Surveying

Trigonometric Levelling

Trigonometric levelling (also called indirect levelling) is the method of determining the difference in elevation between two points by measuring the horizontal distance (or slope distance) between them and the vertical angle (angle of elevation or depression), using a theodolite or total station. The vertical difference is then computed by trigonometry, without measuring vertical heights directly as in direct levelling.

If $D$ = horizontal distance, $s$ = slope distance, and $\theta$ = vertical angle, then:

$V = D \tan\theta \qquad \text{or} \qquad V = s \sin\theta$

The difference in elevation between the instrument station and the target is:

$\Delta h = D \tan\theta + \text{hi} - \text{ht}$

where hi = height of instrument and ht = height of the target/signal above the ground.

Uses:

1. Rough and hilly terrain — where direct levelling is slow, difficult or impossible (steep slopes, rivers, gorges, valleys).
2. Fixing elevations of inaccessible points — tops of hills, towers, spires, across rivers or ravines, where the level staff cannot be held.
3. Topographic mapping and contouring — supplying spot heights over large areas quickly.
4. Control surveys — establishing vertical control for triangulation stations and traverse stations.
5. Engineering projects — route surveys for roads, canals, transmission lines, and reservoir/ dam site surveys.
6. Total station work — modern trigonometric levelling is the basis of all total-station height computations (trig height).

Its main advantage is speed and economy over long distances and rough ground; its limitation is lower accuracy than direct levelling, since errors in distance and vertical angle accumulate.

---

Methods of Contouring

A contour is an imaginary line on the ground joining points of equal elevation. The following are the methods of contouring:

1. Direct Method The contour line is traced on the ground itself. The level is set up, and by taking staff readings at close intervals (usually 5–10 m), points of the same reduced level are located and marked with pegs; these pegs are then joined to give the contour. It is accurate but slow and costly, and unsuitable for hilly or wooded country. Used for small areas of high precision (e.g. reservoir sites, building sites).

2. Indirect Method Spot levels are taken at selected points over the area and the contours are interpolated between them. It is faster and cheaper, and is the usual method for large areas. It has three sub-methods:

• (a) By Cross-Sections — Cross-section lines are run at right angles to a base line (e.g. a road or canal alignment) at regular intervals. Levels are taken along each cross-section, and contours are interpolated between the cross-sections. Suitable for route surveys (roads, railways, canals).

• (b) By Squares (Grid Method) — The area is divided into a grid of squares (e.g. 10 m × 10 m or 20 m × 20 m), the corners are pegged and levelled, and contours are interpolated within each square. Suitable for flat areas and for earthwork/volume computation (borrow pits, tank sites).

• (c) By Tacheometry (Trigonometrical Method) — A tacheometer (or total station) is set up at a station and radial observations are taken to a number of points; horizontal distances and elevations are computed from the readings. Contours are then plotted by interpolation. It is rapid and suitable for hilly, broken and inaccessible country.

3. Interpolation of Contours After spot levels are obtained, contours are drawn between them by estimation, arithmetical calculation, or by drawing horizontal lines (graphical method), assuming the ground slope between adjacent points to be uniform.

4. Contouring by Tracing (Parallel Lines) Levels are taken along parallel lines at fixed intervals and contours are traced between them — a variant of the cross-section method used for large open areas.

Choice of method depends on the purpose, size and nature of the ground, required accuracy, and available time and instruments.