Compass and Plane-Table Surveying

Introduction and context

Compass and plane-table surveying are two traditional field methods used to establish the relative positions of points. Compass surveying determines directions, expressed as bearings, and combines them with measured distances. Plane-table surveying plots observations directly in the field, allowing the surveyor to see the developing map while measurements are being made. Although total stations, GNSS receivers and digital levels are now common, these methods remain important for basic site surveys, reconnaissance, small-area mapping, educational exercises and situations where simple, low-cost equipment is appropriate.

Compass surveying is particularly useful where the area is moderately open and angular precision of about 15′ to 30′ is acceptable. Plane-table surveying is valuable for rapid topographic detail, provided that the board can be held steadily and the plotted points are visible from the instrument station.

Learning objectives

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

1. Explain meridians, bearings, azimuths, included angles and declination.
2. Measure and convert whole-circle and quadrantal bearings.
3. Detect and correct local attraction.
4. Calculate included angles and check a traverse.
5. Describe the principles and field procedures of radiation, intersection, traversing and resection with a plane table.
6. Select suitable methods, identify errors and maintain professional field records.

1. Fundamental directional concepts

1.1 Meridians

A meridian is a reference direction used for measuring horizontal angles. Important types are:

• True meridian: the north–south line through a point determined from the geographic poles.
• Magnetic meridian: the direction indicated by the north-seeking end of a freely suspended magnetic needle.
• Arbitrary meridian: a convenient assumed direction used for a small local survey when connection to true or magnetic north is unnecessary.
• Grid meridian: the north direction of a map projection grid.

The angle between a true meridian and a magnetic meridian is called magnetic declination. Declination is east when magnetic north lies east of true north and west when it lies west of true north. It varies with location, time and magnetic disturbance; therefore, a stated date and reference location are important when declination is used.

1.2 Bearings and azimuths

A bearing is the horizontal angle of a line measured from a specified meridian. In the whole-circle bearing (WCB) or azimuth system, angles are measured clockwise from north and range from 0° to less than 360°.

A quadrantal bearing (QB), also called reduced bearing, is measured from north or south toward east or west and ranges from 0° to 90°. Its form is, for example, N 35° E or S 20° W.

Conversions are:

• NE quadrant: WCB = QB.
• SE quadrant: WCB = 180° − QB.
• SW quadrant: WCB = 180° + QB.
• NW quadrant: WCB = 360° − QB.

For a line observed in both directions, the back bearing differs from the fore bearing by 180°:

Here FB is the fore bearing and BB is the back bearing. The rule applies when both observations refer to the same meridian and no disturbance has affected the instrument. In quadrantal notation, interchange north and south and also east and west; for example, the back bearing of N 40° E is S 40° W.

Figure: A line in the north-east quadrant expressed as both azimuth and quadrantal bearing.

1.3 Included angles

The included angle at a traverse station is the angle between the two lines meeting at that station. If the bearings of the incoming line and outgoing line are known, the angle can be calculated from the appropriate directions. A safe general procedure is to convert both lines to azimuths directed outward from the station, subtract them and take the smaller positive angle:

Here A₁ and A₂ are azimuths of the two rays from the station and θ is the included angle, normally between 0° and 180°. The direction of each ray must be identified; using a back bearing as though it were a fore bearing is a common error.

Example 1. At station B, the bearing of BA is 225° and the bearing of BC is 115°. The included angle ABC is:

θ = |225° − 115°| = 110°.

Thus, angle ABC is 110°. If the bearing of AB rather than BA had been supplied, it would first have to be reversed.

2. The magnetic compass and field procedure

A prismatic compass generally includes a magnetic needle, graduated ring, prism, sighting slit and sighting vane. The graduated ring is commonly attached to the needle and is read through the prism. A surveyor sights the ranging rod through the vanes, allows the needle to settle, and reads the bearing.

A typical compass traverse proceeds as follows:

1. Reconnoitre the site and select stations so that adjacent stations are intervisible.
2. Mark stations with pegs and prepare a sketch showing station names and obstacles.
3. Centre the compass over the station, level it approximately and release the needle.
4. Sight the forward station and record the fore bearing.
5. Measure the line distance with a chain or tape, noting slope, obstacles and offsets.
6. At the next station observe the back bearing of the preceding line and the fore bearing of the next line.
7. Repeat until the traverse closes or reaches a known control point.
8. Observe check lines and repeat questionable readings.

The compass should not be placed near steel ranging rods, vehicles, fences, electric cables, magnets or mobile equipment. The observer should record station, line, bearing, distance, weather and remarks immediately in a field book.

2.1 Local attraction

Local attraction is a deflection of the magnetic needle caused by nearby magnetic materials or electric currents. It is indicated when the difference between fore and back bearings of a line is not 180°.

Let the observed fore bearing and back bearing of a line be F and B. Their discrepancy is:

The sign is selected after reducing the result to the smallest signed angular discrepancy. A discrepancy different from zero indicates disturbance at one or both stations. The amount of attraction at a station cannot always be identified uniquely from one line alone, so correction requires a systematic comparison of connected lines.

A practical correction method is:

1. Find a line whose fore and back bearings differ by exactly 180°, or whose stations are known to be free from attraction.
2. Treat its end stations as unaffected, if justified.
3. Compare the observed bearing of each connected line with the corrected bearing and determine the station correction.
4. Apply the station correction to all bearings observed at that station.
5. Check every corrected line: its fore and back bearings must differ by 180°.

If C is the correction at a station, then:

Both values are reduced to the range 0° to less than 360°. A positive C means the observed reading is increased. This is an observational correction, not a universal constant.

Example 2. The observed fore bearing of AB is 64° and its observed back bearing BA is 247°. The expected back bearing is 244°, so the discrepancy is +3°. Suppose station A is established as unaffected. The observed bearing at B is therefore 3° greater than its correct value, and the correction at B is −3°. If the observed bearing BC at B is 132°, its corrected value is 129°.

Local attraction should not be confused with declination. Declination affects the relation between magnetic and true north over a region; local attraction is a local instrumental or environmental disturbance.

2.2 Declination and true bearings

If magnetic bearings are corrected for declination, the conventional relationship is:

The signs assume bearings are expressed as clockwise azimuths and declination is represented algebraically, east positive and west negative. This correction is valid only when local attraction has first been removed and the declination value corresponds to the survey date and location. In precise work, the current official magnetic model or an astronomical/GNSS reference should be used.

3. Compass traverse computation and checks

A traverse may be open or closed. An open traverse does not return to its starting point and is difficult to check internally. A closed traverse returns to its starting point or ends at a known control point, permitting angular and positional checks.

For a closed polygon with n stations, the theoretical sum of interior angles is:

This relation assumes a simple plane polygon with no crossing sides. The angular misclosure is the observed sum minus the theoretical sum. Distribution of any correction must follow the survey specification; equal distribution is appropriate only when observations have approximately equal reliability.

For coordinate computation, an azimuth A measured clockwise from north gives:

where L is the horizontal length, latitude is the northing component and departure is the easting component. North and east components are positive; south and west components are negative. Angles must be evaluated in degrees if the field bearings are in degrees.

For a closed traverse:

In practice, small residuals occur because of measurement errors. A common proportional adjustment is the Bowditch rule:

Here Lᵢ is the length of side i and ΣL is the total traverse length. The rule assumes angular and linear errors are distributed approximately in proportion to line length. It should not replace an adjustment method required by a project specification.

4. Plane-table surveying

A plane table is a drawing board mounted on a tripod. It is used with an alidade, plumbing fork with plumb bob, spirit level, trough compass and drawing sheet. The alidade provides a straightedge for sighting and drawing rays. Unlike compass surveying, the principal observations are plotted graphically and directions are established by orienting the board.

The essential operations are:

• Setting: fixing the board over the station, levelling it and attaching the sheet.
• Centring: placing the plotted station point directly over the ground station.
• Orientation: rotating the board so that lines on the sheet are parallel to the corresponding ground lines.
• Sighting and plotting: placing the alidade on the station point, sighting a target and drawing a ray; a measured or scaled distance then locates the target.

Orientation may be achieved by backsighting a previously plotted line or by using a trough compass. Backsighting is generally more reliable because it reproduces the established direction and is less affected by magnetic disturbance.

4.1 Radiation

In radiation, all details are observed from one table station. The alidade is placed on the station point, each object is sighted, and a ray is drawn. Distances from the station to the objects are measured and plotted to scale along the rays.

Radiation is rapid and suitable for small areas with a central commanding station. Its limitations are that all points must be visible from one station and distance errors increase with poor ranging or unsuitable scale.

4.2 Intersection

In intersection, the position of an inaccessible point is fixed by rays drawn from two known table stations. The distance between the stations is measured or already known. After plotting the first station and orienting at it, rays are drawn to the unknown point. The board is then shifted to the second station, centred and oriented by backsighting the first station; a second ray is drawn. Their intersection gives the plotted position.

Intersection is useful for locating towers, trees, buildings or across-water points where direct measurement is impossible. The angle of intersection should preferably be neither very acute nor nearly 180°; an angle near 90° normally gives stronger graphical geometry.

4.3 Traversing

Plane-table traversing is used when a series of connected stations is required. At each station, the line to the next station is drawn by sighting, the distance is plotted to scale, and the table is moved forward. The previous line is used for backsight orientation. A closing line or known control point provides a check.

Errors may accumulate through inaccurate centring, imperfect orientation, unstable supports and poor distance measurement. For a closed traverse, the plotted endpoint should coincide with the starting point or known closing point within the specified tolerance.

4.4 Resection

Resection determines the position of the plane table at an unknown station by sighting known points whose positions are already plotted. It is useful when the table is set up at a point that was not previously surveyed.

• Two-point problem: two known points are sighted. The solution requires an additional geometric construction or auxiliary station because two rays alone may not establish a unique, reliable orientation.
• Three-point problem: three known points are sighted. Their rays should meet at the unknown station point when the table is correctly oriented. The graphical solution may use trial orientation, tracing-paper methods, or the Lehmann method.

In the Lehmann method, the plotted position of the instrument station is adjusted by considering the triangle formed by the three known points and the corresponding sight rays. The “triangle of error” indicates incorrect orientation: its size and position guide small rotations and translations until the rays nearly concur. The method requires careful, small adjustments rather than arbitrary movement of the board.

Resection is sensitive to the geometry of known points. If the station lies near the circumcircle of the three known points, the solution becomes weak and small sighting errors can cause a large position error. A well-conditioned arrangement surrounds the unknown station with known points at useful angular separation.

5. Plane-table errors and precautions

Errors are classified as:

• Instrumental: warped board, defective alidade edge, loose tripod, inaccurate level or faulty plumbing fork.
• Personal: inaccurate centring, poor sighting, incorrect orientation, thick lines and careless scale plotting.
• Natural: wind, rain, unstable ground, heat distortion and poor visibility.

Best practice includes using a stable tripod, keeping the sheet dry and firmly attached, checking the alidade edge, using a sharp pencil, avoiding excessive line thickness, backsighting whenever possible, and closing the work on known control. The scale must suit the required detail; for example, a 1:1,000 plan represents 1 cm as 10 m, whereas a 1:5,000 plan represents 1 cm as 50 m.

Plane-table work is graphical and should not be used where high angular precision, legal boundary definition or strict digital-coordinate compatibility is required without independent control. Compass work is also unsuitable near strong magnetic interference or where precise orientation is essential.

6. Applied case: preliminary mapping of a drainage improvement site

Background. A civil works team must prepare a preliminary plan for a 0.8-hectare drainage improvement site containing a shallow channel, access road, trees and a small footbridge. Dense vegetation prevents a complete direct measurement, but most features are visible from two open locations.

Problem. The team needs a rapid plan showing the channel alignment, bridge, road edges and trees before detailed design. A total station is unavailable for the first reconnaissance visit.

Analysis. A compass traverse can establish a basic control line along the accessible road. A plane table can then plot visible details. Magnetic interference is checked by comparing fore and back bearings near the site; a metal fence is found to affect one station.

Approach. The surveyors establish a short closed compass traverse around the open perimeter, reject the disturbed station for primary orientation, and correct bearings using an unaffected line. They select a plane-table station with clear visibility to the bridge and channel. Radiation is used for nearby trees and road corners. Intersection from two stations locates the inaccessible bridge ends. The board is oriented by backsighting, and the final plan is checked against the closing traverse.

Outcome. The preliminary plan identifies the drainage centreline, access constraints and bridge position sufficiently for a detailed survey design. The team records that the plan is reconnaissance-grade, not a legal boundary survey. Later, the control is verified with a total station before construction setting-out.

Learning points. Method selection depends on visibility, required precision, accessibility and environmental interference. Independent checks are essential. A simple instrument can support sound engineering decisions when its limitations are understood and documented.

7. Contemporary practice and professional responsibility

Modern field crews often combine conventional methods with GNSS, total stations, digital field books and geographic information systems. A compass or plane table can provide a useful independent check on a digital survey, while digital coordinates can improve orientation and final drafting. AI-assisted mapping may help classify imagery or detect inconsistent measurements, but it cannot replace field verification, control checks or professional judgement.

Surveyors must protect boundary evidence, preserve original observations, identify assumptions and report uncertainty. Altering field notes, concealing local attraction or presenting reconnaissance plotting as precise construction control is professionally unacceptable. Equipment should be calibrated, records dated and coordinate references clearly stated.

Chapter Summary

Compass surveying measures directions relative to a meridian and combines bearings with distances. Whole-circle bearings run clockwise from north, while quadrantal bearings are measured from north or south toward east or west. Fore and back bearings differ by 180° when observations are undisturbed. Declination relates magnetic and true north; local attraction is a local disturbance and must be detected and corrected separately.

Included angles require correct identification of the two rays at a station. Closed traverses provide angular and coordinate checks, and latitude and departure calculations permit plotting and adjustment. Plane-table surveying plots observations directly in the field. Radiation, intersection, traversing and resection serve different geometric and accessibility conditions. Accurate centring, levelling, orientation, stable equipment, suitable scale and independent closure checks are central to reliable work. These methods remain valuable when applied at an appropriate precision and with transparent professional records.

Key Terms

Meridian: A reference north–south direction for measuring horizontal angles.

True meridian: The geographic north–south direction.

Magnetic meridian: The direction indicated by a magnetic needle.

Arbitrary meridian: An assumed reference direction for a local survey.

Bearing: The horizontal direction of a line measured from a meridian.

Azimuth: A whole-circle bearing measured clockwise from north.

Quadrantal bearing: A bearing measured from north or south toward east or west.

Fore bearing: Bearing of a line in the direction of survey.

Back bearing: Bearing of the same line in the reverse direction.

Declination: The angle between true and magnetic meridians.

Local attraction: Local magnetic disturbance affecting a compass needle.

Traverse: A connected series of survey lines.

Latitude: North–south component of a traverse line.

Departure: East–west component of a traverse line.

Plane table: A field drawing board used for direct plotting.

Alidade: A sighting device and straightedge used with a plane table.

Radiation: Plane-table plotting from one station by drawing rays.

Intersection: Locating a point by intersecting rays from two stations.

Resection: Fixing the table position by sighting known plotted points.

Orientation: Aligning the plane table with previously established directions.

Review Questions

Short Answer Questions

1. Define true meridian and magnetic meridian.
2. What is local attraction, and how can it be detected?
3. State the relationship between a fore bearing and its back bearing.
4. What is the purpose of orienting a plane table?
5. Name two situations in which plane-table intersection is useful.

Long Answer Questions

1. Explain whole-circle and quadrantal bearings, including their conversion rules.
2. Describe a systematic procedure for detecting and correcting local attraction.
3. Explain compass traversing, including field observations and closure checks.
4. Describe radiation, intersection, traversing and resection in plane-table surveying.
5. Discuss the principal sources of error in plane-table surveying and the precautions used to control them.

Analytical / Application Questions

1. The fore bearing of AB is 72° and the back bearing is 254°. Determine whether local attraction is indicated and explain what additional observation is required before assigning station corrections.
2. At station C, the bearings of CA and CD are 238° and 46°, respectively. Calculate the smaller included angle ACD and state the directional rays used.
3. A bridge cannot be reached because of a flooded channel, but it is visible from two accessible plane-table stations whose plotted separation is known. Select the appropriate method, describe the field sequence and identify two checks that should be made.

Multiple Choice Questions with Answers

Question 1

A whole-circle bearing is measured clockwise from:

A. South

B. East

C. North

D. West

Correct Answer: C. North

Question 2

The back bearing of 137° is:

A. 43°

B. 227°

C. 317°

D. 497°

Correct Answer: B. 227°

Question 3

The quadrantal bearing equivalent to a whole-circle bearing of 225° is:

A. S 45° W

B. S 45° E

C. N 45° W

D. N 45° E

Correct Answer: A. S 45° W

Question 4

A fore bearing of 64° and back bearing of 247° indicate that the expected back bearing differs by:

A. 0°

B. 3°

C. 7°

D. 13°

Correct Answer: B. 3°

Question 5

Which condition most directly indicates local attraction?

A. A line is longer than the scale allows

B. Fore and back bearings do not differ by 180°

C. The traverse has four stations

D. The board is not perfectly horizontal

Correct Answer: B. Fore and back bearings do not differ by 180°

Question 6

For an azimuth A and horizontal distance L, the departure is:

A. L cos A

B. L tan A

C. L sin A

D. L ÷ sin A

Correct Answer: C. L sin A

Question 7

Which plane-table method locates a point by rays from two known stations?

A. Radiation

B. Intersection

C. Resection

D. Levelling

Correct Answer: B. Intersection

Question 8

In plane-table radiation, the plotted position of a detail is obtained by:

A. Drawing a ray and plotting the measured distance to scale

B. Using only a magnetic declination table

C. Measuring vertical angles only

D. Setting the board at an unknown point and sighting three controls

Correct Answer: A. Drawing a ray and plotting the measured distance to scale

Question 9

Which operation aligns a plane table using a previously plotted line and a backsight?

A. Centring

B. Levelling

C. Orientation

D. Chaining

Correct Answer: C. Orientation

Question 10

The theoretical sum of interior angles of a five-sided closed traverse is:

A. 360°

B. 540°

C. 720°

D. 900°

Correct Answer: B. 540°