Chain, compass & plane table surveying

Principle of chain surveying

Chain surveying is the simplest method of surveying in which only linear measurements are made in the field. The area is divided into a network of triangles whose sides are measured; details are located by offsets from the survey lines.

• The principle is triangulation in its simplest form — a triangle is the only figure that can be plotted from its sides alone.
• Triangles should be well-conditioned — no angle less than about 30° or greater than about 120° (ideally equilateral) — so that plotting errors are small.

Suitability: small, fairly level, open areas with simple details. Unsuitable for large areas, crowded or wooded areas, or undulating ground.

Instruments

Chains

Chains are robust and easily repaired but change length (stretching, bending of links, wear) and are heavy.

Tapes

Accessories

• Arrows (chaining pins) — mark ends of each chain length.
• Ranging rods — 2 m or 3 m long, painted in alternate red and white (or black and white) bands of 20 cm, with a pointed shoe — mark stations and range lines.
• Offset rods — with a hook to pull the chain through obstructions and notches for setting out right angles.
• Pegs — mark survey stations.
• Plumb bob — transfer points and chaining on slopes.
• Cross staff (open, French, adjustable), optical square and prism square — to set out perpendicular offsets (optical and prism squares use reflection to give 90°).
• Line ranger — to locate intermediate points on a line between two stations.
• Clinometer / Abney level — to measure slopes.

Survey stations and lines

• Main stations — at the ends of main survey lines; subsidiary (tie) stations — on main lines, to run tie lines.
• Base line — the longest line running through the middle of the area, on which the framework is built; measured most accurately.
• Main survey lines — sides of the main triangles.
• Check (proof) lines — run from a station to a point on the opposite side (or between points on two sides) to check the accuracy of the framework.
• Tie lines — join tie stations to locate interior details and provide checks.

Selection of stations: mutually visible adjacent stations; well-conditioned triangles; minimum number of lines; lines through level ground and close to details (short offsets); avoid obstacles to chaining and ranging; lines not crossing busy roads frequently.

Ranging

Ranging is the process of establishing intermediate points on a straight line between two end points.

• Direct ranging — when the ends are intervisible: intermediate ranging rods are aligned by eye (or line ranger).
• Indirect (reciprocal) ranging — when the ends are not intervisible (e.g. due to a rise in ground): two persons with rods on intermediate points alternately align each other with the far end until both are in line.

Chaining on sloping ground

1. Direct method (stepping): the tape is held horizontal in short steps and the end is transferred to the ground by a plumb bob.
2. Indirect methods: measure the slope distance $L$ and:
   • Angle measured: horizontal distance $=L\cos \theta$; correction $=L(1-\cos \theta )$.
   • Difference in level $h$ measured: horizontal distance $=\sqrt{L^2-h^2}$; slope correction $\approx \frac{h^2}{2L}$.
   • Hypotenusal allowance: at each chain length on a slope $\theta$, the arrow is placed ahead by $100(\sec \theta -1)$ links per chain, so that the horizontal distance recorded is directly correct.

Errors in chaining

Sources: incorrect length of chain or tape (standardisation), bad ranging (always makes measured length too long), bad straightening (too long), non-horizontality on slopes (too long), sag (too long), temperature variations (either sign), variation in pull (either sign), displacement of arrows, and personal mistakes (miscounting chains, wrong reading of tallies, wrong booking).

Errors such as bad ranging, bad straightening, sag and slope are positive (cumulative) errors — the measured length is too long.

Tape corrections

Erroneous length of chain

If the true (designated) length of a chain is $L$ and its actual length is $L^{\prime }$:

Offsets

Offsets are lateral measurements from survey lines to locate details.

• Perpendicular offsets — at right angles to the chain line (set by eye, cross staff, optical/prism square).
• Oblique offsets — at any angle; two oblique offsets from two points on the chain line fix a detail by intersection.
• Long offsets need accurate right angles; short offsets can be set by eye. The limiting length of offsets depends on the scale and the accuracy of setting the right angle.

Obstacles in chaining

Field book

The field book records measurements. In single-line booking a single central line represents the chain line; in double-line booking the chainages are written between two parallel lines. Records include station names, chainages, offsets with sketches of details, and remarks. Entries start from the bottom of the page and go upwards.

Worked examples

Frequently tested points

• Chain surveying: only linear measurements; well-conditioned triangles (30°–120°); small open level areas.
• Metric chains 20 m (100 links) and 30 m (150 links); Gunter's chain 66 ft; engineer's chain 100 ft; revenue chain 33 ft.
• Invar tape — very low thermal expansion — base line measurement.
• Ranging rods 2–3 m with 20 cm bands; optical/prism squares for right angles.
• Base line, check lines (for checking), tie lines (details and checks).
• Reciprocal ranging when ends are not intervisible.
• Hypotenusal allowance $100(\sec \theta -1)$ links per chain.
• Bad ranging, bad straightening, sag, slope → measured length too long.
• Corrections: standardisation ±, temperature ±, pull ±, sag −, slope −, MSL −; normal tension balances pull and sag.
• True length $(L^{\prime }/L)\times$ measured; area $(L^{\prime }/L{)}^2$; volume $(L^{\prime }/L{)}^3$.

Compass surveying

Compass surveying is a method of traversing in which directions (bearings) of survey lines are measured with a magnetic compass and lengths with a chain or tape.

Suitability: large areas where details are few, rough or preliminary surveys, wooded or crowded areas where triangulation by chain is impossible, and where speed is more important than high accuracy. Unsuitable near magnetic disturbances (iron structures, power lines, ore bodies).

Traverse — a series of connected survey lines whose lengths and directions are measured:

• Closed traverse — starts and ends at the same point, or at points of known position (provides a check).
• Open traverse — starts and ends at different unknown points (roads, rivers, pipelines); no inherent check.

Meridians and bearings

Bearing of a line is its horizontal angle with respect to a reference direction called the meridian.

Bearings accordingly are true, magnetic, grid or arbitrary bearings.

Systems of bearings

Fore bearing and back bearing

• Fore bearing (FB) of a line — bearing measured in the direction of progress of the survey (e.g. AB measured at A towards B).
• Back bearing (BB) — bearing in the opposite direction (BA measured at B towards A).

(+ if FB < 180°, − if FB > 180°.) In the QB system, the back bearing has the same numerical value with opposite letters (N ↔ S, E ↔ W): e.g. FB = N 30° E → BB = S 30° W.

Included angles from bearings

The angle between two lines meeting at a point is the difference of their bearings measured from that point:

(add 360° if negative). For a closed traverse of $n$ sides, the sum of interior angles $=(2n-4)\times 90°$ and of exterior angles $=(2n+4)\times 90°$.

Computing bearings from included angles: bearing of the next line = bearing of the previous line + included angle ± 180° (for clockwise angles), adjusting by 360° as necessary.

Compasses

Prismatic compass

• A graduated aluminium ring attached to a broad magnetic needle rotates with it; the ring is graduated in WCB with 0° at the south end (so that the reading through the prism gives the bearing of the line sighted), figures written inverted.
• Sighting is done through the prism slit and the object vane (with a horse hair); the reading is taken simultaneously through the prism.
• Can be used on a tripod or held in hand.

Surveyor's compass

• The graduated ring is attached to the box (does not rotate); the needle moves freely over it.
• Graduated in quadrantal bearings with E and W interchanged (so that the reading at the north end of the needle gives the correct quadrant).
• Sighting and reading are not simultaneous; reading taken directly by looking at the needle from the top; must be used on a tripod.

Magnetic declination and dip

Declination

Magnetic declination is the horizontal angle between the true meridian and the magnetic meridian at a place.

• East declination — magnetic north is east of true north.
• West declination — magnetic north is west of true north.

Isogonic lines — lines joining places of equal declination; agonic line — zero declination.

Variations in declination:

• Secular variation — slow, long-period change (over centuries) — most important in surveying (old surveys must be corrected).
• Annual variation — small yearly periodic change.
• Diurnal (daily) variation — daily swing, larger in summer and near the poles.
• Irregular variation — due to magnetic storms, earthquakes, solar flares.

Dip

The magnetic needle, if freely suspended, dips from the horizontal towards the nearer magnetic pole. Angle of dip is 0° at the magnetic equator and 90° at the magnetic poles. To keep the needle horizontal, a small sliding rider (weight) is placed on the higher end. Isoclinic lines join points of equal dip; the aclinic line is the magnetic equator.

Principle

Plane table surveying is a graphical method in which field observations and plotting are done simultaneously. The map is prepared in the field, so no field book is needed.

Principle — parallelism: the rays drawn on the drawing sheet from plotted stations to objects are parallel to the corresponding lines on the ground, and the plotted positions are geometrically similar to the ground positions (the table must be correctly oriented at every station).

Suitability: small-scale mapping, filling in topographic details (e.g. between triangulation stations), areas where high accuracy is not required, magnetic areas (orientation by back sighting), and places where compass surveys are unreliable.

Plane table and accessories

A telescopic alidade increases the range and accuracy of sighting and permits measurement of vertical angles and stadia distances.

Temporary adjustments

1. Setting up and levelling — tripod legs spread firmly; board levelled with a spirit level in two directions.
2. Centring — the point on the sheet representing the ground station is brought vertically over the station using the plumbing fork. (Exact centring is less critical on small-scale maps, since a small centring error causes a negligible plotting error.)
3. Orientation — the table is rotated so that lines on the sheet are parallel to the corresponding ground lines. It is essential for accurate work.

Methods of orientation

• By trough compass — the direction of magnetic north is marked on the sheet at the first station; at subsequent stations the table is rotated until the compass needle again lies along that line. Quick, but affected by local attraction.
• By back sighting — the alidade is placed along a line already drawn from a previous station to the present station, and the table is rotated until the line of sight bisects the previous station; then clamped. Most accurate method.
• By resection — when the station occupied is not yet plotted (see below).

Methods of plane tabling

Resection methods

Resection after orientation by compass or back sighting

• By compass: orient the table with the trough compass; draw back rays from two known plotted points through their alidade sightings; the intersection gives the station.
• By back sighting: if a line from a known station towards the new station has been drawn earlier, orient by back sighting along it and then draw a resector from another known point; the intersection locates the station.

Two-point problem

The station occupied is located by sighting two well-defined points already plotted, when the table cannot be oriented directly. An auxiliary station is chosen, and the procedure involves setting up at the auxiliary station, drawing rays, and correcting orientation by comparison of a constructed line with the known line. It requires more work and is less accurate; it is used only when the three-point problem cannot be applied.

Three-point problem

The station occupied is located by sighting three well-defined points already plotted on the sheet. Solutions:

1. Mechanical (tracing paper) method — rays to the three points are drawn on a tracing paper fixed on the board; the tracing is moved until the three rays pass through the plotted points; the station is pricked through.
2. Graphical methods — e.g. Bessel's method (construction of an auxiliary point by drawing rays, which then gives the correct orientation).
3. Trial and error method (Lehmann's method) — the table is oriented approximately, back rays are drawn from the three points, forming a small triangle of error; the true position is estimated by Lehmann's rules and the orientation improved until the triangle reduces to a point.

Lehmann's rules:

1. The point sought lies on the same side of each of the three rays (either to the right of all or to the left of all, when looking towards the respective point).
2. Its distance from each ray is proportional to the distance of the corresponding known point from the station.
3. If the station is inside the great triangle (formed by the three known points), the point lies inside the triangle of error; if outside, it lies outside the triangle of error.

Indeterminate case: if the station lies on the circle passing through the three known points (the great circle), the problem has no unique solution — another point must be chosen.