Assessment of the Accuracy of Transferring Elevation to the Level of Underground Workings Through a Vertical Shaft
Abstract Generated abstract
This paper develops a practical error analysis for transferring elevations to underground workings through a vertical mine shaft using a wire. It decomposes the total error into contributions from wire temperature, thermal expansion coefficients, elastic elongation, pulley friction, longitudinal oscillations, comparator tape length and reading errors, and rod and marking operations, then combines these terms into formulas for single and double measurements. The analysis identifies temperature determination and comparator tape accuracy as the dominant influences, provides numerical examples for shaft depths up to 1000 m and different pulley arrangements, and distinguishes the treatment of independent and dependent observations. It recommends ball-bearing pulleys, known technical characteristics for wires and tapes, careful determination of mean wire temperature, and differential accuracy assessment according to measurement dependence.
Full Text
ASSESSMENT OF THE ACCURACY OF TRANSFERRING ELEVATION TO THE LEVEL OF UNDERGROUND WORKINGS THROUGH A VERTICAL SHAFT
A. I. VOLKOV
The question of the preliminary calculation of the accuracy of work planned for execution, and the ability to evaluate correctly work already performed, is of great practical importance. In confirmation of this it is enough to say that only on the basis of an accuracy analysis can, for example, a program of observations be correctly drawn up and the proper instruments selected for carrying out the work. If the accuracy of work planned for execution is specified in advance, as is almost always the case in practice, then solving this problem without a preliminary calculation of the expected error of the result is altogether impossible. Nevertheless, we are compelled to note the absence of calculation formulas of this kind for the problem of determining the elevation of a bench mark through a vertical mine shaft, even in literature published in recent years [1].
The present work, in this respect, is the first attempt to fill the indicated gap. To this end, the work considers the influence of individual sources of error on the accuracy of transferring elevation to the level of underground workings, and gives a method and formulas for the preliminary calculation of the expected error in determining elevation from a single measurement, the difference of double measurements, and the error of the final result, taking into account the degree of dependence of the individual measurements on one another. A method is also given for determining the error of elevation from a single measurement and the difference from double dependent measurements according to the actual difference of observations already performed.
In addition to general formulas, the work gives, as examples, certain working formulas and numerical values of the corresponding mean and limiting errors for various mine depths.
It should be noted, however, that we do not claim complete impeccability for the parameters of the working formulas we have adopted, since we are convinced that practice will introduce its own corrections into them, although we are far from thinking that the formulas proposed by us can be subjected to substantial alteration.
The method and formulas presented in this work relate to the case of transferring elevation to the level of underground workings with the aid of wire; however, the general method of preliminary error calculation remains valid also for the case of solving this problem with the aid of long and short tapes.
Degree of influence of individual sources of error on the accuracy of transferring elevation to the level of underground workings
- Error due to incorrect estimation of the temperature of the wire at the surface and in the mine. In view of the fact that the temperature regime of the air in a mine shaft changes in [[unclear: continuation cut off on this page]]
in this case according to the law of a curve (especially in the upper part of the shaft), while temperature observations are made at definite depth intervals, the mean temperature of the wire in the mine shaft is practically always determined approximately. The degree of this approximation depends on the character of the temperature curve and on the number of temperature observation points. Moreover, since determination of the mean temperature of the wire in the mine shaft and the marking of the wire along the sighting rays of the levels are not performed simultaneously, the mean temperature of the outside air found by us will in general not correspond to that which the wire had when the marks were placed on it. Finally, the marking of the wire and the measurement of its length on the comparator may be carried out at a moment when the latter has not yet assumed the temperature of the surrounding air. All this, taken together, even with the most careful observations, leads to distortion of the result of measuring the segment \(L\) (the portion of the wire between the horizons of the levels). Taking the mean total error \(m_t\) in determining the temperature of the wire to be equal to \(1\text{--}3^\circ\), we find the error \(M_1\) of measuring \(L\) from the formula:
\[ M_1=\pm \alpha L m_t . \tag{1} \]
For \(L=500\ \text{m}\) and \(m_t=\pm 1^\circ\), this error will be equal to
\[ M_1=\pm 5.5\ \text{mm}. \]
2. Error due to inaccurate knowledge of the value of the coefficient of linear expansion of the wire and of the comparator tape. When computing the correction for the temperature of the wire and the tape, one usually uses the mean value of the coefficient \(\alpha\), taking it to be the same for the wire and for the comparator tape. However, it must be borne in mind that the temperature coefficient of steel \(\alpha\) varies within rather wide limits. Thus, for ordinary steel tapes it varies within the limits from \(11.0\cdot 10^{-6}\) to \(16.0\cdot 10^{-6}\) [2].
If for the comparator tape the value of \(\alpha\) is given in its certificate and is known with an accuracy of \(\pm 5\cdot 10^{-7}\) [2], then for the wire the value of \(\alpha\) is usually conventionally taken equal to the \(\alpha\) of the tape. Bearing in mind the variation of \(\alpha\) within the limits from \(11.0\cdot 10^{-6}\) to \(16.0\cdot 10^{-6}\), it must be considered that the error of the coefficient of linear expansion of the wire in the above case may reach \(5.0\cdot 10^{-6}\). Such an error in the coefficient \(\alpha\) of the wire corresponds to a linear error equal to \(\pm 1\ \text{cm}\) for every \(200\ \text{m}\) of measured length when the temperature difference is \(10^\circ\). Since such a linear error is practically unacceptable, it should be recommended that a wire be used whose coefficient of linear expansion is known; otherwise it must be determined experimentally.
Taking the mean error of \(\alpha\) of the tape and of the wire to be \(\pm 5\cdot 10^{-7}\), the linear error \(M_2\) in measuring the segment \(L\), when the \(\alpha\)'s are equal, is found from the formula:
\[ M_2=\pm 7\cdot 10^{-7}\cdot L(t_{pr}-t_0)\ \text{m}, \tag{2} \]
where \(L\) is the measured value of the wire between the initial marks of the horizons of the levels;
\[ \begin{aligned} t_{pr}&\text{ is the mean temperature of the wire in the mine shaft;}\\ t_0&\text{ is the temperature of comparison of the tape.} \end{aligned} \]
If the values of \(\alpha\) for the tape and for the wire are not equal to each other, then the linear error \(M'_2\) of the measured segment \(L\) is determined from the formula:
\[ M'_2=\pm 5\cdot 10^{-7}\cdot L\sqrt{(t_{pr}-t_{iz})^2+(t_{iz}-t_0)^2}. \tag{2a} \]
3. Error due to inaccurate allowance for the modulus of elasticity of the wire. When computing the correction for the stretching of the wire...
wire under the action of its own weight, the modulus of elasticity \(E\) is usually taken as equal to its mean value for the material of the given grade. This mean value \(E\), as Prof. I. M. Bakhurin [3] indicates, and as other literary sources state, is known to us with an accuracy of \(\pm 10\%\). On this basis, the correction, and consequently also the value \(L\), is computed with an error \(M_3\) equal to1
\[ \left. \begin{aligned} M_3 &= \pm \frac{\gamma \cdot L}{200E}\,[\,2(s+h)+l\,] \\ \text{or}\qquad M_3 &= \pm \frac{\Delta L_q}{10} \end{aligned} \right\} \tag{3} \]
where \(\Delta L_q\) is the correction to the measured value \(L\), due to the elongation of the wire under the action of its own weight \(q\).
The numerical value of the error \(M_3\), even for great depths, is expressed in fractions of a millimeter. Thus, for \(L=1000\ \text{m}\), \(S=l=20\ \text{m}\), \(M_3=\pm 1.2\ \text{mm}\). Thus this error has no practical significance.
4. Error due to friction in the pulleys. From the analysis given in [4] of the influence of friction in pulleys and from the examples considered there, it follows that a completely serviceable pulley, the diameter of whose drum and axle are respectively equal to \(10\ \text{cm}\) and \(1\ \text{cm}\), causes a linear error \(M_4\) in the measured segment \(L\), determined by the formula
\[ M_4=\pm 0.00001\,L, \tag{4} \]
i.e., \(1\ \text{mm}\) for every \(100\ \text{m}\) of measured length. For serviceable pulleys with a drum diameter of \(25\ \text{cm}\) and an axle diameter of \(1\ \text{cm}\), the error \(M'_4\) in the measurement of \(L\) is expressed by the formula
\[ M'_4=\pm 0.000004\,L, \tag{4a} \]
or \(1\ \text{mm}\) for every \(250\ \text{m}\) of measured length, i.e., practically almost zero.
When pulleys on ball bearings are used, the influence of friction is practically equal to zero.
5. Error due to longitudinal oscillations of the wire. A sign of the presence of longitudinal oscillations of the wire at the moment of measuring its length on the comparator is a rapidly damped rhythmic displacement of the intermediate marks to the right and to the left (decrease and increase of the reading). The maximum amplitude of this displacement, at \(l=s=20\ \text{m}\) and \(P=10\ \text{kg}\), may reach \(2\)—\(3\ \text{cm}\). An amplitude of such magnitude can be observed by giving the plumb bob a “free” fall.
If, when measuring the wire on the comparator, no noticeable displacement of the intermediate marks is observed, this means that the wire either experiences no longitudinal oscillations at all, or that they are so small that they lie beyond the limits of observational accuracy, and consequently, in solving the given problem, have no practical significance. Longitudinal oscillations may arise when the wire is raised non-uniformly or when it makes a sharp reverse movement at the moment the raising ceases.
If it is assumed that an amplitude of displacement of the marks of \(0.2\ \text{mm}\) will not be noticed by the observer, then the total error \(M_5\) in measuring the segment \(L\) due to this will be equal to
\[ M_5=\pm 0.0001\sqrt{n}, \tag{5} \]
where \(n\) is the number of measured wire segments.
It follows from formula (5) that the numerical value of this correction is small even for deep mines. Thus, for a depth of 1000 m \((n = 50)\), the error due to this factor will be only 0.8 mm. If the displacement of the marks remains noticeable to the eye and shows no tendency to decay under the influence of any external factors (for example, dripping water), then the reading corresponding to the mean position of the marks also reduces to zero the influence of longitudinal oscillations.
6. Error due to inaccurate knowledge of the length of the tape comparator. The error \(M_6\) of the measured segment \(L\) due to inaccurate knowledge of the length of the tape comparator is expressed by the formula
\[ M_6 = \pm \lambda L, \tag{6} \]
where \(\lambda\) is the relative error of the length of the tape comparator.
Since, in solving the present problem, measuring tapes are used whose length is known approximately with a relative error
\[ \frac{1}{40000} \]
[5], the error in \(L\) in this case will be equal to:
\[ M_6 = \pm 0.000025\,L, \tag{6a} \]
or 1 cm for every 400 m of measured length.
7. Error due to error of reading on the tape comparator. When measuring the length of the wire between the initial marks, we usually have conditions favorable for making readings with increased accuracy: the wire and the tape have constant tension and, as a rule, are immobile.
Under such conditions the accuracy of reading cannot be lower than \(\pm 0.5\) mm; and when using an applied ruler, if the tape itself does not have millimeter graduations, the reading accuracy may be even somewhat higher, especially if magnifiers are used for the readings.
With a reading accuracy of \(\pm 0.5\) mm, the error of any measured segment of the wire will be equal to \(\pm 0.7\) mm, or, rounded, \(\pm 1\) mm. Hence the total error \(M_7\) of the measured segment \(L\) due to reading error will be equal to:
\[ M_7 = \pm 0.001\sqrt{n}, \tag{7} \]
where \(n\) is the number of measured segments of wire.
For \(L\) equal to 500 m \((n = 25)\), \(M_7 = \pm 5\) mm.
8. Error due to the accuracy of readings on the rods and the accuracy of applying the initial marks to the wire. In all, when transferring an elevation to the level of underground workings, two readings are made on the rods and two initial marks are applied to the wire. Both of these operations are performed with approximately the same accuracy. Taking the mean error of applying marks to the wire and of reading on the rods, with allowance for the error of applying the rod divisions and the oscillation of the sights, as \(\pm 1.5\) mm, we obtain the total error \(M_8\) due to this factor equal to
\[ M_8 = \pm 0.003\,\text{m}. \tag{8} \]
This error, as is evident from the formula, does not depend on the depth of the mine.
From the analysis given above of the individual sources of errors it is clear that the greatest influence on the result of transferring an elevation to the level of underground workings is exerted by the error in determining the mean temperature of the wire in the mine shaft and the error of comparison of the tape. Errors due to the remaining factors are either small or can be made small by applying an appropriate procedure for performing the individual measurement operations.
Error in the elevation transferred to the level of underground workings by a single measurement
In view of the fact that all the errors considered above have random signs, the total error of the result of determining the elevation \(H\) will be found from the formula:
\[ M_H=\pm\sqrt{M_1^2+M_2^2+M_3^2+M_4^2+M_5^2+M_6^2+M_7^2+M_8^2}, \tag{9} \]
where
\[ M_1=\pm \alpha L m_t;\quad M_2=\pm 7\cdot 10^{-7}L;\quad M_3=\pm\frac{\Delta L_q}{10}= \]
\[ =\pm \frac{\gamma}{200}\frac{L}{E}\,[2(s\pm h)+l]; \]
\[ M_4= \begin{cases} \pm 10\cdot 10^{-6}L, & \text{for a pulley diameter of }10\text{ cm},\\ \pm 4\cdot 10^{-6}L, & \text{for a pulley diameter of }25\text{ cm},\\ 0, & \text{for a pulley on ball bearings;} \end{cases} \]
\[ M_5=\pm 10^{-4}\sqrt{n};\quad M_6=\pm \lambda L;\quad M_7=\pm 10^{-3}\sqrt{n};\quad M_8=\pm 3\cdot 10^{-3}. \]
Taking \(\alpha=11\cdot 10^{-6}\), \(m_t=\pm 2^\circ\), \(s\pm h=5\text{ m}\), \(l=20\text{ m}\), \(\gamma=7.8\);
\[ \lambda=0.000025, \]
we obtain:
\[ \begin{aligned} M_1&=\pm 22\cdot 10^{-6}L, & M_5&=\pm 10^{-4}\sqrt{n},\\ M_2&=\pm 0.7\cdot 10^{-6}L, & M_6&=\pm 25\cdot 10^{-6}L,\\ M_3&=\pm 0.585\cdot 10^{-6}L, & M_7&=\pm 10^{-3}\sqrt{n},\\ M_4&= \begin{cases} \pm 10\cdot 10^{-6}L,\\ \pm 4\cdot 10^{-6}L,\\ 0, \end{cases} & M_8&=\pm 3\cdot 10^{-3}. \end{aligned} \]
For the given values of \(M_i\), the error \(M_H\) for different shaft depths and different pulley sizes and designs is expressed in Tables 1–3.
Pulleys with drum diameter 10 cm and axle diameter 1 cm
The mean square error in this case is expressed by the formula:
\[ M_H=\pm\sqrt{12.098\cdot 10^{-10}L^2+1.01\cdot 10^{-6}n+9\cdot 10^{-6}} \tag{10} \]
We take the limiting error equal to:
\[ M_{H\,\mathrm{pr}}=\pm 3M_H \tag{11} \]
The numerical values of the errors \(M_H\) and \(M_{H\,\mathrm{pr}}\) for \(L=50,100,200,\ldots\ldots 1000\text{ m}\), calculated as an example by formulas (10) and (11), are given in Table 1.
The values of these errors can be computed with sufficient accuracy by the following approximate formulas:
\[ M_H=\pm(0.033H+2)\ \text{mm} \tag{12} \]
\[ M_{H\,\mathrm{pr}}=\pm(0.1H+6)\ \text{mm}, \tag{13} \]
where \(H\) is the difference in elevation between the initial and determined bench marks, in meters.
Table 1
| $M_H$, mm / $L$, m | 50 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Root-mean-square | 3,8 | 5,2 | 8,2 | 11,5 | 14,9 | 18,3 | 21,8 | 25,2 | 28,7 | 32,2 | 35,6 |
| Limiting | 11,4 | 15,6 | 24,6 | 34,5 | 44,7 | 54,9 | 65,4 | 75,6 | 86,1 | 96,6 | 106,8 |
Table 2
| $M_H$, mm / $L$, m | 50 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Root-mean-square | 3,8 | 5,1 | 8,0 | 11,2 | 14,4 | 17,7 | 21,1 | 24,4 | 27,7 | 31,1 | 34,4 |
| Limiting | 11,4 | 15,3 | 24,0 | 33,6 | 43,2 | 53,1 | 63,3 | 73,2 | 83,1 | 93,3 | 103,2 |
Table 3
| $M_H$, mm / $L$, m | 50 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Root-mean-square | 3,8 | 5,1 | 8,0 | 11,1 | 14,4 | 17,6 | 20,9 | 24,2 | 27,5 | 30,8 | 34,2 |
| Limiting | 11,4 | 15,3 | 24,0 | 33,3 | 43,2 | 52,8 | 62,7 | 72,6 | 82,5 | 92,4 | 102,6 |
Pulleys with a drum diameter of 25 cm and an axle diameter of 1 cm
The general expression for computing the mean square error in this case will be:
\[ M_H{}_M=\pm \sqrt{11{,}238\cdot 10^{-10}L^2+1{,}01\cdot 10^{-6}n+9\cdot 10^{-6}} \tag{14} \]
The values of the errors \(M_H\) and \(M_{H\,pr}\), computed by formulas (11) and (14), are given in Table 2.
The data of Table 2 may be obtained from the approximate formulas:
\[ M_H=\pm(0{,}032H+2)\ \text{mm} \tag{15} \]
\[ M_{H\,pr}=\pm(0{,}096H+6)\ \text{mm} \tag{16} \]
Pulleys with ball bearings
The mean square error \(M_H\) in this case is expressed by the formula:
\[ M_H=\pm \sqrt{11{,}078\cdot 10^{-10}L^2+1{,}01\cdot 10^{-6}n+9\cdot 10^{-6}} \tag{17} \]
The values of the errors \(M_H\) and \(M_{H\,pr}\), computed by formula (17), are given in Table 3.
As approximate formulas in this case, the following may be adopted:
\[ M_H=\pm(0{,}031H+2)\ \text{mm} \tag{18} \]
\[ M_{H\,pr}=\pm(0{,}094H+6)\ \text{mm} \tag{19} \]
The data of Tables 1–3 show that the accuracy of transferring elevation to an underground level depends little on the dimensions and design of the bearings of the pulleys. Nevertheless, in practice it should still be recommended to use pulleys with ball bearings, since the latter, by virtue of their design features, have a stable coefficient of friction, which cannot be said of ordinary pulleys because of the greater possibility of dust penetrating onto their axles and sharply changing the coefficient of friction.
Error of the difference in a twofold transfer of elevation
to the level of underground workings
The accuracy of elevation transfer is usually judged by comparing the actual difference of twofold measurements with the possible theoretical difference. In order to correctly precompute the possible discrepancy of the measurements, it is necessary to know:
- The accuracy of the individual operations associated with the transfer of elevation.
- The degree of dependence of the individual measurements on one another.
Observations associated with an initially measured temperature of the wire (or wires), or made with the same instruments (meaning the wire and the comparator), are dependent measurements. Observations that do not have the above-mentioned dependence are independent. Differences between dependent observations and differences between independent observations are composed of different numbers of errors and, consequently, numerically, all other conditions being equal, will be different.
The latter circumstance is explained by the fact that equal errors, equal in magnitude and identical in sign, enter into both dependent measurements and cancel one another when the difference is formed. For the difference from indepen-
...observations, the mutual cancellation of individual errors mentioned above is unlikely.
These considerations lead us to the following conclusions:
-
A general expression for the error in the difference of two measurements cannot be given for dependent and independent observations. Nor can a general expression be given for the error in the difference of dependent observations having different degrees of dependence.
-
When determining (precomputing) the root-mean-square error in the difference of two dependent measurements, errors due to common factors must be excluded.
-
From the actual difference of two dependent measurements it is impossible to determine the error of one measurement and the error of the final result, unless systematic errors in the dependent measurements are taken into account.
The error \(M_d\) of the difference \(d\) of two independent determinations of the elevation of one and the same point at an underground horizon \((d=z-z')\) will be found from the formula:
\[ M_d=\pm\sqrt{2\left(M_1^2+M_2^2+M_3^2+M_4^2+M_5^2+M_6^2+M_7^2+M_8^2\right)} \tag{20} \]
To determine the error of the difference of two dependent measurements, formula (20) may also be used; however, the errors due to common factors must first be excluded from it.
Guided by the considerations indicated above, let us determine the error of the difference of two measurements for dependent and independent observations. For this purpose, let us consider several cases possible in practice.
Case 1. The first and second determinations of elevation were carried out with the same wire and comparator. The initial temperature was taken as common for both measurements. In this case, from formula (20) the errors due to the accuracy of the initial temperature of the wire, the coefficient of linear expansion, the modulus of elasticity, and comparison are to be excluded.
After excluding from formula (20) the errors indicated above and substituting the numerical values of the remaining ones, we obtain:
a) when using blocks of ordinary design with a drum diameter of 10 cm,
\[ M_d=\pm\sqrt{2.0\cdot 10^{-10}L^2+2.02\cdot 10^{-6}n+1810^{-6}}, \tag{21} \]
b) when using blocks with ball bearings,
\[ M_d=\pm\sqrt{2.02\cdot 10^{-6}n+18\cdot 10^{-6}} \tag{22} \]
The magnitudes of the errors calculated for various values of \(L\) by formula (21) are given in Table 4, and by formula (22) in Table 5.
The data of Table 4 may be obtained from the approximate formulas:
\[ M_d=\pm(0.014H+4)\ \text{mm}, \tag{23} \]
\[ M_{d\,pr}=\pm(0.042H+12)\ \text{mm}. \tag{24} \]
The data of Table 5 may be obtained from the approximate formulas:
\[ M_d=\pm(0.006H+5)\ \text{mm}, \tag{25} \]
\[ M_{d\,pr}=\pm(0.018H+15)\ \text{mm}. \tag{26} \]
Case 2. The first and second determinations of elevation were carried out with the same wire and comparator, but with an independent determination each time of the initial temperature of the wire.
Table 4
| \(M_{d,\mathrm{mm}}\) / \(L\,\mathrm{m}\) | 50 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Mean square . . . . | 4.8 | 5.5 | 6.8 | 8.1 | 9.5 | 10.9 | 12.3 | 13.7 | 15.1 | 16.4 | 17.8 |
| Limiting . . . . . . . . . | 14.4 | 16.5 | 20.4 | 24.3 | 28.5 | 32.7 | 36.9 | 41.4 | 45.3 | 49.2 | 53.4 |
Table 5
| \(M_{d,\mathrm{mm}}\) / \(L\,\mathrm{m}\) | 50 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Mean square . . . . | 4.8 | 5.3 | 6.2 | 6.9 | 7.6 | 8.3 | 8.9 | 9.4 | 9.9 | 10.4 | 10.9 |
| Limiting . . . . . . . . . | 14.4 | 15.9 | 18.6 | 20.7 | 22.8 | 24.9 | 26.7 | 28.2 | 29.7 | 31.2 | 32.7 |
Table 6
| \(M_{d,\mathrm{mm}}\) / \(L\,\mathrm{m}\) | 50 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Mean square . . . . | 5.1 | 6.3 | 9.2 | 12.4 | 15.7 | 19.0 | 22.3 | 25.7 | 29.1 | 32.5 | 35.7 |
| Limiting . . . . . . . . . | 15.3 | 18.9 | 27.6 | 37.2 | 47.1 | 57.0 | 66.9 | 77.1 | 87.3 | 97.5 | 107.1 |
In the present case, from formula (20) the errors due to the inaccuracy of the coefficient of linear expansion, the modulus of elasticity, and comparison are to be excluded.
Proceeding analogously to the preceding case, we obtain:
a) when using pulleys of ordinary design with a drum diameter of 10 cm:
\[ M_d=\pm\sqrt{11{,}68\cdot10^{-10}L^2+2{,}02\cdot10^{-6}n+18\cdot10^{-6}}, \tag{27} \]
b) when using pulleys with ball bearings:
\[ M_d=\pm\sqrt{9{,}68\cdot10^{-10}L^2+2{,}02\cdot10^{-6}n+18\cdot10^{-6}} \tag{28} \]
The values of the errors computed for various \(L\) by formula (27) are given in Table 6, and by formula (28)—in Table 7.
The data of Table 6 may be obtained from the approximate formulas:
\[ M_d=\pm(0{,}032H+3)\ \text{mm}, \tag{29} \]
\[ M_{d\,pr}=\pm(0{,}096H+9)\ \text{mm}. \tag{30} \]
The data of Table 7 may be obtained from the approximate formulas:
\[ M_d=\pm(0{,}03H+3), \tag{31} \]
\[ M_{d\,pr}=\pm(0{,}089H+9). \tag{32} \]
Case 3. The first and second determinations of elevation were made with different wires, with different comparators, and with different determinations of the initial temperature of the wire.
The error of interest to us in this case is computed directly from formula (20), without excluding any of the quantities entering into it.
After substituting into formula (20) the numerical values of the quantities entering into it, we obtain:
a) when using pulleys of ordinary design with a drum diameter of 10 cm:
\[ M_d=\pm\sqrt{24{,}2\cdot10^{-10}L^2+2{,}02\cdot10^{-6}n+18\cdot10^{-6}}, \tag{33} \]
b) when using pulleys with ball bearings:
\[ M_d=\pm\sqrt{22{,}16\cdot10^{-10}L^2+2{,}02\cdot10^{-6}n+18\cdot10^{-6}} \tag{34} \]
The values of the errors for various \(L\), computed by formula (33), are given in Table 8, and those computed by formula (34)—in Table 9.
The approximate formulas corresponding to the data of Table 8 will be:
\[ M_d=\pm(0{,}048H+2)\ \text{mm}, \tag{35} \]
\[ M_{d\,pr}=\pm(0{,}144H+6)\ \text{mm}. \tag{36} \]
The approximate formulas corresponding to the data of Table 9 will be:
\[ M_d=\pm(0{,}046H+2), \tag{37} \]
\[ M_{d\,pr}=\pm(0{,}138H+6). \tag{38} \]
In all the examples considered above for computing the error in the difference of two independent determinations of elevation, the mean error of the initial temperature of the wire was taken as equal to \(\pm2^\circ\), and the limiting error, consequently, to \(\pm6^\circ\). In practice, the mean initial temperature of the wire can be determined considerably more accurately by increasing the number of points of ob—
Table 7
| \(M_d\), mm / \(L\), m | 50 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Mean square | 5,1 | 6,1 | 8,8 | 11,6 | 14,6 | 17,6 | 20,7 | 23,7 | 26,8 | 29,8 | 32,9 |
| Limiting | 15,3 | 18,3 | 26,4 | 34,8 | 43,8 | 52,8 | 62,1 | 71,1 | 80,4 | 89,4 | 98,7 |
Table 8
| \(M_d\), mm / \(L\), m | 50 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Mean square | 5,4 | 7,2 | 11,6 | 16,3 | 21,1 | 26,0 | 30,8 | 35,7 | 40,6 | 45,5 | 50,4 |
| Limiting | 16,2 | 21,6 | 34,8 | 48,9 | 63,3 | 78,0 | 92,4 | 107,1 | 121,8 | 136,5 | 151,2 |
Table 9
| \(M_d\), mm / \(L\), m | 50 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Mean square | 5,4 | 7,1 | 11,3 | 15,7 | 20,3 | 24,9 | 29,5 | 34,2 | 38,9 | 43,6 | 48,7 |
| Limiting | 16,2 | 21,3 | 33,9 | 47,1 | 60,9 | 73,7 | 88,5 | 102,6 | 116,7 | 130,8 | 146,1 |
observations, using the weighted-mean method [4]. Especially good results can be obtained when the difference between the air temperature in the shaft and on the surface is insignificant. In addition, the accuracy of determining the mean temperature of the wire is considerably increased for dry shafts, especially when the air current has a high velocity. Taking the mean error of temperature determination as \(\pm 1^\circ\), we obtain the following expression for determining the error in the difference of elevation values from duplicate independent measurements (blocks on ball bearings):
\[ M_d=\pm \sqrt{14.94\cdot 10^{-10}L^2+2.02\cdot 10^{-6}n+18\cdot 10^{-6}} . \tag{39} \]
The values of the errors, computed for various \(L\) by formula (39), are given in Table 10.
The approximate formulas corresponding to this case will be:
\[ M_d=\pm(0.036H+3)\ \text{mm}, \tag{40} \]
\[ M_{d\ \text{pr}}=\pm(0.108H+9)\ \text{mm}. \tag{41} \]
In concluding the consideration of the question of the error in the difference in a duplicate transfer of elevation to the level of underground workings, we consider it necessary to point out once more that the magnitude of this difference, for one and the same accuracy of observations, depends to a considerable degree on the character of the observations, i.e., on whether the observations are dependent or not. Thus, for a mine depth of 500 m, depending on the character of the observations, we obtain the following values of the limiting errors (when using ordinary blocks with a drum diameter of 10 cm): \(\pm 32.7\) mm, \(\pm 57.0\) mm, \(\pm 78.0\) mm, although the accuracy of the observations in all cases was the same. Thus, the method currently used for assessing the accuracy of the difference in a duplicate determination of elevation, without taking into account the character of the observations, is incorrect. According to this method, the three cases of determining one and the same elevation cited above would be deemed unequally accurate. It should also be pointed out that the limiting value of the difference obtained by us, even for independent observations, is considerably smaller than that permitted at present [5]. Thus, the question arises of the need to differentiate the allowable error of the difference and to establish corresponding standards according to the degree of dependence of the observations.
Mean Error of the Transfer of Elevation Obtained from Duplicate Measurements
Since the transfer of elevation to an underground level is always performed at least twice, the final result is taken to be the mean of two measurements. The error of the mean value of the elevation from two measurements is determined by the formula:
\[ M_{H\ \text{av}}=\pm \frac{M_H}{\sqrt{2}}, \tag{42} \]
where \(M_H\) is the error of one measurement due to all factors affecting the accuracy of elevation determination,
or
\[ M_{H\ \text{av}}=\pm \sqrt{\frac{1}{2}\left(M_1^2+M_2^2+M_3^2+M_4^2+M_5^2+M_6^2+M_7^2+M_8^2\right)} \tag{43} \]
Table 10
| \(M_d\), mm / \(L\), m | 50 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Root-mean-square | 5,2 | 6,5 | 9,9 | 13,5 | 17,2 | 21,0 | 24,9 | 28,6 | 32,5 | 36,3 | 40,2 |
| Limiting | 15,6 | 19,5 | 29,7 | 40,5 | 51,6 | 63,0 | 74,7 | 85,8 | 97,5 | 108,9 | 120,6 |
However, it should be borne in mind that the application of these formulas can be extended only to independent observations. If the measurements are dependent, then the error of the mean value of the elevation must be computed by the formula:
\[ M_{H\,\mathrm{av}}=\pm\sqrt{\frac{(M'_H)^2}{2}+(M''_H)^2}, \tag{44} \]
where \(M'_H\) is the error of one measurement due to independent factors;
\(M''_H\) is the error of one measurement due to dependent factors.
The value \(M'_H\) is computed by formula (9), from which the errors due to dependent factors have first been excluded. \(M''_H\) is also computed by formula (9); however, in this case, on the contrary, the errors due to independent factors must be excluded from it. Formulas (42), (43), and (44) may be used for a preliminary calculation of the expected error of the result when the accuracy of the individual operations is known. From their comparison it follows that independent observations make it possible to determine the mean value of the elevation more accurately than dependent ones. Thus, for independent observations at a depth of 500 m, \(M_{H\,\mathrm{av}}=\pm 13\) mm (or the limiting error is \(\pm 39\) mm), whereas for dependent observations (when determining the elevation with the same wire and tape-comparator and with an initial wire temperature common to both measurements) \(M_{H\,\mathrm{av}}=\pm 17.5\) mm (or the limiting error is \(\pm 52.5\) mm).
When the actual difference of two independent observations is available, the error of one measurement and the error of the mean of two measurements are determined, respectively, by the formulas:
\[ M_H=\pm\frac{d_H}{\sqrt{2}}, \tag{45} \]
\[ M_{H\,\mathrm{av}}=\pm\frac{d_H}{2}, \tag{46} \]
where \(d_H\) is the actual difference of two independent measurements.
If the observations are dependent, then from their difference it is impossible to determine either the error of one measurement or the error of the mean, since in this case the difference is formed only by the errors of independent factors. Thus, the use in this case of formulas (45) and (46) would lead to the determination of the corresponding errors only on account of the errors of some factors (the independent ones), which, from the point of view of an overall assessment of the accuracy of solving the given problem, is of no interest, although, as will be seen below, it is not without some practical meaning.
The error \(M_H\) of the depth \(H\) from a single measurement and the error \(M_{H\,\mathrm{av}}\) of the mean \(H_{\mathrm{av}}\) from two dependent observations, when the actual difference of the measurements is available, can be determined, respectively, by the formulas:
\[ M_H=\pm\sqrt{\frac{(d'_H)^2}{2}+(M''_H)^2}, \tag{47} \]
\[ M_{H\,\mathrm{av}}=\pm\sqrt{\frac{(d'_H)^2}{4}+(M''_H)^2}, \tag{48} \]
where \(d'_H\) is the actual difference of two dependent measurements.
From consideration of this question two main conclusions follow:
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The mean value of the elevation from independent measurements is more accurate than from dependent ones.
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An assessment of the accuracy of the measurement and of the result from the actual difference of dependent observations is impossible. The error of the measurement and of the result can be determined in this case only theoretically, taking into account the actual difference, by formulas (47) and (48).
However, these conclusions should not be understood as implying the necessity of completely excluding dependent observations from the practice of transferring elevation to the level of underground workings. On the contrary, when, for example, making a connection between opposite headings driven from two different shafts, it should be recommended that the transfer of elevations to the level of the first and second shafts be carried out with the same wire and tape, since in this case the error due to errors of common factors will not affect the result of the connection, and the discrepancy in elevation of the headings being connected will be smaller than if the observations were completely independent.
The errors of connections due to errors in transferring elevation, for dependent and independent measurements, respectively, will be:
\[ M_1=\pm\sqrt{\frac{(M'_{H_1})^2}{2}+\frac{(M'_{H_2})^2}{2}}, \]
\[ M_2=\pm\sqrt{\frac{M_{H_1}^2}{2}+\frac{M_{H_2}^2}{2}}, \]
where \(M'_{H_1}\) and \(M'_{H_2}\) are the errors in transferring elevation to the level of the first and second shafts due only to errors of independent factors (the measurements are dependent);
\(M_{H_1}\) and \(M_{H_2}\) are the same due to all factors (the measurements are independent).
If the dependent measurements for individual shafts are independent of one another, then the error of the connection in elevation due to errors in transferring elevation will be the greatest and equal to:
\[ M=\pm\sqrt{\frac{(M'_{H_1})^2}{2}+\frac{(M'_{H_2})^2}{2}+(M''_{H_1})^2+(M''_{H_2})^2}, \]
where \(M''_{H_1}\) and \(M''_{H_2}\) are the errors in transferring elevation to the level of the first and second shafts due to dependent (common) factors.
This second case of dependent measurements should indeed be excluded from the practice of transferring elevation, whereas the first should be recommended.
Conclusions
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When measuring the depth of a shaft, blocks on ball bearings should be recommended, as having a stable coefficient of friction and, consequently, ensuring a more accurate measurement result.
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Special attention must be paid to determining the mean temperature of the wire, since the temperature factor has the greatest influence on the accuracy of the measurements. For this purpose, the initial temperature of the wire should be determined by the method set forth in work [4].
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It is inadmissible to use tapes and wires for which their technical characteristics are unknown.
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The accuracy assessment of elevation transfer should be carried out differentially, taking into account the degree of dependence of the measurements. For this purpose one could recommend, for example, formulas (24), (30), (36).
References
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D. A. Kazakovskii—Mine Surveying and Geometry of Mineral Deposits, 1948.
M. I. Gusev—Course in Mine Surveying, 1, 1948.
F. I. Vydrin—Geodesy and Mine Surveying, 1948. -
Proceedings of the All-Union Scientific Research Mine-Surveying Institute.
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Prof. I. M. Bakhurin—Problems of Mine-Surveying Art. Supplement to the course “Mine-Surveying Art,” 1936.
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A. I. Volkov—On the question of measuring the depth of a shaft with a wire (printed in this volume of the Proceedings of TPI).
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Technical instruction for carrying out mine-surveying work, 1939.
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The notation used in formula (3) is given in [4], published in the present volume of the Proceedings of the TPI. ↩