Mine Surveying: Levelling, Traversing, Errors, Correlation, Underground Surveys, Curves, Tacheometry, Photogrammetry and Modern Instruments
1. Levels and levelling, above and below ground
Levelling finds differences of height. A level set up between two points reads a back sight (BS) on a point of known reduced level (RL) and a fore sight (FS) on the next. By the height-of-instrument method, HI = RL + BS and the new RL = HI − FS. By the rise-and-fall method, each difference of consecutive readings is a rise (if the reading falls) or a fall. Both carry an arithmetic check: ΣBS − ΣFS = ΣRise − ΣFall = last RL − first RL; the rise-and-fall method also checks intermediate sights, which is its advantage.
Underground, stations are often fixed in the roof, and the staff is held inverted against them. An inverted staff reading is entered with a negative sign: for a roof station, RL = HI + reading, not HI − reading. So a level at HI 51.200 m reading 1.650 m on an inverted staff puts the roof station at 52.850 m. Over long sights the earth’s curvature makes a staff read too high and refraction bends the line of sight back down; the combined correction is −0.0673 d² m, with d in km (curvature −0.0785 d², refraction +0.0112 d²). Reciprocal levelling across a river or a sump cancels both, and collimation error too, by averaging readings taken from both sides.
2. Traversing, triangulation, and errors and their adjustment
A traverse is a chain of survey lines whose lengths and directions are measured. Each line of length L and bearing θ has a latitude L cos θ (north–south) and a departure L sin θ (east–west). In a closed traverse both sums should be zero; what remains is the closing error e = √(ΣL² + ΣD²), and the relative precision is e divided by the perimeter, written 1 in N. The interior angles of a closed n-sided traverse must add to (2n − 4) × 90°.
| Rule | Correction to a line’s latitude | When it is right |
|---|---|---|
| Bowditch (compass) rule | total latitude error × (length of the line / perimeter) | angles and lengths measured with equal precision |
| Transit rule | total latitude error × (|latitude of the line| / Σ|latitudes|) | angles more precise than lengths |
Triangulation fixes a network of stations by measuring the angles of well-conditioned triangles (no angle much below 30°) from a precisely measured base line; the strength of figure expresses how angle errors propagate into lengths. Trilateration measures sides instead, which EDM has made practical. Survey errors are of three kinds: gross errors (blunders — a misread staff), removed by checks; systematic errors, which follow a law and can be corrected (tape temperature, curvature); and random (accidental) errors, which remain and obey probability. The best estimate from repeated measurements is the most probable value; with unequal precision each observation is weighted in inverse proportion to the square of its standard error, and a station adjustment makes the angles round a station sum to 360°.
3. Correlation and underground surveying
Correlation transfers the surface coordinate system and, above all, the surface bearing underground, so that the underground plan can be laid on the surface plan. Through an adit or incline it is simply a traverse carried in. Through a single vertical shaft it is hard, because the only link is a pair of plumb wires a few metres apart, and a small error in their direction becomes a large error at the end of a long underground traverse. The methods are:
- Weisbach triangle: the theodolite is set almost in line with the two wires, and the very small angle at the instrument with the measured sides solves the triangle.
- Co-planing: the instrument is moved until the line of sight passes through both wires exactly.
- Weiss quadrilateral and other two-wire variants; two-shaft correlation, with one wire in each shaft and an underground traverse joining them, which gives a far longer base.
- Gyro-theodolite (gyro-azimuth): the spinning gyroscope seeks true north directly underground, so no wires are needed — now the standard for long workings. Magnetic methods are unreliable near iron and in magnetic ores.
Underground surveying works in darkness, confined space and moving air: stations are set in the roof so that traffic does not disturb them, sights are short, the instrument is centred under a roof station by a plumb bob or an optical plummet, and targets are illuminated. Steep workings are surveyed with a theodolite fitted with an eccentric (side or top) telescope, whose readings need an eccentricity correction. The underground traverse gives the positions of development headings, the depth and extent of workings plotted against the lease boundary, and the check on how close workings approach old flooded galleries — a legal and safety duty in its own right.
4. Tacheometry, curves and contouring
Tacheometry measures distance optically. A theodolite with stadia hairs reads a staff intercept s; with multiplying constant K (usually 100) and additive constant C (zero for an anallactic telescope), on a line of sight inclined at θ: horizontal distance H = K s cos² θ + C cos θ and vertical component V = K s sin 2θ/2 + C sin θ. With s = 1.25 m, K = 100, C = 0 and θ = 10°, H = 125 × 0.9698 = 121.23 m.
A simple circular curve of radius R joining two straights that deflect by Δ has tangent length T = R tan(Δ/2), curve length L = π R Δ/180°, long chord 2R sin(Δ/2), external distance R(sec(Δ/2) − 1) and mid-ordinate R(1 − cos(Δ/2)). The degree of curve is the angle subtended by a standard chord (or arc) of 30 m or 20 m. Haul roads and railway sidings in mines are laid out by these, often with transition curves where speed is high.
A contour is a line joining points of equal elevation, and the contour interval is the constant height between successive ones. Contours never cross except at an overhang, close on themselves within or beyond the map, crowd together on steep ground and spread apart on gentle ground, cross a ridge or valley at right angles, and form V’s pointing upstream in a valley and downhill on a ridge. Contours are plotted by direct or indirect (grid, cross-section, tacheometric) methods, and they give the earthwork of a dump or pit, the route of a haul road at a fixed gradient, and a catchment area for mine drainage.
5. Photogrammetry, EDM, total station, GPS, remote sensing and astronomical survey
In a vertical aerial photograph taken with focal length f from flying height H above datum, the scale at ground elevation h is f/(H − h). A point at height h above datum is displaced radially outward from the principal point by the relief displacement d = r h/H, where r is the radial distance of the displaced (top) image. Stereoscopic cover uses about 60% forward overlap and about 30% side overlap, and the parallax of a point between two overlapping photographs gives its height. A drone (UAV) survey is photogrammetry at low altitude, and is now routine for stockpile volumes and pit progress.
| Instrument | Principle | Mining use |
|---|---|---|
| Theodolite | horizontal and vertical angles on graduated circles; face-left and face-right means cancel instrumental errors | traverses, setting out, correlation |
| EDM | phase comparison (or pulse timing) of a modulated electromagnetic wave reflected from a prism | long lines, trilateration |
| Total station | an electronic theodolite and EDM in one, computing coordinates and storing data | underground and pit surveys, monitoring slopes |
| GPS / GNSS | ranges to satellites; four are needed for a 3-D fix because the receiver clock error is a fourth unknown; DGPS and RTK correct errors with a base station | surface control, fleet dispatch, drill navigation |
| Remote sensing | reflected visible and near-infrared, emitted thermal infrared, and active radar from satellites; spatial, spectral, radiometric and temporal resolution | mineral targeting, land-use change, subsidence by InSAR, coal-fire mapping |
| Astronomical survey | observing the sun or Polaris and solving the astronomical (PZS) triangle | true meridian, azimuth of a base line, latitude |
Key takeaways
- HI = RL + BS and RL = HI − FS; an inverted roof reading enters with a negative sign, so the roof RL is HI + reading.
- The combined curvature-and-refraction correction is −0.0673 d² m (d in km), and reciprocal levelling cancels it.
- Closing error = √(ΣL² + ΣD²); Bowditch distributes it in proportion to line length; weights go as 1/σ².
- Correlation through one shaft uses two plumb wires (Weisbach, co-planing) or a gyro-theodolite; two shafts give a longer base.
- T = R tan(Δ/2), L = πRΔ/180; H = Ks cos²θ; photo scale f/(H − h) and relief displacement rh/H; GPS needs four satellites.
Practice questions (14)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
A back sight of 1.525 m is taken on a bench mark of RL 100.000 m, and the fore sight on the next point is 2.315 m. The RL of that point, in m, to two decimal places, is ____.
Numerical answer — type the value.
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Answer: 99.21
HI = 100.000 + 1.525 = 101.525 m, and RL = 101.525 − 2.315 = 99.210 m. Adding the fore sight instead of subtracting it gives 103.84; the larger fore sight means the new point is lower than the bench mark.Underground, a level reads 1.200 m on a staff held on a floor bench mark of RL 50.000 m, then 1.650 m on a staff held inverted against a roof station. The RL of the roof station, in m, to two decimal places, is ____.
Numerical answer — type the value.
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Answer: 52.85
HI = 50.000 + 1.200 = 51.200 m. The inverted reading counts as −1.650, so RL = 51.200 − (−1.650) = 52.850 m, above the line of sight as a roof must be. Subtracting 1.650 as an ordinary fore sight gives 49.55, which would put the roof below the instrument.The combined correction for curvature and refraction, in m, for a sight of 2 km (magnitude, to four decimal places) is ____.
Numerical answer — type the value.
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Answer: 0.2692
Combined correction = 0.0673 d² = 0.0673 × 2² = 0.2692 m, subtracted from the staff reading. Curvature alone is 0.0785 × 4 = 0.314 m; refraction reduces it by a seventh. Using d in metres rather than kilometres makes the formula meaningless.In the rise-and-fall method of reducing levels, the arithmetic check is:
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Answer: A — ΣBS − ΣFS = ΣRise − ΣFall = last RL − first RL
Each rise or fall is the difference of two consecutive readings, so their net sum equals the net of back and fore sights and equals the total change of level, last minus first. The reversed sign in the last option would be true only if RLs were read downwards; intermediate sights cancel out of the sum and are not equated to it.A closed traverse of perimeter 2500 m has ΣLatitude = +0.30 m and ΣDeparture = −0.40 m. If the relative precision is written 1 in N, N is ____.
Numerical answer — type the value.
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Answer: 5000
Closing error e = √(0.3² + 0.4²) = 0.5 m, and precision = 0.5/2500 = 1/5000, so N = 5000. Adding the two errors arithmetically (0.7 m) gives about 1 in 3571, and using one component alone gives 1 in 8333 or 1 in 6250.In the traverse of the previous kind (perimeter 2500 m, total latitude error 0.30 m), the Bowditch correction to the latitude of a 500 m line, in m (magnitude), is ____.
Numerical answer — type the value.
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Answer: 0.06
Bowditch: correction = total error × line length/perimeter = 0.30 × 500/2500 = 0.06 m, applied with the sign that reduces the misclosure. The transit rule would instead use the line’s latitude over the sum of absolute latitudes, which the question does not give.An angle is measured by two observers with standard errors of 2″ and 4″. In finding its most probable value, the weights given to the two measurements are in the ratio:
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Answer: C — 4 : 1
Weight is inversely proportional to the square of the standard error: w₁ : w₂ = 1/2² : 1/4² = 1/4 : 1/16 = 4 : 1, so the more precise observer counts four times as much. The ratio 2 : 1 uses the errors without squaring them, and 1 : 4 inverts the rule.Which of the following can be used to transfer the surface bearing underground through a single vertical shaft? (Select all that apply.)
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Answer: B — The Weisbach triangle with two plumb wires; C — Co-planing with two plumb wires; D — A gyro-theodolite
Both wire methods and the gyro-theodolite work through one shaft — the wires carry a line down, the gyroscope finds north directly. A traverse through an adit is correlation through a different kind of opening, so it cannot be used when the only connection is the vertical shaft.A tacheometer with multiplying constant 100 and additive constant 0 gives a staff intercept of 1.25 m on a line of sight at an angle of elevation of 10°. The horizontal distance, in m, to two decimal places, is ____.
Numerical answer — type the value.
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Answer: 121.23
H = K s cos² θ = 100 × 1.25 × cos² 10° = 125 × 0.96985 = 121.23 m. Using cos θ instead of cos² θ gives 123.10 m, and ignoring the inclination gives 125 m. The vertical component would be 125 × sin 20°/2 = 21.38 m.A haul road curve of radius 300 m joins two straights that deflect by 40°. The tangent length, in m, to two decimal places, is ____.
Numerical answer — type the value.
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Answer: 109.19
T = R tan(Δ/2) = 300 × tan 20° = 300 × 0.36397 = 109.19 m. Using tan 40° gives 251.7 m; the curve length would be πRΔ/180 = π × 300 × 40/180 = 209.44 m, which is a different quantity.A vertical photograph is taken from 3000 m above datum. The top of a 150 m high chimney standing on the datum appears 80 mm from the principal point. The relief displacement of the chimney top on the photograph, in mm, is ____.
Numerical answer — type the value.
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Answer: 4
d = r h/H = 80 × 150/3000 = 4 mm, outward from the principal point, with r the distance of the displaced top image. Dividing by H − h (2850 m) gives 4.21 mm, which uses the formula for a base above datum incorrectly; the displacement grows with r and h and falls with flying height.The minimum number of satellites a GPS receiver needs for a three-dimensional position fix is:
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Answer: C — 4
Four unknowns — x, y, z and the receiver clock offset — need four ranges. Three satellites suffice only if the height is known or the clock is perfect; more than four improve accuracy and geometry but are not the minimum.On a contour map, contour lines of different elevations can cross each other only where there is:
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Answer: B — an overhanging cliff
Only an overhang places a higher point directly above a lower one set further out, so the contours cross. At a vertical cliff they merge into one line without crossing, and at valleys and ridges they bend into V’s but stay separate.Which of the following statements are correct? (Select all that apply.)
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Answer: A — An EDM measures distance from the phase difference of a modulated electromagnetic wave; B — A total station combines an electronic theodolite with an EDM; D — The azimuth of a line can be found by an astronomical observation of Polaris or the sun
EDMs compare the phase of the returned modulation (or time a pulse), a total station is exactly the theodolite-plus-EDM package, and astronomical observation fixes the true meridian and hence azimuth. Thermal infrared records radiation emitted by the ground, which is why it maps coal fires at night; reflected sunlight is the visible and near-infrared part of the spectrum.