HIGH-PRECISION GEOMETRIC LEVELING OVER SHORT DISTANCES

V. M. ELIZAROV, R. V. BUZUK

With the development of technology, it is increasingly necessary to install foundations for various units with very high accuracy in height, on the order of $\pm 0.1$–$0.2\ \text{mm}$.

To carry out such work, hydrostatic leveling is used, based on the principle of communicating vessels. This type of leveling has high accuracy, but is very cumbersome and takes much time. Hydrostatic leveling cannot be applied to points with a large difference in elevation. The question arises: is it possible, for this purpose, to use ordinary geometric leveling with a sight length, say, up to $10\ \text{m}$? For this it is necessary to have a precision level with a plane-parallel plate and a contact level vial, of the NB or NA-1 type, and a special rod. An ordinary rod with half-centimeter graduations is not suitable for such leveling, since the accuracy of marking the graduations is insufficient and the strokes at the graduations are too thick ($1\ \text{mm}$).

Let us calculate the thickness of the stroke $t$ on the rod for a sight length $d = 10\ \text{m}$, using Reingertz’s formula ([1], p. 353)

\[ t = \frac{l \cdot d}{0.25 \cdot v}, \tag{1} \]

where $l$ is the thickness of the stroke on the rod as seen through the telescope,
$0.25\ \text{m}$ is the distance of best vision,
$v$ is the angular magnification of the telescope.

Taking in (1) $l = 0.1\ \text{mm}$ and $v = 40$, we find that $t = 0.1\ \text{mm}$.

At short distances from the instrument to the rod, it is advisable to have the intervals between strokes on the rod equal to $1\ \text{mm}$. This will make it possible, with one setup of the level, to sight not at the two nearest strokes, as is the case with an ordinary rod, but at several adjacent strokes. This will undoubtedly increase the accuracy of leveling.

Let us estimate the accuracy of geometric leveling for our case. In leveling over short distances, it is meaningful to take into account only the following errors: the pointing error $m_n$, the error of reading the micrometer drum $m_b$, the error of setting the bubble of the instrument level $m_y$, the error due to inclination of the rod $m_p$, the error—

… of the staff graduation interval \(m_d\). The error due to nonparallelism of the line of sight to the level axis will not be of substantial significance if the level is set exactly midway between the points being leveled. However, this error must be reduced to a minimum.

The sighting error \(m_n\) (the introduction of a staff line into the bisector of the reticle threads) is computed by Reingertz’s formula ([1], p. 353)

\[ m_n=\pm \frac{0.04\sqrt{t}\sqrt{d}}{\sqrt{v}}, \tag{2} \]

where the notation is the same as in (1). Taking here \(t=0.1\ \mathrm{mm}\), \(d=10\ \mathrm{m}\), \(v=40\), we find that \(m_n=\pm 0.006\ \mathrm{mm}\).

The error of reading on the drum of the plane-parallel plate, \(m_b\), may be taken as equal to \(\pm 0.05\ \mathrm{mm}\).

The error of bringing the ends of the bubble of the contact level into coincidence, \(m_y\), will be computed by Chebotarev’s formula ([2], p. 70)

\[ m_y=\pm(0.14+0.012\tau)\frac{d}{206265}, \tag{3} \]

where \(\tau\) is the value of one division of the level. Taking for the HA-1 level the value \(\tau=10''\), we find that \(m_y=\pm0.013\ \mathrm{mm}\).

The error due to inclination of the staff will depend on the length of the staff and on the division value of the circular level. If the staff length is taken as \(S=1\ \mathrm{m}\), and the division value of the level as \(\tau_0=25'\), then the error due to inclination of the staff \(m_p\) will be equal to:

\[ m_p=-2\cdot S\cdot \sin^2\frac{\tau_0}{2} =-2\cdot1\cdot\sin^2\frac{25'}{2} =-0.026\ \mathrm{mm}. \]

We shall take the error of marking the staff graduations \(m_d\) as equal to \(\pm0.02\ \mathrm{mm}\).

We now compute the root-mean-square error of taking a reading on the staff, \(m_o\), which will be equal to

\[ m_o=\pm\sqrt{m_n^2+m_b^2+m_y^2+m_p^2+m_d^2}= \]

\[ =\pm\sqrt{(0.006)^2+(0.05)^2+(0.013)^2+(0.026)^2+0.02^2} =\pm0.06\ \mathrm{mm}. \]

Readings on the staff will be made on three adjacent lines, and the mean value will be derived from the readings taken; consequently, for the mean of three readings, \(\overline{m}_o\) we find:

\[ \overline{m}_o=\pm\frac{0.06}{\sqrt{3}}. \tag{4} \]

The root-mean-square error of the elevation difference \(m_h\), obtained as the difference of two readings, will be

\[ m_h=\pm\frac{0.06}{\sqrt{3}}\sqrt{2}. \tag{5} \]

Since the elevation difference will be determined twice, with a change in the height of the instrument, the final error of the elevation difference \(\overline{m}_h\) will be

\[ \overline{m}_h=\pm \frac{0.06\sqrt{2}}{\sqrt{3}\sqrt{2}}=\pm 0.035\ \text{mm}. \tag{6} \]

Consequently, the limiting error \((\overline{m}_h)_{\text{lim}}\) of the elevation difference \(h\) will be equal to:

\[ (\overline{m}_h)_{\text{lim}}=3\cdot \overline{m}_h=\pm 0.1\ \text{mm}. \tag{7} \]

From the above calculations it is evident that geometric leveling with a length of the line of sight up to 10 m can quite well be carried out with an error not exceeding \(\pm 0.1\) mm.

We carried out a production assignment to set a cast-iron foundation plate into a horizontal position, with an installation accuracy requirement of \(\pm 0.2\) mm. The setting was performed using an HA-1 level, No. 00078, which had been carefully examined before the work. The staff used was a standard brass ruler 1.2 m long, with millimeter intervals between marks. This ruler was fastened to a base staff having a circular level and metal fittings (Fig. 1). For leveling the foundation plate, a special leveling shoe was made in the form of a hexagonal prism with a steel ball pressed into the upper base of the prism (Fig. 2). The shoe was placed on points on the plate that had been selected and marked in advance. The setting was carried out in a shop where a constant temperature was maintained. For better visibility the staff was illuminated with a portable lamp. Before the work was performed, special investigations were carried out to determine the total error \(m_{\text{sum}}\) due to errors of pointing, bringing the ends of the level bubble into coincidence, and reading the staff. This total error proved to be equal to

Fig. 1.

Fig. 2.

\[ m_{\text{sum}}=\pm 0.04\ \text{mm}. \]

To determine the error of the ruler divisions, the latter was compared by decimeter, centimeter, and millimeter divisions, as a result of which the following values of the corresponding errors were found:

\[ m_{\text{dec}}=\pm 0.05\ \text{mm};\quad m_{\text{cm}}=\pm 0.04\ \text{mm};\quad m_{\text{mm}}=\pm 0.01\ \text{mm}. \]

Then the root-mean-square error of a staff division \(m_d\) is determined as follows:

\[ m_d=\pm\sqrt{m_{\mathrm{del}}^{2}+m_{\mathrm{sm}}^{2}+m_{\mathrm{em}}^{2}} =\pm\sqrt{0.05^{2}+0.04^{2}+0.01^{2}} =\pm 0.07\ \mathrm{mm}. \]

Taking into account the total error found, \(m_{\mathrm{sum}}=\pm 0.04\ \mathrm{mm}\), and the error due to the inclination of the rod, \(m_r=\pm 0.02\ \mathrm{mm}\), we obtain the root-mean-square error of taking a reading on the rod, \(m_o\):

\[ m_o=\pm\sqrt{m_{\mathrm{sum}}^{2}+m_d^{2}+m_r^{2}} =\pm\sqrt{0.04^{2}+0.07^{2}+0.02^{2}} =\pm 0.08\ \mathrm{mm}. \]

The error of the elevation difference, computed according to (6), proved to be equal to

\[ \overline{m}_h=\pm\frac{0.08\sqrt{2}}{\sqrt{3}\sqrt{2}} =\pm 0.05\ \mathrm{mm}, \]

and then the limiting error will be equal to

\[ (\overline{m}_h)_{\mathrm{lim}}=\pm 0.15\ \mathrm{mm}. \]

The computed limiting error turned out to be somewhat greater than the preliminary calculated one because the accuracy of drawing the graduations on the ruler used does not correspond to the calculated value.

As check measurements with the aid of hydrostatic leveling showed, all points of the foundation slab were set within the prescribed accuracy of \(\pm 0.2\ \mathrm{mm}\).

References

1. F. N. Krasovskii. Manual of Higher Geodesy, Part 1, Issue 2, Redburo, GUGK, 1939.

2. A. S. Chebotarev. Geodesy, Part II, Geodezizdat, 1949.