Principles and Instruments of Levelling

Introduction

Levelling is the branch of surveying concerned with determining the elevations of points and the differences in elevation between them. It provides the vertical control needed for roads, drainage systems, canals, foundations, pipelines, earthwork, building construction and topographic mapping. While horizontal surveying establishes position in plan, levelling establishes position in height.

The central practical task is to determine a point’s reduced level (RL) relative to an accepted reference surface. This is achieved by observing staff readings through a precisely adjusted line of sight. The reliability of the result depends not only on arithmetic, but also on instrument adjustment, staff handling, choice of observation points, atmospheric conditions and appropriate field procedures.

Learning Objectives

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

1. Explain level surfaces, horizontal lines, datums and benchmarks.
2. Define reduced level, line of collimation, backsight, foresight and intermediate sight.
3. Apply the height-of-instrument and rise-and-fall methods.
4. Explain curvature and atmospheric refraction corrections.
5. Set up and use automatic, digital and laser levels safely and accurately.
6. Test an automatic level using a two-peg test and recognize common errors.
7. Select and use levelling staffs correctly.
8. Check field-book arithmetic and assess the quality of levelling work.

1. Fundamental Concepts of Levelling

1.1 Level surface and level line

A level surface is a curved surface everywhere normal to the direction of gravity. In practical surveying, a level surface is commonly approximated by a surface parallel to mean sea level. Because gravity points toward the centre of the Earth approximately, a level surface is curved rather than plane.

A level line is a line lying on a level surface. A horizontal line is a straight line tangential to a level surface at a particular point. For ordinary construction work over short distances, the difference between a level surface and a horizontal plane is small; for precise or long-distance work, it must be considered.

A line perpendicular to the direction of gravity is horizontal at the point under consideration. The vertical direction is the direction of a plumb line.

1.2 Datum

A datum is the reference surface or level from which elevations are measured. The most important national datum is usually related to mean sea level, established through a network of permanent control stations. A project may also use an assumed datum, such as 100.000 m, when connection to a national datum is unnecessary.

All reduced levels in a survey must refer to the same datum. Changing the datum changes every RL by the same constant amount but does not change differences in elevation.

1.3 Reduced level

The reduced level of a point is its elevation above or below the adopted datum. It is normally expressed in metres to the nearest millimetre for ordinary engineering levelling, although the required precision depends on the specification.

If point A has RL 52.360 m and point B has RL 54.115 m, then B is higher than A by:

Here, Δh is the difference in elevation. A positive result indicates that B is higher; a negative result indicates that B is lower.

1.4 Benchmarks

A benchmark (BM) is a fixed, identifiable point whose RL is known or established. Benchmarks provide starting and checking points for levelling.

A benchmark should be stable, accessible, clearly described and protected from disturbance. A nail driven into loose soil is unsuitable as a permanent benchmark, whereas a marked point on sound concrete or masonry is generally preferable.

2. The Levelling Instrument and Line of Collimation

A levelling instrument creates a nearly horizontal line of sight. Its principal parts include the telescope, objective lens, eyepiece, focusing system, spirit bubble or electronic compensator, levelling screws, tribrach and tripod connection.

The line of sight is the line joining the optical centre of the objective lens and the intersection of the cross-hairs. The line of collimation is the line of sight when the instrument is correctly adjusted. In a properly functioning automatic level, the compensator makes the line of collimation horizontal within its operating range after the instrument has been approximately levelled.

The staff reading is the vertical distance from the point on which the staff stands to the line of collimation. Therefore:

where HI is the height of instrument, meaning the RL of the line of collimation.

The line of collimation should not be confused with the centre line of the telescope tube. It is an optical reference determined by the instrument’s adjustment.

3. Levelling Staffs

A levelling staff is a graduated vertical rod used to measure the intercept between a point and the line of sight. Common staffs are telescopic, folding or solid-pattern staffs. Metric staffs are often graduated in metres, decimetres, centimetres and millimetres, with alternating colour blocks to improve visibility.

The staff must be held vertical. A staff that leans toward or away from the instrument usually gives a reading larger than the true vertical intercept. The staff person should rock the staff gently forward and backward; the smallest observed reading is the reading when the staff is vertical.

The staff datum or zero is the bottom contact point. The graduations must be checked for damage, fading, incorrect assembly and dirt. A telescopic staff must be fully seated and locked, because an incompletely extended section can cause a systematic error.

Staff reading conventions

A staff reading may be recorded to the nearest millimetre, for example 1.425 m. The three principal observation types are:

• Backsight (BS): the first reading after setting up the instrument, usually taken on a point of known RL or a change point.
• Intermediate sight (IS): a reading on any point whose RL is required before the instrument is moved.
• Foresight (FS): the last reading from an instrument position, generally taken on a change point or closing benchmark.

4. Methods of Reducing Levels

4.1 Height-of-instrument method

The height-of-instrument method is rapid and convenient, particularly when many intermediate points are observed from one setup.

First calculate the height of instrument:

Then calculate each unknown RL:

The method assumes that all readings used with a particular HI are taken before the instrument is moved and that the staff remains on the same point when a foresight and subsequent backsight are observed.

#### Worked example

A benchmark has RL = 100.000 m. The first backsight is 1.325 m. Intermediate readings are 0.955 m and 2.180 m. A change point has a foresight of 2.745 m. After moving the instrument, the backsight on the same change point is 1.865 m, and the closing benchmark reading is 0.925 m.

First setup:

RLs from the first setup are:

Change-point RL:

Second setup:

Closing benchmark RL calculated from the observations is:

The result must be compared with the known RL of the closing benchmark. If the known closing RL is not 99.520 m, the difference is the closing misclosure, subject to the survey tolerance.

4.2 Rise-and-fall method

The rise-and-fall method compares consecutive staff readings. If the next reading is smaller, the ground has risen; if it is larger, the ground has fallen.

The next RL is then obtained from:

The method provides a strong arithmetic check because:

The equality applies when readings are correctly classified and all observations are included. A failure of this check indicates an arithmetic, transcription or classification error, although it does not by itself prove that the field observations are free from systematic error.

4.3 Choosing between methods

The height-of-instrument method is efficient for profiles and many points from each setup. The rise-and-fall method is more detailed and provides a useful independent check between successive points. In professional work, the field book should show sufficient information for another person to verify the reduction.

5. Curvature and Refraction

5.1 Curvature

Because the Earth is curved, a horizontal line of sight is tangent to the Earth at the instrument but lies above the level surface at a distant staff. If curvature alone is considered, the staff reading appears too large by the curvature correction:

where C_c is curvature correction in metres, d is the horizontal distance from instrument to staff in metres, and R is the mean radius of the Earth, approximately 6.37 × 10⁶ m. The formula assumes a spherical Earth and distances small compared with Earth’s radius.

The correction to the observed staff reading is negative because the observed reading is too large:

5.2 Refraction

Atmospheric refraction bends the line of sight slightly downward because air density normally increases toward the ground. This partially offsets curvature. If the coefficient of refraction is represented by k, the combined correction is commonly written:

For ordinary atmospheric conditions, k is often taken as approximately 0.13, but it varies with temperature, pressure, humidity and terrain. The combined correction to the staff reading is negative.

For d = 100 m, R = 6.37 × 10⁶ m and k = 0.13:

At 100 m the effect is small, but at long sights it becomes significant. The best field practice is to keep backsight and foresight distances approximately equal. Their curvature and refraction effects then nearly cancel, avoiding reliance on uncertain atmospheric assumptions.

6. Setting Up and Using an Automatic Level

A safe and accurate setup follows a consistent sequence:

1. Select firm ground with a clear view to the staff points. Avoid loose soil, vibration, traffic and heat shimmer.
2. Extend the tripod legs to a comfortable height and press the shoes firmly into the ground.
3. Mount the instrument securely and tighten the central fixing screw without forcing it.
4. Approximately level the instrument using the circular bubble and levelling screws.
5. Allow the automatic compensator to settle. Do not use the instrument if the compensator is outside its operating range or behaves erratically.
6. Adjust the eyepiece until the cross-hairs are sharp. This removes parallax for the observer’s eye.
7. Focus on the staff until the graduations are sharp and no relative movement occurs between staff image and cross-hairs when the eye is moved slightly.
8. Read the staff at the horizontal cross-hair and record it immediately.

Parallax is a focusing error in which the staff image appears to move relative to the cross-hairs when the observer’s eye moves. It can produce incorrect readings and must be eliminated before observation.

Optical checks during use

The circular bubble should remain close to the centre. The tripod should not be touched while reading. The instrument should be shaded in hot conditions where possible. Staff readings should not be taken across strong air currents or directly above hot surfaces when precision is required.

7. Digital Levels

A digital level uses an electronic sensor and a coded staff, often called a bar-coded or invar staff. The instrument automatically recognizes the staff pattern and displays the staff reading and, where configured, the distance and elevation.

Advantages include faster recording, reduced transcription errors, data storage and the ability to perform programmed checks. Limitations include dependence on batteries, software settings, staff compatibility, sensor visibility and electronic calibration.

A digital-level procedure remains fundamentally the same as optical levelling: establish a stable setup, level the instrument, ensure a readable staff image, observe balanced sights and record point identification. The operator must confirm the units, datum, instrument constant, observation mode and stored point names before starting.

Electronic results should not be accepted blindly. The operator should compare selected digital observations with an independent optical reading or a second setup, inspect the raw data and verify the closing check.

8. Laser Levels

A laser level projects a visible or invisible laser reference. Common types include rotary lasers, line lasers and dot lasers. Rotary lasers are widely used for grading, formwork, suspended ceilings and machine control. A receiver detects the beam when visibility is poor or when the beam is outside the user’s direct view.

Laser levels are efficient for local construction control but are not automatically more accurate than optical levels. Beam visibility, receiver calibration, vibration, tripod settlement, temperature and range affect performance. A rotating laser must be placed on stable ground and allowed to settle before readings are taken.

Laser safety is essential. Never look directly into the beam, aim it at roads or aircraft, or permit untrained persons to operate it. Use the classification and protective procedures stated by the manufacturer and applicable regulations. A laser plane is useful for transferring a common elevation, but a benchmark and independent check are still required for controlled construction work.

9. Testing an Automatic Level

9.1 Two-peg test

The two-peg test checks whether the line of collimation is truly horizontal when the bubble or compensator indicates a level condition. It does not replace manufacturer calibration, but it is a practical field test.

1. Set two firm points A and B approximately 30–60 m apart.
2. Set the instrument midway between A and B. Take readings a and b. Because the sight lengths are equal, the line-of-collimation error has approximately equal effect on both readings. The true difference is:

1. Move the instrument close to A, while retaining a clear view to B. Take readings c on A and d on B.
2. The correct reading at B from the second setup is:

If the observed d differs materially from d_correct, the line of collimation is out of adjustment or the instrument has another defect. The adjustment should be carried out only by an authorized technician unless the manufacturer provides a specific field-adjustment procedure.

#### Example

From the midpoint setup, readings are a = 1.455 m and b = 2.105 m.

From the setup near A, c = 1.220 m. Therefore:

If the observed reading at B is 1.878 m, the discrepancy is 0.008 m. Whether this is acceptable depends on the project tolerance and sight length. The instrument should be checked before precision work.

9.2 Other checks

The circular bubble, tripod head, optical clarity, compensator response and staff condition should be inspected regularly. A simple bubble test can reveal whether the bubble remains centred after rotating the instrument through 180°. Significant movement indicates adjustment is needed.

10. Field Procedure and Booking Practice

A levelling party normally consists of an instrument operator, a staff person and, for larger work, a recorder or survey assistant. Before observation, the team should confirm the benchmark description, datum, units, required accuracy, route, turning points and safety hazards.

A good field book records date, weather, instrument identity, observer, station, point description, BS, IS, FS, HI or rise/fall, RL, remarks and checks. Readings should be called clearly and repeated when uncertain. Erasing a doubtful value is poor practice; the original should be crossed out and the corrected value entered with an explanation.

Change points should be firm, well defined and protected. Avoid soft ground, loose stones, vehicle surfaces and objects that may move between the foresight and backsight. When a change point is unavoidable, use a turning plate or similar stable device.

11. Precision, Errors and Best Practice

Errors may be classified as:

• Instrumental errors: collimation error, defective compensator, damaged optics or incorrect staff graduations.
• Personal errors: incorrect focusing, misreading, poor staff verticality, wrong point identification or transcription error.
• Natural errors: curvature, refraction, wind, temperature gradients, vibration and settlement.
• Random errors: small unpredictable variations that can be reduced by repeated observations and balanced procedures.

Balanced backsight and foresight distances are one of the most important controls. They reduce the effects of collimation error and largely cancel curvature and refraction. Work should proceed from a known benchmark to a second known benchmark where possible. A loop that closes on its starting benchmark is also valuable.

The closing error is:

The acceptance limit must come from the survey specification or governing standard. A commonly used engineering convention is a tolerance proportional to the square root of distance, such as ±C√K, where K is distance in kilometres and C is a project-specified constant. This is not universal and must not be applied without the relevant specification.

If adjustment is authorized, a misclosure may be distributed according to distance or number of setups, depending on the adopted surveying standard. Adjustment should never conceal a blunder. First identify and correct mistakes in observations, point identification and arithmetic.

12. Applied Case: Establishing Construction Levels

Background

A small drainage project requires a constant-fall invert line between an existing benchmark and a proposed manhole. The contractor needs reliable RLs at several excavation points.

Problem

The existing benchmark has RL 250.000 m. The level route is 420 m long and crosses a busy access road. A single long sight would be vulnerable to traffic vibration, unequal sight distances and refraction.

Analysis and approach

The surveyor divides the route into short setups, keeping BS and FS distances approximately equal. Stable change points are selected on concrete edges away from traffic. An automatic level is tested using the two-peg test before work begins. A digital level is then used for the main run, with a conventional staff and independent check observations at selected points.

The benchmark is observed with a backsight, intermediate points are recorded, and each change point receives a foresight before the instrument is moved. After relocation, the same change point receives a backsight. The route closes on a second known benchmark. The field book is reduced by the height-of-instrument method and checked by the BS–FS relationship.

Outcome

The closing discrepancy is within the project tolerance. The RLs are transferred to durable site marks, which are labelled and protected. Excavation levels are set from these marks rather than from temporary laser readings alone. The use of balanced sights, independent checks and documented benchmarks gives the construction team a defensible vertical control system.

Learning points

The case illustrates that accurate levelling is a complete process: datum selection, instrument testing, stable point selection, careful observation, systematic booking, arithmetic checking and secure transfer of results. Instrument technology improves efficiency, but it does not remove the need for surveying principles.

13. Contemporary Practice and Professional Responsibility

Modern instruments may combine digital levelling, satellite positioning, robotic total stations and cloud-based field records. Satellite positioning is useful for establishing broad vertical control, but satellite-derived heights and conventional datum elevations are not automatically interchangeable. A geoid model and appropriate reference system may be required.

Field data can be synchronized with building-information models and machine-control systems. These workflows increase productivity but also increase the consequences of an incorrect datum, wrong coordinate system or unverified instrument setting. Surveyors should preserve raw observations, maintain instrument calibration records, protect project data and make changes traceable.

Professional responsibility includes protecting the public from unsafe construction levels, reporting suspected errors, refusing to certify unchecked results and following relevant national standards, manufacturer instructions and workplace safety regulations.

Chapter Summary

Levelling determines elevations relative to a datum by observing vertical staff readings from a controlled line of sight. A level surface follows gravity and is curved, while a horizontal line is tangent to that surface at a point. Reduced levels may be computed using the height-of-instrument method or the rise-and-fall method, with arithmetic checks based on backsights, foresights, rises and falls.

Benchmarks provide known elevation control, and change points permit the transfer of elevation between instrument setups. The line of collimation must be horizontal; its accuracy can be assessed by the two-peg test. Curvature and refraction affect long sights, so balanced backsight and foresight distances are essential. Automatic, digital and laser levels each have useful applications, but all require stable setup, correct focusing or sensing, suitable staffs, calibration and independent checks. Accurate levelling is therefore both an observational and a quality-control discipline.

Key Terms

Datum: Reference surface from which elevations are measured.

Level surface: Curved surface everywhere normal to gravity.

Horizontal line: Straight line tangent to a level surface at a point.

Reduced level: Elevation of a point above or below the adopted datum.

Benchmark: Fixed point with a known or established RL.

Temporary benchmark: Project benchmark established for short-term use.

Change point: Stable point used to transfer elevation between setups.

Line of collimation: Corrected line of sight of a levelling instrument.

Height of instrument: RL of the line of collimation.

Backsight: First staff reading from an instrument setup.

Intermediate sight: Reading taken between the backsight and foresight.

Foresight: Final reading from an instrument setup.

Rise: Increase in elevation between consecutive points.

Fall: Decrease in elevation between consecutive points.

Automatic level: Optical level with a compensator that makes the sight line horizontal.

Digital level: Instrument that electronically reads a coded staff.

Laser level: Instrument that projects a visible or detectable reference beam or plane.

Parallax: Apparent movement between staff image and cross-hairs caused by incorrect focusing.

Misclosure: Difference between observed and known closing elevation.

Two-peg test: Field test for line-of-collimation error.

Review Questions

Short Answer Questions

1. What is meant by a datum in levelling?
2. Distinguish between a backsight and a foresight.
3. Why should a levelling staff be held vertical?
4. State the purpose of a change point.
5. What does the two-peg test investigate?

Long Answer Questions

1. Explain level surfaces, horizontal lines, vertical lines and reduced levels, and describe their relationships in surveying.
2. Describe the complete procedure for setting up and using an automatic level.
3. Explain the height-of-instrument method, including its formulas and a suitable field-book arrangement.
4. Discuss curvature and atmospheric refraction and explain how balanced sights reduce their effects.
5. Compare automatic, digital and laser levels in terms of principle, applications, advantages and limitations.

Analytical / Application Questions

1. A benchmark has RL 75.000 m and the backsight is 1.460 m. From the same setup, an intermediate point has a reading of 2.115 m. Calculate the HI and RL of the intermediate point.
2. In a two-peg test, midpoint readings are 1.250 m and 1.890 m. From a setup near the first peg, the first reading is 1.105 m. Calculate the correct reading at the second peg.
3. A levelling route closes on a benchmark with a known RL of 128.640 m, but the observed closing RL is 128.652 m. Explain what the discrepancy represents, list possible causes, and describe the checks that should be completed before any adjustment is made.

Multiple Choice Questions with Answers

Question 1

A benchmark is primarily used to:

A. Measure horizontal angles

B. Provide a point of known elevation

C. Determine magnetic bearing

D. Test staff graduations

Correct Answer: B. Provide a point of known elevation

Question 2

If the RL of a point is 50.000 m and the backsight is 1.250 m, the height of instrument is:

A. 48.750 m

B. 49.250 m

C. 51.250 m

D. 52.500 m

Correct Answer: C. 51.250 m

Question 3

A reading taken first from a new instrument setup is normally called the:

A. Intermediate sight

B. Foresight

C. Backsight

D. Change sight

Correct Answer: C. Backsight

Question 4

If a consecutive staff reading decreases from 1.850 m to 1.220 m, the ground has:

A. Risen by 0.630 m

B. Fallen by 0.630 m

C. Risen by 3.070 m

D. Not changed elevation

Correct Answer: A. Risen by 0.630 m

Question 5

The main purpose of balancing backsight and foresight distances is to:

A. Increase staff length

B. Reduce the effects of collimation, curvature and refraction

C. Eliminate the need for a benchmark

D. Make the staff reading larger

Correct Answer: B. Reduce the effects of collimation, curvature and refraction

Question 6

In the curvature-only correction formula C_c = d² ÷ (2R), d represents:

A. The staff graduation interval

B. The vertical difference between points

C. The horizontal distance from instrument to staff

D. The diameter of the staff

Correct Answer: C. The horizontal distance from instrument to staff

Question 7

A digital level commonly reads a:

A. Coded staff

B. Magnetic staff

C. Plumb bob

D. Steel angle plate

Correct Answer: A. Coded staff

Question 8

The two-peg test is mainly used to check:

A. Tripod leg length

B. Line-of-collimation error

C. Benchmark location

D. Laser colour

Correct Answer: B. Line-of-collimation error

Question 9

A staff held tilted generally produces a reading that is:

A. Always zero

B. Smaller than the true vertical intercept

C. Larger than the true vertical intercept

D. Independent of staff position

Correct Answer: C. Larger than the true vertical intercept

Question 10

If the observed closing RL is 128.652 m and the known closing RL is 128.640 m, the misclosure is:

A. −0.012 m

B. +0.012 m

C. +0.120 m

D. +12.000 m

Correct Answer: B. +0.012 m