Traverse and Triangulation

The theodolite

A theodolite measures horizontal and vertical angles accurately. It is also used for prolonging lines, setting out angles and curves, levelling (trigonometric) and, with stadia hairs, measuring distances (tacheometry).

Types

• Transit theodolite — the telescope can be revolved through 180° in a vertical plane about its horizontal axis (standard today).
• Non-transit theodolite — telescope cannot be transited (obsolete).
• By reading system: vernier theodolites (least count commonly 20″), micrometer/optical theodolites (1″ or better), electronic digital theodolites (with digital angle display); total stations combine electronic theodolites with EDM.

Main parts

Trivet and tribrach with levelling screws, lower plate (carrying the horizontal circle) with lower clamp and tangent screw, upper plate (carrying verniers/reading system) with upper clamp and tangent screw, plate levels, standards (A-frame) supporting the horizontal (trunnion) axis, telescope, vertical circle with vertical circle vernier/index and altitude bubble, plumb bob or optical plummet, tripod.

Technical terms

Temporary adjustments

1. Setting up over the station with tripod.
2. Centring — plumb bob or optical plummet over the station mark; shifting head for fine centring.
3. Levelling up — using plate levels and foot screws: bubble central parallel to two foot screws, then perpendicular using the third screw; repeat until central in all positions.
4. Focusing — eyepiece for cross hairs, objective for the object — elimination of parallax.

Measurement of angles

Horizontal angle (simple method)

Set the vernier to 0°, sight the first station using the lower clamp; release the upper clamp, sight the second station, read the angle. Repeat on the other face and take the mean.

Repetition method

Used to measure a single horizontal angle to a finer degree of accuracy than the least count.

1. Measure the angle once and do not reset the vernier; with the lower clamp, sight the first station again.
2. Using the upper clamp, sight the second station — the reading accumulates the angle twice.
3. Repeat for a set number of repetitions (e.g. 3 on face left and 3 on face right, with left and right swings).
4. Angle = (final reading) ÷ (number of repetitions) — adding full circles if the reading passes 360°.

Errors eliminated/reduced: errors of eccentricity of verniers, errors due to inadequate least count (reading error distributed), errors of graduations (different parts of the circle used), errors of collimation and trunnion axis (face left + face right), errors due to slip partly — accuracy improves.

Reiteration (direction) method

Used when several angles are to be measured at a station: directions of all stations are read successively from a reference station, closing back on it (the horizon is closed; sum of angles = 360°). Repeated with the circle set at different initial readings on both faces.

Vertical angles

Angle of elevation (above the horizontal) or depression (below). Measured using the vertical circle with the altitude bubble central; face left and face right readings are averaged to eliminate index error.

Other operations

• Magnetic bearing of a line — using a trough or tubular compass attached to the theodolite.
• Deflection angle — angle a line makes with the prolongation of the preceding line (right or left) — used in route surveys.
• Prolonging a straight line — by double sighting (face left and face right, taking the mean point) to eliminate collimation error.
• Setting out angles, ranging a line, locating the intersection of two lines.

Fundamental lines and permanent adjustments

Errors eliminated by face-left and face-right observations

Theodolite traversing

Methods

Checks for a closed traverse: sum of interior angles $=(2n-4)\times 90°$; sum of exterior angles $=(2n+4)\times 90°$; sum of deflection angles (right − left) $=360°$. Angular misclosure is distributed equally among the angles (if all measured with equal care).

Traverse computations

For a closed traverse, $\sum L=0$ and $\sum D=0$ ideally.

Balancing the traverse

Other methods: graphical (Bowditch) adjustment, third rule, Crandall's method and least squares adjustment.

Coordinates

• Consecutive coordinates — latitude and departure of each line relative to its starting point.
• Independent (total) coordinates — coordinates of each station relative to a common origin; obtained by cumulative addition of corrected consecutive coordinates.
• Gale's traverse table — a standard tabular format for computing bearings, latitudes, departures, corrections and independent coordinates.

Omitted measurements

When some measurements of a closed traverse are missing (not measured or lost), they can be computed since $\sum L=0$ and $\sum D=0$ provide two equations:

1. Length and bearing of one line omitted — the missing line closes the traverse: $L=-\sum L_{known}$, $D=-\sum D_{known}$.
2. Length of one line and bearing of another omitted, or lengths of two lines omitted, or bearings of two lines omitted — solved by trigonometric relations (sometimes after joining known points with a closing line).

Worked examples

Frequently tested points

• Transit theodolite — telescope revolves 180° vertically; vernier least count commonly 20″.
• Face left = vertical circle on the left of the observer.
• Temporary adjustments: setting up, centring, levelling up, focusing (parallax).
• Repetition method — single angle, higher accuracy; reiteration — several angles at a station.
• Face left and face right eliminate collimation, trunnion axis and index errors; not the vertical axis error.
• Spire test — horizontal axis perpendicular to vertical axis.
• Deflection angles for open (route) traverses; included angles for closed traverses.
• Latitude $=l\cos \theta$, departure $=l\sin \theta$; closing error $\sqrt{(\sum L{)}^2+(\sum D{)}^2}$.
• Bowditch's rule ∝ length of line (equal angular and linear accuracy); transit rule ∝ latitude/departure (angles more precise).
• Consecutive vs independent coordinates; Gale's traverse table.
• Omitted measurements from $\sum L=0$ and $\sum D=0$.

Principle

Triangulation is a method of establishing horizontal control in which the area is covered by a network of triangles. Only one side (the base line) is measured directly with high precision, and all angles of the triangles are measured; the other sides are computed by the sine rule, and the coordinates of stations follow.

Purposes:

• Establishing accurate control points for topographic and engineering surveys over large areas.
• Geodetic surveys — determining the size and shape of the earth.
• Locating inaccessible points; setting out long tunnels, bridges, dams.

Triangulation vs traversing: triangulation suits hilly and undulating areas where long lines can be sighted and chaining is difficult; traversing suits flat, built-up or wooded areas. Trilateration measures all sides (with EDM) instead of angles; modern control uses trilateration, combined networks and GNSS.

Classification of triangulation

• First-order — geodetic, national control, very high accuracy.
• Second-order — fills in the primary network.
• Third-order — provides control for topographic and engineering surveys.

Triangulation figures

Well-conditioned triangle

The accuracy of a computed side depends on the angles opposite to known and unknown sides (errors in sines of small angles are large). A triangle is well-conditioned if no angle is smaller than about 30° or greater than about 120°. The best shape is an equilateral triangle; mathematically, the error in the computed side of an isosceles triangle is minimum when the base angles are about 56°14′.

Strength of figure

Field work in triangulation

1. Reconnaissance — examination of terrain, selection of stations and base line, checking intervisibility, heights of signals, access, materials.
2. Station marking and erection of signals and towers.
3. Measurement of the base line (and base net extension).
4. Measurement of horizontal angles (and vertical angles for heights).
5. Astronomical observations at selected stations — azimuth, latitude and longitude.
6. Computations and adjustment.

Selection of stations

• Stations must be intervisible with adjacent stations; form well-conditioned triangles.
• Easily accessible, on firm ground, requiring low signal heights, providing a line of sight well clear of the ground (to reduce refraction), with minimum clearing of vegetation.

Intervisibility and height of stations

Because of the earth's curvature (and refraction), stations far apart may not be visible even over flat ground.

Signals and towers

• Signals mark stations for sighting: luminous signals — heliotropes and heliographs (reflect sunlight), lamps and lights for night observations; opaque signals — pole signals, target signals, pole and brush, stone cairns, beacons.
• Towers — elevate the instrument and signal above obstacles: scaffolds, masonry pillars, and steel Bilby towers (two independent towers — the inner one for the instrument and the outer for the observer — so that the observer's movements do not disturb the instrument).
• Phase correction — for cylindrical opaque signals illuminated by the sun on one side, the observer tends to bisect the bright portion, introducing an error; a correction is applied to the observed direction depending on the observation method (bright portion or bright line).

Base line measurement

• Site requirements: fairly level or uniformly sloping ground, free from obstructions, well-conditioned connection to the network, ends intervisible, suitable length.
• Equipment: historically invar tapes/wires under standard tension with corrections for temperature, pull, sag, slope and reduction to MSL; now precise EDM.
• Base net (base extension) — the measured base is expanded through a series of well-conditioned triangles to the length of main triangulation sides.
• Check bases are measured at intervals to control error accumulation.

Satellite stations and reduction to centre

When a triangulation station cannot be occupied (e.g. a church spire, chimney or lighthouse) or visibility is blocked, observations are made from a nearby satellite (eccentric) station S, a short distance $d$ from the true station C. The angles observed at S are reduced to the centre C.

Spherical excess

The sum of the angles of a spherical triangle exceeds 180° by the spherical excess:

($A$ = area of the triangle, $R$ = radius of the earth.) It is about 1″ for every 196 km² of area — significant only in geodetic triangulation. Each angle is reduced by one-third of the spherical excess before plane computations (Legendre's theorem).