Basic control surveys
Geodetic surveys involve such extensive areas that allowance must be made for the Earth’s curvature. Baseline measurements for classical triangulation (the basic survey method that consists of accurately measuring a base line and computing other locations by angle measurement) are therefore reduced to sea-level length to start computations, and corrections are made for spherical excess in the angular determinations. Geodetic operations are classified into four “orders,” according to accuracy, the first-order surveys having the smallest permissible error. Primary triangulation is performed under rigid specifications to assure first-order accuracy.
Efforts are now under way to extend and tie together existing continental networks by satellite triangulation so as to facilitate the adjustment of all major geodetic surveys into a single world datum and determine the size and shape of the Earth spheroid with much greater accuracy than heretofore obtained. At the same time, current national networks will be strengthened, while the remaining amount of work to be done may be somewhat reduced. Satellite triangulation became operational in the United States in 1963 with observations by Rebound A-13, launched that year, and some prior work using the Echo 1 and Echo 2 passive reflecting satellites. The first satellite specifically designed for geodetic work, Pageos 1, was launched in 1966.
A first requirement for topographic mapping of a given area is an adequate pattern of horizontal and vertical control points, and an initial step is the assembly of all such existing information. This consists of descriptions of points for which positions (in terms of latitude and longitude) and elevations above mean sea level have been determined. They are occasionally located at some distance from the immediate project, in which case it is necessary to expand from the existing work. This is usually done on second- or third-order standards, depending upon the length of circuits involved.
The accuracy of survey measurements can be improved almost indefinitely but only at increased cost. Accordingly, control surveys are used; these consist of a comparatively few accurate measurements that cover the area of the project and from which short, less accurate measurements are made to the objects to be located. The simplest form of horizontal control is the traverse, which consists of a series of marked stations connected by measured courses and the measured angles between them. When such a series of distances and angles returns to its point of beginning or begins and ends at stations of superior (more accurate) control, it can be checked and the small errors of measurement adjusted for mathematical consistency. By assuming or measuring a direction of one of the courses and rectangular coordinates of one of the stations, the rectangular coordinates of all the stations can be computed.
A system of triangles usually affords superior horizontal control. All of the angles and at least one side (the base) of the triangulation system are measured. Though several arrangements can be used, one of the best is the quadrangle or a chain of quadrangles. Each quadrangle, with its four sides and two diagonals, provides eight angles that are measured. To be geometrically consistent, the angles must satisfy three so-called angle equations and one side equation. That is to say the three angles of each triangle, which add to 180°, must be of such sizes that computation through any set of adjacent triangles within the quadrangles will give the same values for any side. Ideally, the quadrangles should be parallelograms. If the system is connected with previously determined stations, the new system must fit the established measurements.
When the survey encompasses an area large enough for the Earth’s curvature to be a factor, an imaginary mathematical representation of the Earth must be employed as a reference surface. A level surface at mean sea level is considered to represent the Earth’s size and shape, and this is called the geoid. Because of gravity anomalies, the geoid is irregular; however, it is very nearly the surface generated by an ellipse rotating on its minor axis—i.e., an ellipsoid slightly flattened at the ends, or oblate. Such a figure is called a spheroid. Several have been computed by various authorities; the one usually used as a reference surface by English-speaking nations is (Alexander Ross) Clarke’s Spheroid of 1866. This oblate spheroid has a polar diameter about 27 miles (43 kilometres) less than its diameter at the Equator.
Because the directions of gravity converge toward the geoid, a length of the Earth’s surface measured above the geoid must be reduced to its sea-level equivalent—i.e., to that of the geoid. These lengths are assumed to be the distances, measured on the spheroid, between the extended lines of gravity down to the spheroid from the ends of the measured lengths on the actual surface of the Earth. The positions of the survey stations on the Earth’s surface are given in spherical coordinates.
Bench marks, or marked points on the Earth’s surface, connected by precise leveling constitute the vertical controls of surveying. The elevations of bench marks are given in terms of their heights above a selected level surface called a datum. In large-level surveys the usual datum is the geoid. The elevation taken as zero for the reference datum is the height of mean sea level determined by a series of observations at various points along the seashore taken continuously for a period of 19 years or more. Because mean sea level is not quite the same as the geoid, probably because of ocean currents, in adjusting the level grid for the United States and Canada all heights determined for mean sea level have been held at zero elevation.
Because the level surfaces, determined by leveling, are distorted slightly in the area toward the Earth’s poles (because of the reduction in centrifugal force and the increase in the force of gravity at higher latitudes), the distances between the surfaces and the geoid do not exactly represent the surfaces’ heights from the geoid. To correct these distortions, orthometric corrections must be applied to long lines of levels at high altitudes that have a north–south trend.
Trigonometric leveling often is necessary where accurate elevations are not available or when the elevations of inaccessible points must be determined. From two points of known position and elevation, the horizontal position of the unknown point is found by triangulation, and the vertical angles from the known points are measured. The differences in elevation from each of the known points to the unknown point can be computed trigonometrically.
The National Ocean Service in recent years has hoped to increase the density of horizontal control to the extent that no location in the United States will be farther than 50 miles (80 kilometres) from a primary point, and advances anticipated in analytic phototriangulation suggest that the envisioned density of control may soon suffice insofar as topographic mapping is concerned. Existing densities of control in Britain and much of western Europe are already adequate for mapping and cadastral surveys.
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The techniques used to establish the positions of reference points within an area to be mapped are similar to those used in navigation. In surveying, however, greater accuracy is required, and this is attainable because the observer and the instrument are stationary on the ground instead of in a ship or aircraft that is not only moving but also subject to accelerations, which make it impossible to use a spirit level for accurate measurements of star elevations.
The technique of locating oneself by observations of celestial objects is rapidly going out of date. In practicing it, the surveyor uses a theodolite with a spirit level to measure accurately the elevations of the Sun at different times of the day or of several known stars in different directions. Each observation defines a line on the Earth’s surface on which the observer must be located; several such lines give a fix, the accuracy of which is indicated by how closely these lines meet in a point. For longitude it is necessary also to record the Greenwich Mean Time of each observation. This has been obtained since 1884 by using an accurate chronometer that is checked at least once a day against time signals transmitted telegraphically over land lines and submarine cables or broadcast by radio.
A more recent procedure for global positioning relies on satellites, whose locations at any instant are known precisely because they are being continuously observed from a series of stations in all parts of the world. The coordinates of these stations were established by very large scale triangulation based on a combination of radar observations of distances and measurements of the directions of special balloons or flashing satellites, obtained by photographing them at known instants of time against the background of the fixed stars.
The principal method of using satellites for accurate positioning is based on an application of the Doppler effect. A radio signal is transmitted at a steady frequency by the satellite, but a stationary observer detects a higher frequency as the satellite approaches and a lower one as it recedes. The speed of the frequency drop depends on the distance of the observer from the satellite’s track, so a determination of this speed provides a measure of that distance. At the instant of the satellite’s closest approach, the observed frequency is the same as that transmitted, so at that time the observer must be located somewhere along the line at right angles to the satellite’s track. Since this track over the Earth’s surface is accurately known at all times, these data define the observer’s position.
Establishing the framework
Most surveying frameworks are erected by measuring the angles and the lengths of the sides of a chain of triangles connecting the points fixed by global positioning. The locations of ground features are then determined in relation to these triangles by less accurate and therefore cheaper methods. Establishing the framework ensures that detail surveys conducted at different times or by different surveyors fit together without overlaps or gaps.
For centuries the corners of these triangles have been located on hilltops, each visible from at least two others, at which the angles between the lines joining them are measured; this process is called triangulation. The lengths of one or two of these lines, called bases, are measured with great care; all the other lengths are derived by trigonometric calculations from them and the angles. Rapid checks on the accuracy are provided by measuring all three angles of each triangle, which must add up to 180 degrees.
In small flat areas, working at large scales, it may be easier to measure the lengths of all the sides, using a tape or a chain, rather than the angles between them; this procedure, called trilateration, was impractical over large or hilly areas until the invention of electromagnetic distance measurement (EDM) in the mid-20th century. This procedure has made it possible to measure distances as accurately and easily as angles, by electronically timing the passage of radiation over the distance to be measured; microwaves, which penetrate atmospheric haze, are used for long distances and light or infrared radiation for short ones. In the devices used for EDM, the radiation is either light (generated by a laser or an electric lamp) or an ultrahigh-frequency radio beam. The light beam requires a clear line of sight; the radio beam can penetrate fog, haze, heavy rain, dust, sandstorms, and some foliage. Both types have a transmitter-receiver at one survey station. At the remote station the light type contains a set of corner mirrors; the high-frequency type incorporates a retransmitter (requiring an operator) identical to the transmitter-receiver at the original station. A corner mirror has the shape of the inside of a corner of a cube; it returns light toward the source from whatever angle it is received, within reasonable limits. A retransmitter must be aimed at the transmitter-receiver.
In both types of instrument, the distance is determined by the length of time it takes the radio or light beam to travel to the target and back. The elapsed time is determined by the shift in phase of a modulating signal superimposed on the carrier beam. Electronic circuitry detects this phase shift and converts it to units of time; the use of more than one modulating frequency eliminates ambiguities that could arise if only a single frequency had been employed.
EDM has greatly simplified an alternative technique, called traversing, for establishing a framework. In traversing, the surveyor measures a succession of distances and the angles between them, usually along a traveled route or a stream. Before EDM was available, traversing was used only in flat or forested areas where triangulation was impossible. Measuring all the distances by tape or chain was tedious and slow, particularly if great accuracy was required, and no check was obtainable until the traverse closed, either on itself or between two points already fixed by triangulation or by astronomical observations.
In both triangulation and traversing, the slope of each measured line must be allowed for so that the map can be reduced to the horizontal and referred to sea level. A measuring tape may be stretched along the ground or suspended between two tripods; in precise work corrections must be applied for the sag, for tension, and for temperature if these differ from the values at which the tape was standardized. In work of the highest order, known as geodetic, the errors must be kept to one millimetre in a kilometre, that is, one part in 1,000,000.
Though for sketch maps the compass or graphic techniques are acceptable for measuring angles, only the theodolite can assure the accuracy required in the framework needed for precise mapping. The theodolite consists of a telescope pivoted around horizontal and vertical axes so that it can measure both horizontal and vertical angles. These angles are read from circles graduated in degrees and smaller intervals of 10 or 20 minutes. The exact position of the index mark (showing the direction of the line of sight) between two of these graduations is measured on both sides of the circle with the aid of a vernier or a micrometer. The accuracy in modern first-order or geodetic instruments, with five-inch glass circles, is approximately one second of arc, or 1/3,600 of a degree. With such an instrument a sideways movement of the target of one centimetre can be detected at a distance of two kilometres. By repeating the measurement as many as 16 times and averaging the results, horizontal angles can be measured more closely; in geodetic surveying, measurements of all three angles of a triangle are expected to give a sum of 180 degrees within one second of arc.
In the most precise long-distance work, signaling lamps or heliographs reflecting the Sun are used as targets for the theodolite. For less demanding work and work over shorter distances, smaller theodolites with simpler reading systems can be used; targets are commonly striped poles or ranging rods held vertical by an assistant.
An extensive set of these measurements establishes a network of points both on the map, where their positions are plotted by their coordinates, and on the ground, where they are marked by pillars, concrete ground marks, bolts let into the pavement, or wooden pegs of varying degrees of cost and permanence, depending on the importance and accuracy of the framework and the maps to be based on it. Once this framework has been established, the surveyor proceeds to the detail mapping, starting from these ground marks and knowing that their accuracy ensures that the data obtained will fit precisely with similar details obtained elsewhere in the framework.
The actual depiction of the features to be shown on the map can be performed either on the ground or, since the invention of photography, aviation, and rocketry, by interpretation of aerial photographs and satellite images. On the ground the framework is dissected into even smaller areas as the surveyor moves from one point to another, fixing further points on the features from each position by combinations of angle and distance measurement and finally sketching the features between them freehand. In complicated terrain this operation can be slow and inaccurate, as can be seen by comparing maps made on the ground with those made subsequently from aerial photographs.
Ground survey still has to be used, however, for some purposes; for example, in areas where aerial photographs are hard to get; under the canopy of a forest, where the shape of the ground—not that of the treetops—is required; in very large scale work or close contouring; or if the features to be mapped are not easily identifiable on the aerial photographs, as is the case with property boundaries or zones of transition between different types of soil or vegetation. One of two fundamental differences between ground and air survey is that, as already mentioned, the ground survey interpolates, or sketches, between fixed points, while air survey, using semiautomatic instruments, can trace the features continuously, once the positions of the photographs are known. One effect of this is to show features in uniform detail rather than along short stretches between the points fixed in a ground survey.
The second difference is that in ground survey different techniques and accuracies may be adopted for the horizontal and vertical measurements, the latter usually being more precise. Accurate determinations of heights are required for engineering and planning maps, for example, for railway gradients or particularly for irrigation or drainage networks, since water in open channels does not run uphill.
The methods used for fixing locations within the horizontal detail framework are similar to, but less accurate than, those used for the primary framework. Angles may be measured with a hand-held prismatic compass or graphically with a plane table, or they may be estimated as right angles in the case of points that are offset by short distances from straight lines between points already fixed. Detail points may be located by their distances from two fixed points or by distance and bearing from only one.
The surveyor may record measurements made in the field and plot them there on a sketch board or in the office afterward, but if the country is open and hilly, or even mountainous, the plane table offers the best way of recording the data. A disadvantage of plane-table work is that it cannot be checked in the office, and so it requires greater intelligence and integrity of the surveyor. The plane table reached its most efficient form of use in the Survey of India, begun in 1800, in which large areas were mapped with it by dedicated Indian surveyors. It consists of a flat board that is mounted on a tripod so that it can be fixed or rotated around a vertical axis. It is set up over a framework point or one end of a measured baseline with its surface (which is covered with paper or other drawing medium) horizontal. It is turned until the line joining its location with another framework point or the other end of the baseline is parallel to the same line as drawn on the paper. This alignment is performed with the aid of an alidade, or sight rule, a straightedge fitted with simple sights. The alidade is then directed toward points on features that are to be fixed, and pencil rays are drawn along the sight rule toward them. The procedure is repeated at the other framework point or the other end of the baseline; the points where the rays intersect on the table will be the map positions of the features.
In surveying for engineering projects, more sophisticated instruments are employed to maximize accuracy. For example, distances may be measured by EDM or by tachymetry, a geometric technique in which the vertical distance on a graduated vertical staff, seen between two stadia hairs in the theodolite eyepiece, is a measure of the horizontal distance between the theodolite and the staff—usually 100 times the difference between the two readings. This method requires at least one assistant to move the staff from place to place. Modern surveying instruments combine a theodolite, EDM equipment, and a computer that records all the observations and calculates the height differences obtained by measuring vertical angles.
Aviation and photography have revolutionized detailed mapping of features visible from the air. An aerial photograph, however, is not a map. In the case of the House of Parliament and Westminster Bridge, London, for example, the tops of the towers would coincide with the corners of the foundations when mapped. In an aerial photograph, however, they would not, being displaced radially from the centre. An important property of vertical aerial photographs is that angles are correctly represented at their centres, but only there. Similar distortions are present in photographs of hilly ground. This problem may be dealt with in two principal ways, depending on the relative scales of the map and the photographs and on whether contours are required on the map. The older method, adequate for planimetric maps at scales smaller than the photographs, was used extensively during and after World War II to map large areas of desert and thinly populated country; mountainous areas could be sketched in, but the relief was not accurately shown.
As in ground survey, a framework of identified points is necessary before detailed mapping can be carried out from the air. The photographs are ordinarily taken by a vertically aligned camera in a series of strips (see Figure 1) in which each picture overlaps about 60 percent of the preceding one; adjacent strips overlap only slightly. The overlaps make it possible to assemble a low-order framework or control system based on small, recognizable features that appear in more than one photograph. In the simplest form of this procedure each photograph is replaced by a transparent template on which rays are drawn (or slots are cut) from the centre of the picture to the selected features. The angles between these rays or slots are correct, and slotted templates can be fitted together by inserting studs, which represent the features, into the appropriate slots and sliding the templates so that each stud engages the slots in all the pictures showing the corresponding feature. This operation ensures that the centres of the pictures and the selected features are in the correct relationship. The array of overlapping photographs can be expanded or contracted by sliding them about on the work surface as long as the studs remain engaged in the slots, so the assemblage can be positioned, oriented, and scaled by fitting it to at least two—preferably several—ground-control points identified on different photographs.
Figure 1: Photogrammetric photographs from two short, overlapping flight strips arranged for supplying mapping details. Photo-control points are shown on only one photograph; shading indicates a typical terrain feature such as a lake (see text).
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This technique may be extended by using two additional cameras, one on each side, aimed at right angles to the line of flight and 30 degrees below the horizontal. The photographs taken by the side cameras overlap those taken by the vertical one and also include the horizon; the effect is to widen the strip of ground covered and thus to reduce the amount of flying required. Points in the backgrounds of the oblique photographs can be incorporated in the overlapping array as before to tie the adjacent flight paths together. Photography from high-flying jet aircraft and satellites has rendered this technique obsolete, but before those advances took place it greatly facilitated the mapping of underdeveloped areas.
For the production of maps with accurate contours at scales five or six times that of the photographs, a more sophisticated approach is necessary. The ground-survey effort must be expanded to provide the heights as well as the positions of all the features employed to establish the framework.
In this technique the details within each segment of the map are based not on individual photographs but on the overlap between two successive ones in the same strip, proceeding from the positions and heights of features in the corners of each area. A three-dimensional model can be created by viewing each pair of consecutive photographs in a stereoscope; by manipulation of a specially designed plotting instrument, the overlapping area can be correctly positioned, scaled, and oriented, and elevations of points within it can be derived from those of the four corner points. These photogrammetric plotting instruments can take several forms. In projection instruments the photographs are projected onto a table in different colours so that, through spectacles with lenses of complementary colours, each eye sees only one image, and the operator visualizes a three-dimensional model of the ground. A table or platen, with a lighted spot in the middle, can be moved around the model and raised or lowered so that the spot appears to touch the ground while the operator scans any feature, even if it is located on a steep hillside. A pencil directly beneath the spot then plots the exact shape and position of the feature on the map. For contouring the platen is fixed at the selected height (at a scale adjusted to that of the model), and the spot is permitted to touch the model surface wherever it will; the pencil then draws the contour.
With more complex mechanical devices, rays of light reaching the aircraft taking the two photographs are represented by rods meeting at a point that represents the position of the feature of the model being viewed. With a complicated system of prisms and lenses the operator, as with projection instruments, sees a spot that can be moved anywhere in the overlap and up or down to touch the model surface, as described above. A mechanical or electronic system moves a pencil into the corresponding position on a plotting table to which the map manuscript is fixed.
With computerized analytic instruments the mechanical operation is limited to measuring coordinates on the two photographs, and the conversion to a three-dimensional model is performed entirely by the computer. It is possible with the most precise plotting instruments of either type to draw a map at four to six times the scale of the photographs and to plot contours accurately at a vertical interval of about one one-thousandth of the height from which the photographs were taken. With such analytic instruments the record can be stored in digital as well as graphic form to be plotted later at any convenient scale.
All these methods produce a line or drawn map; some of them also create a data file on disk or tape, containing the coordinates of all the lines and other features on the map. On the other hand, aerial photographs can be combined and printed directly to form a photomap. For flat areas this operation requires simply cutting and pasting the photographs together into a mosaic. For greater accuracy the centres of the photographs may be aligned by the use of slotted templates as described above to produce a photomap called a controlled mosaic.
A much more precise technique is based on the use of an orthophotoscope. With this device, overlapping photographs are employed just as in the stereoscopic plotter already described, but the instrument, rather than the manual tracing of the features and contours, scans the overlap and produces an orthophotograph by dividing the area into small sections, each of which is correctly scaled. This procedure is best applied to areas of low relief without tall buildings; the resulting maps can then be substituted for line maps in rural areas where they are particularly useful in planning resettlement in agricultural projects. Because no fair drawing is required, the final printed map can be produced much more quickly and cheaply than would otherwise be possible.
Learn how hydrographic surveyors use sonar technology and GPS to survey the topography of the seafloor for safe navigation in the North Sea
The use of hydrographic surveying to ensure safe navigation in the North Sea.
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Surveying of underwater features, or hydrographic surveying, formerly required techniques very different from ground surveying, for two reasons: the surveyor ordinarily was moving instead of stationary, and the surface being mapped could not be seen. The first problem, making it difficult to establish a framework except near land or in shoal areas, was dealt with by dead reckoning between points established by astronomical fixes. In effect a traverse would be run with the ship’s bearing measured by compass and distances obtained either by measuring speed and time or by a modern log that directly records distances. These have to be checked frequently, because however accurate the log or airspeed indicator and compass, the track of a ship or aircraft is not the same as its course. Crosscurrents or winds continually drive the craft off course, and those along the course affect the speed and the distance run over the ground beneath.
The only way a hydrographer could chart the seabed before underwater echo sounding and television became available was to cast overboard at intervals a sounding line with a lead weight at the end and measure the length of the line paid out when the weight hit the bottom. The line was marked in fathoms, that is, units of one one-thousandth of a nautical mile, or approximately six feet (1.8 metres).
Sounding by lead line is obviously very slow, especially in deep waters, and the introduction of echo sounding in the early 20th century marked a great improvement. It was made possible by the invention of electronic devices for the measurement of short intervals of time. Echo sounding depends on timing the lapse between the transmission of a short loud noise or pulse and its return from the target—in this case the bottom of the sea or lake. Sound travels about 5,000 feet (1,500 metres) per second in water, so that an accuracy of a few milliseconds in measurements of the time intervals gives depths within a few feet.
The temperature and density of water affect the speed at which sound waves travel through it, and allowances have to be made for variations in these properties. The reflected signals are recorded several times a second on a moving strip of paper, showing to scale the depth beneath the ship’s track. The echoes may also show other objects, such as schools of fish, or they may reveal the dual nature of the bottom, where a layer of soft mud may overlie rock. Originally only the depth that was directly beneath the ship was measured, leaving gaps between the ship’s tracks. Later inventions, which include sideways-directed sonar and television cameras, have made it possible to fill these gaps. While measurements of depths away from the ship’s track are not so accurate, the pictures reveal any dangerous objects such as rock pinnacles or wrecks, and the survey vessel can then be diverted to survey them in detail.
Modern position-fixing techniques using radar have made the whole process much simpler, for the ship’s location is now known continuously with reference to fixed stations on shore or to satellite tracks. Another modern technique is the use of pictures taken from aircraft or satellites to indicate the presence and shape of shoal areas and to aid the planning of their detailed survey.
An alternative to the use of radar or satellite signals for continuous and automatic recording of a ship’s position is the employment of inertial guidance systems. These devices, developed to satisfy military requirements, detect every acceleration involved in the motion of a craft from its known starting point and convert them and the elapsed time into a continuous record of the distance and direction traveled.
For studying the seabed in detail, the bottom of the sounding lead was hollowed to hold a charge of grease to pick up a sample from the sea floor. Today television cameras can be lowered to transmit pictures back to the survey ship, though their range is limited by the extent to which light can penetrate the water, which often is murky. Ordinary cameras also are used in pairs for making stereoscopic pictures of underwater structures such as drilling rigs or the wreckage of ancient ships.
Heights of surface features above sea level are determined in four main ways: by spirit leveling, by measuring vertical angles and distances, by measuring differences in atmospheric pressure, and, since the late 20th century, by using three-dimensional satellite or inertial systems. Of these the first is the most accurate; the second is next in accuracy but faster; the third is least accurate but can be fastest if heights are to be measured at well-separated points. The last two techniques require sophisticated equipment that is still very expensive.
In spirit leveling the surveyor has for centuries used a surveying level, which consists of a horizontal telescope fitted with cross hairs, rotating around a vertical axis on a tripod, with a very sensitive spirit level fixed to it; the instrument is adjusted until the bubble is exactly centred. The reading on a graduated vertical staff is observed through the telescope. If such staffs are placed on successive ground points, and the telescope is truly level, the difference between the readings at the cross hairs will equal that between the heights of the points. By moving the level and the staffs alternately along a path or road and repeating this procedure, differences in height can be accurately measured over long horizontal distances.
In the most precise work, over a distance of 100 kilometres the error may be kept to less than a centimetre. To achieve this accuracy great care has to be taken. The instrument must have a high-magnification telescope and a very sensitive bubble, and the graduated scale on the staff must be made of a strip of Invar (an alloy with a very small coefficient of thermal expansion). Moreover, the staffs must be placed on pegs or special heavy steel plates, and the distance between them and the level must always be the same to cancel the effects of aerial refraction of the light.
In less precise work a single wooden staff can be used; for detailed leveling of a small area, the staff is moved from one point to another without moving the level so that heights can be measured within a radius of about 100 metres. The distances of these points from the instrument can be measured by tape or, more commonly, by recording not only the reading at the central cross hair in the field of view of the telescope but also those at the stadia hairs, that is by tachymetry, as described above. The bearing of each point is observed by compass or on the horizontal circle of the level so that it can be plotted or drawn on the map.
Since the 1950s levels have been introduced in which the line of sight is automatically leveled by passage through a system of prisms in a pendulum, thus removing the need to check the bubble. The disadvantage of spirit leveling is the large number of times the instrument has to be moved and realigned, particularly on steep hills; it is used primarily along practically flat stretches of ground.
For faster work in hilly areas, where lower accuracies usually are acceptable, trigonometric height determination is employed using a theodolite to measure vertical angles and measuring or calculating the distances by triangulation. This procedure is particularly useful in obtaining heights throughout a major framework of triangulation or traverse where most of the points are on hilltops. To increase precision, the observations are made simultaneously in both directions so that aerial refraction is eliminated; this is done preferably around noon, when the air is well mixed.
The third method of height determination depends on measurements of atmospheric pressure differences with a sensitive aneroid barometer, which can respond to pressure differences small enough to correspond to a foot or two (0.3 to 0.6 metre) in height. The air pressure changes constantly, however, and to obtain reliable results it is necessary to use at least two barometers; one at a reference point of known height is read at regular intervals while the surveyor proceeds throughout the area, recording locations, times, and barometer readings. Comparison of readings made at the same time then gives the height differences.
An alternative to the barometer for pressure measurement is an apparatus for measuring the boiling point of a liquid, because this temperature depends on the atmospheric pressure. Early explorers determined heights in this way, but the results were very rough; this technique was not accurate enough for surveyors until sensitive methods for temperature measurement were developed. The airborne profile recorder is a combination of this refined apparatus with a radar altimeter to measure the distance to the ground below an aircraft.
Analysis of the signals received simultaneously from several satellites gives heights as accurately as positions. Heights determined in this way are useful in previously unmapped areas as a check on results obtained by faster relative methods, but they are not accurate enough for mapping developed areas or for engineering projects. All-terrain vehicles or helicopters can carry inertial systems accurate enough to provide approximate heights suitable for aerial surveys of large areas within a framework of points established more accurately by spirit leveling.