Levelling Notes

Last reviewed 16 Sept 2026 · 11 min read

Terms

Term	Meaning
Level surface	A surface parallel to the mean spheroidal surface of the earth (e.g. still water surface) — every point is equidistant from the earth's centre
Horizontal plane	Tangent to the level surface at a point
Datum	Reference surface to which elevations are referred — commonly mean sea level (MSL)
Reduced level (RL)	Elevation of a point above (or below) the datum
Benchmark (BM)	A fixed point of known RL — GTS benchmarks (Great Trigonometrical Survey, established by the Survey of India), permanent BMs (by government departments), arbitrary BMs (assumed RL for small works), temporary BMs (at the end of a day's work)
Line of collimation (line of sight)	Line through the intersection of cross hairs and the optical centre of the objective
Height of instrument (HI)	RL of the line of collimation when the instrument is levelled
Back sight (BS)	First staff reading after setting up the instrument — taken on a point of known RL (BM or change point)
Fore sight (FS)	Last staff reading before shifting the instrument
Intermediate sight (IS)	Any reading between BS and FS at the same setup
Change point (turning point, CP)	A point on which both an FS (from one setup) and a BS (from the next setup) are taken

Instruments

Levels

Level	Features
Dumpy level	Telescope rigidly fixed to the vertical spindle; simple, stable, retains adjustments; widely used
Wye (Y) level	Telescope rests in Y-shaped supports and can be removed/rotated — easy to test and adjust, but wears
Tilting level	Telescope can be tilted slightly about a horizontal axis with a fine screw to centre the bubble for each sight — quicker accurate levelling
Automatic (auto) level	A compensator (suspended prisms) automatically makes the line of sight horizontal after rough levelling with a circular bubble — fast and accurate; most common now
Digital level	Reads a bar-coded staff electronically and stores data
Laser level	Rotating laser beam defines a horizontal (or inclined) plane — construction work

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)

Setting up — tripod firmly set, instrument fixed, approximate levelling by tripod legs.
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.
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)

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

Two-peg test

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).
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.
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

Type	Purpose / method
Simple levelling	Difference in level of two nearby points from one setup
Differential (compound) levelling	Difference in level of points far apart or not visible from one setup — series of setups with change points
Fly levelling	Approximate levelling (long sights, few readings) to carry levels roughly or check benchmark values
Check levelling	Levelling back to the starting benchmark (or to another BM) to check the work
Profile (longitudinal) levelling	RLs of points at regular intervals along a line (road, canal, pipeline) to draw a longitudinal section
Cross-sectioning	RLs along lines perpendicular to the centre line — for earthwork quantities
Reciprocal levelling	Accurate difference in level of two points far apart with an obstacle between (river, valley) — eliminates collimation, curvature and refraction errors
Precise levelling	High-accuracy levelling with special instruments and procedures — establishing benchmarks
Trigonometric levelling	From measured vertical angles and distances (theodolite/total station)
Barometric levelling	From differences in atmospheric pressure — rough, exploratory
Hypsometric levelling	From the boiling point of water (which falls with altitude) — rough

Reduction of levels

FormulaHeight of instrument (collimation) method

H I = R L of BM or CP + B S; R L of a point = H I - (I S or F S)

Arithmetic check: $\sum B S - \sum F S = last RL - first RL$

Quick and less laborious; suitable for profile levelling with many intermediate sights; no check on intermediate RLs.

FormulaRise and fall method

Compare each reading with the previous reading at the same setup:

Previous reading − present reading > 0 → rise; < 0 → fall.
$R L = previous RL + rise$ (or − fall).

Arithmetic checks: $\sum B S - \sum F S = \sum rise - \sum fall = last RL - first RL$

More laborious but provides a complete check on all readings (including intermediate sights) — preferred for accurate work such as fly and check levelling.

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.

FormulaCurvature and refraction corrections (

d

in km, corrections in m)

Curvature: $C_{c} = - \frac{d ^{2}}{2 R} = - 0.07849 d^{2}$
Refraction: $C_{r} = + \frac{1}{7} \times 0.07849 d^{2} = + 0.01121 d^{2}$
Combined: $C = - 0.06728 d^{2}$

Distance to the visible horizon from a height $h$ (m):

d = \frac{h}{0.06728} = 3.855 h km

Reciprocal levelling

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

Set the level near A; read the staff at A ( $a_{1}$ ) and at B ( $b_{1}$ ).
Set the level near B; read the staff at A ( $a_{2}$ ) and at B ( $b_{2}$ ).

True difference in level = \frac{( b _{1} - a _{1} ) + ( b _{2} - a _{2} )}{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.

FormulaSensitivity of bubble tube

α = \frac{s}{n D} radians = \frac{s}{n D} \times 206265 seconds

R = \frac{n l D}{s}

$s$ = difference in staff readings for bubble movement of $n$ divisions; $D$ = distance from instrument to staff; $l$ = length of one bubble division; $R$ = radius of curvature of the bubble tube.

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

Errors in levelling

Type	Examples
Instrumental	Line of collimation not parallel to bubble axis (collimation error — eliminated by equal back and fore sight distances), sluggish bubble, defective staff graduations, loose tripod
Natural	Curvature and refraction, wind (vibration), sun (unequal expansion of instrument and bubble), settlement of tripod or staff on soft ground
Personal	Imperfect levelling, parallax, staff not held vertical (reading too large), wrong reading or booking, bubble not central at the time of reading, change point not firm

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

Worked examples

Worked ExampleExample 1 — height of instrument and rise and fall methods

The following readings were taken with a level (BM = 100.000 m): 1.585 (BS on BM), 1.965 (IS at P1), 2.325 (FS at CP1), 1.215 (BS at CP1), 0.855 (IS at P2), 1.035 (FS at P3). Find the RLs by both methods and apply checks.

Solution — HI method.

Station	BS	IS	FS	HI	RL
BM	1.585			101.585	100.000
P1		1.965			99.620
CP1	1.215		2.325	100.475	99.260
P2		0.855			99.620
P3			1.035		99.440

Check: $\sum B S - \sum F S = 2.800 - 3.360 = - 0.560 = 99.440 - 100.000$ ✓

Rise and fall method: BM→P1 fall 0.380 (99.620); P1→CP1 fall 0.360 (99.260); CP1→P2 rise 0.360 (99.620); P2→P3 fall 0.180 (99.440). Check: $\sum rise - \sum fall = 0.360 - 0.920 = - 0.560$ ✓

Worked ExampleExample 2 — curvature and refraction

Find the combined correction for a sight of 2 km, and the distance to the visible horizon from a lighthouse 25 m high.

Solution. $C = - 0.06728 \times 2^{2} = - 0.269 m$ $d = 3.855 25 = 19.3 km$

Worked ExampleExample 3 — reciprocal levelling

With the level near A: staff at A = 1.625, at B = 2.545. With the level near B: staff at A = 0.920, at B = 1.810. Find the true difference in level and the combined error.

Solution. True difference $= \frac{( 2.545 - 1.625 ) + ( 1.810 - 0.920 )}{2} = \frac{0.920 + 0.890}{2} = 0.905 m$ (B lower than A) Combined error $= \frac{0.920 - 0.890}{2} = 0.015 m$

Worked ExampleExample 4 — sensitivity of bubble tube

With the staff 100 m away, the bubble was moved through 2 divisions (each 2 mm) and the staff readings changed by 0.010 m. Find the sensitivity and radius of the bubble tube.

Solution. $α = \frac{0.010}{2 \times 100} \times 206265 = 10. 3^{''}$ per division $R = \frac{2 \times 0.002 \times 100}{0.010} = 40 m$

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: $R L = H I - r e a d in g$ ; check $\sum B S - \sum F S$ only; rise and fall checks all readings.
Curvature $- 0.07849 d^{2}$ ; refraction $+ 0.01121 d^{2}$ (1/7); combined $- 0.06728 d^{2}$ ; visible horizon $3.855 h$ 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 $α = s / (n D)$ rad; increases with radius of the tube.

Common MistakeCommon mistakes

Applying the arithmetic check of the HI method to intermediate sights (it does not check them).
Adding the curvature correction to staff readings (it is subtractive).
Taking the difference of readings in reciprocal levelling from one setup only.

Revision SummaryChapter summary

Levelling determines relative elevations with respect to a datum using levels and staves, with benchmarks as references.
Temporary adjustments (levelling, parallax removal) are made at each setup; permanent adjustments are checked by tests such as the two-peg test.
Differential, fly, check, profile, cross-section, reciprocal, precise and trigonometric levelling serve different purposes.
RLs are reduced by the height of instrument or rise and fall methods, with arithmetic checks.
Curvature, refraction, collimation and other errors are minimised by balanced sights, reciprocal observations and careful procedure.

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Levelling