ADJUSTMENT OF AN UNDERGROUND CONNECTING TRAVERSE WITH ORIENTATION THROUGH TWO VERTICAL MINE SHAFTS

V. I. Akulov

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

The orientation of underground surveys by the method “through two vertical mine shafts” is widely used in practice.

The Technical Instructions for the Performance of Mine-Surveying Work, 1959 edition, § 210, require:

“...When two communicating vertical mine shafts are present, orientation must necessarily be carried out by the method through two vertical mine shafts.”

When orienting underground surveys by the method “through two vertical mine shafts,” there is one redundantly measured element, and in connection with this the problem arises of adjusting the underground connecting traverse.

Adjustment, on the one hand, ensures that the elements of the underground connecting traverse are brought into conformity with the initial data—the coordinates of the plumb lines from the surface survey—and, on the other hand, increases the accuracy of orientation of underground surveys.

The question of adjusting an underground connecting traverse has not been sufficiently developed in the mine-surveying literature; in particular, no general method has been developed for adjusting a previously oriented underground connecting traverse and a network of underground traverses closed on two fixed points (plumb lines).

§1. Analysis of Known Methods for Adjusting an Underground Connecting Traverse

In the domestic mine-surveying and geodetic literature, the question of adjusting an underground connecting traverse is considered in works [2, 3, 4, 5, 6, 7].

Prof. N. G. Kell’ [6], when adjusting an underground connecting traverse (Fig. 1) by the method of conditional observations, proceeding from the condition of equality of the distances between plumb lines from the underground and surface surveys, obtained the condition equation

\[ \sum_{1}^{n}\left(\frac{R_i''}{\rho}\right)\varepsilon_{\beta_i} + \sum_{1}^{n+1}\cos \delta_i\, \varepsilon_{l_i} + \Delta C = 0; \tag{1} \]

\[ \Delta C = C_{\sh} - C_{n}; \tag{2} \]

\[ \delta_i = \alpha'_i - (AB)', \tag{3} \]

where
\(\varepsilon_\beta\) is the correction to the angle, in seconds;
\(\varepsilon_l\) is the correction to the length of the side;
\(R_i''\) is the shortest distance from the vertex of the angle to the plumb-line alignment;
\((AB)'\), \(\alpha'\) are the conditional direction angle of the plumb-line alignment and of the side with respect to the conditional \(x\)-axis, coinciding with the first side of the underground connecting traverse;
\(C_{\sh}\) is the distance between the plumb lines from the underground connecting traverse;
\(C_n\) is the distance between the plumb lines from the surface survey;
\(\Delta C\) is the difference of the distances between the plumb lines from the underground and surface surveys;
\(\rho = 206265\).

From the solution of conditional equation (1), under the condition \([p_\beta \varepsilon_\beta^2] + [p_l \varepsilon_l^2] = \min\), the most probable corrections to the measured angles and sides of the traverse are found.

Prof. N. G. Kell recommends computing the direction angles of the sides of the adjusted underground connecting traverse in a single coordinate system on the basis of the direction angle of the first side, found from the formulas:

Fig. 1. Diagram of an underground connecting traverse when oriented through two vertical mine shafts.

\[ \alpha_1 = (AB) - (AB)'; \tag{4} \]

\[ \tg (AB)' = \frac{y'_B}{x'_B}, \tag{5} \]

where \((AB)\) is the direction angle of the plumb-line alignment from the surface survey;
\(x'_B, y'_B\) are the conditional coordinates of plumb line \(B\) with respect to the conditional origin of coordinates, coinciding with plumb line \(A\), computed from the measured elements of the underground connecting traverse.

The method recommended by Prof. N. G. Kell for computing the direction angles of the sides of an adjusted underground connecting traverse is incorrect, since the direction angle of the first side, as established in [2], receives no correction as a result of the adjustment.

Associate Professor T. A. Bui [3] recommends adjusting the underground connecting traverse by the method of conditional observations, proceeding from the condition equation

\[ \sum_{1}^{n}\left(\frac{R_i'}{\rho}\right)\varepsilon_{\beta_i} + \sum_{1}^{n+1}\cos \delta_i\,\varepsilon_{l_i} + \omega_x\cos(AB_1) + \omega_y\sin(AB_1) =0, \tag{6} \]

where \(\omega_x, \omega_y\) are the coordinate misclosures of the underground connecting traverse, computed from the approximate value of the connecting angle at plumb \(A\);

\((AB_1)\) is the approximate direction angle of the closing side of the underground connecting traverse.

The condition equations (1) and (6) differ only in their free terms.

After transformation, the free term of condition equation (6) takes the form:

\[ \omega_x\cos(AB_1)+\omega_y\sin(AB_1) = \Delta C+2C_n\sin^2\frac{t_1}{2}, \tag{7} \]

where \(t_1\) is the error in determining, by the graphical method, the approximate value of the connecting angle at plumb \(A\).

It follows from equality (7) that in all cases when the approximate value of the connecting angle is determined by the graphical method, as recommended by Associate Professor T. A. Bui, instead of an adjustment of the underground connecting traverse there will be a distortion of it.

In work [4], strict and approximate methods are given for adjusting an underground connecting traverse.

In the strict method of adjustment, the authors of work [4] proceed from condition equation (1), which they obtained from the geometric condition

\[ \sum_{1}^{n+1} l_i\cos \alpha_i''-C_n=0, \tag{8} \]

where \(\alpha''\) is the conditional direction angle of a side relative to the conditional \(x\)-axis, coinciding with the direction of the plumb-line alignment.

In deriving, from the geometric condition (8), the condition equation (1), it is assumed in work [4] that the conditional direction angle of the first side relative to the conditional \(x\)-axis, coinciding with the direction of the plumb-line alignment, does not depend on the measured elements of the traverse, which does not correspond to reality.

The shortest distances from the traverse vertices to the plumb-line alignment are recommended by the authors of work [4] to be determined analytically. For purposes of adjustment, the shortest distances can be determined with sufficient accuracy by a more productive graphical method.

The simplified adjustment method recommended in work [4] was first proposed by Engineer B. A. Rostkovskii [7].

Engineer B. A. Rostkovskii [7] recommends distributing the entire linear misclosure of the underground connecting traverse, regardless of its shape, among the sides of the traverse.

The Technical Instruction for the Performance of Mine-Surveying Work, edition of 1959, recommends distributing the coordinate misclosures of the underground connecting traverse in proportion to the lengths of the sides.

Associate Professor Zh. S. Erzhanov [5] criticizes the method recommended by the Technical Mine-Surveying Instruction for distributing the coordinate misclosures of the underground connecting traverse in proportion to the lengths of the sides.

Assoc. Prof. Zh. S. Erzhanov believes that, when distributing coordinate discrepancies in proportion to the lengths of the sides, the entire linear discrepancy of the underground connecting traverse is eliminated at the expense of the sides of the traverse. In this connection Assoc. Prof. Zh. S. Erzhanov proposes his “method” for distributing the linear discrepancy of an underground connecting traverse between the sides and angles, which is not only incorrect, but impossible in its very conception.

Assoc. Prof. Zh. S. Erzhanov’s opinion that, when distributing coordinate discrepancies in proportion to the lengths of the sides, the entire linear discrepancy of the underground connecting traverse is eliminated at the expense of the sides does not correspond to reality.

From an analysis of the literature it follows that the rigorous method of adjusting orientation through two vertical mine shafts requires an additional theoretical investigation, which is set forth below.

§2. Adjustment of an Underground Connecting Traverse by the Method of Conditional Observations

An underground connecting traverse, as a traverse closed in coordinates, is computed in a unified coordinate system without one measured side, for example, without the side numbered \(k\), as established by the author [1], by the formulas:

\[ z_{1_k}=(AB)+\varphi_k-z'_k; \tag{9} \]

\[ \sin \varphi_k=\frac{X'_B \sin z'_k-Y'_B \cos z'_k}{C_n}, \tag{10} \]

where \(z_{1_k}\) is the direction angle, in the unified coordinate system, of the first side, computed without the side numbered \(k\);

\(z'_k\) is the conditional direction angle of the side without which the traverse is computed, relative to the conditional abscissa axis coinciding with the first side of the traverse;

\(\varphi_k\) is the angle between the line of the plumb-line alignment and the side without which the traverse is computed.

The side without which the underground connecting traverse is computed is an redundantly measured quantity.

Consequently, any side of the underground connecting traverse may be taken as an redundantly measured quantity.

Let us take the side numbered \(k\) (Fig. 1) as the redundantly measured quantity.

The redundantly measured side requires fulfillment, in the underground connecting traverse, of the following geometric condition:

\[ l'_k-l_k=0, \tag{11} \]

in which

\[ l'_k=\sqrt{(X_k-X_{k-1})^2+(Y_k-Y_{k-1})^2}, \tag{12} \]

where \(l_k\) is the measured length of the side numbered \(k\);

\(l'_k\) is the computed length of the side numbered \(k\);

\(X_{k-1}, Y_{k-1}, X_k, Y_k\) are the coordinates of the vertices of the traverse bounding the side numbered \(k\), i.e., the side without which the traverse has been computed.

\(^{1)}\) The angle \(\varphi_k\), computed by formula (10), does not depend on the side without which the traverse is computed.

Owing to errors of measurement, the geometric condition (11) will not be satisfied; in connection with this the problem of adjustment arises.

Because of the higher accuracy of the surface surveys, we shall distribute the misclosure of the geometric condition (11) among the measured elements of the underground connecting traverse.

From the geometric condition (11) there follows the following condition equation:

\[ \sum_{1}^{n}\left(\frac{R_i''}{\rho}\right)\varepsilon_{\beta_i} + \sum_{1}^{n+1}\cos\varphi_i\,\varepsilon_{l_i} - \Delta l_k \cos\varphi_k = 0; \tag{13} \]

\[ \Delta l_k = l'_k - l_k . \]

The relation between \(\Delta l\) and \(\Delta C\) is expressed by the formula:

\[ \Delta l_k = -\,\frac{\Delta C}{\cos\varphi_k}. \tag{14}^{2)} \]

Taking equality (14) into account, the condition equation (13) assumes the form of condition equation (1).

The shortest distances \(R''\) from the vertices of the underground connecting traverse to the plumb-line alignment, entering into condition equation (1), are determined graphically from a plan at a scale of \(1:500\) or \(1:1000\), with their sign taken into account.

For left-hand angles the shortest distances \(R''\) have a plus sign for vertices located to the right of the plumb-line alignment, and a minus sign for vertices located to the left of the plumb-line alignment.

Solving condition equation (1) under the condition \([p_{\beta}\varepsilon_{\beta}^{2}] + [p_l\varepsilon_l^{2}] = \min.\), we obtain the normal equation of the correlate

\[ \left\{ \left[\frac{R_i''^{\,2}}{\rho^2 p_{\beta_i}}\right] + \left[\frac{\cos^2\varphi_i}{p_{l_i}}\right] \right\}K + \Delta C = 0. \tag{15} \]

The weights of the measured angles and side lengths of the underground connecting traverse are determined from the relation:

\[ m_{\beta_i}^{2}p_{\beta_i} = m_{l_i}^{2}p_{l_i} = m^{2}. \tag{16} \]

Hence

\[ p_{\beta_i} = \frac{m^{2}}{m_{\beta_i}^{2}}; \tag{17} \]

\[ p_{l_i} = \frac{m^{2}}{m_{l_i}^{2}}, \tag{18} \]

where \(m_{\beta_i},\, m_{l_i}\) are the mean square errors of the measurements of the angle and of the side length, respectively;

\(m\) is the mean square error of a measurement whose weight is equal to unity;

\(p_{\beta},\, p_l\) are the weights of the angle and of the side length, respectively.

\(^{2)}\) Formula (14) is valid for sides not perpendicular to the plumb-line alignment.

The most probable corrections to the angles and side lengths and their adjusted values are computed by the formulas:

\[ \varepsilon_{\beta_i}=\frac{R_i''}{\rho p_{\beta_i}}K; \tag{19} \]

\[ \varepsilon_{l_i}=\frac{\cos \delta_i}{p_{l_i}}K; \tag{20} \]

\[ \overline{\beta}_i=\beta_i+\varepsilon_{\beta_i}; \tag{21} \]

\[ \overline{l}_i=l_i+\varepsilon_{l_i}, \tag{22} \]

where \(\beta,\overline{\beta}\) are the measured and adjusted angle;
\(l,\overline{l}\) are the measured and adjusted side length.

To compute the adjusted underground connecting traverse in a unified coordinate system, it is first necessary, in this same coordinate system, to determine the adjusted direction angle of its first side, which can be computed in two ways.

First method

From the adjusted angles and side lengths, the underground connecting traverse is recomputed in a conditional coordinate system in which the conditional axis of abscissae coincides with the first side of the traverse, and the conditional coordinate origin with plumb line \(A\).

The adjusted direction angle of the first side of the traverse in the unified coordinate system is computed by the formulas:

\[ \overline{\alpha}_1=(AB)-\overline{(AB)}'; \tag{23} \]

\[ \tg \overline{(AB)}'=\frac{\overline{Y}'_B}{\overline{X}'_B}, \tag{24} \]

where \(\overline{X}'_B,\overline{Y}'_B\) are the adjusted conditional coordinates of plumb line \(B\);
\(\overline{(AB)}'\) is the adjusted conditional direction angle of the plumb-line alignment.

The correctness of the adjustment is checked by the equality of the distances between the plumb lines from the adjusted underground connecting traverse and from the surface survey.

Second method

Using differential formulas, corrections to the individual coordinate increments are computed, and then corrections to the conditional coordinates of plumb line \(B\) due to the adjustment of the underground connecting traverse.

The corrections are computed by the formulas:

\[ \varepsilon_{\Delta x_i}=\varepsilon_{l_i}\cos\alpha'_i-\frac{\varepsilon\alpha'_i}{\rho}\Delta y'_i; \tag{25} \]

\[ \varepsilon_{\Delta y_i}=\varepsilon_{l_i}\sin\alpha'_i+\frac{\varepsilon\alpha'_i}{\rho}\Delta x'_i. \tag{26} \]

\[ \varepsilon_{\alpha_i}'=\sum_{1}^{i-1}\varepsilon_{\beta_i}; \tag{27} \]

\[ \varepsilon_{x_B}'=\sum_{1}^{n+1}\varepsilon_{\Delta x_i}; \tag{28} \]

\[ \varepsilon_{y_B}'=\sum_{1}^{n+1}\varepsilon_{\Delta y_i}, \tag{29} \]

where \(\varepsilon_{\Delta x}, \varepsilon_{\Delta y}\) are corrections to the conditional coordinate increments;
\(\varepsilon_{\alpha'}\) is the correction to the conditional direction angle of a side;
\(\varepsilon_{x_B}', \varepsilon_{y_B}'\) are corrections to the conditional coordinates \(x_B, y_B\) of plumb line \(B\).

The correctness of the adjustment of the underground connecting traverse is checked by the equality:

\[ \varepsilon_{x_B}'\cos(AB)' + \varepsilon_{y_B}'\sin(AB)'=-\Delta C. \tag{30} \]

The correction to the direction angle of the first side, computed by formula (4), due to the adjustment of the underground connecting traverse is determined by the formula:

\[ \varepsilon_{\alpha_1}=-\rho\,\frac{U}{C}; \tag{31} \]

\[ U=\varepsilon_{y_B}'\cos(AB)'-\varepsilon_{x_B}'\sin(AB)', \tag{32} \]

where \(U\) is the transverse displacement of plumb line \(B\) relative to the alignment line of the plumb lines due to the adjustment of the traverse;
\(C\) is the distance between the plumb lines.

The adjusted direction angle of the first side of the traverse in the unified coordinate system is computed by the formula

\[ \bar{\alpha}_1=\alpha_1+\varepsilon_{\alpha_1}. \tag{33} \]

The adjusted direction angles of the remaining sides of the underground connecting traverse are computed from the adjusted direction angle of the first side.

From the adjusted direction angles and the side lengths, the coordinate increments and the coordinates of all vertices of the underground connecting traverse in the unified system are computed.

The correctness of the computation of the adjusted underground connecting traverse in the unified coordinate system is checked by the equality of the coordinates of plumb line \(B\) from the adjusted traverse and from the surface survey.

The root-mean-square error \(M\alpha_k\) of the adjusted direction angle of the side with number \(k\) is determined by the formulas:

\[ M\alpha_k=\pm m\sqrt{\frac{1}{P\alpha_k}}; \tag{34} \]

\[ \frac{1}{P\alpha_k}=\left[\frac{ff}{p}\right]- \frac{\left[\frac{af}{p}\right]^2}{\left[\frac{aa}{p}\right]}. \tag{35} \]

\[ \left[\frac{ff}{p}\right] = \left[\frac{f_{\beta} f_{\beta}}{p_{\beta}}\right] + \left[\frac{f_l f_l}{p_l}\right]; \tag{36} \]

\[ \left[\frac{af}{p}\right] = \left[\frac{a_{\beta} f_{\beta}}{p_{\beta}}\right] + \left[\frac{a_l f_l}{p_l}\right]; \tag{37} \]

\[ \left[\frac{aa}{p}\right] = \left[\frac{R_i^{\prime 2}}{\rho^2 p_{\beta_i}}\right] + \left[\frac{\cos^2 \delta_i}{p_{l_i}}\right]. \tag{38} \]

The quantities \(f_{\beta}\) and \(f_l\) denote the partial derivatives with respect to the angle and the side, respectively, of the function:

\[ \alpha_k = \alpha_1 + \sum_{1}^{k-1} \beta_i \pm (k-1)180^\circ . \tag{39} \]

From (39) we have:

\[ f_{\beta_i(i<k)} = 1 - \frac{R'_i}{C}; \tag{40} \]

\[ f_{\beta_i(i\geq k)} = - \frac{R'_i}{C}; \tag{40a} \]

\[ f_{l_i} = - \frac{\rho \sin \delta_i}{C}, \tag{41} \]

where \(R'\) is the projection of the distance connecting the vertex of the traverse angle with plumb line \(B\) onto the alignment line of the plumb lines.

The projections \(R'\) are determined graphically, to an accuracy of one meter, from the plan of the underground connecting traverse drawn at a scale of \(1:500\) or \(1:1000\); in doing so, the sign of the projections \(R'\) must be taken into account.

For left-hand angles the projection \(R'\) has a plus sign if its direction, counted from the vertex of the angle toward plumb line \(B\), coincides with the direction of the alignment line of the plumb lines \(AB\), and a minus sign when it does not coincide.

The mean square error \(m\) of a measurement with weight 1 is computed by the formula

\[ m = \pm \sqrt{\left[p_{\beta_i}\varepsilon_{\beta_i}^{2}\right] + \left[p_{l_i}\varepsilon_{l_i}^{2}\right]} . \tag{42} \]

§3. Adjustment of an Extended Underground Connecting Traverse

The condition equation (1) for an extended underground connecting traverse will take the form

\[ \sum_{1}^{n+1} \varepsilon_{l_i} + \Delta C = 0. \tag{43} \]

Taking as the weight of a side a quantity inversely proportional to its length, i.e. \(p_l = \dfrac{1}{l}\), from the condition equation (43) we obtain the normal equation of the correlate

\[ [l]K + \Delta C = 0. \]

Hence

\[ K=-\frac{\Delta C}{[l]}, \]

where \([l]\) is the length of the traverse.

The most probable corrections to the lengths of the sides are computed by the formula

\[ \varepsilon_{l_i}=-\frac{\Delta C}{[l]}l_i . \tag{44} \]

For an elongated traverse \([l]=C_{\mathrm{sh}}\), therefore formula (44) takes the form

\[ \varepsilon_{l_i}=-\frac{\Delta C}{C_{\mathrm{sh}}}l_i \tag{44a} \]

or

\[ \varepsilon_{l_i}=-\frac{C_{\mathrm{sh}}-C_n}{C_{\mathrm{sh}}}l_i . \tag{44б} \]

The adjusted length of a side is equal to

\[ \bar l_i=l_i+\varepsilon_{l_i} \tag{45} \]

or

\[ \bar l_i=\frac{C_n}{C_{\mathrm{sh}}}l_i . \tag{45a} \]

Formula (45a) is recommended by Eng. B. A. Rostkovskii [7] for any form of underground connecting polygon.

The corrections to the coordinate increments due to the adjustment of the polygon are:

\[ \varepsilon_{\Delta x_i}=\varepsilon_{l_i}\cos(AB); \]

\[ \varepsilon_{\Delta y_i}=\varepsilon_{l_i}\sin(AB), \]

or

\[ \varepsilon_{\Delta x_i}=-\frac{\Delta C\cos(AB)}{[l]}l_i; \]

\[ \varepsilon_{\Delta y_i}=-\frac{\Delta C\sin(AB)}{[l]}l_i. \]

Since

\[ f_x=\Delta C\cos(AB); \]

\[ f_y=\Delta C\sin(AB), \tag{46} \]

then

\[ \varepsilon_{\Delta x_i}=-\frac{f_x}{[l]}l_i; \]

\[ \varepsilon_{\Delta y_i}=-\frac{f_y}{[l]}l_i, \tag{47} \]

where \(f_x, f_y\) are the coordinate misclosures of the underground connecting polygon.

Consequently, in rigorous adjustment of an elongated underground connecting polygon, the coordinate misclosures are distributed proportionally to the lengths of the sides, which corresponds to the method of adjustment, re-

recommended by the Technical Instruction for the performance of mine-surveying work.

Thus, the simplified methods of adjustment recommended by Eng. B. A. Rostkovskii and by the Technical Mine-Surveying Instruction correspond to the rigorous method of adjustment only when the underground connecting traverse has an elongated form.

§4. Adjustment of a Preliminarily Oriented Underground Connecting Traverse

If, in an underground connecting traverse (Fig. 1), there is some side numbered \(k\), whose direction angle has been established from a previous orientation through one or two vertical shafts, or has been determined from magnetic or gyroscopic orientation, then, in adjusting the traverse by the method of conditional observations, there will be two conditional equations following from the geometrical conditions:

\[ C_{\ш}=C_{n}=0; \]

\[ \alpha_{k}=\alpha^{\circ}_{k}=0; \tag{48} \]

where \(\alpha_{k}\), \(\alpha^{\circ}_{k}\) are the direction angles of the \(k\)-th side, respectively, from the previous orientation and from the orientation through two vertical shafts.

Since the direction angle, both from a geometrical orientation (through one or two vertical shafts) and from a non-geometrical orientation (gyroscopic or magnetic), is usually determined with an error exceeding the error of measurement of a horizontal angle, then, when composing the conditional equations from the geometrical conditions (48), it is necessary to regard the direction angle from the previous orientation as a measured quantity subject to correction through the adjustment of the underground connecting traverse.

From the geometrical conditions (48) we obtain the conditional equations:

\[ \sum_{1}^{n}\left(\frac{R_i''}{\rho}\right)\varepsilon_{\beta_i} +\sum_{1}^{n+1}\cos\gamma_i\,\varepsilon_{l_i} +\Delta C=0; \]

\[ \varepsilon_{\alpha_k} -\sum_{1}^{k-1}\left(\frac{R'_{A i}}{C}\right)\varepsilon_{\beta_i} +\sum_{k}^{n}\left(\frac{R'_i}{C}\right)\varepsilon_{\beta_i} -\frac{\rho''}{C}\sum_{1}^{n+1}\sin\gamma_i\,\varepsilon_{l_i} +v_{\alpha_k}=0, \tag{49} \]

where \(\varepsilon_{\alpha_k}\) is the correction to the direction angle from the previous orientation;

\(R'_A\) is the projection of the distance from the vertex of the angle to plumb line \(A\) onto the line of the plumb-line alignment;

\(v_{\alpha_k}\) is the misclosure in the direction angles of the \(k\)-th side, i.e., the free term of the second conditional equation.

The free term of the second conditional equation is determined by the formula

\[ v_{\alpha_k}=a_k-a^{\circ}_{k}. \tag{50} \]

If the underground connecting traverse has been preliminarily computed in a single coordinate system, proceeding from the direction angle, by...

obtained from the preceding orientation, and the coordinates of plumb line \(A\), then the condition equations (49) may be written in the following form:

\[ -\frac{Y_B-Y_A}{\rho}\,\varepsilon_{\alpha_k} -\frac{1}{\rho}\sum_{1}^{k-1}(Y_A-Y_i)\varepsilon_{\beta_i} -\frac{1}{\rho}\sum_{k}^{n}(Y_B-Y_i)\varepsilon_{\beta_i} + \]

\[ +\sum_{1}^{n+1}\cos \alpha_i\,\varepsilon_{l_i}+f_x=0; \]

\[ \frac{X_B-X_A}{\rho}\,\varepsilon_{\alpha_k} +\frac{1}{\rho}\sum_{1}^{k-1}(X_A-X_i)\varepsilon_{\beta_i} +\frac{1}{\rho}\sum_{k}^{n}(X_B-X_i)\varepsilon_{\beta_i} + \]

\[ +\sum_{1}^{n+1}\sin \alpha_i\,\varepsilon_{l_i}+f_y=0, \tag{49a} \]

where \(f_x, f_y\) are the coordinate misclosures of the underground connecting traverse.

Solving the system of condition equations (49) or (49a) under the condition

\[ \{p_{\alpha_k}\varepsilon_{\alpha_k}^{2}+[p_{\beta}\varepsilon_{\beta}^{2}]+[p_l\varepsilon_l^{2}]\}=\min, \]

we find the most probable corrections both to the direction angle from the preceding orientation and to the measured angles and sides of the underground connecting traverse.

The weight of the direction angle from the preceding orientation is determined by the formula

\[ P_{\alpha_k}=\frac{m^{2}}{m_{\alpha_k}^{2}}. \tag{51} \]

The mean square error \(m_{\alpha_k}\) of the direction angle from the preceding orientation is established either from the results of double independent orientations or by preliminary computation.

If, from the preceding orientation, the direction angle of the first side of the underground connecting traverse is known, then the condition equations (49a) take the form:

\[ -\frac{Y_B-Y_A}{\rho}\,\varepsilon_{\alpha_1} -\frac{1}{\rho}\sum_{1}^{n}(Y_B-Y_i)\varepsilon_{\beta_i} +\sum_{1}^{n+1}\cos \alpha_i\,\varepsilon_{l_i} +f_x=0; \]

\[ \frac{X_B-X_A}{\rho}\,\varepsilon_{\alpha_1} =\frac{1}{\rho}\sum_{1}^{n}(X_B-X_i)\varepsilon_{\beta_i} +\sum_{1}^{n+1}\sin \alpha_i\,\varepsilon_{l_i} +f_y=0. \tag{52} \]

When in the underground connecting traverse there are two sides, for example the \(k\)-th and \(r\)-th (Fig. 1), with direction angles known from preceding orientations, then the number of condition equations will be

is equal to three; of these, one is for abscissas, another for ordinates, and the third for direction angles. The last equation has the form

\[ \varepsilon_{\alpha_k}+\sum_k^{r-1}\varepsilon_{\beta_i}-\varepsilon_{\alpha_r}+v_{\alpha_r}=0 . \tag{53} \]

The free term of conditional equation (53) is determined by the formula

\[ v_{\alpha_r}=\alpha_k+\sum_k^{r-1}\beta_i+(r-k)180^\circ-\alpha_r, \tag{54} \]

where \(\alpha_k,\alpha_r\) are the values of the direction angles of the sides numbered \(k\) and \(r\), obtained from the preceding orientations.

The conditional equations of abscissas and ordinates have the form, respectively, of the first and second equations of system (49a).

If, from the preceding orientations, the direction angles of the first and last sides are known and the underground connecting traverse is computed in a single coordinate system, proceeding from the direction angle of the first side and the coordinates of plumb line \(A\), then the conditional equations of abscissas, ordinates, and direction angles will have the form:

\[ -\frac{Y_B-Y_A}{\rho}\,\varepsilon_{\alpha_1} -\frac{1}{\rho}\sum_1^n (Y_B-Y_i)\varepsilon_{\beta_i} +\sum_1^{n+1}\cos\alpha_i\,\varepsilon_{l_i} +f_x=0; \]

\[ \frac{X_B-X_A}{\rho}\,\varepsilon_{\alpha_1} -\frac{1}{\rho}\sum_1^n (X_B-X_i)\varepsilon_{\beta_i} +\sum_1^{n+1}\sin\alpha_i\,\varepsilon_{l_i} +f_y=0; \tag{55} \]

\[ \varepsilon_{\alpha_1}+\sum_1^n\varepsilon_{\beta_i} -\varepsilon_{\alpha_{n+1}}+v_{\alpha_{n+1}}=0. \]

The free term of the third conditional equation is equal to

\[ v_{\alpha_{n+1}}=\alpha_1+\sum_1^n\beta_i+n\cdot180^\circ-\alpha_{n+1}, \tag{56} \]

where \(\alpha_1,\alpha_{n+1}\) are the values of the direction angles of the first and last sides, known from the preceding orientations.

§5. Adjustment of a Network of Underground Connecting Traverses

When orienting underground surveys by the method “through two vertical mine shafts,” on the oriented horizon between plumb lines \(A\) and \(B\) sometimes not one connecting traverse is run, but a network of connecting traverses (Figs. 2, 3, 4, and 5).

In the network of underground connecting traverses shown in Figs. 2 and 4, four conditional equations arise: one for the fixed side \(AB\), having the form of conditional equation (1), one for abscissas, one for ordinates, and one for the figure. The last three conditional equations are composed for the closed traverse \(1\text{-}2\text{-}3\text{-}4\text{-}5\text{-}6\text{-}1\).

The condition equation of the rigid side \(AB\) is formed for the polygon \(A-1-6-5-B\).

In the network shown in Fig. 3, seven condition equations arise, of which: one is for the rigid side \(AB\), two are for abscissas, two for ordinates, and two for figures. The condition equations of abscissas, ordinates, and figures are formed for the closed polygons \(1-2-3-7-6-1\) and \(3-4-5-6-7-3\).

Fig. 2. One-sided network with one closed polygon.

Fig. 3. One-sided network with two closed polygons.

In the network presented in Fig. 5, there are eight condition equations: one for the rigid side \(AB\), one for the figure of the closed polygon \(1-2-3-4-5-6-1\), three for abscissas, and three for ordinates. The condition equations of abscissas and ordinates are formed for the closed polygons \(A-1-6-A\), \(B-3-4-B\), and \(1-2-3-4-5-6-1\).

Fig. 4. Two-sided network with one closed polygon.

Fig. 5. Complex two-sided network.

The total number of condition equations in a network of underground connecting traverses resting on two fixed points (plumb lines) is determined by the formula

\[ R = N - 2p + 4; \tag{57} \]

where \(N\) is the number of measured angles and sides in the network;
\(p\) is the number of all points in the network, including the fixed points (plumb lines).

As an example, let us write the form of the condition equations for the network shown in Fig. 5.

After adjustment, eight geometric conditions must be satisfied in the network:

a) in the polygon \(A-1-2-3-B\):

\[ C_{\sh} - C_n = 0; \]

b) in the polygon \(1-2-3-4-5-6-1\):

\[ \sum_{1}^{6} \beta_i - 720^\circ = 0; \]

\[ \sum_{1}^{6} \Delta x'_i = 0; \]

\[ \sum_{1}^{6} \Delta y'_i = 0; \]

c) in the polygon \(A-1-6-A\):

\[ \sum_{i=6,7,10} \Delta x'_i = 0; \]

\[ \sum_{i=6,7,10} \Delta y'_i = 0; \]

d) in the polygon \(B-3-4-B\):

\[ \sum_{i=8,3,9} \Delta x'_i = 0; \]

\[ \sum_{i=8,3,9} \Delta y'_i = 0. \]

Here \(\Delta x'\) and \(\Delta y'\) are conditional coordinate increments relative to a conditional abscissa axis coinciding with side “\(A-1\).”

From the geometric conditions follow the following conditional equations:

\[ 1.\quad \sum_{i=7,2,8} \left(\frac{R_i''}{\rho}\right)\varepsilon_{\beta_i} + \sum_{i=7,1,2,8} \cos \vartheta_i \varepsilon_{l_i} + \Delta C = 0; \]

\[ 2.\quad \sum_{i=1}^{6} \varepsilon_{\beta_i} + v_\beta = 0; \]

\[ 3.\quad -\sum_{1}^{6} \frac{(Y'_1 - Y'_i)}{\rho}\,\varepsilon_{\beta_i} + \sum_{1}^{6} \cos \alpha'_i \varepsilon_{l_i} + f_{x_1} = 0; \]

\[ 4.\quad \sum_{1}^{6} \frac{(X'_1 - X'_i)}{\rho}\,\varepsilon_{\beta_i} + \sum_{1}^{6} \sin \alpha'_i \varepsilon_{l_i} + f_{y_1} = 0; \]

\[ 5.\quad \frac{Y'_A - Y'_1}{\rho}\varepsilon_{\beta_1} + \frac{Y'_A - Y'_6}{\rho}\varepsilon_{\beta_6} - \frac{Y'_A - Y'_1}{\rho}\varepsilon_{\beta_7} - \frac{Y'_A - Y'_6}{\rho}\varepsilon_{\beta_{10}} + \sum_{i=7,6,10} \cos \alpha'_i \varepsilon_{l_i} + f_{x_2} = 0. \tag{58} \]

\[ 6.\quad -\frac{X'_A - X'_1}{\rho}\varepsilon_{\beta_1} - \frac{X'_A - X'_6}{\rho}\varepsilon_{\beta_6} + \frac{X'_A - X'_1}{\rho}\varepsilon_{\beta_7} + \frac{X'_A - X'_6}{\rho}\varepsilon_{\beta_{10}} + \]

\[ +\sum_{i=7,6,10}\sin \alpha'_i \varepsilon_{l_i}+f_{y_2}=0; \]

\[ \begin{aligned} 7.\quad &-\frac{Y'_B-Y'_3}{\rho}\varepsilon_{\beta_3} -\frac{Y'_B-Y'_4}{\rho}\varepsilon_{\beta_4} +\frac{Y'_B-Y'_3}{\rho}\varepsilon_{\beta_8} +\frac{Y'_B-Y'_4}{\rho}\varepsilon_{\beta_9} \\ &+\sum_{i=8,3,9}\cos \alpha'_i \varepsilon_{l_i}+f_{x_3}=0; \\[1.2em] 8.\quad &\frac{X'_B-X'_3}{\rho}\varepsilon_{\beta_3} +\frac{X'_B-X'_4}{\rho}\varepsilon_{\beta_4} -\frac{X'_B-X'_3}{\rho}\varepsilon_{\beta_8} -\frac{X'_B-X'_4}{\rho}\varepsilon_{\beta_9} \\ &+\sum_{i=8,3,9}\sin \alpha'_i \varepsilon_{l_i}+f_{y_3}=0; \end{aligned} \tag{58} \]

where \(v_\beta\) is the angular misclosure of the closed polygon \(1\text{-}2\text{-}3\text{-}4\text{-}5\text{-}6\text{-}1\);

\(f_{x_1}, f_{y_1}\) is the misclosure in the coordinate increments of the closed polygon \(1\text{-}2\text{-}3\text{-}4\text{-}5\text{-}6\text{-}1\);

\(f_{x_2}, f_{y_2}\) is the misclosure in the coordinate increments of the polygon \(A\text{-}1\text{-}6\text{-}A\);

\(f_{x_3}, f_{y_3}\) is the misclosure in the coordinate increments of the polygon \(B\text{-}3\text{-}4\text{-}B\);

\(x', y'\) are the conditional coordinates of the point.

To determine the constant terms and the coefficients of the condition equations, the network is computed in a conditional coordinate system in which the conditional abscissa axis coincides with the side “\(A\text{-}1\),” and the conditional origin of coordinates is the plumb line \(A\).

The conditional direction angles of the sides of the closed polygon \(A\text{-}1\text{-}6\text{-}A\) are computed, starting from the direction angle of the side “\(A\text{-}1\),” by the formulas:

\[ (1-6)'=(\beta_7-\beta_1)\pm 180^\circ; \]

\[ (6-A)'=(1-6)'+(\beta_{10}-\beta_6)\pm 180^\circ. \]

Solving the system of condition equations (58) under the condition

\[ [p_{\beta_i}\varepsilon_{\beta_i}^{2}]+[p_{l_i}\varepsilon_{l_i}^{2}]=\min, \]

we find the most probable corrections to the measured angles and side lengths.

To compute the adjusted network in a single coordinate system, the adjusted connecting polygon \(A\text{-}1\text{-}2\text{-}3\text{-}B\) is computed in the conditional coordinate system, and by formula (23) the adjusted direction angle of the side “\(A\text{-}1\)” is determined; from it, the adjusted direction angles of the sides of the network are computed in the single coordinate system.

From the adjusted side lengths, the coordinate increments of all vertices of the adjusted network are computed.

The correctness of the network adjustment is checked by the satisfaction of the geometric conditions in the closed polygons and by the equality of the coordinates of plumb line \(B\) from the underground and surface surveys.

In conclusion, we note that underground connecting polygons, as well as networks of underground connecting polygons run within near-shaft yards, must be adjusted by the rigorous method.

References

1. Akulov V. I. A method for computing orientation through two vertical shafts. Ugol’, No. 12, 1959.

2. Belolikov A. N. Adjustment of orientation through two vertical shafts by the method of least squares. Studies on Problems of Mine Surveying, Collection XXVI, VNIMI, 1952.

3. Bui T. A. On the question of orienting an underground survey through two or three vertical shafts. Collection of Scientific Works of the Department of Mine Surveying and Geodesy of the Donetsk Industrial Institute, issue 1, Metallurgizdat, 1950.

4. Baranov A. N., Egunov K. I., et al. Geodesy in Tunnel Construction, Part II. Geodezizdat, 1953.

5. Erzhanov Zh. S. On orientations through two vertical shafts. Ugol’, No. 3, 1955.

6. Kell’ N. G. Graphical adjustment. Proceedings of TsNIMB, No. 10, 1939.

7. Rostkovskii B. A. Orientation of an underground survey through two vertical shafts. Proceedings of TsNIMB, issue II, 1940.