A Method for Computing Orientation Through Two Vertical Mine Shafts Under the Condition of Minimal Orientation Error in Underground Surveys

V. I. Akulov

(Presented by the scientific seminar of the departments of mine surveying and geodesy)

For each side of an underground connecting traverse consisting of $N$ sides, from a single orientation through two vertical mine shafts, as established by the author [1], “$N+1$” values of the directional angle are determined, with different accuracy, in the coordinate system adopted on the surface.

As the probable value of the directional angle of a side from among its “$N+1$” values, naturally, that value must be adopted which has the minimal error.

The surveying of the mine workings of the oriented horizon usually begins from some one side of the underground connecting traverse; therefore there is no need to establish probable directional angles for all its sides; it is sufficient to establish them only for the initial side, i.e., for the side from which the underground survey begins.

Determination of the probable value of the directional angle of the initial side from among its “$N+1$” values is practically reduced to establishing that side of the orientation traverse without which the directional angle of the initial side of the underground connecting traverse is determined with the minimal error.

The side of the orientation traverse without which the directional angle of the initial side of the underground connecting traverse is determined with the minimal error can be established by analytical and grapho-analytical methods.

In what follows we shall consider only the grapho-analytical method, which, in comparison with the analytical one, does not require extensive computations and, at the same time, makes it possible:

1) to construct the curve of the mean errors of orientation;
2) to determine the root-mean-square errors for all “$N+1$” values of the directional angle of the initial side.

Let us take the side numbered “$q$” of the underground connecting traverse (Fig. 1) as the initial side of the underground surveys of the horizon being oriented.

The directional angle of the initial side and its root-mean-square error, depending on the errors of measurements of the angles and lengths of the sides of the underground connecting traverse, for the case when the orien-

the orientation is computed without side “\(K\)” of the orienting traverse, are determined by formulas [1]:

\[ \alpha_{q_k}=\alpha_{1K}+\alpha'_q, \tag{1} \]

\[ \alpha_{1K}=\alpha_{1AB}+\frac{\rho\,\Delta C\,\tg \varphi_k}{c}, \tag{2} \]

\[ \alpha_{1AB}=(AB)-(AB)', \tag{3} \]

\[ \tg(AB)'=\frac{y'_B}{x'_B}, \tag{4} \]

\[ \varphi_k=\alpha'_k-(AB)', \tag{5} \]

\[ M_{\alpha_{q_k}}=\pm \frac{\rho}{c\cos\varphi_k}\sqrt{m_{Ak}^{\prime 2}+m_{Bk}^{\prime 2}}, \tag{6} \]

where \(\alpha_{1K}, \alpha_{1AB}\) are the direction angle of the first side of the underground connecting traverse, computed without side “\(K\)” and without the side of the orienting traverse between the plumb lines;

\((AB)\) is the direction angle of the alignment of the plumb lines;

\(\alpha'_k, \alpha'_q, (AB)'\) are the conditional direction angles of the side without which the orientation is computed, of the initial side, and of the line of the alignment of the plumb lines relative to a conditional abscissa axis coinciding with the direction of the first side of the underground connecting traverse;

\(x'_B, y'_B\) are the conditional coordinates of plumb line \(B\) relative to a conditional origin of coordinates coinciding with plumb line \(A\);

\(\Delta C\) is the difference between the distances between the plumb lines from the underground and surface surveys;

\(c\) is the distance between the plumb lines;

\(m'_{Ak}\) is the error of plumb line \(A\) in the direction perpendicular to side “\(K\),” from the suspended traverse from vertex “\(q-1\)” to plumb line \(A\), taking the direction angle of side “\(q\)” as fixed;

\(m'_{Bk}\) is the error of plumb line \(B\) in the direction perpendicular to side “\(K\),” from the suspended traverse from vertex “\(q-1\)” to plumb line \(B\), taking the direction angle of side “\(q\)” as fixed;

\(\rho=206265''\).

We shall write formula (6) in the following form:

\[ M_{\alpha_{q_k}}=\pm \frac{\rho P_k}{c\cos\varphi_k}, \tag{6a} \]

where \(P_k\) is the radius vector of the resultant curve of mean errors, drawn perpendicular to side “\(K\).”

The resultant curve of mean errors in a polar coordinate system has the equation:

\[ P^2=A^2\cos^2\Theta+B^2\sin^2\Theta, \tag{7} \]

where \(P\) is the radius vector of the resultant curve of mean errors;

\(A, B\) are, respectively, the major and minor semiaxes of the resulting error ellipse;

\(\Theta\) is the angle formed by the radius vector with the major axis of the resulting error ellipse.

— resulting curve of mean errors

— curve of mean errors of orientation

Fig. 1

The elements of the resulting error ellipse are determined by the formulas:

\[ \tan 2\gamma_1=\frac{\omega_y}{\omega_x}; \tag{8} \]

\[ \omega_x=(A_1^2-B_1^2)\cos 2\alpha_1+(A_2^2-B_2^2)\cos 2\alpha_2; \tag{9} \]

\[ \omega_y=(A_1^2-B_1^2)\sin 2\alpha_1+(A_2^2-B_2^2)\sin 2\alpha_2; \tag{10} \]

\[ A^2=\frac{1}{2}\left[(A_1^2+B_1^2)+(A_2^2+B_2^2)\right]+\frac{1}{2}\omega; \tag{11} \]

\[ B^2=\frac{1}{2}\left[(A_1^2+B_1^2)+(A_2^2+B_2^2)\right]-\frac{1}{2}\omega; \tag{12} \]

\[ \omega=\sqrt{\omega_x^2+\omega_y^2}, \tag{13} \]

where \(\gamma_1\) is the direction angle of the minor axis of the resulting error ellipse;

\(A_1, B_1, \alpha_1\) are the major and minor semiaxes and the direction angle of the minor axis of the mean-square error ellipse for plumb line \(A\), from the hanging traverse from vertex “\(q-1\)” to plumb line \(A\), taking the direction angle of side “\(q\)” as fixed;

\(A_2, B_2, \alpha_2\) are the major and minor semiaxes and the direction angle of the minor axis of the mean-square error ellipse for plumb line \(B\), from the hanging traverse from vertex “\(q-1\)” to plumb line \(B\), taking the direction angle of side “\(q\)” as fixed.

The elements of the mean-square error ellipses for plumb lines \(A\) and \(B\) are determined by the formulas of Prof. F. F. Pavlov [2].

On the orientation scheme (Fig. 1), taking plumb line \(B\) as the center of the resultant error ellipse, on the basis of equation (7) the resultant curve of mean errors is constructed.

For each side of the orientation traverse, including also the side between the plumb lines, the value of the perpendicular radius vector of the resultant curve of mean errors is determined graphically.

Substituting successively into formula (6a) the values of the perpendicular radius vectors of the curve of mean errors, we find, for all \(N+1\) values of the directional angle of the initial side, the mean square errors; from these we determine the side of the orientation traverse without which the directional angle of the initial side is determined with the minimum error, i.e., we determine the side without which the orientation should be computed under the condition of minimum error in orienting underground surveys.

The angle \(\varphi_k\), entering into formula (6a), is determined graphically from the orientation scheme.

The side of the orientation traverse without which the orientation should be computed under the condition of minimum error in orienting underground surveys can also be established from the curve of mean errors of the directional angle of the initial side, which we shall call the curve of mean errors of orientation.

The curve of mean errors of orientation in a polar coordinate system has the equation

\[ R_k=\pm \frac{P_k}{\cos \varphi_k} \tag{14} \]

or

\[ R_k=\pm \frac{\sqrt{A^2\sin^2(\alpha_k-\gamma_2)+B^2\cos^2(\alpha_k-\gamma_2)}} {\cos[\alpha_k-(AB)]}, \tag{14a} \]

where \(R_k\) is the radius vector of the curve of mean errors of orientation, perpendicular to side “\(K\)”;

\(P_k\) is the radius vector of the resultant curve of mean errors (7), perpendicular to side “\(K\)”;

\(\alpha_k\) is the directional angle of side “\(K\)”;

\(\gamma_2\) is the directional angle of the major axis of the resultant error ellipse.

From expressions (6a) and (14) we obtain:

\[ M_{\alpha_q k}=\pm K R_k; \tag{15} \]

\[ K=\frac{\rho}{c}. \tag{16} \]

It follows from formula (15) that the directional angle of the initial side is determined with the minimum error without the side of the orientation traverse that has the smallest perpendicular radius vector of the curve of mean errors of orientation. Therefore, in order to decide without which side of the orientation traverse the orientation should be computed under the condition of minimum error in orienting underground surveys, it is necessary, on the orientation scheme, taking plumb line \(B\) as the center of the resultant error ellipse, to construct the curve of mean errors of orientation.

From the study of the curve of mean orientation errors (14a) it follows:

1) The curve is tangent to the resultant curve of mean errors (7) at the points of its intersection with the line drawn through the center of the resultant error ellipse perpendicular to the plumb-line alignment, i.e., at points \(C\) and \(D\) (Fig. 1).

2) The curve has two minimum vertices with polar coordinates \(e(\alpha_e, R_e)\) and \(f(\alpha_f, R_f)\).

3) The center of the curve coincides with the center of the resultant error ellipse.

4) The asymptote of the curve is the line of the plumb-line alignment \(AB\).

The polar coordinates of the minimum vertices are determined by the formulas:

\[ \alpha_e = \gamma_2 + \Theta_0 + 90^\circ; \tag{17} \]

\[ \alpha_f = \alpha_e + 180^\circ; \tag{18} \]

\[ \tg \Theta_0 = \frac{B^2}{A^2}\tg\left[(AB)-\gamma_2\right]; \tag{19} \]

\[ R_e = R_f = \frac{A\cdot B}{P_{AB}}, \tag{20} \]

where \(P_{AB}\) is the radius vector of the resultant curve of mean errors along the line of the plumb-line alignment.

To construct the curve of mean orientation errors, it is necessary, using polar coordinates, to plot the minimum vertices “\(e\)” and “\(f\)” (Fig. 1), after which, on the basis of the resultant curve of mean errors, a number of points belonging to one branch of the curve of mean orientation errors are determined graphically (as is shown in Fig. 1 for point \(4_0\)). By connecting the indicated points with a smooth curve, one obtains one of the branches of the curve of mean orientation errors; from it, symmetrically with respect to the center of the resultant error ellipse, its second branch is constructed.

To each side of the orientation traverse, including the side between the plumb lines, perpendicular radius vectors of the curve of mean orientation errors are drawn.

Of all the perpendicular radius vectors, the minimum one is determined, and from it the side of the orientation traverse is established without which the orientation must be computed under the condition of minimum error of orientation of underground surveys.

By formula (2), the value of the direction angle of the first side is determined, computed under the condition of minimum error of the direction angle of the initial side; on the basis of this, the direction angles of all sides of the underground connecting traverse, including the initial side, are determined.

The root-mean-square error of the direction angle of the initial side, computed under the condition of minimum error of orientation of underground surveys, is determined by formula (15), on the basis of the minimum perpendicular radius vector of the curve of mean orientation errors.

By the graphoanalytic method it is also possible to establish the side of the orientation traverse without which the direction angle of the initial side of the underground connecting traverse is determined with the minimum error depending on the errors of the initial control points po-

of polygonometry or triangulation, the errors of measuring the angles and lengths of the sides both of the surface theodolite traverses to the plumb lines \(A\) and \(B\), and of the underground connecting polygon. In this case the resultant curve of mean errors is constructed from the elements of the resultant error ellipse obtained by the quadratic addition of the mean square error ellipses for the initial points of the polygonometry or triangulation \(Q\) and \(S\) (Fig. 1), for the plumb lines \(A\) and \(B\) from the approach surface theodolite traverses \(Q—1'—2'—A\) and \(S—1''—2''—B\), and of the underground traverses respectively from vertex “\(q—1\)” to plumb line \(A\) and from vertex “\(q—1\)” to plumb line \(B\), considering the direction angle of side “\(q\)” in these polygons to be fixed.

The computation of the orientation through two vertical mine shafts under the condition of the minimum error of the direction angle of the initial side increases the accuracy of orientation of underground surveys by \(t(\%)\), determined from the formula

\[ t = 100(1 - K_q); \tag{21} \]

\[ K_q = \frac{M\alpha_q}{M\alpha_{qAB}}, \tag{22} \]

where
\(M\alpha_q\) is the error of the direction angle of the initial side, computed under the condition of the minimum error of orientation of underground surveys;

\(M\alpha_{qAB}\) is the error of the direction angle of the initial side, computed without the side of the orientation polygon between the plumb lines, i.e., by the method recommended in mine-surveying literature.

For the case when the major axis of the resultant error ellipse makes an angle of \(45^\circ\) with the line of the plumb lines, and in the orientation polygon there is a side perpendicular to the minimum radius vector \(Be\) or \(Bf\) of the curve of mean orientation errors, the greatest possible increase in the accuracy of orientation of underground surveys due to the method of computing the orientation will be

\[ t_{\max} = 100 \frac{(\lambda - 1)^2}{1 + \lambda^2}; \tag{23} \]

\[ \lambda = \frac{A}{B}. \tag{24} \]

In Table 1, on the basis of formula (23), the greatest possible increase in orientation accuracy due to the computation method has been calculated for various ratios of the semiaxes of the resultant error ellipse.

Table 1

In conclusion, let us consider increasing the accuracy of orienting underground surveys by means of repeated independent orientations. If one and the same orientation was carried out independently \(n\) times, then the accuracy of orienting underground surveys, due to repeated independent orientations, will increase by \(t'\) (%), determined by the formula:

\[ t' = 100 \left( 1 - \frac{1}{\sqrt{n}} \right). \tag{25} \]

The increase in orientation accuracy due to repeated independent orientations, calculated by formula (25) for various integer values of \(n\), is given in Table 2.

Table 2

For each orientation, proceeding from Table 2, it is possible to establish the economic expediency of computing the orientation under the condition of a minimum orientation error for underground surveys.

Thus, for example, increasing the accuracy of orientation by 29% by means of the computation method is equivalent to a repeated orientation.

Orientation for control is always carried out twice; therefore, for the same conditions (\(t = 29\%\)), in order to obtain the direction angle of the initial side when computing the orientation by the method recommended in surveying literature, with the same accuracy with which it is determined from two independent orientations computed under the condition of a minimum orientation error for underground surveys, it is necessary to carry out four independent orientations.

LITERATURE

1. Akulov V. I. A general method for computing and estimating the accuracy of orientation through two vertical mine shafts (in the present collection).

2. Pavlov F. F. Prediction of errors in basic mine-surveying works. Ugletekhizdat, 1950.