APPLICATION OF THE GEODETIC INTERSECTION METHOD IN ESTABLISHING HORIZONTAL CONTROL FOR SURVEYS OF OPEN-PIT MINES AND GEOLOGICAL EXPLORATION WORKINGS

M. V. POSTNIKOV

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

As the horizontal geodetic control for open-pit mines, special triangulation or polygonometric traverses are used (also for special purposes). Both can successfully be replaced by constructing networks by the method of geodetic intersections [1].

The accuracy of the geodetic control at open-pit workings must not be lower than the limiting graphical accuracy of the working plan, according to which problems of mining and geological exploration are solved (calculation of reserves, volume of work performed, design of mining and exploration operations, etc.).

The limiting accuracy of the position of a determined point of the main geodetic network on the ground, proceeding from the limiting graphical accuracy of the plan, will be:

\[ T_{np}=P_{np}\cdot M, \]

where

\[ P_{np} \]
is the graphical accuracy of the linear scale in mm;

\[ M \]
is the denominator of the numerical scale.

Taking into account that working plans of open-pit mines are compiled at a scale of 1:500 or 1:1000, we shall have:

\[ T_{np}\text{—for scale }1:500=0.1\ \text{m}. \]

\[ T_{np}\text{—for scale }1:1000=0.2\ \text{m}\ [2]. \]

For the production of current mine-surveying surveys, the points of the geodetic control determined by triangulation or polygonometry methods are usually insufficient. Additional networks of control points are created, which are called networks of working control for mine-surveying surveys.

Depending on the size of the quarry field, the conditions of occurrence of the mineral deposit, and the nature and rate of mining operations, the following are used as working control:

a) microtriangulation networks; b) third-class polygonometry and theodolite traverses; c) individual points determined from triangulation and polygonometry points by the intersection method; d) operating grids.

Microtriangulation networks are constructed with sides 100–200 m long, and the angles in them are observed with an error of ±10″.

The accuracy indicators for polygonometric and theodolite traverses laid out for survey purposes are given in Table 1.

Table 1

Distinguished by the considerable labor-intensiveness of field work (especially linear measurements), polygonometry of the 1st and 2nd classes has not found extensive application in open workings. Polygonometry of the 3rd class and theodolite traverses, as a type of control, are used rather widely in open pits.

However, the laying out of theodolite and polygonometric traverses of the 3rd class requires a good terrain profile (for linear measurements). Fulfillment of this condition in an open pit is often difficult, takes much time, and increases the cost of the control.

Construction of working control by the triangulation method from continuous triangles under open-pit conditions is very complex, owing to the considerable volume of angular measurements.

Obtaining the coordinates of control points constructed by the methods indicated above is possible only after completion of all field work. Under conditions of intensive mining operations, however, it is often necessary to obtain the coordinates of control points without waiting for the completion of all the work.

Many of the shortcomings cited in constructing working control in open workings can be eliminated by using the method of geodetic intersections. In the method of geodetic intersections, rapid execution of field work, high accuracy in determining points, and a reduction in the cost of the control are ensured by the following circumstances:

a) In the intersection method only angular observations are required, as in triangulation, but the number of observation points, compared with triangulation, is reduced by one third.

b) In the intersection method, labor-intensive and cumbersome linear measurements, performed in polygonometry, are reduced to a minimum.

c) The speed of field work, compared with theodolite traverses and triangulation, increases considerably, and the cost of the control decreases by approximately 20%.

d) The computation of the coordinates of control points in the intersection method can be carried out without waiting for completion of all the work, as the traverse line advances.

e) In the method of geodetic intersections, the control of field measurements is more reliable than in triangulation and in polygonometry. The control is…

is performed at each point of the traverse line by comparing the length and the direction angle of one and the same side, determined from two transfers.

e) Field work by the method of geodetic intersections may be carried out under summer and winter conditions, in the daytime and at night. For night work it is necessary to have the appropriate equipment.

g) Theoretical investigations and experience show that the accuracy of networks constructed by the intersection method is twice as high as in theodolite traverses (1:3000–1:5000, instead of 1:1500–1:2000), with the same accuracy of angular measurements (2 repetitions, 30″ theodolite, with an angular measurement error of ±20″).

The author’s work, carried out in 1953 in Kuzbass on the laying of traverse lines by the intersection method, confirms the conclusions indicated.

Schemes for Constructing a Geodetic Control by the Intersection Method

We shall give a number of typical schemes for constructing a working control by the intersection method for surveys of open workings.

In Fig. 1 the points \(A^\circ\), \(B^\circ\), \(C^\circ\), and \(D^\circ\) are points of a geodetic control of triangulation or polygonometry of a higher class than the traverses themselves; \(A\), \(B\), \(C\), \(D\), \(E\ldots\) are points of the traverse line established on the bottom of the open pit; \(1—1'\), \(2—2'\ldots\), \(5—5'\) are auxiliary points located along the edges of the open pit. The lengths of the initial sides of the traverse line and their direction angles may be obtained by laying an additional tie traverse. The accuracy of transferring the lengths of the initial sides and the direction angles will depend on the technical requirements for the survey traverse.

Let us note that the scheme for constructing the traverse shown in Fig. 1 may be applied when providing control for surveys of individual sites (benches) of considerable width.

When constructing a working control by the intersection method, it is necessary to take into account that the accuracy of the network will depend on the configuration of the triangles.

To determine the best geometric form for constructing a bundle of intersections, let us assume that the triangles located along the traverse line are constructed from symmetrical bundles of intersections (Fig. 2). Such a construction of the intersection is possible if the condition of equality of the angles \(B = B' = 90^\circ\) is satisfied.

Substituting the value of the cotangent of the angle \(B = 90^\circ\) into the known formula [1],

\[ Q=\left[2\operatorname{ctg}^{2}(A+B)+\operatorname{ctg}^{2} A\, 2\operatorname{ctg} A\,\operatorname{ctg}(A+B)\right]\left(\frac{10^{6}\mu}{\rho}\right)^{2}, \]

we obtain:

\[ Q=\left(\operatorname{ctg}^{2} A+2\operatorname{tg}^{2} A+2\operatorname{tg}^{2} A+2\right)\left(\frac{10^{6}\mu}{\rho}\right)^{2}. \tag{1} \]

The form of the intersection will be most advantageous when the inverse geometric weight \(Q\) is equal to a minimum, which leads to the equality

\[ \frac{dQ}{dA}=0; \]

or

\[ \frac{2\sin^{4}A-\cos^{4}A}{\sin^{2}A\cdot\cos^{2}A}=0, \]

whence

\[ \operatorname{ctg} A = \sqrt[4]{2} = 1.189 \]

\[ A^\circ = 40^\circ 03'. \tag{2} \]

Fig. 1

Consequently, the best construction of the intersection bundle, for a symmetrical form, is an intersection in which angle \(A\) is close in value to \(40^\circ\).

When the traverse line is placed along a bench of the open pit (Fig. 3), it is possible to construct the traverse line with longer sides, which will increase the accuracy of the traverse being laid out.

With a curvilinear direction of the open pit (Fig. 4), the traverse line will have bends, and the network will be constructed by means of multi-figure intersections.

Fig. 2.

Bends in the traverse line increase the accuracy with which side lengths are transferred. Thus, for example, a bend of the traverse from a straight line by \(15^\circ\) increases the accuracy of transferring the side lengths of the traverse by a factor of 2 in comparison with a straight traverse; longitudinal errors are 1.5–2 times smaller, and transverse errors 1.2–1.5 times smaller than the errors in a rectilinear traverse.

Fig. 3.

In the absence of a regular bench system of workings, the working control can be constructed according to the scheme shown in Fig. 5. The traverse line \(ABCEK\) is placed along both sides of the open pit with the auxiliary points \(1—1'\ldots 2—2'\) arranged on one side. Such an arrangement of the traverse line (a bend), as indicated above, noticeably increases the accuracy of determining the points.

If it is necessary to have additional survey points in the open pit, their position may be determined as points of the second order (items I and II). These points are established near the sides of the traverse line, and the corresponding conditional measurements are made at them. The computation of the coordinates of the additional points should be carried out after adjustment of the main traverse. Such a construction increases the accuracy of the traverse and creates additional control over the transfer of the direction angle along the traverse.

In large open pits, with a one-sided front of mining operations, by the intersection method it is possible simultaneously to construct both an approach network to the open workings and the working control for current mine-surveying surveys.

In Fig. 6, points \(A\), \(B\), \(C\), \(E\), and \(K\) are points of the traverse line laid along the edge of the open pit on the opposite side

from the active benches. The points \(P, P', M\), and \(M'\), which are auxiliary points and are determined simultaneously with the traverse points, serve as the immediate working control. We note that the points \(P, P', M\), and \(M'\) will be obtained with somewhat lower accuracy than the points of the traverse line.

Studies show that the loss in the accuracy of determining auxiliary points, as compared with the points of the traverse line, amounts to \(10\text{–}15\%\).

The indicated scheme for the simultaneous construction of the approach network and the working control can be widely used in the practice of mine-surveying surveys.

Remarks on constructing geodetic control in open-pit workings by the intersection method

a) Reconnaissance of points

The traverse line must be selected in the required directions with respect to the intended purpose of the network. Auxiliary points are established with a view to ensuring visibility to them from the points of the traverse line, in locations convenient for their subsequent use and for ensuring the least reciprocal geometric weight of transmission.

Fig. 4.

After reconnaissance of the traverse line, the error of the last side of the traverse can be computed in advance by formula [1]:

\[ m_{s_r}=m''\sqrt{\frac{1}{2P_{s_r}}}, \tag{3} \]

where \(m''\) is the root-mean-square error of measuring an angle,

\[ \frac{1}{P_{s_r}} \]

is the reciprocal weight of the transmission of the length of the traverse side.

Fig. 5.

The value \(m''\) is established depending on the requirements imposed by the technical instruction on the working control.

Let us note that the selection of points in the field presents no difficulty. The intersection method has exceptional flexibility and makes it possible successfully to overcome obstacles in the terrain.

b) Monumenting the points

The points of the traverse line and the auxiliary points of the main traverses of the first transfer are monumented as control points in accordance with the existing technical instructions. All other points are monumented with wooden stakes.

Fig. 6

c) Measurement of horizontal angles

Depending on the purpose and accuracy of the traverses developed by the intersection method, optical theodolites or theodolites of 30″ accuracy may be used for angular measurements.

After reconnaissance and computation of the reciprocal geometrical weight of the traverse by formula (3), one can calculate the required accuracy of angle measurement, and then establish the number of sets (or repetitions) with which the angles in the traverse line should be measured. If the angular measurements are made with a 30-second theodolite, then the number of sets (or repetitions) may be established from Table 2 \[3\] of the errors of measuring an angle with a 30-second theodolite, caused by reading and sighting errors.

Table 2

The measurement of angles in traverses replacing class-3 polygonometry, with a traverse length of 1.5–2 km, may be carried out with a 30-second theodolite in 2–3 sets.

When measuring the angles of a traverse line, it is necessary to keep strict watch over the precise centering of the instrument and of the sighting targets, since inaccurate centering with short traverse sides entails errors that sometimes exceed the accuracy of angle measurement.

With an average traverse-side length of 150 m, the centering error of the instrument and sighting targets should not be allowed to exceed 3 mm; for shorter sides (50–75 m), 1–2 mm.

Bearing in mind that the traverse line will be constructed from relatively short lines, in order to eliminate errors due to the centering of the theodolite and of the signal reduction, when measuring angles it is especially desirable to use a set consisting of three angle-measuring instruments of the TB-1 type.

If it is impossible to use the method of measuring angles with a set of several instruments or with a three-tripod system, then, when measuring angles, rigid plummets should be used, set up at the points of the traverse line and at auxiliary points. Rigid plummets and tripods for them can be manufactured at low cost in local machine shops. The design of the rigid plummets and tripods was developed by the author when creating a set of apparatus for measuring the lines and angles of polygonometric traverses. A description of the design of the rigid plummet and the tripod for it is given below.

Tripod

The tripod head is cast from cast iron. The shape of the tripod head is shown in Fig. 7.

Fig. 7.

The opening of the tripod head (1) has a diameter of 6 cm. Parallel to the opening in the head, two circular cutouts (2) are made. A metal ruler (3), having in its middle part an enlargement (4) with a threaded screw-shaped opening cut in it, and at the edges straight longitudinal cutouts (5), is placed on the tripod head. The cutouts in the ruler and in the tripod head make it possible to move the ruler in various directions. After the corresponding setting, the position of the ruler on the tripod head is secured by clamping screws (6).

Rigid Plumb Lines (Signals)

A rigid plumb line is made of round steel 1–1.5 cm in diameter and 2 m long (Fig. 8). In the lower part the plumb line is turned down to a cone. For suspending the plumb line from a tripod, a device is placed on the tripod consisting of two rings at a coupling, fastened to each other by two mutually perpendicular metal axes (Fig. 9). For suspending the plumb line on the tripod, the coupling has a clamping screw. The stability of the suspended plumb line is ensured by means of a metal cylinder weight of 1.5–2.0 kg, which is placed on the lower part of the plumb line and secured to it with a clamping screw. The outer ring of the indicated device, together with the plumb line, is inserted into the round opening of the tripod plate. Having adjusted the height of the plumb line and moved the plate in the slots of the tripod and of the plate, the plumb line can be set exactly over the center of the point. After the plumb line has been set, the plate is secured on the tripod by means of clamping screws.

Fig. 8.

Fig. 9.

For better visibility, the plumb line is painted in alternating black or red and white bands every 10–12 cm. If necessary, a light shield with a mark is placed on the upper end of the signal. To ensure the required accuracy of angular observations, it is necessary to make sure that the signal has no curvature.

Computational Work

The procedure for processing and adjusting a network constructed by the intersection method for mine-surveying surveys of an open pit is established as follows:

1. The field journal of angle measurements is checked.
2. The lengths of the sides of the traverse line are computed along both transmissions, and the accuracy of transmitting the length of the line is determined (scheme 1).
3. The adjustment of the line lengths is carried out under the condition of the initial sides (scheme 2).
4. The computation and adjustment of the coordinates of the points of the traverse line are performed.

1. The coordinates of auxiliary points and second-order points are computed (Scheme 3).

Scheme 1. Computation of the side lengths of the traverse line

Fig. 10

Computation formulas

\[ BC=\frac{AB\sin 1\sin(3+4)}{\sin 4\sin(1+2)}. \]

\[ BC=\frac{AB\sin 1'\sin(3'+4')}{\sin 4'\sin(1'+2')}. \]

\[ \Delta = 2_{\text{units}} \lg \text{ of the 5th digit} \]

\[ \lg BC = 2{,}40793 \]

Methodological instructions for Scheme 1

1. The side lengths of the traverse line are computed by both transfers from the initial side \(AB\). Having obtained the difference of the logarithms of the lengths of one and the same side of the traverse line, \(\Delta\), and assuming that this quantity is equal to the error of the logarithm of the side multiplied by \(\sqrt{2}\), i.e. \(m_{\lg s}\sqrt{2}\), the relative error in transferring the length of the side may be computed by the formula:

where

\[ \frac{\Delta_s}{S}=\frac{\Delta}{\mu\cdot 10^n\sqrt{2}}, \]

\(\mu\) is the modulus of the common logarithm, equal to \(0{,}43\);

\(\Delta\) is expressed in units of the \(n\)-th decimal place of the logarithm.

Example: \(\Delta = 2\) (the fifth decimal place of the logarithm),

\[ \frac{\Delta_s}{S}=\frac{2}{0{,}43\cdot 10^5\cdot \sqrt{2}}=\frac{1}{30\,000}. \]

The required accuracy in transferring the length of the line will depend on the requirements imposed on the traverse being laid out.

1. After determining the accuracy of the line-length transfer, the length of the side of the traverse line is computed as the arithmetic mean of the two transfers.

In traverses of increased accuracy, line lengths are computed using six-place logarithms, and the final length is determined as a weighted mean from the two transfers.

For this purpose, the values of the reciprocal weight \(Q\) of each triangle are selected from a special table; then, for each transfer, the weights are computed [7]:

\[ Q' = Q'_1 + Q'_2;\qquad Q'' = Q''_1 + Q''_2; \]

for the 1st transfer

\[ P'=\frac{100}{Q'}; \]

for the 2nd transfer

\[ P''=\frac{100}{Q''}. \]

The weight of both transfers is

\[ P_m=P'+P''=\frac{Q'+Q''}{Q'\cdot Q''}. \]

The reciprocal weight of the transfers is

\[ Q_m=\frac{1}{P_m}=\frac{Q'\cdot Q''}{Q'+Q''}. \]

Example:

\[ Q'=Q'_1+Q'_2=24+44=68;\qquad Q''=Q''_1+Q''_2=23+36=59; \]

\[ P'=\frac{100}{Q'}=\frac{100}{68}=1{,}5;\qquad P''=\frac{100}{Q''}=\frac{100}{59}=1{,}7; \]

\[ P_m=P'+P''=3{,}2. \]

\[ Q_m=\frac{68\cdot 59}{68+59}=31. \]

The computed logarithms of side \(BC\):

from the 1st transfer — \(2{,}407944\);

from the 2nd transfer — \(2{,}407922\).

The final value of the logarithm of side \(BC\) is obtained as the weighted mean value

\[ \lg BC = 2{,}407920+\frac{(24\cdot 1{,}5)+(2\cdot 1{,}7)}{1{,}5+1{,}7}=2{,}407932 \]

1. Having obtained the length of the second side in the traverse, \(BC\), the next side of the traverse line is computed, and so on.

Scheme 2. Adjustment of the sides of the traverse line by the condition of the bases

Fig. 11.

The misclosure is \(w_s = +31\) units of the 6th decimal place of the logarithm.

\[ \Sigma Q_m = 62. \]

The relative error of the last side (before adjustment) is

\[ \frac{\Delta_s}{S} = \frac{31}{0.43 \cdot 10^6} = \frac{1}{14000}. \]

Methodological Instructions for Scheme 2

1. The logarithms of the traverse-side lengths obtained as a result of the computations are entered in the scheme, and the free term of the base condition, \(w_s\), is determined as the difference between the computed and known values of the logarithm of the terminal side:

\[ w_s = 2.361\,889 - 2.361\,858 = +31 \]

units of the 6th decimal place of the logarithm.

1. The relative error of the most distant side of the traverse (before adjustment) is computed:

\[ \frac{\Delta_s}{s} = \frac{31}{0.43 \cdot 10^6} = \frac{1}{14,000}. \]

We note that, after adjustment of the traverse, the accuracy of the mean side of the traverse will increase approximately twofold.

1. The obtained misclosure is distributed with the opposite sign, in proportion to the values \(Q_m\), over each side of the traverse.

For the first side the correction will be:

\[ \delta_{s_1} = \frac{-w_s}{\Sigma Q_m}\cdot Q_{m_1} = \frac{-31}{62}\cdot 28 = -14; \]

for the second side:

\[ \delta_{S_2}=\frac{-w_s}{\sum Q_m}\cdot Q_{m_i} =\frac{-31}{62}\cdot 14=-7\ \text{etc.}, \]

then the logarithms of the adjusted traverse sides are computed.

1. The direction angles of the traverse lines are computed from the mean values of the angles measured at the traverse-line stations.

The permissible angular misclosure of the traverse is determined by the formula:

\[ f_{\beta\,\text{perm.}}=\pm m''\sqrt{n}, \]

where \(m''\) is the mean square error of angle measurement adopted for the given traverse;
\(n\) is the number of measured angles.

Notes: 1. The procedure indicated in item 3 for adjusting the lengths of traverse lines under the base condition is performed only for main traverses.
2. For working-control traverses, the obtained misclosure is distributed equally, cumulatively, to each line.
3. The computation of the lengths of the sides of a traverse line may also be carried out according to the scheme recommended by N. A. Kuzin and N. N. Lebedev [4].

Methodological notes on computing the coordinates of traverse stations

1. The computation of coordinate increments is carried out from the adjusted sides and adjusted direction angles.

2. The misclosures in the coordinate increments \(f_x\) and \(f_y\), after checking their permissibility, are distributed proportionally to the lengths of the sides.

For main traverses, the misclosures \(f_x\) and \(f_y\) may be distributed proportionally to \(Q_m\), the inverse weight of each transfer, according to the formulas:

\[ \delta_{\Delta x_i}=\frac{-f_x}{\sum Q_m}\cdot Q_{m_i} \]

\[ \delta_{\Delta y_i}=\frac{-f_y}{\sum Q_m}\cdot Q_{m_i} \]

Scheme 3. Computation of the coordinates of auxiliary point No. 2 (Fig. 12)

Fig. 12

Formulas

\[ X_3=\frac{X_1 \cdot \operatorname{ctg} 2+X_2 \operatorname{ctg} 1-y_1+y_2}{\operatorname{ctg} 1+\operatorname{ctg} 2}, \]

\[ y_3=\frac{y_1 \operatorname{ctg} 2+y_2 \operatorname{ctg} 1+X_1-X_2}{\operatorname{ctg} 1+\operatorname{ctg} 2}. \]

Methodological Notes on Scheme 3

1. The coordinates of auxiliary points are computed by the formulas of the cotangents of the angles.
2. Angles 1 and 2, (1) and (2), are taken as measured in the network.
3. Of all the obtained coordinate values of the auxiliary point, the arithmetic mean is taken as the final value.

Conclusion

1. The use of polygonometric and theodolite traverses as the geodetic control for open-pit workings is associated with known difficulties, since linear measurements are carried out under poor profile conditions of the terrain. Field control of the work when laying traverses is very weak, and field work in polygonometry can be carried out only by day; under winter conditions, measurement of lines is difficult. The construction of control by triangulation from solid triangles under open-pit conditions is complicated. Considerable difficulties arise in reconnaissance of points, and the number of points at which angular measurements are made increases. The coordinates of points of the geodetic control developed by methods of triangulation and polygonometry can be obtained only after completion of all field work.

2. Development of the geodetic control by the intersection method significantly reduces the difficulties of constructing a geodetic network under open-pit conditions. In the intersection method, linear measurements are reduced to a minimum, and the number of observation points, in comparison with triangulation, is reduced by one third. The accuracy of traverses laid by the method of geodetic intersections is considerably higher than that of theodolite traverses, and corresponds to the accuracy of triangulation, construc-

...than that of a series of equilateral triangles at the same accuracy of angular measurements. The speed of field work with the intersection method increases almost twofold, and the cost of establishing control is reduced by approximately 20–30%. Field work can be carried out throughout the entire year, by day and by night. The coordinates of control points can be computed as the traverse line is advanced. The method of geodetic intersections is flexible and easily adapted to any terrain conditions.

1. The method of geodetic intersections can be used successfully when tying operating networks, laid out in open-pit workings for mine-surveying surveys, to points of the geodetic control. For this purpose it is necessary, at certain points lying on one of the axes of the network, to establish traverse points (Fig. 1), or to determine their positions as auxiliary points of the traverse line (Fig. 6).

2. The construction of control networks for mine-surveying surveys of failures that have formed as a result of mining operations should be carried out by the intersection method. The method of geodetic intersections makes it possible to construct the network in the required direction and to obtain the necessary accuracy, independently of relief conditions.

3. The method of geodetic intersections can be widely used for establishing control for surveys of geological-exploration workings (prospecting pits, boreholes, trenches, etc.). As auxiliary points of the traverse line it is recommended to choose the most important exploration workings (prospecting pits, boreholes, and other objects), whose positions must be obtained with higher accuracy.

4. When the traverse line is of considerable length, in order to increase the accuracy of traverses, control bases may be measured, moreover in places convenient for linear measurements. To improve the accuracy of transferring the directional angle at points of the traverse line, in addition to measuring angles to auxiliary points, the angles of deflection of the traverse line should be measured at a separate station.

REFERENCES

1. Durnev, A. I. New Systems for Constructing Geodetic Networks, Geodezizdat, 1952.

2. Peregudov, M. A. Mine-Surveying Survey of Open-Pit Workings, Ugletekhizdat, 1950.

3. Pavlov, V. F. Instructions for Tying Photographs in Aerial Surveying, Geodezizdat, 1941.

4. Kuzin, N. A.; Lebedev, N. N. Practical Manual on Urban and Engineering Polygonometry, 1954.

5. Rudakov, M. L. Mine-Surveying Work in Open-Pit Workings, Ugletekhizdat, 1950.

6. Burianov, G. N. Open-Pit Mining Operations, Ugletekhizdat, 1951.

7. Postnikov, M. V. “A Graph-Analytical Method of Preliminary Accuracy Computation for a Chain of Triangles,” Izvestiya TsNIIGAiK, vol. 64, 1956.