# Traversing

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## Traversing

a method of determining the relative positions of points on the earth’s surface in order to construct a geodetic control network for such purposes as topographic surveys, the planning and building of cities, and the laying out of engineering structures. In this method the positions of the points in a given coordinate system are determined by measurement of the lengths of the lines that successively connect the points and by measure

Figure 1. Traverse

ment of the horizontal angles between the lines; the series of lines is called a traverse. Thus, if the points 1, 2, 3, …, n, n + 1 are selected, measurements are made of the lengths s1, s2,…., sn of the lines between them and also of the angles β2,, β3, …, between the lines (Figure 1).

As a rule, point 1 of a traverse is made to coincide with a reference point PH, whose coordinates xH, yH are known and at which the direction angle αH of the direction to some adjacent point pHʹ is also known. The adjacent angle β1 between the first leg of the traverse and the initial direction PH PHʹ is also measured at the initial point of the traverse— that is, at the station PH. The direction angle αi of leg i and the coordinates xi+1, yi+1 of station i + 1 can then be calculated from the formulas

The accuracy of traverse measurements is checked and estimated by having the final points n + 1 of the traverse coincide with a second reference point Pk, whose coordinates xk, yk are known and at which the direction angle αk of the direction to an adjacent point Pkʹ is also known. This method permits calculation of the angular and linear errors of closure, which depend on the errors associated in the measurement of the line lengths and angles and are expressed by the formulas

fα = αn+1 – αk

fx = xn+1 – xk

fy = yn+1 – yk

These errors are eliminated by adjusting the measured angles and leg lengths with corrections determined from balancing computations that use the least squares method.

If the area over which a geodetic control network is to be constructed is large, interconnecting traverses that form a traverse net are laid out (Figure 2).

Figure 2. Traverse net

Traverse stations are stabilized in place by setting into the ground concrete monuments or metal pipes with anchors and by erecting bench marks in the form of pyramidal towers made of wood or metal.

Traverse angles are measured by means of theodolites and transits; the objects of observation are usually special marks set up at the stations being sighted. The lengths of the legs of traverses and traverse nets are measured with steel or invar tapes or wires. Suitable corrections are made in the length and angle measurements in order to obtain the measurement results in the coordinate system in which it is required that the positions of the traverse stations be determined.

Figure 3. Determination of the length of a traverse leg by the trig-traverse method

When local conditions do not permit direct measurement of lines, the lengths of traverse legs can be determined indirectly by the trig-traverse method. In this case, the length of the leg IK is determined by measuring the length b of the short base line AB and the angles Φ1 and Φ2 that are located at the end points of IK and are subtended by AB. AB and IK are perpendicular to each other and intersect at their midpoints; the magnitudes of Φ1 and Φ2 are usually about 3° to 6° (Figure 3). The length of IK is calculated by the formula

Other indirect methods for the measurement of traverse legs are also used, depending on the local conditions.

Traverses and traverse nets are divided into classes corresponding to triangulation classes, in accordance with the degree of precision and the sequence of construction. The precision indexes characterizing the various classes of state traverse nets are given in Table 1. Somewhat different precision indexes may be had by nets constructed, for example, for engineering purposes or for surveys in urban areas.

Table 1
ClassAngle errorLeg error
1±0.4±1:300,000
2±1.0±1:250,000
3±1.5±1:200,000
4±2.0±1:150,000

The origin of the traverse method is not known. In the past, use of the method was limited because traversing required a great number of linear measurements that were difficult to perform owing to local conditions, the unwieldiness of the necessary equipment, and the impossibility of checking results before the completion of the survey. For this reason, the traverse method was in the past used only for surveys in urban areas and for local adjustment of control networks constructed by triangulation.

With the appearance of invar tapes in the early 20th century, linear measurements were made easier, their precision was increased, and they became less dependent on local conditions. As a result, traversing has become as important and precise as triangulation. An important role in the development of traversing was played by the Russian geodesist V. V. Danilov, who worked out in detail the trig-traverse method, which had been outlined by V. Ia. Struve as early as 1836. The invention of electronic distance-measuring instruments that make use of radio or light waves made possible highly precise direct measurements of traverse lines. Traversing was thereby freed of its principal drawback and is now used as often as triangulation. The work of the Soviet geodesists A. S. Chebotarev and V. V. Popov has been of great importance in the development of the theory and techniques of traversing. These two scientists have developed efficient methods of conducting various types of traverse surveys at different levels of precision, as well as methods for computing and estimating the error of the results.

### REFERENCE

Spravochnik geodezista. Edited by V. D. Bol’shakov and G. P. Levchuk. Moscow, 1966.
Danilov, V. V. Tochnaia poligonometriia, 2nd ed. Moscow, 1953.
Krasovskii, F. N., and V. V. Danilov. Rukovodstvo po vysshei geodezii, part 1, fasc. 2. Moscow, 1939.
Chebotarev, A. S., V. G. Selikhanovich, and M. N. Sokolov. Geodeziia, part 2. Moscow, 1962.
Chebotarev, A. S. Uravnitel’nye vychisleniia pri poligonometricheskikh rabotakh. Moscow-Leningrad, 1934.
Popov, V. V. Uravnoveshivanie poligonov, 9th ed. Moscow, 1958.
Kuzin, N. A., and N. N. Lebedev. Prakticheskoe rukovodstvo po gorodskoi i inzhenernoi poligonometrii, 2nd ed. Moscow, 1954.
Instruktsiia 0 postroenii gosudarstvennoi geodezicheskoi seti SSSR, 2nd ed. Moscow, 1966.

A. A. IZOTOV

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