Compared with the height-of-instrument method, the rise-and-fall method of reducing levels

provides an arithmetic check on the intermediate sights as well

needs no arithmetic check at all

cannot be used when there are change points

is faster when there are many intermediate sights

The fore bearing of line AB is 45°30′ and the back bearing observed at B is 226°15′. This indicates

local attraction at A or B, since the bearings differ by 180°45′

a correct observation, since the difference is close to 180°

Which of the following errors are eliminated by keeping the back-sight and fore-sight distances equal?

Atmospheric refraction (if uniform)

Which of the following statements about compass surveying are correct?

Local attraction does not affect the included angle between two lines observed from the same station

The agonic line joins places of zero magnetic declination

A prismatic compass reads whole-circle bearings

Local attraction does not affect the included angle between two lines observed from the same station

The agonic line joins places of zero magnetic declination

A prismatic compass reads whole-circle bearings

Magnetic declination at a place is constant over the years

Which of the following statements about levels and levelling are correct?

An automatic level uses a compensator to keep the line of sight horizontal

Parallax is removed by focusing the eyepiece on the cross-hairs and then the objective on the staff

A more sensitive bubble tube has a larger radius of curvature

An automatic level uses a compensator to keep the line of sight horizontal

Parallax is removed by focusing the eyepiece on the cross-hairs and then the objective on the staff

A more sensitive bubble tube has a larger radius of curvature

Reciprocal levelling is used because it avoids the need for any fore sights

Land Surveying I: Levelling Instruments and Methods, and the Compass

This is a section of Part B1, Surveying and Mapping, of the Geomatics Engineering (GE) paper. B1's Land Surveying heading names “Levelling instruments, Levelling methods, Compass, Theodolite, Total Station and COGO functions, Tachometer, Trigonometric levelling, Traversing, Triangulation and Trilateration” — nine items across three kinds of work, so it takes three chapters. This first one is levelling and the compass: the instruments from the dumpy level to the digital level, their temporary adjustments and the two-peg test of the permanent one, the sensitivity of the bubble tube; the methods — differential, fly, profile, cross-section, check and reciprocal levelling; the height-of-instrument and rise-and-fall booking with their arithmetic checks; the curvature and refraction correction; and the compass, with whole-circle and reduced bearings, magnetic declination and local attraction. The numericals are a level book, the 0.0673 D² correction, reciprocal levelling and bearings.

1. Levelling terms and instruments

Levelling finds differences in height. A level surface is everywhere perpendicular to gravity; a datum is the reference level surface (mean sea level); a bench mark (BM) is a fixed point of known reduced level (RL). The instrument's line of collimation (line of sight) is made horizontal; the first reading after setting up is a back sight (BS) on a point of known RL, the last before moving is a fore sight (FS), and any others are intermediate sights (IS). A change point (CP) carries both an FS and a BS. The height of instrument (HI) is the RL of the line of collimation: HI = RL + BS.

Levels
Instrument	How the line of sight is made horizontal
Dumpy level	telescope rigidly fixed to the vertical axis; levelled with foot screws until the bubble stays central in all directions
Tilting level	approximately levelled, then the telescope is tilted by a fine screw until the bubble is central for each sight
Automatic level	a pendulum compensator keeps the line of sight horizontal once the circular bubble is roughly central
Digital level	an automatic level that reads a bar-coded staff electronically and records the reading and distance

Temporary adjustments at every set-up: setting up (tripod firm, instrument approximately level), levelling up, and elimination of parallax (focus the eyepiece on the cross-hairs, then the objective on the staff, until the hairs do not move against the staff image as the eye moves). The key permanent adjustment is that the line of collimation is parallel to the bubble axis, checked by the two-peg test: readings with the level midway between pegs A and B give the true difference (equal sight lengths cancel any collimation error); readings from a station near one peg reveal the error. Sensitivity of the bubble tube: if moving the bubble through n divisions changes a staff reading at distance D by s, the angle per division is α = s/(nD) radians (× 206 265 for seconds) and the tube's radius is R = n·l·D/s for a division length l. A difference of 0.02 m for 5 divisions at 100 m is α = 0.02/500 rad = 8.25″; with 2 mm divisions, R = 5 × 0.002 × 100/0.02 = 50 m.

2. Levelling methods and booking

Differential (simple and compound): the height difference between two points, through change points if far apart. Fly levelling: a quick run to carry an approximate level or check a BM. Check levelling: a return run to close on the starting BM.
Profile (longitudinal section) along a centre line and cross-sectioning at right angles to it, for earthwork quantities.
Reciprocal levelling across a river or valley where the instrument cannot be midway: readings on both staffs from near A, then from near B. The true difference is the mean of the two apparent differences, h = ((b₁ − a₁) + (b₂ − a₂))/2, which cancels collimation, curvature and refraction errors.

A level book by the height-of-instrument method (BM RL 100.000 m)
Station	BS	IS	FS	HI	RL
BM	1.525	—	—	101.525	100.000
P	—	2.110	—	—	99.415
CP	1.750	—	0.985	102.290	100.540
Q	—	1.200	—	—	101.090
R	—	—	2.345	—	99.945

Checks. HI method: ΣBS − ΣFS = last RL − first RL: 3.275 − 3.330 = −0.055 = 99.945 − 100.000. It does not check the intermediate RLs. Rise-and-fall method: each consecutive pair of readings gives a rise (previous reading larger) or a fall; ΣBS − ΣFS = ΣRise − ΣFall = last RL − first RL, and because every reading enters a rise or fall, the intermediate sights are checked too. Here the successive rises and falls are −0.585, +1.125, +0.550 (BS 1.750 then IS 1.200) and −1.145: ΣRise − ΣFall = (1.125 + 0.550) − (0.585 + 1.145) = −0.055. A closed circuit's misclosure is judged against an allowable error of the form C√K (K in km).

⚠️ A bigger staff reading means a lower point

The staff is read downward from the line of sight, so a reading that increases from one point to the next is a fall. Swapping rise and fall is the commonest level-book error, and only the arithmetic check catches it.

3. Curvature and refraction

A horizontal line of sight leaves the curved level surface. Over a distance D the curvature correction is c = D²/(2R); with R = 6370 km and D in km, c = 0.0785 D² m, and it makes staff readings too large, so it is subtracted. Refraction bends the ray down toward the Earth, taken as one-seventh of curvature: r = 0.0112 D² m, added. The combined correction is −0.0673 D² m (6/7 of curvature). At 1 km it is 67 mm; at 2 km, 0.0673 × 4 = 0.269 m. Inverting, the distance to the visible horizon from a height h metres is d = √(h/0.0673) = 3.855√h km.

🎯 Why equal sight lengths are the surveyor's best friend

Collimation error, curvature and refraction all grow with sight length. If the back sight and fore sight are equal, each error is the same on both readings and cancels in their difference. Reciprocal levelling achieves the same cancellation when balanced sights are impossible.

4. The compass: bearings, declination and local attraction

The prismatic compass (graduated ring rotating with the needle, read through a prism, in whole-circle bearings) and the surveyor's compass (graduated box, quadrantal bearings) measure the magnetic bearing of a line. A whole-circle bearing (WCB) runs 0° to 360° clockwise from north; a reduced (quadrantal) bearing (RB) is the acute angle from north or south toward east or west. Conversion: WCB 0–90° → N θ E; 90–180° → S (180° − θ) E; 180–270° → S (θ − 180°) W; 270–360° → N (360° − θ) W. So WCB 235°30′ is S 55°30′ W. The back bearing of a line differs from its fore bearing by exactly 180° when both ends are free of local disturbance. The included angle between two lines from a station is the difference of their bearings.

Magnetic declination is the horizontal angle between true (geographic) north and magnetic north; it varies with place and time (secular, annual and daily variation). True bearing = magnetic bearing + declination when the declination is east, and − declination when it is west. With 2°30′ W declination, a magnetic bearing of 48°15′ is a true bearing of 45°45′. Lines of equal declination are isogonic lines; the line of zero declination is the agonic line. Local attraction is the deflection of the needle by nearby iron, steel, rails or power lines. It is detected when the fore and back bearings of a line do not differ by 180°; a station whose bearings are consistent is taken as free, and corrections are carried round the traverse from it (or the included angles, which local attraction does not affect, are used).

🧠 Local attraction cancels in an included angle

Local attraction deflects both bearings observed at one station by the same amount, so the angle between them is unaffected. That is why a compass traverse with local attraction is corrected through its included angles.

Key takeaways

HI = RL + BS; RL = HI − FS (or IS); check ΣBS − ΣFS = last RL − first RL, and rise-and-fall also checks intermediate sights.
The two-peg test checks that the line of collimation is parallel to the bubble axis; equal sight lengths cancel collimation, curvature and refraction.
Curvature 0.0785 D², refraction 0.0112 D², combined −0.0673 D² m (D in km); horizon distance 3.855√h km.
Reciprocal levelling: true difference = mean of the two apparent differences.
WCB ↔ RB by quadrant; FB and BB differ by 180° without local attraction; true = magnetic ± declination (E +, W −); included angles are free of local attraction.

Practice questions (14)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

From a bench mark of RL 100.000 m, the readings in order are: BS 1.525 (on the BM), IS 2.110, FS 0.985 (change point), BS 1.750, IS 1.200, FS 2.345. The RL of the last point, in metres, is ____.
Numerical answer — type the value.
Show answer
Answer: 99.945
HI₁ = 100.000 + 1.525 = 101.525; CP RL = 101.525 − 0.985 = 100.540; HI₂ = 100.540 + 1.750 = 102.290; last RL = 102.290 − 2.345 = 99.945 m. Check: ΣBS − ΣFS = 3.275 − 3.330 = −0.055 = 99.945 − 100.000.
In the same level book (BM RL 100.000 m, BS 1.525), the RL of the first intermediate point, whose staff reading is 2.110, in metres, is ____.
Numerical answer — type the value.
Show answer
Answer: 99.415
HI = 100.000 + 1.525 = 101.525 m; RL = HI − IS = 101.525 − 2.110 = 99.415 m. The larger reading than the back sight means the point is lower than the BM — a fall of 0.585 m.
Compared with the height-of-instrument method, the rise-and-fall method of reducing levels
1. provides an arithmetic check on the intermediate sights as well
2. needs no arithmetic check at all
3. cannot be used when there are change points
4. is faster when there are many intermediate sights
Show answer
Answer: A — provides an arithmetic check on the intermediate sights as well
Every reading enters a rise or a fall, so ΣRise − ΣFall = last RL − first RL checks the intermediate RLs too; the HI method's check ΣBS − ΣFS involves only back and fore sights. The price is more arithmetic, so HI is faster for profiles with many intermediate sights.
Taking R = 6370 km and refraction as one-seventh of curvature, the combined curvature and refraction correction for a sight of 2 km, in metres (to three decimal places), is ____ (give the magnitude).
Numerical answer — type the value.
Show answer
Answer: 0.269
Curvature = D²/(2R) = (2000)²/(2 × 6 370 000) = 0.3140 m; refraction = 0.3140/7 = 0.0449 m; combined = 0.3140 − 0.0449 = 0.269 m, i.e. 0.0673 × 2² = 0.269 m, subtracted from the staff reading. Curvature alone is 0.314 m.
An observer's eye is 9 m above sea level. Allowing for curvature and refraction (combined correction 0.0673 D² m, D in km), the distance to the visible horizon, in km (to two decimal places), is ____.
Numerical answer — type the value.
Show answer
Answer: 11.56
The horizon is where the combined drop equals the eye height: 0.0673 D² = 9, D = √(9/0.0673) = √133.7 = 11.56 km (3.855√9 = 11.57 km with the rounded constant). Ignoring refraction (0.0785 D²) gives 10.71 km.
In reciprocal levelling between A and B, with the level near A the staff readings are 1.625 on A and 2.545 on B; with it near B they are 0.725 on A and 1.405 on B. The true difference in level between A and B, in metres, is ____.
Numerical answer — type the value.
Show answer
Answer: 0.8
Apparent differences: 2.545 − 1.625 = 0.920 and 1.405 − 0.725 = 0.680. True difference = (0.920 + 0.680)/2 = 0.800 m, B lower than A. Half their difference, (0.920 − 0.680)/2 = 0.120 m, is the combined collimation, curvature and refraction error on the long sight.
In a two-peg test, with the level midway between pegs A and B the readings are 1.520 on A and 1.830 on B. With the level close to A the readings are 1.410 on A and 1.740 on B. The reading on B that a correctly adjusted level would have given from the second station, in metres, is ____.
Numerical answer — type the value.
Show answer
Answer: 1.72
From midway, the equal sight lengths cancel the collimation error, so the true difference is 1.830 − 1.520 = 0.310 m (B lower). From near A, the reading on A is essentially error-free, so the correct reading on B is 1.410 + 0.310 = 1.720 m. The observed 1.740 is 0.020 m too high: the line of sight points upward.
Moving the bubble of a level through 5 divisions changes the staff reading at 100 m by 0.020 m. The sensitivity of the bubble tube, in seconds of arc per division (to two decimal places, 1 rad = 206 265″), is ____.
Numerical answer — type the value.
Show answer
Answer: 8.25
Angle for 5 divisions = 0.020/100 = 2 × 10⁻⁴ rad; per division = 4 × 10⁻⁵ rad = 4 × 10⁻⁵ × 206,265 = 8.25″. Forgetting to divide by the 5 divisions gives 41.25″.
The whole-circle bearing of a line is 235°30′. Its reduced (quadrantal) bearing is
1. S 55°30′ W
2. N 55°30′ W
3. S 34°30′ W
4. S 55°30′ E
Show answer
Answer: A — S 55°30′ W
235°30′ lies between 180° and 270°, the south-west quadrant, so RB = S (235°30′ − 180°) W = S 55°30′ W. S 34°30′ W measures the acute angle from west instead of from south.
The magnetic bearing of a line is 48°15′ and the magnetic declination is 2°30′ W. The true bearing of the line is
1. 45°45′
2. 50°45′
3. 48°15′
4. 228°15′
Show answer
Answer: A — 45°45′
With west declination, magnetic north lies west of true north, so a bearing measured from magnetic north is larger than from true north: true = 48°15′ − 2°30′ = 45°45′. Adding the declination (50°45′) is correct only for an east declination.
The fore bearing of line AB is 45°30′ and the back bearing observed at B is 226°15′. This indicates
1. local attraction at A or B, since the bearings differ by 180°45′
2. a correct observation, since the difference is close to 180°
3. magnetic declination of 45′
4. a gross error in chaining
Show answer
Answer: A — local attraction at A or B, since the bearings differ by 180°45′
Free of local attraction, FB and BB differ by exactly 180°; here the difference is 226°15′ − 45°30′ = 180°45′, so the needle was deflected at one end (or both). Declination shifts both bearings equally and cannot produce this discrepancy; distance errors do not affect bearings.
Which of the following errors are eliminated by keeping the back-sight and fore-sight distances equal?
1. Collimation error of the level
2. Curvature of the Earth
3. Atmospheric refraction (if uniform)
4. A staff held out of plumb
Show answer
Answer: A — Collimation error of the level; B — Curvature of the Earth; C — Atmospheric refraction (if uniform)
Collimation error, curvature and uniform refraction each produce the same error on two sights of equal length, which cancels in the difference. A tilted staff always reads too high, by an amount that depends on the reading, not on the sight length, so balancing sights does not remove it.
Which of the following statements about compass surveying are correct?
1. Local attraction does not affect the included angle between two lines observed from the same station
2. The agonic line joins places of zero magnetic declination
3. A prismatic compass reads whole-circle bearings
4. Magnetic declination at a place is constant over the years
Show answer
Answer: A — Local attraction does not affect the included angle between two lines observed from the same station; B — The agonic line joins places of zero magnetic declination; C — A prismatic compass reads whole-circle bearings
Both bearings at a disturbed station are shifted equally, so their difference is unaffected; the agonic line is the isogonic line of zero declination; and the prismatic compass's ring is graduated 0–360°. Declination changes with time — secular, annual and daily variations — so old magnetic bearings must be updated.
Which of the following statements about levels and levelling are correct?
1. An automatic level uses a compensator to keep the line of sight horizontal
2. Parallax is removed by focusing the eyepiece on the cross-hairs and then the objective on the staff
3. A more sensitive bubble tube has a larger radius of curvature
4. Reciprocal levelling is used because it avoids the need for any fore sights
Show answer
Answer: A — An automatic level uses a compensator to keep the line of sight horizontal; B — Parallax is removed by focusing the eyepiece on the cross-hairs and then the objective on the staff; C — A more sensitive bubble tube has a larger radius of curvature
The pendulum compensator is what makes a level automatic; parallax is removed in that order; and R = n·l·D/s, so a tube whose bubble moves more for a given tilt (more sensitive) has a larger radius. Reciprocal levelling still uses readings on both staffs from both sides; its purpose is to cancel collimation, curvature and refraction where the level cannot be set midway.