Levelling and Contouring

Terms

Instruments

Levels

Levelling staves

• Self-reading staves — read directly by the observer at the instrument: solid, folding (e.g. 4 m in two parts) and telescopic (Sopwith) staves, graduated commonly to 5 mm divisions.
• Target staff — a sliding target is moved by the staffman to the line of sight and read by him (long sights, precise work).
• Bar-coded (invar) staves for digital levels.

Adjustments of a level

Temporary adjustments (at every setup)

1. Setting up — tripod firmly set, instrument fixed, approximate levelling by tripod legs.
2. Levelling up — using foot screws (three-screw head: turn two screws in opposite directions for one axis, then the third screw for the perpendicular axis) until the bubble remains central in all positions.
3. Elimination of parallax — focus the eyepiece on the cross hairs first (against a light background), then focus the objective on the staff until there is no apparent movement of the image relative to the cross hairs.

Permanent adjustments (dumpy level)

1. Axis of the bubble tube perpendicular to the vertical axis (so the bubble stays central through a full rotation).
2. Line of collimation parallel to the axis of the bubble tube — checked by the two-peg test.
3. Horizontal cross hair perpendicular to the vertical axis.

Two-peg test

1. Set two pegs A and B about 50–100 m apart. Set the level midway; the difference of staff readings gives the true difference in level (collimation error cancels as sight lengths are equal).
2. Set the level near one peg (or beyond one peg) and read both staves; the apparent difference differs from the true difference if the line of collimation is inclined.
3. Compute the error and the correct staff reading on the far peg; adjust the cross hairs (dumpy level) or the bubble (tilting level) accordingly.

Types of levelling

Reduction of levels

Curvature and refraction

Over long sights, the horizontal line of sight departs from the level surface.

• Curvature makes staff readings too large (the level line curves downward away from the line of sight) — correction is negative.
• Refraction bends the line of sight downward towards the earth, making readings smaller — correction is positive; about one-seventh of the curvature correction.

Reciprocal levelling

Used when the instrument cannot be placed midway between two points (e.g. across a wide river).

1. Set the level near A; read the staff at A ($a_1$) and at B ($b_1$).
2. Set the level near B; read the staff at A ($a_2$) and at B ($b_2$).

(positive → B is lower than A.) The method eliminates errors due to collimation, curvature and refraction (if refraction is the same at both times). The combined error $e=\frac{(b_1-a_1)-(b_2-a_2)}{2}$.

Sensitivity of the bubble tube

The sensitivity is the angle through which the line of sight tilts when the bubble moves by one division — a more sensitive bubble moves more for a small tilt.

Sensitivity increases with larger radius of curvature, larger diameter of the tube, longer bubble, lower viscosity and smooth interior surface.

Errors in levelling

Permissible closing errors (commonly quoted, $K$ = distance in km): rough levelling about $\pm 100\sqrt{K}$ mm; ordinary levelling about $\pm 24\sqrt{K}$ mm; accurate levelling about $\pm 12\sqrt{K}$ mm; precise levelling about $\pm 4\sqrt{K}$ mm. The closing error is distributed to intermediate points in proportion to distance (or number of setups).

Worked examples

Frequently tested points

• RL relative to datum (MSL); GTS benchmarks by Survey of India.
• BS — first reading after setup (known RL); FS — last reading before shifting; CP has both FS and BS.
• Parallax removed by focusing the eyepiece first, then the objective.
• Two-peg test checks line of collimation parallel to bubble axis.
• Automatic level — compensator; tilting level — tilting screw; digital level — bar-coded staff.
• HI method: $RL=HI-reading$; check $\sum BS-\sum FS$ only; rise and fall checks all readings.
• Curvature $-0.07849d^2$; refraction $+0.01121d^2$ (1/7); combined $-0.06728d^2$; visible horizon $3.855\sqrt{h}$ km.
• Reciprocal levelling eliminates collimation, curvature and refraction errors.
• Equal BS and FS distances eliminate collimation and curvature errors.
• Staff not vertical → reading too large.
• Sensitivity $\alpha =s/(nD)$ rad; increases with radius of the tube.

Definitions

Choice of contour interval

The contour interval is kept constant on a map and depends on:

1. Nature of the ground — flat ground needs a small interval; steep hilly ground a larger interval.
2. Scale of the map — large-scale maps use small intervals; small-scale maps larger intervals.
3. Purpose and extent of the survey — detailed design (e.g. building sites, irrigation) needs small intervals; route location or reconnaissance can use larger intervals.
4. Time and expense — smaller intervals need more field work.

Typical intervals range from about 0.5 m for detailed engineering plans on flat land to 10–20 m or more on small-scale topographic maps of hilly country.

Characteristics of contours

1. All points on a contour have the same elevation.
2. Two contours of different elevations cannot cross or meet each other — except for an overhanging cliff or cave, where they appear to cross.
3. Contours merge into a single line where the ground is a vertical cliff.
4. Closely spaced contours indicate steep slopes; widely spaced contours indicate gentle slopes; equally spaced contours indicate a uniform slope; straight, parallel, equally spaced contours indicate a plane surface.
5. A series of closed contours with higher values inside represents a hill (summit); with lower values inside represents a depression (pond, lake).
6. Contours crossing a ridge form U- or V-shapes with the convex side (apex) pointing towards lower ground; contours crossing a valley form V-shapes pointing towards higher ground (uphill).
7. Contours are perpendicular to the line of steepest slope (and to ridge and valley lines).
8. Every contour must close on itself, either within or beyond the limits of the map; a contour cannot end abruptly.
9. A single contour cannot split into two.
10. The same contour appears on both sides of a ridge or a valley.

Methods of locating contours

Direct method

Points on the contours themselves are located on the ground and then surveyed:

1. Establish a benchmark and set up the level; the HI is known.
2. For a contour of RL $X$, the required staff reading $=HI-X$.
3. The staffman moves until the staff reading equals this value; the point is marked and its horizontal position fixed by plane table, tape/compass or total station.
4. The procedure is repeated for other contours.

Most accurate but slow and tedious; suitable for small areas and where high accuracy is needed (e.g. reservoir sites, building layouts).

Indirect methods

Spot levels of selected guide points are observed, and contours are drawn by interpolation between them.

Indirect methods are quicker, cheaper and adequate for most purposes.

Interpolation of contours

Assumes the slope between two guide points is uniform.

1. Estimation — positions judged by eye (rough work, small scales).
2. Arithmetic calculation — distance of a contour from a point:

3. Graphical method — a tracing paper with equally spaced parallel lines representing contour values is placed over the two points and rotated until the points lie on lines of their RLs; intermediate crossings are pricked through.

Uses of contour maps

• Nature of the ground — assessing topography for planning.
• Drawing longitudinal sections and cross-sections along any line.
• Intervisibility between stations — by drawing the section between them.
• Locating route alignments (roads, railways, canals, pipelines) at a given gradient — by stepping off the horizontal equivalent (with dividers) from contour to contour.
• Catchment area — delineating the watershed (ridge line) of a drainage basin.
• Reservoir capacity — from areas enclosed by successive contours.
• Earthwork quantities — for cuts and fills, and choosing formation levels.
• Site selection — dams, bridges, buildings, drainage.

Volume from contour areas

Modern methods

Digital elevation models (DEM) and digital terrain models (DTM) are created from total station and GNSS points, LiDAR (airborne laser scanning), photogrammetry and satellite data. Software generates triangulated irregular networks (TIN) or grids and interpolates contours automatically, and computes slopes, sections, catchments and volumes.

Worked examples

Frequently tested points

• Contour: line of equal elevation; CI constant on a map; HE varies with slope.
• CI depends on nature of ground, scale, purpose, time and cost.
• Contours never cross except at overhanging cliffs; merge at vertical cliffs.
• Close spacing → steep slope; wide → gentle; equal → uniform slope.
• Closed contours: higher inside → hill; lower inside → depression.
• Valley contours point uphill (V towards higher ground); ridge contours point downhill.
• Contours are perpendicular to the steepest slope and close on themselves.
• Direct method — accurate but slow; indirect — grid (flat areas), cross-section (routes), radial lines/tacheometry (hilly areas).
• Interpolation assumes uniform slope; methods: estimation, arithmetic, graphical.
• Uses: sections, intervisibility, route at given gradient, catchment area, reservoir capacity, earthwork.
• Prismoidal formula needs an odd number of contour areas.