Tacheometry and Indirect Measurement

Introduction

Surveying commonly requires the determination of horizontal distance, difference in elevation, and the position of points. Direct measurement with a chain or tape is reliable on accessible, fairly level ground, but it becomes slow or difficult across steep slopes, rivers, ravines, marshes, thick vegetation, and rough terrain. Tacheometry provides a rapid indirect method in which distance and elevation are obtained from angular observations and a staff reading.

The word tacheometry is derived from Greek terms meaning “rapid measurement.” It is especially useful for preliminary route surveys, contouring, topographic mapping, reconnaissance, and checking inaccessible distances. It does not normally replace precise chaining, electronic distance measurement, or differential levelling where high accuracy is required.

Learning Objectives

After studying this chapter, the learner should be able to:

• explain the principle and uses of tacheometry;
• identify the principal instruments and observations used in stadia and tangential methods;
• calculate horizontal distances and elevations from staff readings and vertical angles;
• distinguish between horizontal, inclined, and vertical angles;
• apply additive and multiplying constants in stadia surveying;
• reduce field observations systematically and check their quality;
• select an appropriate method for terrain and survey purpose;
• recognise major sources of error and apply suitable field controls.

1. Principle of Indirect Measurement

In direct measurement, distance is obtained by physically comparing a line with a standard unit, such as a metre tape. In indirect measurement, distance is calculated from another observable quantity. In tacheometry, the observable quantities are generally:

1. the intercept made by the instrument’s stadia hairs on a graduated staff; and
2. the angle of elevation or depression of the line of sight.

A telescope fitted with two additional horizontal hairs is called a stadia telescope. If the upper and lower hairs cut staff readings r₁ and r₂, the staff intercept is:

where s is the intercept in metres and r₁ and r₂ are staff readings in metres. The central hair reading is normally used to determine the elevation of the point sighted.

The basic geometry is that a known angular separation between the stadia hairs corresponds to a staff intercept. For a fixed instrument, the ratio between the distance from the instrument to the staff and the intercept is approximately constant. This allows distance to be calculated without moving along the measured line.

2. Instruments and Terminology

A typical tacheometric observation uses:

• a theodolite or total station with stadia hairs;
• a graduated levelling or tacheometric staff;
• a tripod and plumb or optical centring device;
• a field book or electronic data collector;
• a tape for instrument-height checks and control measurements.

Important terms include:

• Instrument station: the point at which the instrument is set up.
• Staff station: the point where the staff is held vertically.
• Height of instrument (HI): the elevation of the instrument line of sight when the sight is horizontal, calculated as station elevation plus instrument height.
• Line of sight: the straight line through the telescope’s optical axis.
• Staff intercept: the difference between the upper and lower stadia readings.
• Vertical angle (θ): the angle made by the line of sight with the horizontal.
• Slope distance: the distance measured along the inclined line of sight.
• Horizontal distance (D): the horizontal projection of the slope distance.
• Reduced level (RL): the elevation of a point relative to an adopted datum.
• Multiplying constant (K): the stadia constant, usually the ratio of focal length to stadia-hair interval.
• Additive constant (C): the constant arising from the telescope’s optical geometry.

The staff must ordinarily be held vertical. A tilted staff introduces an error because the observed intercept is then not the true vertical staff interval corresponding to the line of sight.

3. Stadia Method

3.1 Horizontal sight

For a horizontal line of sight, the general stadia equation is:

where:

• D = horizontal distance from the instrument’s vertical axis to the staff station;
• K = multiplying constant, dimensionless;
• s = staff intercept, in metres;
• C = additive constant, in metres.

The equation assumes a vertical staff, correctly read stadia hairs, and a calibrated instrument. In many instruments K is approximately 100. C may be small, but it should not be assumed zero unless the instrument is an anallactic instrument or calibration establishes that condition.

For an anallactic telescope:

and therefore:

If K = 100, C = 0, and s = 1.26 m, then:

3.2 Inclined line of sight

When the line of sight makes a vertical angle θ with the horizontal, the standard stadia formula for a vertical staff is:

The corresponding vertical component from the instrument’s line of sight to the central staff reading is:

Here V is positive when the line of sight rises from the instrument toward the staff and negative when it descends. The horizontal distance is D, not the slope distance.

The derivation follows from the inclined geometry. The distance along the line of sight associated with the stadia interval is approximately (Ks + C) when the staff is normal to the line of sight. Resolving this distance horizontally and accounting for the vertical staff projection gives the stated equations. The form is applicable to ordinary stadia observations with a vertical staff and a line of sight inclined at θ.

The reduced level of the staff station is:

where:

• RL_A = reduced level of the instrument station A;
• HI = height of the instrument above A;
• V = vertical component from the instrument line of sight to the central staff point;
• r = central-hair staff reading at B;
• RL_B = reduced level of B.

For a downward sight, V is negative. The equation is therefore safer than memorising separate “rise” and “fall” cases.

3.3 Special case when C = 0

For an anallactic instrument, the equations simplify to:

and

These equations are widely used in practical examples, but the condition C = 0 must be stated. Applying the simplified equations to an instrument with a significant additive constant can produce systematic errors.

4. Stadia Constants and Instrument Calibration

The multiplying constant is theoretically:

where f is the telescope focal length and i is the stadia-hair interval. The additive constant is commonly represented by:

where d is the distance from the objective’s optical centre to the instrument’s vertical axis, subject to the particular optical arrangement. In practice, the constants are best obtained by calibration rather than by relying only on nominal instrument dimensions.

A practical calibration procedure is:

1. Select a level, unobstructed line.
2. Mark several accurately measured distances from the instrument station.
3. Hold the staff vertically at each mark.
4. Observe the horizontal staff intercept s at each distance.
5. Use the relationship D = Ks + C.
6. Determine K and C from the best-fitting straight line of D against s.

With two observations, the constants can be calculated as:

and

The distances must be in the same unit. More than two observations are preferable because they reveal reading mistakes and reduce the influence of random error.

5. Worked Example: Horizontal Stadia Observation

A tacheometer has K = 100 and C = 0.30 m. The upper and lower staff readings on a horizontal sight are 2.865 m and 1.425 m. The reduced level of the instrument station is 250.000 m, the instrument height is 1.500 m, and the central reading is 2.145 m. Determine the horizontal distance and staff-station RL.

First calculate the intercept:

Distance:

Since the sight is horizontal, V = 0. The instrument line-of-sight elevation is:

Therefore:

The staff station is 144.30 m horizontally from the instrument station and has an RL of 249.355 m.

6. Worked Example: Inclined Stadia Observation

An anallactic tacheometer has K = 100. The staff readings are 2.740 m, 1.880 m, and 2.310 m for the upper, lower, and central hairs respectively. The vertical angle is +6°30′. The instrument station RL is 312.400 m and the instrument height is 1.450 m. Find the horizontal distance and RL of the staff station.

The intercept is:

Since C = 0:

Using cos 6°30′ ≈ 0.99357:

The vertical component is:

Since sin 13° ≈ 0.22495:

The height of the instrument line of sight is:

Thus:

The results are approximately D = 84.897 m and RL_B = 321.213 m. The positive vertical angle indicates that the staff point observed is above the instrument line of sight, before subtraction of the staff reading.

7. Tangential Method

The tangential method does not require stadia hairs. A normal theodolite is used to observe two points on a staff, usually separated by a known staff interval. The vertical angles to the two points are measured. Distance is then calculated from trigonometric relationships.

Let the lower and upper staff points be separated vertically by a known interval h. Let their vertical angles from the instrument be θ₁ and θ₂. If both angles are measured from the horizontal and the staff is vertical, the horizontal distance is:

This form applies when the upper point has the larger angle and both observations are taken with consistent sign convention. If one sight is downward, the signed tangent must be used.

If the elevation of the lower staff point is required and its staff reading above the station base is y₁, the vertical difference between the instrument axis and the lower point is:

The lower point RL can then be found by combining V₁, instrument height, and the staff geometry. In practice, the staff readings and the instrument’s vertical circle observations must be clearly recorded so that the sign convention is not lost.

The tangential method is useful when:

• the instrument has no stadia hairs;
• a staff intercept cannot be read reliably;
• the staff must be held at a particular known interval;
• an independent check on stadia results is needed.

Its major weakness is that distance becomes highly sensitive when θ₂ − θ₁ is small. Small angular-reading errors can then create large distance errors. The staff interval should be sufficiently large, and the vertical angles should be observed carefully.

8. Reduction of Field Observations

A professional field-book entry should include station, target point, upper reading, central reading, lower reading, vertical angle, face, instrument height, and remarks. A reduction sheet should show the equations and intermediate quantities rather than only final answers.

A typical stadia reduction sequence is:

1. Compute s = upper reading − lower reading.
2. Identify K and C.
3. Calculate D using the correct inclined or horizontal formula.
4. Calculate V, retaining its sign.
5. Calculate HI from the known station RL and instrument height.
6. Calculate the target RL.
7. Compare check observations or closing results.

A compact reduction table may be organised as follows:

Angles should not be mixed between degrees and radians in calculator settings. Distances and staff readings should use consistent units. Rounding should normally be postponed until the final reported result.

9. Errors, Checks, and Accuracy

Important errors include:

• staff not held vertical;
• incorrect identification of upper, central, or lower readings;
• parallax caused by poor focusing;
• inaccurate vertical-angle observation;
• wrong face or circle-reading interpretation;
• incorrect sign for a depression;
• poorly determined K and C;
• atmospheric refraction over long sights;
• movement of the staff or tripod;
• incorrect instrument height;
• transcription and arithmetic errors.

The following controls are recommended:

• centre and level the instrument carefully;
• focus the cross-hairs and staff separately to remove parallax;
• observe on both faces where accuracy matters;
• keep sights reasonably balanced where possible;
• use reciprocal or check observations across difficult terrain;
• verify that the central reading is approximately midway between upper and lower readings;
• close a route or level loop on a known point;
• compare selected tacheometric distances with tape or EDM measurements;
• record units, signs, station names, and instrument constants in the field.

For a vertical staff, an observed central reading should be close to (r₁ + r₂)/2. A substantial discrepancy suggests a reading, staff, or instrument problem. This is a practical check, not a substitute for calibration.

10. Selection of Method and Applications

Stadia method

The stadia method is generally preferred for rapid topographic work because one sight supplies both distance and elevation information. It is efficient for contouring, reconnaissance, road alignment, canal investigations, and preliminary construction surveys.

Tangential method

The tangential method is appropriate when the instrument lacks stadia hairs or when an independent observation is useful. It can be slower because two vertical-angle observations are required, and its precision may be poor for small angular separations.

Electronic alternatives

Total stations and GNSS receivers have largely replaced traditional tacheometry for many production surveys. A total station directly measures slope distance and angles, then computes coordinates and elevations. Nevertheless, understanding stadia and tangential principles remains important because:

• many instruments retain stadia functions;
• field data must be checked intelligently;
• older survey records may use these methods;
• rapid visual estimates remain valuable in reconnaissance;
• the underlying geometry supports understanding of modern surveying systems.

Digital instruments improve storage and calculation but do not eliminate errors in setup, target identification, instrument constants, atmospheric conditions, or survey control.

11. Illustrative Case: Preliminary Access-Road Survey

Background

A civil engineering team must investigate a possible access road across rolling ground. A chain survey is impractical because of bushes, short ravines, and several private boundaries. The purpose is preliminary alignment and approximate earthwork assessment, not final construction setting-out.

Problem

The team requires horizontal positions and approximate elevations for points along a 600 m corridor. Some points are inaccessible from the proposed centreline, and the ground includes slopes of up to approximately 10°.

Analysis

A stadia theodolite is selected because each observation can provide distance and elevation. Known control points are established at intervals, and the instrument is calibrated before fieldwork. Staff stations are selected where the staff can be held securely and seen clearly.

Approach

At each instrument station, the crew records upper, central, and lower readings, vertical angle, face, instrument height, and station identification. Observations are made on both faces at control points. Horizontal distances and RLs are reduced daily, and selected points are checked with a total station measurement.

For a typical sight with K = 100, C = 0, s = 1.10 m, and θ = −8°, the horizontal distance is calculated from D = Ks cos²θ. The negative angle is retained in the vertical calculation, giving a negative V. This sign correctly indicates that the target is below the instrument line of sight.

Outcome

The survey produces a preliminary longitudinal profile and sufficient cross-section information to compare alternative road corridors. The team identifies a short steep section requiring possible realignment. Because the work is preliminary, the speed and coverage of tacheometry justify its use. Before detailed design, the alignment and levels must be verified using calibrated total-station or GNSS observations and appropriate construction control.

Learning points

The method should be selected according to purpose, required accuracy, terrain, visibility, and available equipment. Rapid indirect measurement is valuable when it saves time without compromising the decision being made. It should not be used to conceal inadequate control or to replace precise measurements required by a contract or regulation.

12. Professional Practice and Safety

Surveyors must respect property boundaries, traffic controls, public access restrictions, and site safety procedures. Instrument operators should not occupy live traffic lanes, unstable slopes, or areas near plant without suitable controls. Staff holders need communication procedures, especially where they cannot see the instrument operator directly.

Survey data are professional records. Observations should be traceable to control points, instrument identification, date, weather conditions, and operator. Digital calculations should preserve raw observations so that reductions can be independently audited. Artificially adjusting observations to obtain a desired alignment or level is unethical and can create serious construction and safety consequences.

Chapter Summary

Tacheometry determines distance and elevation indirectly from staff readings and angular observations. In the stadia method, the staff intercept is the difference between upper and lower stadia readings. For a horizontal sight, D = Ks + C; for an inclined sight with a vertical staff, D = Ks cos²θ + C cosθ and V = (Ks sin 2θ)/2 + C sinθ. The staff-station RL is obtained from the station RL, instrument height, signed vertical component, and central staff reading.

The tangential method uses two vertical angles to staff points separated by a known interval, with D = h/(tanθ₂ − tanθ₁) under the stated sign convention. Stadia is generally faster, while tangential surveying is useful with ordinary theodolites but is sensitive to small angular differences. Accurate calibration, careful observation, correct signs, vertical staff holding, field checks, and closure are essential. Tacheometry is appropriate for rapid preliminary and topographic surveys, but precise control and construction surveys generally require more accurate methods.

Key Terms

Tacheometry: Rapid indirect measurement of distance and elevation using angular and staff observations.

Stadia method: Tacheometric method using the intercept between stadia hairs.

Tangential method: Method using vertical angles to two staff points with known separation.

Stadia hair: Additional telescope cross-hair used to obtain a staff intercept.

Staff intercept: Difference between upper and lower stadia readings.

Vertical angle: Angle between a line of sight and the horizontal.

Horizontal distance: Horizontal projection of the line from instrument to staff.

Slope distance: Distance along an inclined line of sight.

Multiplying constant: Instrument constant K relating staff intercept to distance.

Additive constant: Instrument constant C added to the stadia distance equation.

Anallactic telescope: Telescope for which the additive constant is zero.

Instrument station: Point where the surveying instrument is set up.

Staff station: Point where the graduated staff is held.

Height of instrument: Elevation difference between the station point and instrument line of sight.

Reduced level: Elevation of a point relative to a selected datum.

Vertical component: Signed elevation difference between the instrument line of sight and the observed staff point.

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

Calibration: Determination or verification of instrument constants.

Indirect measurement: Determination of a quantity from related observations rather than direct comparison.

Review Questions

5 Short Answer Questions

1. What is meant by tacheometry?
2. Define staff intercept and state how it is obtained.
3. What is the difference between a horizontal distance and a slope distance?
4. Why must a staff normally be held vertical in stadia surveying?
5. State two practical uses of tacheometry.

5 Long Answer Questions

1. Explain the principle of the stadia method and derive or describe the meaning of the constants K and C.
2. Present the general equations for horizontal distance and vertical component for an inclined stadia sight, defining every variable.
3. Explain the tangential method, including its assumptions, equation, advantages, and limitations.
4. Describe a practical procedure for calibrating the multiplying and additive constants of a tacheometer.
5. Discuss the principal sources of error in tacheometric surveying and the field procedures used to control them.

3 Analytical / Application Questions

1. A tacheometer has K = 100 and C = 0.25 m. The upper and lower readings are 2.950 m and 1.650 m, and the vertical angle is +5°. Determine the horizontal distance using the inclined stadia equation. State all assumptions.
2. A preliminary road survey crosses rough, wooded terrain. Compare stadia tacheometry, tangential surveying, and total-station measurement for the reconnaissance stage and recommend a method with reasons.
3. During reduction, a calculated staff-station RL differs greatly from a check observation, although the distance appears reasonable. Develop a systematic investigation covering field notes, instrument setup, signs, staff readings, constants, and closure checks.

Multiple Choice Questions with Answers

Question 1

For a horizontal stadia sight, which equation gives the distance when K is the multiplying constant and C is the additive constant?

A. D = Ks + C

B. D = Ks cos²θ + C cosθ

C. D = h/(tanθ₂ − tanθ₁)

D. D = Ks sin 2θ

Correct Answer: A. D = Ks + C

Question 2

A stadia observation gives upper and lower readings of 2.865 m and 1.425 m. What is the staff intercept?

A. 1.240 m

B. 1.440 m

C. 3.290 m

D. 4.290 m

Correct Answer: B. 1.440 m

Question 3

For an anallactic instrument, which condition applies?

A. K = 0

B. C = 1

C. C = 0

D. θ = 90°

Correct Answer: C. C = 0

Question 4

In the inclined stadia equation D = Ks cos²θ + C cosθ, D represents which quantity?

A. Slope distance

B. Horizontal distance

C. Vertical component

D. Staff intercept

Correct Answer: B. Horizontal distance

Question 5

If the vertical angle to a staff station is negative, what does this indicate?

A. The staff station is below the instrument line of sight

B. The staff is necessarily upside down

C. The horizontal distance is negative

D. The additive constant is zero

Correct Answer: A. The staff station is below the instrument line of sight

Question 6

Which observations are required by the tangential method?

A. Only the central staff reading

B. Two vertical angles to staff points with known separation

C. Only the instrument height

D. Two horizontal angles to unrelated points

Correct Answer: B. Two vertical angles to staff points with known separation

Question 7

If K = 100, C = 0.30 m, and s = 1.440 m for a horizontal sight, what is D?

A. 14.43 m

B. 100.30 m

C. 144.00 m

D. 144.30 m

Correct Answer: D. 144.30 m

Question 8

Which field condition most directly causes an incorrect stadia intercept?

A. The staff is not held vertically

B. The station has a known RL

C. The instrument is centred carefully

D. The central reading is recorded

Correct Answer: A. The staff is not held vertically

Question 9

What is the principal advantage of stadia tacheometry over direct taping in rough terrain?

A. It always provides construction-grade accuracy

B. It avoids all angular observations

C. It rapidly obtains distance without physically measuring the ground line

D. It requires no instrument calibration

Correct Answer: C. It rapidly obtains distance without physically measuring the ground line

Question 10

In the RL equation RL_B = RL_A + HI + V − r, what must be retained correctly for an inclined sight?

A. The sign of V

B. The colour of the staff

C. The tape length only

D. The weather forecast only

Correct Answer: A. The sign of V