Tables for Determining the Root-Mean-Square Errors of Horizontal Angle Measurements Due to Improper Centering of the Theodolite and Signals
Abstract Generated abstract
The paper addresses the repeated calculation of root-mean-square errors in horizontal angle measurements arising from improper centering of a theodolite and survey signals, a task important in assessing mine-surveying accuracy. It reformulates the standard error equation into tabulated components and describes two tables: one for the numerator terms depending on side lengths, signal distance, and centering errors, and another for the product of squared angle-side lengths. The procedure permits interpolation for intermediate values and scaling for other ranges, and an example demonstrates the computation of the angular error. The author reports that the tables allow determination of the error within one unit of the third significant figure in about two to three minutes.
Full Text
TABLES FOR DETERMINING THE ROOT-MEAN-SQUARE ERRORS OF MEASURING HORIZONTAL ANGLES DUE TO IMPROPER CENTERING OF THE THEODOLITE AND SIGNALS
G. F. LYSOV
(Presented by the scientific seminar of the departments of mine surveying and geodesy)
When assessing the accuracy of the most critical mine-surveying operations, the squares of the mean errors of measuring horizontal angles due to improper centering of the theodolite and signals must repeatedly be computed from the formula
\[ m_y^2=\frac{\rho^2 e_c^2}{2a^2}+\frac{\rho^2 e_c^2}{2b^2}+\frac{\rho^2 e_T^2 c^2}{2a^2b^2}, \tag{1} \]
where \(e_c, e_T\) are the mean linear errors of centering the signal and the theodolite,
\[ \begin{aligned} a, b&\text{ are the lengths of the sides of the measured angle,}\\ c&\text{ is the distance between the signals,}\\ \rho&\text{ is the radian in seconds }(206265''). \end{aligned} \]
Calculation of the quantity \(m_y^2\) by formula (1) is too complicated. In connection with this, the mine-surveying literature indicates a number of methods that make it possible to find the quantity \(m_y^2\) with less expenditure of labor and time.
Meanwhile, to solve this problem for any possible values of the variables \(e_c, e_T, a, b\), and \(c\), special tables compiled by the author of this article are used at the Department of Mine Surveying of the Tomsk Polytechnic Institute, on the basis of the considerations set forth below.
Let us present formula (1) in the following form
\[ m_y^2=\frac{A_1+A_2+A_3}{a^2b^2}, \tag{2} \]
where
\[ A_1=\frac{\rho^2 e_c^2 a^2}{2}, \tag{3} \]
\[ A_2=\frac{\rho^2 e_c^2 b^2}{2}, \tag{4} \]
\[ A_3=\frac{\rho^2 e_T^2 c^2}{2}. \tag{5} \]
In accordance with formulas (3), (4), (5), Table 1 (Appendix 1) contains the values of \(A_1\), \(A_2\), and \(A_3\) (reduced by a factor of \(10^4\)) within the range of variation of the variables \(a\), \(b\), \(c\) from 10 to 100 m at intervals of two meters, and of \(e_c\), \(e_T\)—from 0.1 mm to 0.3 mm at intervals of 0.02 mm and from 0.3 mm to 0.95 mm at intervals of 0.05 mm. The quantities \(A_1\), \(A_2\), \(A_3\) corresponding to intermediate values of the variables are found by interpolation in the mind. This table may be used for any values of the variables, since increasing one variable by a factor of 10 increases the function by a factor of 100.
Table 2 (Appendix 2) contains the quantities \(a^2b^2\) (reduced by a factor of \(10^4\)) within the range of variation of the variables \(a\) and \(b\) from 10 to 100 m at intervals of two meters. The quantities \(a^2b^2\) corresponding to intermediate values of the variables are found by interpolation. This table may likewise be used for any values of the variables, since increasing one variable by a factor of 10 increases the function by a factor of 100.
The determination of \(m_y^2\) by means of the tables described is carried out in the following order:
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Using Table 1, we successively find the values \(A_1\), \(A_2\), and \(A_3\), looking for the numbers at the intersections of the rows corresponding to the given \(a\), \(b\), \(c\) (in meters) and the columns corresponding to the given \(e_c\) and \(e_T\) (in mm); we compute the sum \((A_1 + A_2 + A_3)\).
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From Table 2, at the intersection of the row and column corresponding to the given \(a\) and \(b\) (in meters), we find the value \(a^2b^2\).
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By formula (2) we compute the value of \(m_y^2\).
Example. Determine \(m_y^2\), if \(a = 91\) m, \(b = 70\) m, \(c = 140\) m, \(e_c = 1.5\) mm, \(e_T = 2\) mm.
Solution: From Table 1 we find: \(A_1 = 40000\), \(A_2 = 23500\), \(A_3 = 167000\). \((A_1 + A_2 + A_3) = 230500\). From Table 2 we find \(a^2b^2 = 4060\).
Therefore,
\[ m_y^2 = \frac{230500}{4060} = 56.7\ \text{sec}^2;\quad m_y = \pm 7.5\ \text{sec}. \]
In conclusion, let us note that these tables make it possible to determine \(m_y\) with an error of no more than one unit of the third significant figure, with an average time expenditure of 2–3 minutes per determination.