Surveying and Levelling: Chains, Traverses, Plane Table, Levels, Contours, Total Station, GPS, Areas and Volumes

A bearing is the horizontal angle of a line from a reference meridian. The whole-circle bearing (WCB) runs from 0° to 360° clockwise from north; the quadrantal (reduced) bearing (QB) is measured from north or south towards east or west, never exceeding 90°. WCB 210° is S30°W, WCB 120° is S60°E, WCB 300° is N60°W. The back bearing differs from the fore bearing by exactly 180°: the back bearing of a line of WCB 75° is 255°. When the fore and back bearings of a compass line do not differ by 180°, a station is under local attraction from iron or electric lines nearby; the included angles computed from bearings at a station are unaffected, which is how the errors are corrected.

A traverse is a series of connected lines whose lengths and directions are measured — with a compass for rough work, with a theodolite (horizontal and vertical angles read on graduated circles, with temporary adjustments of centring, levelling and focusing, and face-left and face-right readings averaged to cancel instrumental errors) for precise work. It is closed when it returns to its start or ends on a known point. Each line’s latitude is L cos θ and departure L sin θ (θ the bearing): a 100 m line at WCB 60° has a latitude of 50 m and a departure of 86.6 m. In a closed traverse ΣL and ΣD should be zero; their residuals give the closing error e = √((ΣL)² + (ΣD)²), and the relative precision is e divided by the perimeter. The error is distributed by the Bowditch (compass) rule, correcting each line in proportion to its length, or by the transit rule. The interior angles of a closed polygon of n sides should sum to (2n − 4) × 90°.

In plane tabling the map is drawn in the field on a drawing board set up on a tripod, with an alidade to sight and draw rays, a spirit level, a trough compass for orientation and a plumbing fork to centre the table over a ground point. Because plotting and observation happen together, nothing is forgotten and errors show at once; the price is lower accuracy, dependence on weather and bulky equipment. Orientation — setting the board so that every line on the paper is parallel to its line on the ground — is done by back-sighting along a known line or by the trough compass.

Levelling finds heights relative to a datum. A dumpy or automatic level gives a horizontal line of sight, and a graduated staff is read on it. The first reading after setting up is a back sight (BS) on a point of known level (a benchmark); the last before moving is a fore sight (FS); readings between are intermediate sights (IS). A point where both an FS and a BS are taken is a change point. Reduced levels are worked by the height-of-instrument method — HI = RL + BS, then RL = HI − FS (or − IS) — or by the rise-and-fall method, which offers a complete arithmetic check. Either way ΣBS − ΣFS = last RL − first RL. From a benchmark at 100.000 m, a BS of 1.525 m gives HI = 101.525 m, and an FS of 2.340 m puts the next point at 99.185 m.

The types of levelling are simple (one set-up), differential (a series of set-ups between distant points), fly levelling (a quick check run back to a benchmark), profile (longitudinal section) and cross-section levelling for channels and bunds, reciprocal levelling across a river or valley, and trigonometric levelling from vertical angles. Over long sights the level line departs from the horizontal: the curvature correction is −0.0785d² m and the refraction correction +0.0112d² m (one-seventh of curvature), giving a combined correction of −0.0673d² m with d in km — 0.269 m at 2 km. Reciprocal levelling, taking readings from both banks and averaging, cancels curvature, refraction (if the air is the same both times) and any collimation error of the instrument.

A contour is a line joining points of equal elevation; the contour interval is the vertical distance between successive contours and is constant on a map. Contours close on themselves, never cross (except at an overhang), are closely spaced on steep slopes and widely spaced on gentle ones, cross ridges and valleys at right angles, and form V-shapes pointing uphill in valleys and downhill on ridges. They are surveyed directly (locating points on each contour in the field — slow but accurate) or indirectly (levelling a grid or radial lines and interpolating). Contour maps give the alignment of contour bunds and graded channels and the capacity of ponds.

A total station combines an electronic theodolite, an EDM and a data recorder: it measures horizontal and vertical angles and slope distance in one sighting to a prism and computes coordinates and elevations directly, which makes detail and contour surveys fast and digital. GPS (GNSS) fixes position by trilateration from satellite ranges; since the receiver clock is an unknown alongside the three coordinates, at least four satellites are needed for a three-dimensional fix. A single receiver is accurate to metres; differential GPS and RTK, correcting against a base station of known position, reach centimetres, enough for land levelling and layout.

For offsets of 3, 5, 6, 4 and 2 m at 10 m intervals, Simpson’s rule gives (10/3)[(3 + 2) + 4(5 + 4) + 2(6)] = (10/3)(53) = 176.7 m², against 175 m² by the trapezoidal rule; Simpson is exact for a boundary made of parabolic arcs and needs an odd number of ordinates. Cross-sections of 20, 30 and 44 m² at 10 m spacing give a prismoidal volume of (10/3)(20 + 4 × 30 + 44) = 613.3 m³.