The science and art of surveying



Fritz van der Merwe University of Pretoria Copyright reserved


The purpose of this paper is to sketch, and I emphasise SKETCH, the realities surveyors work with every day. The theory that a surveyor has to keep in mind while he plies his trade, is implicit in most modern survey instruments. Implicit in the sense that the instrument will automatically calculate corrections and transformations to facilitate a survey operation, without the operator being aware of it.

The implicit knowledge that you do not see in the operation, has to be embedded in every surveyor’s mind, to enable him to know when something goes wrong.

For surveyors, there are no excuses. There are no “I didn’t know”. When you produce the results of a survey, they must be correct and they must reflect the situation on the ground. If there is a mistake, an error, or some unsound measure of judgment, the surveyor is liable. The implicit theory in the software does not, EVER, exonerate the surveyor from his responsibility for the survey and the results.

The art of a survey is not to over-simplify the environment or over complicate the mathematics, yet obtain the desired results. The science is in the mathematics and the capabilities and limitations of the instruments.

The devil is in the detail, where it always sits, as I’m sure my engineering, planning, and architectural colleagues will confirm. The rest of this paper will highlight some of the details surveyors have to have in the back of their minds, as tacit knowledge, to plan and evaluate their own work, as they go about a survey.

The Mathematical Environment Surveyors Work Surveyors work in both the natural environment and in a mathematical environment. The mathematical environment is necessary to simplify and describe the natural environment.

This is also about the theory embedded in the more modern electronic surveying and computing equipment, but which the surveyor must always keep track of in the execution and computation of his survey.

The Shape of the Earth

The earth is, in our terms, a huge ball of rock, molten material, and metal, hurtling through space at a vast speed. It has no fixed shape since it is essentially a fluid body. There is a crust over it, but that is thinner than the crust of a milk tart in proportion. It only comprises about 0,8% of the earth’s radius, or 30 to 50 km. Below that crust, for about 6000 km or so, the earth is fluid. It is the top of that crust, and maybe up to 4 km into it (, that the surveyor has to map because that is where we live, build and construct our spaces.

The Ellipsoid

It is, however, possible to determine some workable shape and size of the earth, which are constant enough for a long enough time, to enable us to map the present surface. (Thanks to Newton.)

The shape used for horizontal mapping is an ellipsoid—the solid that one gets when an ellipse is rotated around its short axis. When viewed perpendicular to the plane of the equator, the elevation is an ellipse. Viewed perpendicular to the equator, along the axis of rotation, the plan view is a circle.

Figure 1: Ellipsoid from

Figure 2: An exaggerated ellipsoidal shape of the earth

The bi-axial ellipsoid is fully described by two of the following parameters: a, its semi-major axis, b, its semi-minor axis and the inverse of its flattening, 1/f = a/(a-b). See figure 3.

Over the centuries, since Newton’s definition of the shape of the earth, various Figure 3.: Bi-axial Ellipsoid (from Knippers, 2006)

ellipsoids have been calculated. The following are examples of them.

Ellipsoid                a              b              Unit        Used

Mod. Clarke 1880

6378249.145 6356514.967



Botswana Zimbabwe

WGS 84
Bessel Clarke 1866

6378137.000 6356752.314



Globally Namibia Mazambique

6377397.155 6356078.963


German Legal

Globally Namibia Mazambique

6378206.400 6356584.467



Table 1: Some ellipsoid parameters. (Chief Directorate Surveys and Mapping, n.d.)

The Geoid

That is not the whole story though. Due to the fact that the earth has an irregular density, the gravity field is also irregular. If a surface of equal gravity potential is calculated close to the surface of the ellipsoid, it is seen to cross the ellipsoid in several places. Such a surface is called a Geoid and it coincides with Mean Sea Level. It is used as the vertical reference surface for mapping and it also defines the horisontal plane for a survey.

Measurements made on the earth with conventional survey instruments will follow along the shape of the geoid. Horisontal measurements, therefore, have to be reduced mathematically to the ellipsoid for mapping purposes. Instruments using satellites refer all measurements, including heights, directly to the ellipsoid.

Figure 4: Undulations of the Geoid from

Figure 5: Geoid model for Africa – AGP2003 (contour interval: 2 metres) from Merry, 2003

Coordinate Systems

In order to be able to indicate a position on the surface of the earth, or a map, a coordinate system is needed.

Spherical and Ellipsoidal Coordinate Systems
The conventional system is one of latitudes and longitudes with the equator and the Greenwich meridian as the reference circles. The position of point P in figure 6 is described in terms of the angles f, l and the height above the reference ellipsoid, h. Such a coordinate system is called a graticule.

Figure 6: Latitude (f), Longitude (l) and Height (h) above Ellipsoid. (From Knippers, 2006)

When working with gravity, satellites or datums (defined in the next section) a convenient coordinate system is a three dimensional, right handed, Cartesian coordinate system, with its origin at the centre of gravity of the earth. Because it rotates with the earth it is also called an Earth Centred, Earth Fixed (ECEF) system.

Figure 7: 3-dimensional Cartesian coordinate system. ECEF (Knippers, 2006)

Plane and Mapping Coordinate Systems

For mapping purposes a two dimensional Cartesian coordinate system is adopted. The mapping coordinate system is a mirror image of the conventional Cartesian mathematical coordinate system and it is rotated through 90° anti-clockwise internationally and clockwise in Southern Africa. Position is described by an X- and Y- coordinate and the system is referred to as a grid. To maintain mathematical and navigational conventional integrity, the angles are measured from the positive x-axis in the direction of the positive y-axis.

Figure 8: 2-Dimensional Cartesian and Polar Coordinate Systems

In the polar coordinate system, the position of the point is described by an angle with respect to the X-axis and a distance from the origin of the system.


“A National geodetic co-ordinate system is defined by a Geodetic Datum, which consists of two parts: a) A defined geodetic reference ellipsoid, in terms of the a, b or a, f parameters. b) A defined orientation, position, and scale of the Geodetic system in space.” (Chief Directorate Surveys and Mapping, n.d.)

The survey of a large area of the earth, like a country, starts at an arbitrary convenient point and necessarily follows the geoid, because of the construction of the survey instruments. The horisontal coordinates are, however, reduced to the surface of the ellipsoid, which effectively fits an ellipsoid of choice onto the geoid. This has the effect of fixing the origin of the ellipsoid and therefore its orientation in space. In theory an infinite number of ellipsoids can be fitted to the earth, all with different orientations and scale, which is a function of the surveying equipment and methods used.

Figure 9:Different Datums (Chief Directorate Surveys and Mapping, n.d.)

As from 1 January 1999, South Africa has adopted the WGS 84 ellipsoid as its national datum. It is a satellite derived datum with the origin at the centre of gravity of the earth. Prior to that all surveys were calculated in the Clark 1880 datum with an arbitrary origin like those in figure 9.

Map Projections

In order to transfer positions from the ellipsoid to a map, use have to be made of map projections. A map projection is a way of mathematically and mechanically transferring the geodetic, ellipsoidal latitudes and longitudes, the graticule, to a flat surface for mapping purposes.

Surfaces that can be used for map projections are planes, cones and cylinders, because they can all be developed into a flat surface. The class of the projection is named after the surface.

Figure 10: Projection Surfaces or Classes of Projection (Knippers, 2006)

The orientations of the projection surfaces in figure 10 are referred to as the normal aspect projections. Aspects of projections can also be transverse or oblique. See figure 11.

Figure 11: Transverse and Oblique Projections (Knippers, 2006)

Some of the relationships between features on the earth can be kept true on a map, but in general a map is always distorted, because it is impossible to represent a round surface truly on a flat surface without distortion.

The following relationships can be introduced mathematically into projections:

  • Conformality – Shapes and angles remain true, but areas and distances are distorted and the scale of the map is only correct in some places.
  • Equivalence – Areas remain correct, but shape distance, and scale become distorted. Equidistant – Distances are correct in some directions, but not all. Shapes, areas, and scales are distorted.
  • Map scale and Scale Factor

Map scale is the nominal scale of a map. It refers to the scale of the projected reference ellipsoid which is a scaled model of the real sized ellipsoid. It is a ratio of the distance on the scaled model to the distance on the real ellipsoid. The only lines, or, points, where the scale is true on the map, the projection surface, is where the projected surface touch the projection surface.

The scale is, in modern maps, expressed as a representative fraction (RF), i.e., like 1:50 000. That means that the reference ellipsoid from which the projection was done is 1/50 000th of the size of the real ellipsoid. It does not mean for one instant that the size of features, all over the map, is 1/50 000th of their size on the earth. The only place on the map where that is true, is where the projection surface, be it a cylinder, cone or plane, touches the reference ellipsoid.

The rest of the map is subject to a scale factor. That is a multiple of the scale of the map that indicates the measure of distortion induced in the map by the projection method and the contributing mathematics. Whenever measurements are made on the surface of the earth, they have to be reduced to the equivalent measurements on the map.

The most used projection in the world today for surveying purposes is a Transverse, Conformal, Cylindrical projection, also known as the Transverse Mercator (TM) projection, because it is derived from the normal, conformal cylindrical projection which is known as Mercator’s Projection (ca 1569). That is, the projection surface is a cylinder, the aspect is transverse and shapes and angles remain true. The only line along which the scale of the reference ellipsoid remains true on the map, is where the elliptical cylinder touches the ellipsoid along a curve of longitude. The wider line in figure 12. This is called the Central Meridian.

Figure 12: The Transverse Mercator Projection Surface for the Ellipsoid

In South Africa this is called the Gauss Conformal Projection and in Germany the Gauss Krüger projection. It is the projection of choice for surveyors, because it is conformal (we measure angles) and the way it is projected causes a minimal scale factor. The scale factor,

i.e. the multiple of the nominal scale of the map, cannot however be ignored in any survey purporting to have any precision at all.

A representation of the Transverse Mercator projection is given in figure 13. Although the measurements on the map is done on the grid, they can be mathematically related to the graticule. That is exactly what happens on a topographical map of the 1:50 000 series in South Africa where the 2-dimensional grid coordinates, as well as the ellipsoidal graticule coordinates are given outside the neatline or margin.

Figure 13: The Transverse Mercator (TM) Projection: Grid and Graticule

In South Africa, where the stars go around the south pole clockwise and angles are measured clockwise, it just made sense to turn the projection through 180°. The result is seen in figure 14.

Figure 14: TM From a Southern Perspective. The Gauss Conformal Projection.

The ellipsoid, projection, grid and graticule are related in the way shown in Figure 15. A surveyor must be able to recognise and transform between these coordinates with competent speed.

Figure 15: Grid, Projection, Ellipsoid and 3-D Coordinates (Chief Directorate Surveys and Mapping, n.d.)

The Conduct of a Survey

Given the above mathematical environment and discounting the physical environment of the surveyor, the heat or cold he has to cope with, the terrain, mountainous or flat, the land cover, forest or plain, the accessibility of the terrain and numerous other factors, not excluding the populace, it becomes clear that a survey and the resultant maps, diagrams or coordinate lists, become undertakings of no small effort, physically or mentally.

Survey Instruments

Survey instruments, by nature of their design, have certain capabilities and, more important, limitations.

Conventional Instruments

Conventional instruments are the Theodolite, used to measure horizontal and vertical angles, the Level, used to measure vertical height differences and the Tape measure, used to measure distances. These are inherently bound to the geoid and the local terrain and climate.

Theodolites and levels are “leveled”, i.e. aligned with the horizon, by means of a bubble in a liquid in a tube. The axis of the tube is aligned perpendicular to the gravity field and in the horizontal plane. All angles are measured with respect to the horizontal plane, which is not necessarily parallel to the surface of the ellipsoid.

This is not in itself a problem, since most of what we consider to be upright structures are parallel to the gravity field and water must flow downhill, or, along the gravity gradient.

Another consideration is that the angles are measured in a plane, whilst the observations were in actual fact done on the curve of the earth. All observations have to be corrected for that curvature before calculations on any grid can be done.

Conventional instruments and trigonometry served surveyors, and engineers, well for well over a millennium. Their limitations are the time it takes to conduct, calculate, and produce the results of a survey.

Modern Instruments

With the recent and present advances in technology, the results of surveys were expected to be produced faster and faster. So more, and sometimes better, instruments were developed. Aerial photography, RADAR, Measurement on aerial photographs, Electronic Distance Measurement, Total Stations, Autonomous Stations, and ultimately GPS, (Courtesy of Ben Remondi), and LIDAR. It’s beautiful—we can now survey faster and more than ever before, but we can also make mistakes faster and more than ever before. The reason for that is that, as in Geographical Information Systems and cartography, anyone with a license to drive a computer can survey, map and create databases. No knowledge of the shape of the earth or the origin of coordinate systems and how they function or the gravity field of the earth needs to be known. The instruments will do all!

Errors and Uncertainty

In any survey, the biggest concern becomes the error that can be introduced into the observations. These errors are a function of the instrument precision, observation precision, and the precision with which values can be read off the instruments.

Uncertainties are introduced by the modeling of the shape of the earth and the map projections used to portray the surveyed area.

A prime concern in any survey is to keep the errors as small as possible and eliminate uncertainties as far as possible. A further concern is to know the probable magnitude of the errors and uncertainties left after adjustment of the observations.

Height Measurements

As explained earlier on, the geoid represents Mean Sea Level (MSL). All heights on maps are referred to MSL and thus to the geoid. The fact that the gravity field is not the same al over is not a concern for conventional surveying. With instruments working with positioning from satellites, like the Global Positioning System (GPS), it becomes a problem, since the heights calculated with those instruments are heights above the ellipsoid, which does not necessarily coincide with the geoid as was shown earlier. The separation between the geoid and ellipsoid is called the geoid height, normally indicated by the letter “N”. The height of the topography above the ellipsoid is called the ellipsoidal height “h” and the height of the topography above MSL the orthometric height “H”.

Figure 16: The relationship between Geoid height, Ellipsoidal height and Orthometric height. (

Obtaining heights above MSL with GPS becomes a bit of a guessing game. If a geoid model like the one in Figure 5 is available, N can be interpolated for the points surveyed by GPS. The problem with that strategy is the order of accuracy. The best accuracy that can be expected from such a model is in the order of ± 50 cm.

Another strategy is to model the geoid in the area. This is done by determining the MSL heights of a number of well-distributed points in the area, of which the GPS heights are known, with conventional heighting instruments. These are then used as modeling points.

Fortunately the Directorate of Surveys and Mapping, in the Department of Land Affairs, has a fairly well-maintained network of height benchmarks around the country. The MSL heights of these benchmarks have been obtained using precise leveling techniques and their accuracies, if undisturbed, are in the millimeter order.

From these benchmarks, if any are available within a convenient distance, the heights of the points in the area can be derived with careful leveling. By comparing the GPS heights and leveled heights at the points the value of N can be calculated at each modeling point. A best-fit surface for these modeling points can then be calculated. It can be a first, second or third-order surface, depending on the number of points available. The surface equations are then used to interpolate N for all the other points.

Sometimes it may not be necessary to have accurate MSL heights, but precise orthometric height differences. In that case, one point in the area can be assigned a height from a geoid model and GPS height. The same procedure as above can then be followed to interpolate an orthometric height difference correction surface.

Survey Design

The design of a survey is the thought process that goes into the creation of a survey procedure to produce a map of the required accuracy in position and height. It takes cognisance of the optimal layout of control points and observation procedures to minimise error and fit the survey as best as possible to the accepted model of the shape of the earth.

It is not a random process, since all the data that has to be gathered and the position and heights of the mapped features must fit into an integrated picture of the terrain, which in turn must fit into the overall picture of the surrounding area.

The surveyor normally has to work from the whole to the part, i.e. he has to work from the outside of the area that has to be surveyed to the detail inside it. This entails a control network that is fixed into the surrounding areas and used to fill in the details of the survey area. Any other procedure will cause the measurement errors to accumulate without the surveyor knowing their magnitude. By working inwards, the magnitude of the measurement errors can be computed and the observations can be adjusted to eliminate most of the error, while the probable magnitude of the remaining errors can be computed.

Control Points

The surveyor will start his survey from points with known coordinates outside the area of interest from where he will establish new control points in the area where the survey is to be conducted. It is necessary that the control points are evenly spread around the area for any sort of accuracy to be established.

Figure 17: Control Network for Establishing New Control Points A and B

Precision of the Survey

The surveyor needs to know what precision is required in the positioning of points in the area. The higher the precision, the more the control points and the higher must be the standard and precision of the observations.

For deformation surveys, a very high precision is required since very small movements must be detected. In such a case the surveyor might even resort to very fine measuring equipment like micrometers. For road construction surveys, the requirements may be less rigid.

The point is that the survey must be designed in such a way that proof can be obtained that the required precision has been obtained. This is normally the residuals from a least squares solution of the observation equations constructed from the observed measurement values. It is also customary in very precise surveys to do a pre-analysis to determine the expected accuracy.


That then, is the rigorous mathematical environment the surveyor operates in. In short, he must be competent in moving between various models of the shape of the earth, various coordinate systems, various survey designs and procedures with a variety of
instrumentation. Above all he must continuously apply his knowledge of his instruments and his science of the earth processes, like gravity, to anticipate and eliminate errors.


Chief Directorate Surveys & Mapping, n.d.: Hartbeesthoek 94: The new South African Datum,

Knippers, R. 2006: Geometric Aspects of Mapping, Merry, C.L., 2003: The African Geoid Project and Its Relevance to the Unification of African Vertical Reference Frames, Proceedings of the 2nd FIG Regional Conference Marrakech, Morocco, December 2-5, 2003.

* References to picture origins are given with the pictures.

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