps.The deviation between the geoid and an reference ellipsoid is called Geoid undulation (N). Geoid undulations can be used to adjust the ellipsoidal heights (H = h +/- N).
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Ellipsoidal height h above the reference ellipsoid and the orthometric height H above the Geoid for two points on the earth surface. The ellipsoidal height is measured orthogonal to the ellipsoid. The orthometric height is measured orthogonal to the geoid.
3.6 References
Mehlbreuer, A. Geometric Fundamentals of Mapping. Non-published notes. Enschede, ITC.
Knippers, R.A. (1999). Geometric Aspects of Mapping. Non-published notes, Enschede, ITC
Stefanovic, P. (1996) Georeferencing and Coordinate Transformations. Non-published notes. Enschede, ITC.