Levelling Methods and Computations

Introduction

Levelling is the surveying operation used to determine the elevations, or reduced levels (RLs), of points relative to a chosen datum. In civil engineering, reliable levels are required for roads, drainage systems, canals, foundations, pipelines, earthwork quantities and setting out construction works. A levelling survey establishes the vertical relationship between points by observing differences in elevation with a level instrument and a graduated staff.

The central principle is simple: a correctly adjusted level establishes a horizontal line of sight. The staff readings made on points are therefore compared to determine which point is higher or lower. Although the observation itself is straightforward, professional levelling depends on suitable method selection, stable instrument positions, correct booking, arithmetic checks and appropriate adjustment of errors.

This chapter develops differential, fly, check, profile, cross-section, reciprocal and precise levelling. It also explains the height-of-instrument (HI) and rise-and-fall methods, field-book formats, arithmetic checks and the adjustment of misclosure.

Learning Objectives

By the end of this chapter, the learner should be able to:

• explain the purpose and field procedure of common levelling methods;
• distinguish backsight, intermediate sight, foresight, change point and benchmark;
• compute reduced levels by the HI and rise-and-fall methods;
• perform arithmetic checks on levelling books;
• identify collimation, curvature, refraction, staff and booking errors;
• apply reciprocal and precise levelling to reduce systematic errors;
• calculate and distribute a permissible closing misclosure; and
• select an appropriate levelling method for a civil engineering application.

1. Fundamental Terminology and Principle

A datum is a reference surface from which elevations are measured. Mean sea level is commonly used as a national datum, although a project may use an assumed datum. A benchmark (BM) is a fixed point whose RL is known. A temporary benchmark (TBM) is established for use during a project.

A backsight (BS) is the first staff reading taken after setting up the instrument, normally on a point of known RL or on a change point. An intermediate sight (IS) is any reading taken between the backsight and foresight without moving the instrument. A foresight (FS) is the last reading taken before the instrument is moved, normally on a change point or closing point.

A change point (CP), also called a turning point, is a firm point on which a foresight is taken before moving the instrument and a backsight is taken after the move. The same physical point therefore appears twice in the field book, but it has only one RL.

The line of collimation is the line through the intersection of the cross-hairs and the optical centre of the objective. The height of instrument (HI) is the RL of the line of collimation for a particular instrument setup. It is not the height of the tripod or telescope above the ground.

The basic relationship is:

For a point observed from that setup:

Thus, a smaller staff reading indicates a higher point when readings are taken from the same horizontal line of sight.

Levelling staff readings

A staff reading is recorded in metres to the required precision. For ordinary engineering levelling, readings may be recorded to the nearest millimetre, such as 1.735 m. The staff must be held vertical; rocking the staff and taking the minimum reading is a practical method of detecting the vertical position, because a tilted staff produces a reading larger than the true reading.

2. Classification of Levelling Methods

3. Differential Levelling

Differential levelling is used when the difference in elevation between two points cannot be determined from one convenient instrument position. The instrument is placed successively at suitable locations and the line of sight is transferred through change points.

Field procedure

1. Set up and carefully level the instrument.
2. Take a BS on a benchmark or known point.
3. Take any required IS readings.
4. Take an FS on a stable change point.
5. Move the instrument without disturbing the change point.
6. Take a BS on the same change point.
7. Continue until the destination point is reached.
8. Close on a known benchmark where possible.

The instrument should be positioned so that sight lengths are reasonably balanced. Avoid unstable ground, excessive sight lengths, heat shimmer and obstructions.

Worked Example 1: HI method

A benchmark A has RL 100.000 m. The following observations are made:

For setup 1:

Therefore,

For setup 2:

The final point B is therefore 98.600 m above the datum. Notice that CP1 has one RL only; its two readings belong to different instrument setups.

4. Fly Levelling and Check Levelling

Fly levelling

Fly levelling is a rapid form of differential levelling used mainly to transfer a known RL to another location. At each setup, only a BS and FS are generally observed. Intermediate details are omitted. The method is useful before construction begins, when a TBM must be established near a work site, or when control must be transferred quickly.

Because fewer observations are made, fly levelling is efficient but does not provide detailed information about the ground surface. It should be closed on a known benchmark whenever accuracy is important.

Check levelling

Check levelling is undertaken to test the reliability of previous work. A line may be run from BM1 to BM2, where BM2 has a known RL, or a closed circuit may return to the starting benchmark. The observed closing RL is compared with the known RL.

The closing misclosure is:

A positive value means the computed closing RL is too high; a negative value means it is too low. The observed result must be compared with the tolerance specified by the surveying authority, project specification or instrument class. A misclosure outside tolerance should normally be investigated and, if necessary, the work repeated rather than automatically adjusted.

5. Profile Levelling

Profile levelling determines the elevations of points along a selected alignment, such as the centre line of a road, railway, sewer or canal. Readings are taken at regular chainages and at significant changes in ground slope. The resulting longitudinal section supports gradient design and earthwork calculations.

A typical field book includes chainage, BS, IS, FS, HI, RL, remarks and descriptions of points. Points should be observed at:

• regular chainages, for example every 10 m or 20 m;
• abrupt changes in ground level;
• existing roads, drains, streams and property boundaries;
• proposed culvert or crossing positions; and
• benchmarks and change points.

The difference in elevation between two profile points is obtained from their RLs. If a design gradient is specified, the design RL at chainage x may be obtained from:

where g is the gradient expressed as a signed decimal, x is horizontal distance in metres, and the same vertical and horizontal units are used. For a falling gradient, g is negative.

Profile levelling should not be confused with merely taking readings at equal distances. The surveyor must record the location and significance of each point so that the profile can be interpreted during design and construction.

6. Cross-Section Levelling

Cross-section levelling obtains elevations at offsets on both sides of an alignment. It is used for road formation, railway earthworks, canals, pipelines and quantity estimation. Sections are generally taken perpendicular to the centre line, although special terrain may require another direction.

At each selected chainage, the surveyor records the centre-line point and offset points such as 3 m left, 6 m left, 3 m right and 6 m right. Additional points are required wherever the ground changes slope. The cross-section data can be plotted to show the existing ground profile and compared with a proposed formation section.

Cross-section observations may be taken from one instrument setup or several setups. When the instrument is moved, a CP must be used and the HI or rise-and-fall computation continued correctly. A common field error is to record offsets without chainages or to omit whether an offset is left or right. Such omissions can make otherwise accurate readings unusable.

7. Reciprocal Levelling

Reciprocal levelling is used when an obstacle prevents the instrument from being positioned midway between two points. Examples include rivers, wide roads, deep valleys and unsafe or inaccessible ground. The instrument is set near point A and readings are taken on A and B; it is then set near B and readings are repeated.

Let:

• a₁ = reading on A when the instrument is near A;
• b₁ = reading on B when the instrument is near A;
• a₂ = reading on A when the instrument is near B; and
• b₂ = reading on B when the instrument is near B.

The true difference in elevation of B relative to A is:

The observed difference in the first setup is a₁ − b₁, and that in the second setup is a₂ − b₂. Averaging the two differences eliminates, to first order, the effects of collimation error, curvature and refraction, provided the relevant errors are approximately equal and opposite in the two long sight lines.

The error in the first setup may be estimated by:

The reciprocal method does not eliminate staff graduation error, poor staff holding, settlement during observations or errors caused by changing atmospheric conditions. The two setups should be completed promptly and the instrument should be carefully levelled each time.

Worked Example 2: Reciprocal levelling

A and B are separated by a river. The RL of A is 52.460 m. Observations are:

• instrument near A: a₁ = 1.865 m, b₁ = 3.214 m;
• instrument near B: a₂ = 2.745 m, b₂ = 1.506 m.

Therefore,

Thus B is 0.055 m lower than A:

8. Precise Levelling

Precise levelling establishes control elevations to a higher standard than ordinary construction levelling. The exact requirements depend on the specification, but sound practice includes:

• using a calibrated, stable level and suitable precise staff;
• keeping backsight and foresight distances nearly equal;
• limiting sight lengths according to the required accuracy;
• using firm change points;
• allowing the instrument and staff to adapt to temperature where required;
• avoiding observations through heat shimmer, rain or strong wind;
• reading in a consistent order and recording immediately; and
• closing lines and checking misclosure against the prescribed tolerance.

Balanced sight lengths are especially important because many systematic errors depend on sight distance. If the backsight and foresight distances are equal, the effects of a small collimation error tend to cancel when differences are formed. Precise levelling is not achieved merely by taking more decimal places; it requires control of the complete observation procedure.

9. Computation by the Height-of-Instrument Method

The HI method is efficient when many points are observed from each setup. First calculate the HI for each setup using the known RL and BS. Then subtract every staff reading from that HI.

At a change point, the FS determines its RL from the old HI, and the subsequent BS determines the new HI. The method is quick and convenient for profile and cross-section work. However, it provides less direct information about each individual rise or fall and can allow a single omitted or misclassified reading to pass unnoticed unless arithmetic checks are used.

10. Computation by the Rise-and-Fall Method

The rise-and-fall method compares consecutive staff readings. For two consecutive points:

• if the next reading is smaller, the ground has risen;
• if the next reading is larger, the ground has fallen.

For consecutive readings r₁ and r₂:

The RL of the next point is then:

Only consecutive points observed from the same setup should be compared directly. At a change point, the FS and the next BS are not treated as two different ground points: the FS establishes the CP RL and the BS begins the next setup. In a properly booked table, the change-point reading pair is handled so that the CP is not assigned two different RLs.

Arithmetic checks

For the HI method:

For the rise-and-fall method:

A further check for the HI method is that the calculated RLs agree with each HI and its corresponding reading. These checks detect many booking and subtraction errors, but they do not prove that the field observations are free from systematic error.

Worked Example 3: Rise-and-fall computation

A benchmark has RL 200.000 m. Consecutive staff readings, with a change point after the third reading, are:

The displayed table should not be used without completing the arithmetic. Correctly,

Suppose the final FS on R is 0.640 m from the second setup. The rise from Q to R is 1.265 m, so:

This example illustrates an important professional practice: a draft table containing an uncertain or incomplete entry must be corrected before issue. The correct Q RL is 199.000 m, not a value inferred from an incomplete line.

For this sequence, ΣRise = 0.375 + 1.265 = 1.640 m and ΣFall = 0.585 + 0.790 = 1.375 m. Their difference is 0.265 m, equal to RL_R − RL_BM = 200.265 − 200.000 = 0.265 m.

11. Misclosure and Adjustment

A levelling line that begins and ends on known RLs should satisfy the known elevation difference. The misclosure is the difference between the computed and known closing RL. For a closed loop returning to its starting point, the theoretical elevation difference is zero.

Adjustment is justified only when the observations are within the specified tolerance and no blunder is suspected. A blunder, such as an incorrect staff reading or an omitted change point, must be found and corrected rather than concealed by adjustment.

If a line of total length L has misclosure f and the specification permits adjustment proportional to distance, the correction at a point a distance l from the start is:

The adjusted RL is:

Here f, l and L must use the same length unit. The correction at the final point is −f, so the adjusted closing RL agrees with the known RL. Proportional-to-distance adjustment assumes that random error accumulates approximately with distance and that observations have comparable quality.

For sections of unequal precision, a weighted adjustment may be more appropriate. In basic diploma field practice, however, the specified project rule—often proportional to length or number of setups—must be followed.

Worked Example 4: Adjustment

A closed levelling line has total length 1.2 km. At a point 0.3 km from the start, the observed RL is 145.286 m. The line closes 0.012 m high, so f = +0.012 m.

Therefore:

If the point were at the closing station, its correction would be −0.012 m.

12. Sources of Error and Quality Control

Levelling errors are commonly classified as instrumental, personal and natural.

Instrumental errors include collimation error, defective circular or tubular bubbles, maladjusted compensators and staff graduation defects. Regular calibration and an approved two-peg test help identify collimation error.

Personal errors include inaccurate centring of the bubble, incorrect reading, failure to hold the staff vertical, confusing BS and FS columns, misidentifying a change point and transposing digits. The observer should read aloud when appropriate and the recorder should repeat the value.

Natural errors include curvature of the Earth, atmospheric refraction, wind, temperature gradients, vibration, poor visibility and settlement of the instrument or staff. Curvature makes a distant surface appear lower relative to a horizontal line of sight, while refraction partly offsets this effect. Keeping sight lengths balanced reduces their influence in ordinary differential levelling.

A professional field procedure includes:

1. checking the instrument before departure;
2. selecting firm, stable setups and change points;
3. balancing BS and FS distances where possible;
4. recording observations immediately;
5. checking arithmetic before leaving the site;
6. closing on a known point; and
7. retaining original field notes and clearly identifying any corrections.

Original observations should never be erased in a way that hides the first entry. Corrections should remain traceable, because field notes are technical records and may have contractual or legal significance.

13. Applied Case: Establishing Road-Construction Control

Background

This illustrative case concerns a proposed 1.5 km access road. The design team requires a TBM near the construction compound and profile and cross-section levels along the centre line. An existing BM at the site entrance has RL 312.480 m.

Problem

The route crosses a small stream. A direct long sight across the stream would be affected by unequal sight lengths and possible collimation error. The survey team must transfer the RL across the obstacle, continue profile levelling and verify the result at a second known benchmark.

Approach and solution

The team first performs reciprocal levelling between points on opposite banks. The calculated difference is used to establish a TBM on the far bank. A fly line then transfers the TBM to the construction compound, using firm change points and a closing observation on a known control point.

From the compound, differential levelling establishes the centre-line RLs. Profile observations are made at 20 m chainages and at drainage crossings. Cross-sections are taken at 50 m intervals and wherever the ground changes abruptly. The HI method is used in the field office for rapid computation, while the rise-and-fall method is used as an independent check on selected sections.

The final line closes 8 mm high over 1.0 km. The project tolerance allows 12 mm, so the result is accepted and adjusted proportionally to distance. If the misclosure had exceeded tolerance, the team would inspect the field book, check the instrument and repeat the affected section.

Outcome and learning points

The project obtains a traceable vertical control network, reliable design profiles and cross-sections suitable for estimating excavation and embankment. The case demonstrates that reciprocal levelling is appropriate at an obstacle, fly levelling is efficient for transferring control, and closure checks are essential before levels are used for construction.

14. Technology and Contemporary Practice

Automatic levels remain widely used because they are robust and efficient. Digital levels can read coded staffs electronically, reduce reading and recording errors, and store observations for later download. Total stations and GNSS may also provide elevation information, but their suitability depends on required accuracy, satellite visibility, geoid modelling and project control. GNSS ellipsoidal heights are not automatically equivalent to orthometric RLs; a suitable geoid or vertical datum model is required.

Field software can calculate HI, rise and fall, closures and adjustments. Nevertheless, software does not replace surveying judgement. The operator must verify point identification, instrument setup, units, datum, backsight and foresight classification, and closure results. AI-assisted systems may help detect unusual readings or incomplete records, but all engineering decisions and issued levels require competent human review and an auditable record.

Chapter Summary

Levelling determines elevations relative to a datum by comparing staff readings made from a horizontal line of sight. Differential levelling transfers RLs through multiple setups using change points. Fly levelling rapidly transfers control, while check levelling tests reliability. Profile levelling provides longitudinal ground levels along an alignment, and cross-section levelling records levels perpendicular to that alignment. Reciprocal levelling is used across obstacles and reduces the effects of collimation, curvature and refraction. Precise levelling requires calibrated equipment, balanced sights, stable points and strict procedures.

The HI method calculates the line-of-sight elevation and subtracts each staff reading. The rise-and-fall method compares consecutive readings and provides a useful independent check. The essential arithmetic checks are ΣBS − ΣFS = RL_last − RL_first and ΣRise − ΣFall = RL_last − RL_first. Misclosure must be investigated and should be adjusted only when it is within the applicable tolerance. Professional levelling combines correct observation, careful booking, independent checking, appropriate technology and traceable quality control.

Key Terms

Benchmark: A fixed point whose reduced level is known.

Change point: A stable point used to transfer the line of sight between instrument setups.

Check levelling: Levelling performed to verify a known RL or a previous survey.

Collimation: The line through the instrument’s cross-hairs and optical centre.

Cross-section: A section showing ground levels across an alignment.

Datum: The reference surface from which elevations are measured.

Differential levelling: Levelling between separated points using one or more setups.

Fly levelling: Rapid levelling used mainly to transfer or check elevations.

Foresight: The last staff reading before an instrument is moved.

Height of instrument: The RL of the line of collimation for a setup.

Intermediate sight: A reading taken between the backsight and foresight.

Levelling staff: A graduated rod used to measure the vertical distance below the line of sight.

Misclosure: The difference between observed and known closing elevation.

Profile levelling: Levelling along an alignment to obtain a longitudinal ground profile.

Reduced level: The elevation of a point relative to a selected datum.

Reciprocal levelling: Levelling from both sides of an obstacle to reduce systematic errors.

Rise and fall: The increase or decrease in elevation between consecutive points.

Temporary benchmark: A project point established with a transferred RL.

Review Questions

Short Answer Questions

1. Define a benchmark, backsight, foresight and change point.
2. State the two basic equations used in the height-of-instrument method.
3. What is the main purpose of fly levelling?
4. Why are backsight and foresight distances balanced in precise levelling?
5. State the arithmetic check used with the rise-and-fall method.

Long Answer Questions

1. Explain the field procedure and applications of differential levelling.
2. Describe profile and cross-section levelling and explain their use in road construction.
3. Explain reciprocal levelling and derive the expression for the true difference in elevation between two points.
4. Compare the HI and rise-and-fall methods, including their advantages and limitations.
5. Discuss the major sources of levelling error and the quality-control measures used to reduce them.

Analytical / Application Questions

1. A benchmark has RL 250.000 m. From one setup the BS is 1.455 m, the IS on P is 0.885 m and the FS on CP is 2.105 m. After moving the instrument, the BS on CP is 1.765 m and the FS on Q is 1.250 m. Calculate the RLs of P, CP and Q by the HI method.
2. A line closes 0.018 m low over 900 m. Calculate the proportional correction at a point 225 m from the start and explain whether the line should be adjusted before checking the specified tolerance.
3. In a reciprocal levelling operation, a₁ = 1.420 m, b₁ = 2.860 m, a₂ = 2.115 m and b₂ = 1.525 m. If the RL of A is 75.300 m, calculate the RL of B and explain why the method is suitable across a river.

Multiple Choice Questions with Answers

Question 1

A benchmark has RL 120.000 m and the backsight is 1.650 m. What is the height of instrument?

A. 118.350 m

B. 120.000 m

C. 121.650 m

D. 122.650 m

Correct Answer: C. 121.650 m

Question 2

If the height of instrument is 105.800 m and the staff reading on point P is 2.135 m, what is the RL of P?

A. 103.665 m

B. 107.935 m

C. 105.800 m

D. 102.135 m

Correct Answer: A. 103.665 m

Question 3

Which levelling method is specifically suited to crossing a river while reducing collimation effects?

A. Profile levelling

B. Reciprocal levelling

C. Fly levelling

D. Cross-section levelling

Correct Answer: B. Reciprocal levelling

Question 4

In ordinary levelling, a smaller staff reading from the same line of sight indicates that the point is:

A. Lower

B. At the datum

C. Higher

D. At the instrument station

Correct Answer: C. Higher

Question 5

Which equation is the standard arithmetic check for the HI method?

A. ΣBS + ΣFS = RL_last + RL_first

B. ΣBS − ΣFS = RL_last − RL_first

C. ΣBS − ΣFS = RL_first − RL_last

D. ΣIS = ΣRise + ΣFall

Correct Answer: B. ΣBS − ΣFS = RL_last − RL_first

Question 6

A closed line has an observed closing RL 0.010 m higher than the known RL. What is the correction at the closing point?

A. +0.010 m

B. −0.010 m

C. +0.005 m

D. 0.000 m

Correct Answer: B. −0.010 m

Question 7

What is the main purpose of profile levelling?

A. To determine magnetic bearing

B. To obtain elevations along an alignment

C. To measure horizontal angles only

D. To calibrate a staff

Correct Answer: B. To obtain elevations along an alignment

Question 8

In the rise-and-fall method, if a consecutive reading changes from 1.800 m to 1.250 m, the ground has:

A. Fallen 0.550 m

B. Risen 0.550 m

C. Risen 3.050 m

D. Not changed

Correct Answer: B. Risen 0.550 m

Question 9

If a levelling line has misclosure f and total length L, the proportional correction at distance l from the start is:

A. fL/l

B. −f(l/L)

C. −L(f/l)

D. f + l + L

Correct Answer: B. −f(l/L)

Question 10

Why are balanced backsight and foresight distances important in precise levelling?

A. They increase staff graduation length

B. They eliminate the need for a benchmark

C. They help cancel distance-dependent systematic errors

D. They ensure every point has the same RL

Correct Answer: C. They help cancel distance-dependent systematic errors