Simplified Methods for Adjusting an Underground Connecting Traverse When Orienting Through Two Vertical Mine Shafts and Criteria for Their Application

V. I. Akulov

(Presented by the scientific seminar of the Department of Mine Surveying and Geodesy)

In an underground connecting traverse (Fig. 1), two equalities must be satisfied:

\[ [\Delta x] - (X_B - X_A) = 0; \]

\[ [\Delta y] - (Y_B - Y_A) = 0, \tag{1} \]

where \(X_A, Y_A, X_B, Y_B\) are the coordinates of plumb lines \(A\) and \(B\) from the surface survey;
\(\Delta x, \Delta y\) are coordinate increments.

Because of measurement errors, equalities (1) are not satisfied, namely:

\[ f_x = [\Delta x] - (X_B - X_A); \]

\[ f_y = [\Delta y] - (Y_B - Y_A), \tag{2} \]

where \(f_x, f_y\) are the misclosures in the coordinate increments;
\(\Delta x, \Delta y\) are the coordinate increments computed from the measured elements of the underground connecting traverse.

The misclosures in the coordinate increments may also be computed by the formulas

\[ f_x = \Delta c \cdot \cos(AB); \]

\[ f_y = \Delta c \cdot \sin(AB); \tag{3} \]

\[ \Delta c = C_m - C_n, \tag{4} \]

where \(C_m, C_n\) are the distances between the plumb lines from the underground connecting traverse and from the surface survey;
\((AB)\) is the direction angle of the line of the plumb-line alignment.

The relative linear misclosure of the underground connecting traverse, taking into account equalities (3), is

\[ \frac{\sqrt{f_x^2+f_y^2}}{[l]}=\frac{\Delta c}{[l]}, \tag{5} \]

where \([l]\) is the perimeter of the underground connecting traverse.

The misclosures in the coordinate increments of the underground connecting traverse may be distributed in two ways: proportionally to the lengths of the sides or proportionally to the increments, i.e., the underground connecting traverse may be adjusted by two simplified methods.

First method. The misclosures in the coordinate increments are distributed proportionally to the lengths of the sides.

Considering the coordinate increments \(\Delta x\) and \(\Delta y\) as “measured” independent quantities, and equalities (1) as “geometric conditions,” we obtain two conditional equations

\[ [(\Delta x)] + f_x = 0; \]

\[ [(\Delta y)] + f_y = 0, \tag{6} \]

where \((\Delta x)\), \((\Delta y)\) are corrections to the coordinate increments.

Solving the system of conditional equations (6) under the condition \([p_{\Delta x}(\Delta x)^2]=\min\) and \([p_{\Delta y}(\Delta y)^2]=\min\), and considering the weight of the coordinate increments to be inversely proportional to the length, we obtain

\[ (\Delta x_i)=-\frac{f_x}{[l]}\,l_i; \]

\[ (\Delta y_i)=-\frac{f_y}{[l]}\,l_i. \tag{7} \]

The method of distributing the misclosures in coordinate increments proportionally to the lengths of the sides, i.e., in accordance with formulas (7), is recommended by the Technical Instruction for Mine-Surveying Work for all underground connecting traverses, regardless of their form.

In connection with the correction of the coordinate increments, the direction angles and side lengths of the underground connecting traverse receive corrections

\[ (\alpha_i)=\frac{\rho\,\Delta c}{[l]}\sin\varphi_i; \tag{8} \]

\[ (l_i)=-\frac{\Delta c}{[l]}\,l_i\cos\varphi_i; \tag{9} \]

\[ \varphi=\alpha-(AB), \tag{10} \]

where \((\alpha)\), \((l)\) are the corrections to the direction angle and to the side length, respectively;

\(\varphi\) is the conditional direction angle relative to a conditional axis of abscissas coinciding with the direction of the plumb-line alignment \(AB\);

\(\rho=206265''\).

Corrections to the measured angles due to adjustment are found from the formulas:

a) for left angles

\[ (\beta_i) = (\alpha_{i+1}) - (\alpha_i); \tag{11} \]

b) for right angles

\[ (\beta_i) = (\alpha_i) - (\alpha_{i+1}). \tag{12} \]

Let us determine, relative to the plumb-line alignment line, the transverse and longitudinal displacements of the vertex of the side with number \(i\) (Fig. 2) of the underground connecting polygon caused by the adjustment.

Fig. 2. Diagram of the displacement of a polygon vertex due to adjustment.

In Fig. 2:

\(i\), \(i_l\) are the positions of the vertex of the side with number \(i\) before and after adjustment of the polygon, respectively;

\(AB\) is the direction of the plumb-line alignment line.

When the length is corrected by \((l_i)\), the vertex of the side is displaced to point \(i_l\). The transverse and longitudinal displacements of the vertex are equal to

\[ u_{i_l} = iC = (l_i)\sin |z_i - (AB)|; \]

\[ t_{i_l} = iD = (l_i)\cos |z_i - (AB)|. \]

Taking formulas (9) and (10) into account, we obtain:

\[ u_{i_l} = \frac{\Delta c}{[l]}\, l_i \sin z_i \cos z_i; \tag{13} \]

\[ t_{i_l} = -\frac{\Delta c}{[l]}\, l_i \cos^2 z_i. \tag{14} \]

In connection with the correction of the direction angle by \((\alpha_i)\), the vertex \(i_l\) will be displaced to point \(i_\alpha\). The transverse and longitudinal displacements of the vertex are equal to

\[ u_{i_\alpha} = i_lD = i_li_\alpha \cos |z_i - (AB)|, \]

\[ t_{i_\alpha} = i_lD = i_li_\alpha \sin |z_i - (AB)|. \]

Since

\[ i_li_\alpha = \frac{(\alpha_i)l_i}{\rho}, \]

then

\[ u_{i_\alpha} = -\frac{(\alpha_i)l_i}{\rho}\cos |z_i - (AB)|, \]

\[ t_{i_\alpha} = -\frac{(\alpha_i)l_i}{\rho}\sin |z_i - (AB)|, \]

or

\[ u_{i_\alpha} = -\frac{\Delta c}{[l]}\, l_i \sin z_i \cos z_i, \tag{15} \]

\[ t_{i_\alpha} = -\frac{\Delta c}{[l]}\, l_i \sin^2 z_i. \tag{16} \]

The total displacements of a vertex are equal to

\[ u_i = u_{i_1} + u_{i_2} = 0; \tag{17} \]

\[ t_i = t_{i_1} + t_{i_2} = - \frac{\Delta c}{[l]}\, l_i . \tag{18} \]

Consequently, when coordinate misclosures are distributed in proportion to the lengths of the sides, the vertices of the polygon are displaced in directions parallel to the line of alignment of the plumb lines, toward the side opposite to the linear misclosure of the traverse, i.e., to the difference \(\Delta c\).

The total displacement of all vertices of the polygon along the line of alignment of the plumb lines, due to the separate adjustment of angles and sides, is equal to

\[ t_{\mathrm{з}} = - \frac{[l \sin^2 \varphi]}{[l]}\,\Delta c; \tag{19} \]

\[ t_l = - \frac{[l \cos^2 \varphi]}{[l]}\,\Delta c. \tag{20} \]

Hence

\[ t_{\mathrm{з}} + t_l = - \Delta c, \tag{21} \]

where \(t_{\mathrm{з}}, t_l\) are parts of the linear misclosure of the underground connecting polygon, eliminated by means of the angles and the sides.

Thus, when misclosures in coordinate increments are distributed in proportion to the lengths of the sides, the linear misclosure of the polygon is eliminated both by means of the angles and by means of the sides.

Second method. The misclosures in coordinate increments are distributed in proportion to the increments.

Solving the system of conditional equations (6) under the conditions \([p_{\Delta x}(\Delta x)^2] = \min\) and \([p_{\Delta y}(\Delta y)^2] = \min\), and considering the weight of the “measured” increment to be inversely proportional to the increment itself, we obtain:

\[ (\Delta x_i) = - \frac{f_x}{[\Delta x]}\,\Delta x_i, \tag{22} \]

\[ (\Delta y_i) = - \frac{f_y}{[\Delta y]}\,\Delta y_i. \]

The corrections to the angles in connection with the adjustment of the coordinate increments are equal to zero, while those to the lengths of the sides are found from the formula

\[ (l_i) = - \frac{\Delta c}{C_{\mathrm{ш}}}\, l_i. \tag{23} \]

The adjusted lengths of the sides are determined from the formula

\[ l_i^{p} = l_i + (l_i) \tag{24} \]

or

\[ l_i^{p} = \frac{C_n}{C_{\mathrm{ш}}}\, l_i. \tag{25} \]

The transverse and longitudinal displacements of the terminal vertex of a side relative to the plumb-line alignment, due to adjustment, are:

\[ u_i=(l_i)\sin[\alpha_i-(AB)], \]

\[ t_i=(l_i)\cos[\alpha_i-(AB)] \]

or

\[ u_i=-\frac{\Delta c}{C_{\mathrm{ш}}}\,l_i\sin\varphi_i; \tag{26} \]

\[ t_i=-\frac{\Delta c}{C_{\mathrm{ш}}}\,l_i\cos\varphi_i. \tag{27} \]

The total displacements of all vertices of the traverse in the directions perpendicular and parallel to the plumb-line alignment, due to adjustment, are equal to:

\[ u_e=0; \tag{28} \]

\[ t_l=-\Delta c. \tag{29} \]

Thus, in the second method of adjustment the linear misclosure of the traverse is completely eliminated by the sides.

The corrected coordinates of the vertices of the underground connecting traverse, under the second method of adjustment, may be computed in two ways.

First way. The underground connecting traverse is computed in a single coordinate system, on the basis of the measured angles and side lengths reduced to the level surface and to the plane in the Gauss projection. The misclosures in the coordinate increments are distributed in proportion to the increments.

To bring the side lengths into agreement with the corrected coordinates, it is necessary to add to them the corrections computed by formula (23).

Second way. The underground connecting traverse, from the measured angles and side lengths reduced to the level surface and to the plane in the Gauss projection, is computed in a conventional coordinate system. The distance between the plumb lines is found from the underground connecting traverse, and the adjusted side lengths are determined by formula (25).

Using the measured angles and the adjusted side lengths, the underground connecting traverse is computed in a single coordinate system.

The coordinates of plumb line \(B\) from the underground connecting traverse must correspond to the coordinates obtained from the surface survey.

The second way of computing the adjusted coordinates of the vertices of the underground connecting traverse is recommended by engineer B. A. Rostkovskii¹.

Criteria for the Application of Simplified Methods of Adjustment

The criteria for applying simplified methods of adjustment are established on the basis of the law of accumulation of errors in the underground connecting traverse.

The error in the position of the terminal point of the underground connecting traverse, i.e., of plumb line \(B\), as a function of the measurement errors of its elements, as is known, is expressed by the formulas

\[ M_B^2 = M_\beta^2 + M_l^2; \tag{30} \]

\[ M_\beta^2=\left[\left(\frac{\eta}{\rho}\right)^2 m_\beta^2\right]; \tag{31} \]

\[ M_l^2 = a^2 l \cos^2 \varphi + b^2 C^2, \tag{32} \]

where \(M_\beta\), \(M_l\) are the errors in the position of plumb line \(B\), dependent on the errors of measurement, respectively, of the angles and sides of the underground connecting traverse;

\(\eta\) is the conditional ordinate of the traverse vertex with respect to a conditional axis of abscissas coinciding with the line of the plumb-line alignment;

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

\(m_\beta\) is the mean error of angle measurement;

\(a\) is the coefficient of random influence in length measurement;

\(b\) is the coefficient of systematic influence.

The computation of the error in the position of plumb line \(B\) by formula (30) is performed in order to determine the permissible discrepancy of the distances between plumb lines from underground and surface surveys, calculated by the formula

\[ \Delta c_{\mathrm{dop}} = \pm 2 M_B. \tag{33} \]

The correctness of the performance of measurement work when orienting through two vertical mine shafts is checked by the inequality

\[ \Delta c \leq \Delta c_{\mathrm{dop}}. \tag{34} \]

Starting from equality (30), let us write the identity

\[ M_B^2 = \left(\frac{M_\beta}{M_B}\right)^2 M_B^2 + \left(\frac{M_l}{M_B}\right)^2 M_B^2. \tag{35} \]

The coefficients \(\left(\dfrac{M_\beta}{M_B}\right)^2\) and \(\left(\dfrac{M_l}{M_B}\right)^2\) express the relative distribution of the square of the linear misclosure of the underground connecting traverse between the angles and the sides, respectively.

We shall establish the conditions for applying simplified methods of adjustment on the basis of the character of the distribution of the linear misclosure of the traverse between the angles and the sides which they provide. For adjustment, the method adopted is the one that provides a distribution of the linear misclosure in accordance with the law of accumulation of errors.

To establish the criterion for applying the first simplified adjustment method, starting from equality (21), let us write the identity

\[ \Delta c^2 = (t_\beta + t_l)t_\beta + (t_\beta + t_l)t_l \]

or

\[ \Delta c^2 = A \Delta c^2 + B \Delta c^2; \tag{36} \]

\[ A=\frac{[l\sin^{2}\varphi]}{[l]}; \tag{37} \]

\[ B=\frac{[l\cos^{2}\varphi]}{[l]}. \tag{38} \]

The coefficients \(A\) and \(B\) express the relative distribution of the square of the linear misclosure of the traverse between the angles and the sides, respectively, in the first method of adjustment.

From identities (35) and (36) it follows that the first simplified method of adjustment ensures the distribution of the linear misclosure of the traverse between the angles and the sides in accordance with the law of error accumulation when the equalities

\[ \frac{M_{\text{з}}}{M_{B}}=\sqrt{A}, \]

\[ \frac{M_{l}}{M_{B}}=\sqrt{B}. \tag{39} \]

are satisfied.

If one of the equalities (39) is satisfied, then, as a consequence, the other is also satisfied; therefore it is sufficient to use one of the indicated equalities.

In practice, the first simplified method may be used to adjust those underground connecting traverses in which the smaller, in absolute value, equality (39) is satisfied within limits up to \(10\%\), i.e., when the inequality

\[ 0.9\sqrt{A}<\frac{M_{\text{з}}}{M_{B}}<1.1\sqrt{A}\quad (\text{when } M_{\text{з}}<M_{l}) \tag{40} \]

is satisfied, or

\[ 0.9\sqrt{B}<\frac{M_{l}}{M_{B}}<1.1\sqrt{B}\quad (\text{when } M_{l}<M_{\text{з}}). \tag{40a} \]

The second simplified method may be used to adjust those underground connecting traverses in which the linear misclosure is caused exclusively by errors in measuring the lengths of the sides.

In practice, the second method may be used to adjust underground connecting traverses in which the inequality

\[ M_{l}\geq 3M_{\text{з}}. \tag{41} \]

is satisfied.

Underground connecting traverses of complex form, for which inequalities (40) or (41) are not satisfied, must be adjusted by the rigorous method.

Underground connecting traverses of elongated form may be adjusted by either of the simplified methods; in this case the adjustment results will correspond to the rigorous method.

1. Control by (34) does not reveal gross errors in the measurement of sides perpendicular to the alignment lines of the plumb lines, and of angles, in an elongated traverse. ↩