TOPOGRAPHIC SURVEYING WITH A HORIZONTAL RAY

B. I. BOLSHANIN

(Presented by the scientific conference of the Mining and Exploitation Faculty).

In connection with the growth in the volume of hydromelioration works, the construction of industrial and other enterprises, and housing and municipal construction, especially in areas of virgin and fallow lands, large-scale topographic surveys (1:10000 and larger) are acquiring great importance. In the mining industry, large-scale surveys are fundamental.

A special place in these works is occupied by plane-table surveying. Among ground surveys it is the most profitable and gives comparatively high accuracy, especially in depicting relief; consequently, serious attention must be paid to increasing labor productivity in this type of work.

Recently a collection of articles by innovators in topographic-geodetic production [1] was published, in which field topographers share their experience in rationalizing field work in terrain surveying. In particular, attention is given there to the question of applying a horizontal line of sight in topographic surveying.

In the set of field operations for plane-table surveying, a large place is occupied by the so-called taking of stadia points, consisting of sighting the telescope on the rod, determining the distance from the rod, and determining the elevation of the rod point. The process of obtaining the elevations of rod points by means of trigonometric leveling (with an inclined line of sight) is the most labor-intensive. It consists of the following operations: 1) sighting the telescope with the middle thread of the reticle at a definite reading on the rod or at a mark indicating the height of the instrument, 2) bringing the level bubble at the vertical circle to the center, 3) taking a reading on the vertical circle, 4) computing the angle of inclination of the line of sight, 5) computing the difference in elevation from a table or nomogram, and, finally, 6) computing the elevation of the rod point.

Thus, the process of determining point elevations consists of six operations. The recording of readings and the computations are performed by a calculator who has a certain skill.

The use of a horizontal line of sight in taking stadia points makes it possible to reduce the number of operations from six to four, or even to three if a special rod is used. In this case the most labor-intensive and inconvenient stages of the work are eliminated: reading the vertical circle and computing differences in elevation.

In 1954 the Department of Geodesy of the Tomsk Polytechnic Institute carried out experimental work on taking stadia points in plane-table surveying with the use of a horizontal line of sight. The survey was conducted at a scale of 1:2000 on comparatively level terrain. In this work a Kipregel with a level attached to the telescope and a special rod with a movable scale of graduations were used.

This article deals with questions related to the use of a horizontal line of sight in plane-table surveying: checking a Kipregel with a level on the telescope, the use of reversing levels, the design of a special staff, and timing data for taking spot heights.

Setting the line of sight in a horizontal position and the accuracy of leveling

In carrying out the experimental work, a Fennel Kipregel No. 11411 was used, with a reversing (turning) level on the telescope, characterized by the following data: focal length of the objective 250 mm, telescope magnification \(18\times\), value of one division of the telescope level \(33''\), value of one division of the level on the vertical circle \(72''\), and vernier precision \(1'\).

To bring the line of sight into a horizontal position, it is necessary that the condition of parallelism between the line of sight and the axis of the telescope level be satisfied. Satisfying this condition involves carrying out double leveling, which takes much time and, undoubtedly, affects labor productivity.

The use of reversing levels eliminates the need to use double leveling under field conditions at each station. Let us consider how, in this case, the condition of parallelism between the line of sight and the axis of the level is fulfilled.

In a reversing level there are two axes. In principle, these axes should be parallel. However, they may also be nonparallel; therefore, let us consider both cases.

First case: the axes of the level are parallel.

Let in Fig. 1 \(o_1 o_2\) be the axis of the level (either one), \(v_1 v_2\) the line of sight of the telescope, \(b_0\) the staff reading corresponding to the horizontal position of the line of sight, \(b'\) and \(b''\) the staff readings taken at different positions of the vertical circle, and \(x\) the error in the staff reading due to the nonhorizontality of the line of sight.

Fig. 1 a and b.

From the drawing it is evident that

\[ x = \frac{1}{2}(b' - b''), \tag{1} \]

\[ b_0 = \frac{1}{2}(b' + b''). \tag{2} \]

To establish parallelism between the axis of the level and the line of sight, it is sufficient to sight the telescope at the reading \(b_0\) and, with the adjusting screw of the level, bring its bubble to the middle. Such a check can be performed very quickly.

Second case: the axes of the level are nonparallel.

In Fig. 2, \(o_1o_2\) and \(o'_1o'_2\) are the axes of the level, \(\nu\) is the angle between the axes of the level, \(\varepsilon_1\) and \(\varepsilon_2\) are the angles between the line of sight and the axes of the level, \(b'\) and \(b''\) are the readings on the rod set up at point \(B\), \(i_a\) is the height of the instrument above point \(A\), and \(x\) and \(y\) are the errors in the rod readings caused by the non-horizontality of the line of sight when the circle is to the right and to the left (CR and CL).

According to the drawing, we write the following relations:

\[ \text{CL (level above)} \qquad h = i_a - b' + x, \tag{a} \]

\[ \text{CR (level below)} \qquad h = i_a - b'' + y. \tag{b} \]

After moving from point \(A\) to point \(B\), we shall have:

\[ \begin{aligned} \text{CL} \qquad -h &= i_b - a' + x, \tag{c}\\ \text{CR} \qquad -h &= i_b - a'' + y, \tag{d} \end{aligned} \]

where \(i_b\) is the height of the instrument above point \(B\);
\(a'\) and \(a''\) are the readings on the rod set up at point \(A\).

Fig. 2 a and b.

From relations \((a)\) and \((c)\), and from \((b)\) and \((d)\), we obtain the following formulas:

\[ x = \frac{1}{2}(a' + b') - \frac{1}{2}(i_a + i_b), \tag{3} \]

\[ y = \frac{1}{2}(a'' + b'') - \frac{1}{2}(i_a + i_b). \tag{4} \]

Control:

\[ x - y = a' - a'' = b' - b''. \]

The angles between the line of sight and the axes of the level will be equal to

\[ \varepsilon_1 = \frac{x}{d}\rho; \qquad \varepsilon_2 = \frac{y}{d}\rho; \tag{5} \]

\[ \nu = \varepsilon_2 - \varepsilon_1 \]

or

\[ \nu = \frac{\rho}{2d}\left[(a'' + b'') - (a' + b')\right], \tag{6} \]

where \(d\) is the distance from the instrument to the rod.

From formula (6) it is seen that the error in determining the angle between the axes of the level is equal to the error of the rod reading, expressed in angular measure,

\[ m_\nu = m_0. \]

But the error of the rod reading is composed of the sighting error \(m_v\) and the error of bringing the level bubble to the center \(m_y\); therefore,

\[ m_\nu=\sqrt{m_v^2+m_y^2}. \tag{7} \]

Calculations show that the error in determining the angle \(\nu\) by one observation, with telescope magnification \(18\times\) and a level division value of \(33''\), is approximately \(8''\)—\(10''\).

The possibility of using a reversible level, whose axes are nonparallel, will depend first of all on the magnitude of the angle \(\nu\). If the angle \(\nu\) is very large, then the level may be used as an ordinary one. What, then, is the limiting value of the angle \(\nu\) at which the reversible level can be used for its intended purpose?

Fig. 3.

In order to answer this question, let us make a small calculation. Suppose that the sighting axis is parallel to the bisector of the angle between the axes of the level; then the error in the rod reading, according to the drawing (Fig. 3), will be

\[ \Delta b=\frac{\nu}{2\rho}\,d. \]

Bearing in mind that the error in the rod reading must not exceed a certain value, we find at what angle \(\nu\) the nonparallelism may be neglected:

\[ \nu \leq 2\,\frac{\Delta b}{d}\,\rho. \tag{8} \]

Table 1 gives values of the angle \(\nu\), computed by formula (8), for various values of \(\Delta b\).

Table 1

It follows from this table that, before leaving for field work, it is necessary to determine the value of the angle \(\nu\) and establish the limits for using the level as a reversible one.

On the other hand, knowing the angle between the axes of the level, one can, as a convention, shift the zero point on one of its sides by the number of divisions corresponding to this angle, after which the axes of the level will be parallel. A shift of the zero point by 5–10 divisions, with a division value of \(30''\)—\(35''\), will change the position of the level axis by \(150''\)—\(350''\). It is more convenient for work for the zero-

point should be shifted on the side that faces the telescope. Naturally, with a very large nonparallelism of the axes, the zero point can be shifted on both sides of the level.

Let us now compute the error in the elevation difference of points leveled by a horizontal line of sight.

The error of setting the line of sight in a horizontal position, \(m_{\Gamma}\), with parallel axes of the level, will be composed of the error of setting the parallelism of the line of sight with the level axis, \(m_{\Pi}\), and of setting the level bubble to the middle, \(m_y\):

\[ m_{\Gamma}=\sqrt{m_{\Pi}^{2}+m_y^{2}}. \]

The error of setting the parallelism of the line of sight with the level axis depends on the error of setting the line of sight to the mean reading, \(b_o - m_{bo}\), and on the error of bringing the level bubble to the middle,

\[ m_{\Pi}=\sqrt{m_{bo}^{2}+m_y^{2}} \]

and then

\[ m_{\Gamma}=\sqrt{m_{bo}^{2}+2m_y}. \]

But, according to formula (2),

\[ m_{bo}=m_b\sqrt{2}=\sqrt{2m_v^{2}+2m_y^{2}}, \]

where \(m_b\) is the error of the actual rod reading, equal to \(\sqrt{m_v^{2}+m_y^{2}}\);
\(m_v\) is the sighting error,

and then the error of bringing the line of sight into the horizontal position will be

\[ m_{\Gamma}=\sqrt{2m_v^{2}+4m_y^{2}}, \]

while the error in the rod reading caused by the error of setting the line of sight in the horizontal position and by the error of the actual rod reading will be equal to

\[ m_0=\sqrt{m_{\Gamma}^{2}+m_v^{2}} \]

or

\[ m_0=\sqrt{3m_v^{2}+4m_y^{2}}. \tag{9} \]

The error in determining the elevation difference may be computed by the formula

\[ m_h=\sqrt{3m_v^{2}+4m_y^{2}+m_i^{2}}, \tag{10} \]

where \(m_i\) is the error of determining the height of the instrument.

When leveling from the middle this error will be considerably smaller; it will be determined by the formula

\[ m_h=\sqrt{2m_v^{2}+2m_y^{2}}. \tag{11} \]

Table 2 gives data for computing the magnitude of the errors calculated by formulas (9) and (11), in which it is assumed that: \(m_v=3'',3\); \(m_y=0.15\tau\)—\(0.2\tau\), and the factor

\[ \frac{d}{\rho}. \]

is introduced.

For comparison, let us consider what error in the elevation difference we shall have when setting the line of sight in the horizontal position by the vertical circle and the level attached to it.

The error of setting the line of sight in the horizontal position will depend on the errors: of reading the vertical circle, \(m_k\), of determining the position of the zero, \(m_{MO}\), and of bringing the level bubble to the middle, \(m'_y\),

\[ m'_{\Gamma}=\sqrt{m_k^{2}+m_{MO}^{2}+m_y'^{\,2}}. \tag{12} \]

The position of the zero is determined by the formula \(MO=\dfrac{1}{2}(R+L)\), therefore:

\[ m_{MO}=\sqrt{\frac{1}{4}\left(m_R^2+m_L^2\right)}, \]

but

\[ m_R=m_L=\sqrt{m_k^2+m_y'^2+m_v^2} \]

and then

\[ m_{MO}=\sqrt{\frac{1}{2}\left(m_k^2+m_y'^2+m_v^2\right)}. \]

Substituting the value \(m_{MO}\) into (12), we obtain

\[ m'_{\Gamma}=\sqrt{\frac{3}{2}\left(m_k^2+m_y'^2\right)+\frac{1}{2}m_v^2}. \tag{12a} \]

The error in the staff reading is

\[ m_o=\sqrt{m_{\Gamma}'^{\,2}+m_v^2} \]

or

\[ m_o=\sqrt{\frac{3}{2}\left(m_k^2+m_y'^2\right)+m_v^2}. \tag{13} \]

To obtain the error in the elevation, it is sufficient to add the error in determining the height of the instrument to the reading error.

The values of the errors in the reading, calculated by formula (13), are given in Table 2, in which it is assumed that: \(m_k=30''—60''\), \(m_y'=15''\), \(m_v=4''\), and the factor \(\dfrac{d}{\rho}\) has been introduced.

Table 2

The calculation presented in Table 2 shows that the use of a level on the alidade tube considerably increases the accuracy of determining elevations and, when constructing the working control for a survey, may to some extent replace geometric leveling.

Setting the line of sight to the horizontal position by means of the vertical circle and the level attached to it is performed much more roughly. Determining elevations with the line of sight set to the horizontal position by means of the vertical circle will be carried out with the same accuracy as in trigonometric leveling. Nevertheless, surveying with a horizontal ray set by the vertical circle should be recommended, since in this case labor productivity can be significantly increased by reducing certain stages of the work and computations.

Construction and Use of the Stadia Rod

In carrying out the experimental work, a special rod was used, made in the mechanical workshop of the department. It consists of two wooden bars measuring \(1.5 \times 6.5\) cm and 3 m long each—a sliding rod whose total length can be set from 3.1 to 5.8 m. The wooden bars of the rod have no graduations. To determine distances by stadia and to determine elevations, a tape rod with graduations (checker marks) applied to it is stretched over the surface of the bars. At the lower end of the front bar and at the upper end of the rear bar, at a distance of 5 cm from the ends, rollers are set into the body of the bars, over which the tape rod is stretched. The ends of the tape rod are fastened by means of special clamps located near the rollers (Fig. 4).

The tape rod was made by us from two standard four-meter tapes with centimeter checker marks applied to them. The tapes are joined at their zero points, so that on the upper part of the tape the graduation numbers increase upward, and on the lower part downward (Fig. 5). In general, the coloring of the tape rod may be of any type used for stadia rods.

If, when taking pickets, the tape on the sliding wooden rod is fastened so that the zeros of both tapes are at the height of the instrument, then the reading on the rod

Fig. 4.

Roller
Clamp for the tape

Fig. 5.

with a horizontal line of sight will be equal to the elevation of the rod point above the point on which the plane table is set. In this case, if the reading is taken on that part of the tape on which the graduation numbers increase downward (in the telescope upward), the elevation will be positive; but if the reading is taken where the graduations increase upward (in the telescope downward), the elevation will be negative.

To avoid errors in the sign of elevations, tapes should be used with different coloring of the checker marks, or with different shapes of the marks, or, finally, with different backgrounds on which the checker marks are applied. In our experiments, we used tapes with different backgrounds: white and light blue.

The drawbacks of the tape rod include its windage and stretching, accompanied by residual deformation.

The windage of the tape will not be a major hindrance in the work, since the tape is under tension. In order that the upper part of the tape, where it does not touch the wooden bars, not vibrate from the wind, additional devices in the form of small clamps may be made. The ends of the tapes should be rolled up and fastened with special spring clamps.

Obviously, a major drawback of the tape rod will be its capacity to stretch under the influence of load. Stretching of the tape is accompanied by residual deformation, the magnitude of which depends on the size of the load, on the duration of the action of this load, on the moisture content of the tape, and on the time elapsed after removal of the load.

Tests of the tape rod, carried out under laboratory conditions, showed that the stretching of a dry tape is almost uniform and amounts on average to about 1:500 for each kilogram of load (the tests were carried out with a load of up to six kilograms).

Residual deformation, like stretching, is greater if the tape is new and has not been subjected to stretching. The residual deformation of a dry tape that has been subjected to stretching, under a short action of load (10–15 minutes), is about 1:2000–1:3000; the residual deformation of a wet tape is two to three times greater. Under prolonged action of the load (one or two days), the residual deformation of a dry tape is about 1:1500, and of a wet tape likewise two to three times greater.

It follows from the foregoing that the tape rod, stretched over wooden bars, can be recommended as a rangefinder rod in topographic surveying. In doing so, certain precautions must be observed: do not leave the tape unnecessarily for a long time under great tension, especially if it is wet. In the practice of topographic-survey work this condition can be observed without particular difficulty, since each time when moving from one station to another there will be a different instrument height, to which the zero division of the tape must be set. At night the tape should be removed from the bars and dried. For convenience in work, on the front side of the front bar of the wooden rod it is necessary to set off a definite length (2.0–2.5 m)—to make marks and use them to regulate the tension of the tape.

Timing of Work in Taking Pickets

During a production topographic survey at a scale of 1:2000, a timing study was carried out of the work of taking pickets. Plane-table surveying was performed by one topographer both when working with a horizontal line of sight—a Kipregel with a level on the tube—and with an inclined line of sight—trigonometric leveling. A full timing study was carried out at 19 stations; at 12 stations the survey was conducted with a horizontal line of sight, and at 7 with an inclined one. The working conditions were chosen, as far as possible, to be identical both with respect to the terrain and with respect to meteorological and other factors affecting labor productivity. Timing was conducted at stations where almost the same relief was being surveyed—the number of situation points was small, and they were surveyed in the second instance.

Table 3 gives the timing data, which show that surveying with a horizontal line of sight and the use of a rod with a movable tape scale make it possible to increase labor productivity considerably. In the taking of pickets alone, the saving of time averages 0.8 minutes per relief point, or about 20–25%. If it is assumed that the taking of pickets accounts for about 50% of the total time spent

Table 3

for the survey, then the labor productivity of surveying with the use of a horizontal line of sight may be higher by 10–12%. The same result is obtained if the calculation is made in another way. The production norm for surveying at a scale of 1:2000 for terrain of category II and a relief interval of 0.5 m is 11 ha per day [2]. To depict the relief when surveying this area, it is necessary to collect, on average, 60–70 elevation rod points. The saving of time at each rod point is 0.8 minutes; consequently, the total saving of time will be 50–55 minutes, which will give an increase in labor productivity of 11–13%.

Of course, the data on the basis of which the calculation has been made are insufficient—they do not cover the various working conditions, and therefore the conclusions drawn need confirmation. However, the very fact of an increase in labor productivity with the indicated methods of work is beyond doubt, and the expediency of using a horizontal line of sight in topographic surveying where the relief is not sharply expressed is obvious.

Conclusions

The use of a horizontal line of sight in large-scale topographic surveying (1:10000 and larger) is the next step both in increasing labor productivity and in increasing the accuracy of topographic surveying operations.

The use of a level on the alidade telescope and of a rod with a movable scale frees surveyors from taking readings on the vertical circle, from computing angles of inclination, elevations, and horizontal distances from tables or nomograms. All this leads to a significant reduction in errors that are inevitable in computations and readings. An experienced computer—the topographer’s assistant—can be replaced by a less qualified worker, and in individual cases the topographer can manage even without an assistant.

The use of reversible levels on the alidade telescope will spare the topographer the need to carry out double leveling at each station, which takes a great deal of time. With an alidade equipped with a level attached to the telescope, elevations and elevations of points can be obtained with considerably greater accuracy than in trigonometric leveling. The data presented in Table 2 show that leveling with an alidade by the method from the middle gives an error of about 10–15 mm per kilometer of traverse (without taking into account errors due to tape stretching and deformation), which corresponds to the accuracy of fourth-class geometric leveling. Consequently, such leveling may be used to obtain the elevations of points of the control for the work.

An extensible rod with a movable scale of divisions applied to an adhesive tape makes it possible to vary its dimensions from 3.1 to 5.8 m (with rods 3 m long) and to set the zero of the scale easily at the height of the instrument. Setting the zero of the scale at the height of the instrument and increasing the divisions downward and upward from zero has the advantage that no computations need be made, since the reading on the rod will be equal to the elevation of the picket above the station point.

To increase the radius of surveying at a station, the place for setting up the plane table must be chosen on an elevation.

Thus, topographic surveying with a horizontal line of sight, using a special rod, makes it possible to increase labor productivity by 10–15% in comparison with surveying by an inclined line of sight and with an ordinary stadia rod. Surveying with a horizontal line of sight is most profitable where the terrain relief is weakly expressed, but it may also be recommended for surveying terrain with comparatively pronounced relief. In this case, surveying with a horizontal line of sight can be combined with surveying by an inclined line of sight; in the latter case the need to determine the place of zero by the usual method is eliminated—it will be determined much more simply.

REFERENCES

1. Experience of the Work of Leading Workers in Topographic Surveying at a Scale of 1:10000. Collection of articles, Geodezizdat, 1954.

2. Unified Production Norms for Topographic and Geodetic Work. Geodezizdat, 1949.