On Centering Errors in the Observation of Short Directions

B. I. Bolshanin

When measuring angles formed by short sides, centering errors can introduce large errors into the results of observations.

Errors in centering the instrument over a point are divided into random and systematic errors.

Random errors in centering an instrument with an ordinary (plumb-line) plummet may be caused by oscillations of the plummet, eccentricity of its point, and the inconvenience of observing the plummet at ground level. Even under favorable conditions these errors may amount to 1–1.5 mm. When optical centering devices are used, random errors in centering the instrument consist of errors in sighting the point and in leveling the instrument. With a tripod height of 1.1–1.2 m, they amount to 0.2–0.3 mm.

Systematic errors with an ordinary plummet are due to the plumb line not coinciding with the rotation axis of the instrument brought into working position. With an optical centering device they arise from inaccurate setting of the centering device’s reticle.

In the geodetic literature there are no recommendations concerning a check of the mutual position of the plumb line and the instrument’s rotation axis—this question has not arisen. It is recommended, however, that the optical centering device be checked as follows. By means of the telescope of an adjusted instrument, two alignments are set out on the floor, and the point of their intersection is taken as the center of the instrument. The accuracy of such a determination of the true center of the instrument is not high; under the most favorable conditions the error may be 2–3 mm. With such centering accuracy, the error of a short direction (about 5 m) will be 1.5′–2′ and, obviously, in many practical cases will be gross.

Another method of checking an optical centering device is also known. The theodolite is brought into working position, and with a pencil on the tripod head the position of the tribrach—a metal plate—is marked, while on the floor the projection of the centering device’s reticle is marked. Then the tripod is rotated by 120°, the tribrach is aligned with the marks on the tripod head, the instrument is leveled again, and the projection of the reticle is marked on the floor a second time. The projection of the centering device’s reticle is marked a third time after rotating the tripod by another 120° and carrying out the same operations for setting up the instrument. If the reticle of the optical centering device is set not on

together, then on the floor one obtains three points forming an equilateral triangle. The reticle of the centering device is then set at the center of the triangle. With a symmetrical tribrach this method can give good results—the error will not exceed 2 mm.

Both of these methods for checking an optical centering device can be satisfactory only in those cases when a centering error of 3–4 mm may be neglected. When measuring angles formed by short directions, such an error cannot be neglected. The present paper considers the question of how correct results of angular measurements of short directions can be obtained in the presence of an error in setting the reticle of the centering device, and how the centering error can be accurately determined.

If the measurement of angles is carried out in three sets with rotation of the theodolite tribrach after each set by \(120^\circ\), then the mean value will be free from the influence of the systematic centering error.

The centering error can be determined from the results of similar observations of four short (3–5 m) directions arranged approximately at intervals of \(90^\circ\). The linear and angular elements of the centering error are determined from the deviations of the results of measurements of angles close to \(180^\circ\) from their mean value. For the \(i\)-th set we shall have successively:

\[ 1)\quad e^i_{13}=\frac{\Delta \beta^i_{13}}{K_{13}}; \qquad 2)\quad e^i_{24}=\frac{\Delta \beta^i_{24}}{K_{24}}. \]

\[ 3)\quad K_{13}=\frac{\rho}{d_1}+\frac{\rho}{d_3}; \qquad 4)\quad K_{24}=\frac{\rho}{d_2}+\frac{\rho}{d_4}; \]

\[ 5)\quad \Delta \beta^i_{13}=\beta^i_{13}-\beta_{13}; \qquad 6)\quad \Delta \beta^i_{24}=\beta^i_{24}-\beta_{24}. \]

\[ 7)\quad e^i=\sqrt{(e^i_{13})^2+(e^i_{24})^2}; \qquad 8)\quad \operatorname{tg}\Theta'_i=\frac{e^i_{13}}{e^i_{24}}; \qquad 9)\quad \Theta_i=\Theta'_i+A_i \]

\(e^i_{13}\) and \(e^i_{24}\) are the components of the linear element of the centering error, perpendicular to directions 1–3 and 2–4 from the \(i\)-th set (1, 2, 3, and 4 are the numbers of the directions);

\(d_i\) is the distance from the instrument to the sighting points;

\(\beta^i\) is the angle measured in the \(i\)-th set;

\(\beta\) is the mean value of the angle from the three sets;

\(e^i\) is the linear element of the centering error in the \(i\)-th set;

\(\Theta_i\) is the angular element of the centering error;

\(A_i\) is the conventional azimuth of the tribrach, established according to the direction of the telescope of the optical centering device or of its clamping screw.

The signs of the components \(e^i_{13}\) and \(e^i_{24}\) will be the same as the signs of the differences \(\beta^i-\beta\), if \(\beta\) is taken to be the angle directed toward the positive direction of the coordinate axes. The coordinate axis \(x\) should be taken as coinciding with one of the measured directions.

Tables 1 and 2 present the observation data for four directions, obtained with the TT-4 theodolite No. 19669, and the corresponding computed values of the components of the linear element of the centering error.

On the basis of these data, the position of the center of the instrument relative to the observation point can be obtained graphically. This is simple and clear. In this case there is no need to determine the angular element of the centering error.

Table 1

Table 2

Figure 1 shows the layout of the observed points and the position of the coordinate axes. Figures 2, 3, and 4 give the graphical constructions of the positions of the instrument center $I_1$, $I_2$, $I_3$ and of the optical plummet tube $CW_1$, $CW_2$, $CW_3$ for sets I, II, and III. With the given choice of coordinate axes relative to the observed directions, the segments $e_{13}$ are laid off along the $Y$ axis, and $e_{24}$ along the $X$ axis. To compare the results obtained, they should be transferred onto a single waxed sheet (Fig. 5). When transferring, the points $I$ of the position of the optical-plummet tube in sets II and III were made to coincide with the line $CW_1$. The error triangle $(1—2—3)$ shows the accuracy of the work performed: the error in determining the position of the instrument axis in the given case is about $0,15\ \mathrm{mm}$.

Correction of the systematic error in instruments with an optical center is made by shifting the reticle; the projection of the reticle cross should be moved from point $I$ to point $C$. It is difficult to change the suspension point of an ordinary (string) plumb line; therefore, for short directions, observations should be performed in three sets, with the theodolite base rotated by $120^\circ$. When there is a large volume of work, observations should be made at one station in three sets and the elements of the systematic error obtained. At subsequent stations the observations may be made in only one set, introducing the corres-

Fig. 1.

In Fig. 2:

\(C\) — center of the signal
\(J\) — center of the instrument
\(CW\) — direction of the telescope’s optical center

Fig. 2.

Fig. 3.

Labels in the diagram: \(x\), \(y\), \(C\), \(W_2\), \(A_2 = 285^\circ\), \(J_3\), \(e''_{13}\), \(e''_{24}\).

Fig. 4.

Labels in the diagram: \(x\), \(y\), \(C\), \(W_3\), \(A_3 = 45^\circ\), \(J_3\), \(e''_{13}\), \(e''_{24}\).

the corresponding corrections to the observed directions or angles. It should be borne in mind that the linear element of the centering error of the optical plummet varies in direct proportion to the height of the tripod head above the station point.

In conclusion, we note that, in order to achieve high accuracy in centering by means of the optical plummet, the trainer’s clamping screw should be loosened before centering and tightened after centering.

Fig. 5.

Because of the existing play between the theodolite sleeve and the trainer sleeve, after the latter’s screw has been tightened the axis of rotation of the theodolite may tilt slightly. Before observations, the instrument must be leveled. After this, the crosshairs will move off the point over which the instrument is centered, but the centering will not thereby be disturbed.