Adjustment of a Network of Underground Connecting Traverses Oriented Through Three or More Vertical Mine Shafts
Abstract Generated abstract
This paper develops a least squares procedure for adjusting networks of underground connecting traverses oriented through three or more vertical mine shafts, a case identified as insufficiently treated in existing mine surveying and geodetic literature. It formulates condition equations for traverse networks closed on three and four fixed plumb-line points, including variants with closed polygons and previously determined direction angles, and gives rules for determining the required coefficients, misclosures, corrections, and checks against surface-survey coordinates. The paper also discusses the establishment of weights for measured angles and side lengths, deriving error and weighting formulas that account for centering errors and distance-measurement accuracy. It concludes that such multi-shaft underground traverse networks should be adjusted by a rigorous method.
Full Text
ADJUSTMENT OF A NETWORK OF UNDERGROUND CONNECTING TRAVERSES WHEN ORIENTED THROUGH THREE OR MORE VERTICAL MINE SHAFTS
V. I. AKULOV
(Presented by the scientific seminar of the Departments of Mine Surveying and Geodesy)
Mine fields, as is known, may be opened by two or three, and even four, vertical shafts; therefore, in the case when all vertical shafts are connected with one another by mine workings, the orientation of underground surveys may be performed through two, three, or four vertical mine shafts. In particular cases, taking into account the presence of auxiliary vertical winzes, the orientation of underground surveys may also be carried out through more than three or four vertical workings.
The question of adjusting a network of underground connecting traverses when oriented through three or more vertical mine shafts has not been developed either in mine-surveying or in geodetic literature.
When underground surveys are oriented through three vertical mine shafts, the network of underground connecting polygons (Fig. 1) closes on three plumb lines \(A\), \(B\), and \(C\), lowered from the surface through each vertical shaft. Neglecting the linear errors of projecting points from the surface into the mine, we may regard the plumb lines on which the network of underground connecting traverses closes as fixed points with coordinates obtained from surface surveys. Consequently, when orientation is carried out through three vertical mine shafts, the problem arises of adjusting a network of underground connecting traverses closed in coordinates on three fixed points (plumb lines).
When adjusting the network of underground connecting traverses (Fig. 1) by the method of condition observations, there will be three conditional equations of sides:
\[ \frac{1}{\rho}\sum_{AB} R''_{AB}\varepsilon_\beta+\sum_{AB}\cos \delta_{AB}\varepsilon_l+\Delta C_{AB}=0; \tag{1} \]
\[ \frac{1}{\rho}\sum_{AC} R''_{AC}\varepsilon_\beta+\sum_{AC}\cos \delta_{AC}\varepsilon_l+\Delta C_{AC}=0; \tag{2} \]
\[ \frac{1}{\rho}\sum_{BC} R''_{BC}\varepsilon_\beta+\sum_{BC}\cos \delta_{BC}\varepsilon_l+\Delta C_{BC}=0, \tag{3} \]
where \(R''_{AB}\), \(R''_{AC}\), and \(R''_{BC}\) are the shortest distances from the vertex of angle \(\beta\) to the plumb-line alignment lines, respectively \(AB\), \(AC\), and \(BC\);
\(\delta_{AB}\), \(\delta_{AC}\), and \(\delta_{BC}\) are the angles between the side and the corresponding plumb-line alignment line, respectively \(AB\), \(AC\), and \(BC\);
\(\Delta C_{AB}\), \(\Delta C_{AC}\), and \(\Delta C_{BC}\) are the discrepancies between the distances \(AB\), \(AC\), and \(BC\) from the underground connecting traverses \(AДB\), \(AДC\), and \(BДC\) and from the surface surveys;
\(\varepsilon_\beta\), \(\varepsilon_l\) are the most probable corrections to the measured angles and side lengths;
\[ \rho = 206\,265''. \]
Fig. 1
The shortest distances \(R''_{AB}\), \(R''_{AC}\), \(R''_{BC}\) are determined graphically from the scheme of the network of underground connecting traverses, drawn at a scale of 1:500 or 1:1000; in doing so, their sign is taken into account, which is determined on the basis of the following rules:
-
If the angle measured along the course of the connecting traverse is a left-hand angle, then \(R''\) for that angle will have a plus sign if the vertex of the angle is located to the right of the closing line of the connecting traverse, and a minus sign if the vertex of the angle is located to the left of the closing line.
-
If the angle measured along the course of the connecting traverse is a right-hand angle, then \(R''\) for that angle will have a plus sign if the vertex of the angle is located to the left of the closing line of the connecting traverse, and a minus sign if the vertex of the angle is located to the right of the closing line.
The angles \(\delta_{AB}\), \(\delta_{AC}\), \(\delta_{BC}\) are determined graphically from the network scheme or are computed by the formulas:
\[ \delta_{AB}=\alpha-(AB); \]
\[ \delta_{AC}=\alpha-(AC); \tag{4} \]
\[ \delta_{BC}=\alpha-(BC), \]
where \((AB)\), \((AC)\), and \((BC)\) are the direction angles of the plumb-line alignment lines \(AB\), \(AC\), and \(BC\);
\(\alpha\) is the direction angle of the side.
The free terms of the conditional equations are equal to:
\[ \begin{aligned} \Delta C_{AB} &= C'_{AB} - C_{AB};\\ \Delta C_{AC} &= C'_{AC} - C_{AC}; \tag{5}\\ \Delta C_{BC} &= C'_{BC} - C_{BC}, \end{aligned} \]
where \(C_{AB}, C_{AC}, C_{BC}\) are the distances between the plumb lines, respectively, \(A\) and \(B\), \(A\) and \(C\), \(B\) and \(C\), computed from the surface surveys;
\(C'_{AB}, C'_{AC}, C'_{BC}\) are the distances computed from the underground connecting traverses \(ADB\), \(ADC\), and \(BDC\), respectively.
To determine the distances \(C'_{AB}, C'_{AC}\), and \(C'_{BC}\), it is necessary, on the basis of the measured angles and side lengths reduced to the mean level surface and to the projection plane, to compute the entire network of underground connecting traverses in one arbitrary coordinate system, for example, in an arbitrary system in which the arbitrary origin of coordinates coincides with plumb line \(A\), and the arbitrary abscissa axis with the side adjoining plumb line \(A\), i.e., with the side “\(A\)—1.”
Solving the conditional equations (1, 2, 3) under the condition
\[ [p_\beta \varepsilon_\beta^2] + [p_l \varepsilon_l^2] = \text{minimum}, \tag{6} \]
we obtain the normal equations of the correlates:
\[ \left\{\left[\frac{a_\beta^2}{p_\beta}\right] + \left[\frac{a_l^2}{p_l}\right]\right\}K_1 + \left\{\left[\frac{a_\beta b_\beta}{p_\beta}\right] + \left[\frac{a_l b_l}{p_l}\right]\right\}K_2 + \left\{\left[\frac{a_\beta c_\beta}{p_\beta}\right] + \left[\frac{a_l c_l}{p_l}\right]\right\}K_3 + \Delta C_{AB}=0; \]
\[ \left\{\left[\frac{b_\beta^2}{p_\beta}\right] + \left[\frac{b_l^2}{p_l}\right]\right\}K_2 + \left\{\left[\frac{b_\beta c_\beta}{p_\beta}\right] + \left[\frac{b_l c_l}{p_l}\right]\right\}K_3 + \Delta C_{AC}=0; \tag{7} \]
\[ \left\{\left[\frac{c_\beta^2}{p_\beta}\right] + \left[\frac{c_l^2}{p_l}\right]\right\}K_3 + \Delta C_{BC}=0, \]
where
\[ a_\beta=\frac{R''_{AB}}{\rho}; \qquad b_\beta=\frac{R''_{AC}}{\rho}; \qquad c_\beta=\frac{R''_{BC}}{\rho}; \]
\[ a_l=\cos \delta_{AB}; \qquad b_l=\cos \delta_{AC}; \qquad c_l=\cos \delta_{BC}; \]
\(p_\beta\) is the weight of the measured angle;
\(p_l\) is the weight of the measured side.
The weights of the measured angles and side lengths are determined from the relation
\[ m_{\beta}^{2}p_{\beta}=m_{l}^{2}p_{l}=m_{p}^{2}=1 \tag{8} \]
Hence
\[ p_{\beta}=\frac{m_{p}^{2}=1}{m_{\beta}^{2}} . \tag{9} \]
\[ p_{l}=\frac{m_{p}^{2}=1}{m_{l}^{2}} . \tag{10} \]
From the solution of equations (7) we find the values of the correlates \(K_1, K_2\) and \(K_3\).
The most probable corrections \(\varepsilon_\beta\) and \(\varepsilon_l\), and the adjusted angles and side lengths, are computed by the formulas:
\[ \varepsilon_{\beta}=\frac{a_{\beta}}{p_{\beta}}K_1+\frac{b_{\beta}}{p_{\beta}}K_2+\frac{c_{\beta}}{p_{\beta}}K_3, \tag{11} \]
\[ \varepsilon_{l}=\frac{a_{l}}{p_{l}}K_1+\frac{b_{l}}{p_{l}}K_2+\frac{c_{l}}{p_{l}}K_3, \tag{12} \]
\[ \beta^{\circ}=\beta+\varepsilon_{\beta}, \tag{13} \]
\[ l^{\circ}=l+\varepsilon_{l}, \tag{14} \]
where \(\beta,\beta^{\circ}\) are the measured and adjusted value of the angle,
\(l,l^{\circ}\) are the measured and adjusted value of the side length.
To compute the adjusted network of underground connecting traverses in the coordinate system adopted on the surface, it is necessary first, in this same coordinate system, to determine the adjusted direction angle of its right-hand side, i.e., the side whose direction was adopted as the conventional axis of abscissas when computing the network in the conventional coordinate system, for example, side “\(A-I\).”
The adjusted direction angle in the coordinate system adopted on the surface for the first side of the network, for example, side “\(A-I\),” is computed by the formula:
\[ \alpha_{1}^{0}=(AB)-(AB)'_{0}; \tag{15} \]
\[ \operatorname{tg}(AB)'_{0}=\frac{y'_{B}}{x'_{B}}, \tag{16} \]
where \((AB)\) is the direction angle of the plumb-line alignment \(AB\) from the surface survey;
\(x'_{B}, y'_{B}\) are the conventional coordinates of plumb line “\(B\),” computed from the adjusted angles and side lengths of the connecting traverse \(ADB\).
On the basis of the adjusted direction angle of the first side and the adjusted angles and side lengths, and the coordinates of plumb line \(A\) obtained from the surface survey, the direction angles of all sides and the coordinates of all vertices, including the coordinates of plumb lines \(B\) and \(C\), are computed in the coordinate system adopted on the surface.
A check on the correctness of the adjustment of the network of underground connecting traverses is provided by the equality of the coordinates of plumb lines \(B\) and \(C\) from the underground and surface surveys.
For the case when the network of underground connecting traverses, when oriented through three vertical mine shafts, includes a closed polygon (Fig. 2), then to the three conditional equations of the sides (1, 2, 3) there will be added three more conditional equations for the closed polygon, namely: one for the figure, one for the abscissas, and one for the ordinates.
Fig. 2.
When oriented through four vertical mine shafts, the network of underground connecting traverses is closed on four fixed points (plumb lines) \(A\), \(B\), \(C\), and \(D\) (Figs. 3, 4, and 5).
Fig. 3.
When adjusting the network of underground connecting traverses shown in Fig. 3 or 4 by the method of conditional observations, there will be five conditional equations of the sides of the form:
\[ \frac{1}{\rho}\sum_{AB} R''_{AB}\varepsilon_3 + \sum_{AB}\cos \delta_{AB}\, z_1 + \Delta C_{AB} =0; \tag{17} \]
\[ \frac{1}{\rho}\sum_{BC} R'_{BC}\varepsilon_\beta+\sum_{BC}\cos \delta_{BC}\,\varepsilon_l+\Delta C_{BC}=0; \tag{18} \]
\[ \frac{1}{\rho}\sum_{CD} R''_{CD}\varepsilon_\beta+\sum_{CD}\cos \delta_{CD}\,\varepsilon_l+\Delta C_{CD}=0; \tag{19} \]
\[ \frac{1}{\rho}\sum_{DA} R''_{DA}\varepsilon_\beta+\sum_{DA}\cos \delta_{DA}\,\varepsilon_l+\Delta C_{DA}=0; \tag{20} \]
\[ \frac{1}{\rho}\sum_{AC} R''_{AC}\varepsilon_\beta+\sum_{AC}\cos \delta_{AC}\,\varepsilon_l+\Delta C_{AC}=0. \tag{21} \]
Fig. 4.
Instead of the condition equation of side \(AC\), i.e., equation (21), one may take the condition equation of side \(BD\):
\[ \frac{1}{\rho}\sum_{BD} R''_{BD}\varepsilon_\beta+\sum_{BD}\cos \delta_{BD}\,\varepsilon_l+\Delta C_{BD}=0. \tag{21a} \]
Of the condition equations (21) and (21a), it is necessary to take the condition equation that has the smaller number of unknowns, i.e., corrections \(\varepsilon_\beta\) and \(\varepsilon_l\).
When adjusting, by the method of condition observations, the network of underground connecting traverses shown in Fig. 5, there will be eight condition equations, of which five are side equations (17–21), one is for the figure, one for abscissas, and one for ordinates for the closed polygon.
The total number of condition equations in a network of underground connecting traverses closed on two or more fixed points (plumb lines) can be determined by the formula:
\[ R=N-2(P-p), \tag{22} \]
where \(N\) is the number of all measured angles and sides in the network;
\(P\) is the number of all points in the network, including fixed points (plumb lines);
\(p\) is the number of fixed points (plumb lines).
From the total number of condition equations computed by formula (22), the number of condition equations of sides is:
\[ r = 2p - 3 . \tag{23} \]
Fig. 5
If in the network of underground connecting traverses there are \(n\) sides whose direction angles are known from previous orientations, then, when adjusting the indicated network by the method of conditional observations, the number of condition equations will be:
a) equations of abscissas and ordinates
\[ r_{x,y} = 2(p + m) - 2 , \tag{24} \]
b) equations of direction angles
\[ r_\alpha = n - 1 , \tag{25} \]
c) equations of figures
\[ r_\phi = m , \tag{26} \]
d) the total number of condition equations
\[ R = 2p + 3m + n - 3 \tag{27} \]
or
\[ R = N - 2(P - p) + n , \tag{27a} \]
where \(m\) is the number of closed polygons in the network.
The direction angles of sides from previous orientations must be regarded as measured quantities, subject to correction through the adjustment of the network.
The weights of the known direction angles are determined from their mean square errors, established either from the results of double independent orientations or by preliminary calculation.
As an example, let us establish the form of the condition equations for the network shown in Fig. 1, in which the direction angles of the sides “A—1,” “6—7,” and “9—10” are known from previous orientations.
For the case when the indicated network of underground connecting traverses is preliminarily computed in the coordinate system adopted on the surface, on the basis of the direction angle of the side “A—1” obtained from previous orientations and of the coordinates of plumb line \(A\), the condition equations take the form:
\[ -\frac{(y_B-y_A)}{\rho}\,\varepsilon_{\alpha_1} -\frac{1}{\rho}\sum_{AB}(y_B-y_i)\varepsilon_{\beta_i} +\sum_{AB}\cos\alpha_i\,\varepsilon_{l_i}+f_{x_{AB}}=0; \tag{28} \]
\[ \frac{(x_B-x_A)}{\rho}\,\varepsilon_{\alpha_1} +\frac{1}{\rho}\sum_{AB}(x_B-x_i)\varepsilon_{\beta_i} +\sum_{AB}\sin\alpha_i\,\varepsilon_{l_i}+f_{y_{AB}}=0; \tag{29} \]
\[ -\frac{(y_C-y_A)}{\rho}\,\varepsilon_{\alpha_1} -\frac{1}{\rho}\sum_{AC}(y_C-y_i)\varepsilon_{\beta_i} +\sum_{AC}\cos\alpha_i\,\varepsilon_{l_i}+f_{x_{AC}}=0; \tag{30} \]
\[ \frac{(x_C-x_A)}{\rho}\,\varepsilon_{\alpha_1} +\frac{1}{\rho}\sum_{AC}(x_C-x_i)\varepsilon_{\beta_i} +\sum_{AC}\sin\alpha_i\,\varepsilon_{l_i}+f_{y_{AC}}=0; \tag{31} \]
\[ \varepsilon_{\alpha_1}+\sum_{1}^{6}\varepsilon_{\beta_i}-\varepsilon_{\alpha_7}+V_{\alpha_7}=0; \tag{32} \]
\[ \varepsilon_{\alpha_1}+\sum_{i(1,\,2,\,3,\,4,\,4',\,8,\,9)}\varepsilon_{\beta_i} -\varepsilon_{\alpha_{11}}+V_{\alpha_{11}}=0, \tag{33} \]
where \(f_{x_{AB}}, f_{y_{AB}}\) are the coordinate misclosures of the connecting traverse ADB;
\(f_{x_{AC}}, f_{y_{AC}}\) are the coordinate misclosures of the connecting traverse ADC;
\(V_{\alpha_7}\) and \(V_{\alpha_{11}}\) are the misclosures in the direction angles of the sides “6—7” and “9—10,” respectively;
\(\varepsilon_{\alpha}\) is the most probable correction to the direction angle.
Solving the system of conditional equations (28–33) under the condition
\[ \sum\left(p_\alpha \varepsilon_\alpha^2+p_\beta \varepsilon_\beta^2+p_l \varepsilon_l^2\right)=\text{minimum}, \]
we obtain the most probable corrections to the measured angles and side lengths, and to the direction angles obtained from the preceding orientations, bringing the network of underground theodolite traverses into agreement with the data of the surface surveys.
If in the network (Fig. 1) the direction angles of the sides “2—3,” “6—7,” and “9—10” are known from the preceding orientations, and the network has been preliminarily computed in the coordinate system adopted on the surface, proceeding, for example, from the direction angle of the side “2—3,” which does not directly adjoin plumb line “A,” then the conditional equations (28–31) take the form:
\[ -\frac{(y_B-y_A)}{\rho}\,\varepsilon_{\alpha_3} -\frac{1}{\rho}\sum_{1}^{2}(y_A-y_i)\varepsilon_{\beta_i} -\frac{1}{\rho}\sum_{3}^{7}(y_B-y_i)\varepsilon_{\beta_i} +\sum_{AB}\cos\alpha_i\,\varepsilon_i+f_{x_{AB}}=0; \tag{28a} \]
\[ \frac{(x_B-x_A)}{\rho}\,\varepsilon_{\alpha_3} +\frac{1}{\rho}\sum_{1}^{2}(x_A-x_i)\varepsilon_{\beta_i} + \frac{1}{\rho}\sum_{3}^{7}(x_B-x_i)\varepsilon_{\beta_i} + \sum_{AB}\sin \alpha_i\,\varepsilon_{l_i} +f_{yAB}=0; \tag{29a} \]
\[ -\frac{(y_C-y_A)}{\rho}\,\varepsilon_{\alpha_3} -\frac{1}{\rho}\sum_{1}^{2}(y_A-y_i)\varepsilon_{\beta_i} -\frac{1}{\rho}\sum_{i(3,4,4',8,9,10)}(y_C-y_i)\varepsilon_{\beta_i} + \sum_{AC}\cos \alpha_i\,\varepsilon_{l_i} +f_{xAC}=0; \tag{30a} \]
\[ \frac{(x_C-x_A)}{\rho}\,\varepsilon_{\alpha_3} +\frac{1}{\rho}\sum_{1}^{2}(x_A-x_i)\varepsilon_{\beta_i} +\frac{1}{\rho}\sum_{i(3,4,4',8,9,10)}(x_C-x_i)\varepsilon_{\beta_i} + \sum_{AC}\sin \alpha_i\,\varepsilon_{l_i} +f_{yAC}=0. \tag{31a} \]
Establishing the Weights of the Measured Angles and Side Lengths
The results of adjustment by the method of least squares for a network of underground connecting traverses oriented through three or more vertical mine shafts depend mainly on the accuracy with which the weights of the measured angles and side lengths are established.
If the weights of the measured angles and side lengths are established incorrectly, then adjustment of the network by the method of least squares will be rigorous only formally. Therefore, the weights of the measured angles and side lengths must be determined on the basis of the results of measurements of the angles and side lengths of the network being adjusted.
To determine the weights of the measured angles and side lengths by formulas (9) and (10), it is necessary to know, for each angle and for each side of the network, the mean square errors.
Let us establish the mean square errors of the measurements of angles and side lengths from the discrepancies of the distances between plumb lines as obtained from the underground and surface surveys.
The mean square error of the discrepancy of the distances between plumb lines from the underground and surface surveys when orienting through two vertical shafts, as is known, is expressed by the formula:
\[ M_{\Delta C_{AB}} = \pm \sqrt{ \sum_{AB} R_{AB}^{\prime\prime 2} \left(\frac{m_\beta}{\rho}\right)^2 + \sum_{AB}\cos^2\delta_{AB}\,m_l^2 }. \tag{34} \]
Equating the error \(M_{\Delta C_{AB}}\) to the actual discrepancy \(\Delta C_{AB}\), we obtain the following equation:
\[ \sum_{AB} R_{AB}^{\prime\prime 2} \left(\frac{m_\beta}{\rho}\right)^2 + \sum_{AB}\cos^2\delta_{AB}\,m_l^2 - \Delta C_{AB}^{2} =0. \tag{35} \]
In underground connecting traverses, especially those with short sides or in which the angles are measured with high-precision theodolites, the principal source of errors in angle measurements is inaccurate centering of the theodolite and signals. In this connection, we shall determine the error of angle measurement by the formula:
\[ m_\beta=\pm\sqrt{Z_C e_C^2+Z_T e_T^2}; \tag{36} \]
\[ Z_C=\frac{\rho^2}{2a^2b^2}\left(a^2+b^2\right); \tag{37} \]
\[ Z_T=\frac{\rho^2}{2a^2b^2}\left(a^2+b^2-2ab\cos\beta\right), \tag{38} \]
where
\[
a,\ b
\]
are the sides of the angle;
\[
e_C,\ e_T
\]
are the linear errors of centering of the signals and of the theodolite, respectively.
For \(e_T=e_C\sqrt{2}\), formula (36) will take the form:
\[ m_\beta=\pm e_T\sqrt{\frac{1}{2}Z_C+Z_T}. \tag{39} \]
Since
\[ m_{\beta_i}^2\,p_{\beta_i}=m_{\beta_k}^2\,p_{\beta_k}, \]
then
\[ \frac{1}{p_{\beta_i}}=\frac{m_{\beta_i}^2}{m_{\beta_k}^2\,p_{\beta_k}}. \tag{40} \]
We shall take the weight of the angle \(\beta_k\) equal to unity; then:
\[ \frac{1}{p_{\beta_i}}=\frac{m_{\beta_i}^2}{m_{\beta_k}^2}. \tag{41} \]
Hence
\[ m_{\beta_i}=\pm m_{\beta_k}\sqrt{\frac{1}{p_{\beta_i}}}. \tag{42} \]
From formula (39) we have:
\[ m_{\beta_i}=\pm e_T\sqrt{\frac{1}{2}Z_{C_i}+Z_{T_i}}; \tag{43} \]
\[ m_{\beta_k}=\pm e_T\sqrt{\frac{1}{2}Z_{C_k}+Z_{T_k}}. \tag{44} \]
Substituting \(m_{\beta_i}\) and \(m_{\beta_k}\) from (43) and (44) into (41), we obtain:
\[ \frac{1}{p_{\beta_i}}= \frac{\frac{1}{2}Z_{C_i}+Z_{T_i}} {\frac{1}{2}Z_{C_k}+Z_{T_k}}. \tag{45} \]
We shall express the errors of measurement of the side lengths by the formula
\[ m_l=\pm\mu\sqrt{l}, \tag{46} \]
where \(\mu\) is the coefficient of random influence in measuring side lengths.
Substituting \(m_\beta\) and \(m_l\) from (42) and (46) into (35), we obtain:
\[ m_{\beta_k}^{2}\sum_{AB}\frac{1}{p_\beta}\left(\frac{R''_{AB}}{\rho}\right)^2 +\mu^2\sum_{AB}l\cos^2\delta_{AB} -\Delta C_{AB}^{2}=0 . \tag{47} \]
For a network of underground connecting traverses oriented through three shafts (Fig. 1), three equations of the form (47) may be written, namely:
\[ m_{\beta_k}^{2}\sum_{AB}\frac{1}{p_\beta}\left(\frac{R''_{AB}}{\rho}\right)^2 +\mu^2\sum_{AB}l\cos^2\delta_{AB} -\Delta C_{AB}^{2}=0, \]
\[ m_{\beta_k}^{2}\sum_{AC}\frac{1}{p_\beta}\left(\frac{R''_{AC}}{\rho}\right)^2 +\mu^2\sum_{AC}l\cos^2\delta_{AC} -\Delta C_{AC}^{2}=0, \tag{48} \]
\[ m_{\beta_k}^{2}\sum_{BC}\frac{1}{p_\beta}\left(\frac{R''_{BC}}{\rho}\right)^2 +\mu^2\sum_{BC}l\cos^2\delta_{BC} -\Delta C_{BC}^{2}=0 . \]
The differences \(\Delta C_{AB}\), \(\Delta C_{AC}\), and \(\Delta C_{BC}\) are determined by formulas (5). The system of equations (48) makes it possible to determine the error of measuring an angle with weight equal to unity, i.e., the angle \(\beta_k\), and the coefficient \(\mu\) of formula (46).
Let us denote:
\[ x=m_{\beta_k}^{2}, \qquad y=\mu^2; \]
\[ a_1=\sum_{AB}\frac{1}{p_\beta}\left(\frac{R''_{AB}}{\rho}\right)^2; \qquad a_2=\sum_{AC}\frac{1}{p_\beta}\left(\frac{R''_{AC}}{\rho}\right)^2; \]
\[ a_3=\sum_{BC}\frac{1}{p_\beta}\left(\frac{R''_{BC}}{\rho}\right)^2; \]
\[ b_1=\sum_{AB}l\cos^2\delta_{AB}; \qquad b_2=\sum_{AC}l\cos^2\delta_{AC}; \qquad b_3=\sum_{BC}l\cos^2\delta_{BC}; \]
\[ v_1=\Delta C_{AB}^{2}; \qquad v_2=\Delta C_{AC}^{2}; \qquad v_3=\Delta C_{BC}^{2}. \]
With the adopted notation, the system of equations (48) will take the form:
\[ a_1x+b_1y-v_1=0, \]
\[ a_2x+b_2y-v_2=0 \tag{48a} \]
\[ a_3x+b_3y-v_3=0. \]
Solving the inconsistent system of equations (48a) under the condition
\[ \sum (ax + by - v)^2 = \text{minimum}, \]
we obtain the following two normal equations for determining \(x\) and \(y\):
\[ \begin{aligned} [aa]x + [ab]y - [av] &= 0,\\ [ab]x + [bb]y - [bv] &= 0. \end{aligned} \tag{49} \]
Having solved the system of normal equations (49), we obtain \(x\) and \(y\). The error of measuring an angle with weight equal to unity, and the value of the coefficient \(\mu\), are determined from the formulas:
\[ m_{\beta_k} = \pm \sqrt{x}; \tag{50} \]
\[ \mu = \pm \sqrt{y}. \tag{51} \]
The errors of angle measurements are computed by formula (42), and those of side lengths by formula (46).
For the networks shown in Figs. 2, 3, and 4, the system of equations (48) will consist of six equations.
After determining the error \(m_{\beta_k}\) and the coefficient \(\mu\), the weights of the measured angles and side lengths are established on the basis of formulas (9) and (10).
For the error \(m_p = 1\), one may take the error of measurement of some angle, or the error of measurement of a side length equal to one meter.
If the error of measurement of some angle, for example the angle \(\beta_k\), is taken as the error \(m_p = 1\), then the weights of the measured angles and side lengths, on the basis of formulas (9), (10), and (46), will be equal to:
\[ p_{\beta_i} = \frac{m_{\beta_k}^{2}}{m_{\beta_i}^{2}}; \tag{52} \]
\[ p_{l_i} = \frac{m_{\beta_k}^{2}}{\mu^{2} l_i}. \tag{53} \]
Whence:
\[ \frac{1}{p_{\beta_i}} = \frac{m_{\beta_i}^{2}}{m_{\beta_k}^{2}}; \tag{54} \]
\[ \frac{1}{p_{l_i}} = \frac{\mu^{2} l_i}{m_{\beta_k}^{2}}. \tag{55} \]
If the error of measurement of a side length of \(1\) m is taken as the error \(m_p = 1\), then the weights of the measured angles and side lengths, on the basis of formulas (9), (10), and (46), will be equal to:
\[ p_{\beta_i} = \frac{\mu^{2}}{m_{\beta_i}^{2}}; \tag{56} \]
\[ p_{l_i} = \frac{1}{l_i}. \tag{57} \]
Whence:
\[ \frac{1}{p_{r_i}}=\frac{m_{\beta_i}^{2}}{\mu^{2}}; \tag{58} \]
\[ \frac{1}{p_{l_i}}=l_i . \tag{59} \]
If the free terms of the conditional equations of the sides (1–3) or of the abscissas and ordinates (28–31), and the projections \(R''\) or the differences of abscissas and ordinates, are expressed in millimeters, while the side lengths in computing the weights by formula (53) are expressed in meters, then formulas (53) and (55) will take the form:
\[ p_{l_i}=\frac{\Theta}{l_i}; \tag{53a} \]
\[ \frac{1}{p_{l_i}}=\frac{l_i}{\Theta}; \tag{55a} \]
\[ \Theta=\left(\frac{m_3}{1000\mu}\right)^2 . \tag{60} \]
If, simultaneously with the adjustment of the network of underground connecting traverses, an assessment is made of the accuracy of the adjusted directional angles or coordinates, then, for convenience of computation, the weights of the measured angles and side lengths must be determined by formulas (52) and (53a).
In conclusion, we note that underground connecting traverses that have been run and closed within the shaft station over three or more fixed points (plumb lines) must necessarily be adjusted by the rigorous method.