Compass Surveying

Introduction to Compass Surveying

Surveying is the science of determining the relative positions of points on or near the Earth's surface. Among various surveying techniques, compass surveying is one of the simplest and most widely used methods, especially for measuring directions and plotting traverses in civil engineering projects.

Compass surveying involves using a magnetic compass to measure the bearing or direction of survey lines relative to the Earth's magnetic field. This method is essential for tasks such as road alignment, boundary marking, and preliminary site layouts.

Before diving into the practical aspects, it is important to understand the difference between true north and magnetic north. True north refers to the direction along the Earth's surface towards the geographic North Pole. In contrast, magnetic north is the direction that a magnetic compass needle points, towards the Earth's magnetic pole, which is not exactly the same as the geographic pole.

The angle between true north and magnetic north at a particular location is called the magnetic declination. Since compass surveying measures bearings relative to magnetic north, understanding and accounting for this difference is crucial for accurate surveying.

In this chapter, you will learn how to use a compass to measure bearings, conduct traverses, calculate coordinates, and handle errors such as local attraction. These skills form the foundation for more advanced surveying techniques used in civil engineering.

Magnetic Bearings and Compass Use

A bearing is the angle between a survey line and a reference direction, usually magnetic north, measured clockwise from the north line. Bearings are expressed in degrees from 0° to 360°.

In compass surveying, bearings are measured using a prismatic compass, which allows the surveyor to sight a distant object and read the bearing directly from the compass card.

Bearings are categorized as:

Fore Bearing (FB): The bearing of a line when moving from the starting point to the next point.
Back Bearing (BB): The bearing of the same line when moving in the opposite direction, from the next point back to the starting point.

Fore and back bearings are related by adding or subtracting 180°, adjusted for the quadrant. This relationship helps in checking the accuracy of measurements.

How to read a prismatic compass:

Hold the compass level and sight the object through the sighting vane.
Look through the prism to read the bearing indicated by the magnetic needle on the compass card.
Record the bearing as the fore bearing of the line.
When measuring the back bearing, reverse the direction and read the bearing similarly.

Recording both fore and back bearings helps detect errors and local magnetic disturbances.

Traverse Surveying Using Compass

A traverse is a series of connected survey lines whose lengths and bearings are measured to determine positions of points. Traverses can be:

Open Traverse: Starts at a known point and ends at an unknown point without returning to the start.
Closed Traverse: Starts and ends at the same point or at two known points, forming a closed polygon.

Closed traverses are preferred for accuracy checks because the survey should ideally "close" with no misclosure in coordinates.

Procedure for compass traverse surveying:

Establish survey stations at points along the traverse.
Measure the distance between stations using a chain or tape.
Measure the bearing of each line using the compass.
Record fore and back bearings to check for consistency.
Plot the traverse on graph paper or CAD software using bearings and distances.
For closed traverses, calculate the closing error and apply corrections.

Closing the traverse ensures that the survey is accurate and reliable. The difference between the starting and ending coordinates is called the closing error, which should be minimized.

Calculations in Compass Surveying

Once distances and bearings are recorded, the next step is to calculate the position coordinates of each station relative to a starting point. This involves computing the latitude and departure for each traverse leg.

Latitude (L): The north-south component of a line, positive if northward and negative if southward.
Departure (P): The east-west component of a line, positive if eastward and negative if westward.

These are calculated using trigonometric relations based on the bearing angle \( \theta \) and distance \( D \):

Latitude: \( L = D \times \cos \theta \)

Departure: \( P = D \times \sin \theta \)

After calculating latitudes and departures for all traverse legs, the coordinates of each station can be found by cumulative addition. For closed traverses, the sum of latitudes and departures should ideally be zero. Any discrepancy is the closing error.

To adjust for these errors, Bowditch's rule (also called the compass rule) is applied, which distributes the error proportionally to each traverse leg based on its length.

Line	Bearing (°)	Distance (m)	Latitude (L) (m)	Departure (P) (m)	Correction in L (ΔL) (m)	Correction in P (ΔP) (m)	Corrected L (m)	Corrected P (m)
AB	45	100	70.71	70.71	-	-	-	-
BC	135	80	-56.57	56.57	-	-	-	-
CD	225	90	-63.64	-63.64	-	-	-	-
DA	315	110	77.78	-77.78	-	-	-	-

Note: Corrections are calculated after summing latitudes and departures and applying Bowditch's rule.

Error Handling in Compass Surveying

Errors are inevitable in compass surveying due to instrument imperfections, human mistakes, and environmental factors. One common source of error is local attraction, which occurs when nearby magnetic objects distort the compass needle, causing incorrect bearing readings.

Detecting local attraction:

Compare fore and back bearings of the same line; they should differ by 180°. If not, local attraction may be present.
Check bearings at adjacent stations; inconsistent readings indicate attraction.

Correcting bearings affected by local attraction:

Identify the station(s) affected.
Use bearings from unaffected stations to calculate corrected bearings.
Ignore or avoid using data from stations with severe local attraction.

Other common errors include incorrect recording of bearings, misreading the compass, and calculation mistakes. Regular checks and systematic procedures help minimize these errors.

Formula Bank

Latitude

\[ L = D \times \cos \theta \]

where: \( L \) = Latitude (m), \( D \) = Distance (m), \( \theta \) = Bearing angle (°)

Departure

\[ P = D \times \sin \theta \]

where: \( P \) = Departure (m), \( D \) = Distance (m), \( \theta \) = Bearing angle (°)

Bowditch's Rule Correction for Latitude

\[ \Delta L_i = - \frac{L_{total}}{\sum D} \times D_i \]

where: \( \Delta L_i \) = Correction for latitude of leg \( i \), \( L_{total} \) = Sum of latitudes, \( \sum D \) = Sum of distances, \( D_i \) = Distance of leg \( i \)

Bowditch's Rule Correction for Departure

\[ \Delta P_i = - \frac{P_{total}}{\sum D} \times D_i \]

where: \( \Delta P_i \) = Correction for departure of leg \( i \), \( P_{total} \) = Sum of departures, \( \sum D \) = Sum of distances, \( D_i \) = Distance of leg \( i \)

Area of Closed Traverse (Shoelace Formula)

\[ \text{Area} = \frac{1}{2} \left| \sum (x_i y_{i+1} - x_{i+1} y_i) \right| \]

where: \( x_i, y_i \) = Coordinates of point \( i \)

Example 1: Calculating Bearings and Distances in a Closed Traverse Medium

A closed traverse ABCDA has the following data:

AB: Distance = 100 m, Bearing = 45°
BC: Distance = 80 m, Bearing = 135°
CD: Distance = 90 m, Bearing = 225°
DA: Distance = 110 m, Bearing = 315°

Calculate the latitudes and departures for each leg and determine the closing error.

Step 1: Calculate latitude \( L = D \cos \theta \) and departure \( P = D \sin \theta \) for each leg.

Line	Distance (m)	Bearing (°)	Latitude (m)	Departure (m)
AB	100	45	100 x cos 45° = 70.71	100 x sin 45° = 70.71
BC	80	135	80 x cos 135° = -56.57	80 x sin 135° = 56.57
CD	90	225	90 x cos 225° = -63.64	90 x sin 225° = -63.64
DA	110	315	110 x cos 315° = 77.78	110 x sin 315° = -77.78

Step 2: Sum latitudes and departures:

\( \sum L = 70.71 - 56.57 - 63.64 + 77.78 = 28.28 \) m
\( \sum P = 70.71 + 56.57 - 63.64 - 77.78 = -13.14 \) m

Step 3: Calculate closing error:

Since the traverse is closed, sums should be zero. The non-zero sums indicate closing errors:

Closing error in latitude = 28.28 m
Closing error in departure = -13.14 m

Answer: The closing errors are 28.28 m in latitude and -13.14 m in departure, which must be corrected using Bowditch's rule.

Example 2: Correcting Traverse Using Bowditch's Rule Hard

Using the data from Example 1, apply Bowditch's rule to correct the latitudes and departures for each traverse leg.

Step 1: Sum of distances:

\( \sum D = 100 + 80 + 90 + 110 = 380 \) m

Step 2: Calculate corrections for latitude and departure for each leg using:

\[ \Delta L_i = - \frac{L_{total}}{\sum D} \times D_i = - \frac{28.28}{380} \times D_i \]

\[ \Delta P_i = - \frac{P_{total}}{\sum D} \times D_i = - \frac{-13.14}{380} \times D_i \]

Step 3: Calculate corrections for each leg:

Line	Distance (m)	\(\Delta L_i\) (m)	\(\Delta P_i\) (m)
AB	100	\(- \frac{28.28}{380} \times 100 = -7.44\)	\(- \frac{-13.14}{380} \times 100 = 3.46\)
BC	80	\(- \frac{28.28}{380} \times 80 = -5.95\)	\(- \frac{-13.14}{380} \times 80 = 2.77\)
CD	90	\(- \frac{28.28}{380} \times 90 = -6.70\)	\(- \frac{-13.14}{380} \times 90 = 3.11\)
DA	110	\(- \frac{28.28}{380} \times 110 = -8.19\)	\(- \frac{-13.14}{380} \times 110 = 3.80\)

Step 4: Calculate corrected latitudes and departures:

Line	Original L (m)	\(\Delta L_i\) (m)	Corrected L (m)	Original P (m)	\(\Delta P_i\) (m)	Corrected P (m)
AB	70.71	-7.44	63.27	70.71	3.46	74.17
BC	-56.57	-5.95	-62.52	56.57	2.77	59.34
CD	-63.64	-6.70	-70.34	-63.64	3.11	-60.53
DA	77.78	-8.19	69.59	-77.78	3.80	-73.98

Answer: The corrected latitudes and departures ensure the traverse closes accurately with minimal error.

Example 3: Identifying and Correcting Local Attraction Medium

A surveyor measures the following bearings for line AB:
Fore bearing (FB) = 60°, Back bearing (BB) = 230°. Is there any local attraction? If yes, suggest a correction.

Step 1: Calculate the expected back bearing:

Back bearing should be \( FB \pm 180^\circ \). Since FB = 60°, expected BB = 60° + 180° = 240°.

Step 2: Compare expected and measured BB:

Measured BB = 230°, expected BB = 240°, difference = 10°.

Step 3: Since the difference is not 180°, local attraction is likely affecting the compass.

Step 4: Correction:

Check bearings at adjacent stations to identify which station is affected.
Use bearings from unaffected stations to calculate corrected bearings.
If local attraction is at station B, avoid using bearings from that station or apply correction by averaging.

Answer: Local attraction is present. The surveyor should verify adjacent bearings and apply corrections or avoid using affected data.

Example 4: Open Traverse Surveying with Compass Easy

An open traverse starts at point P and proceeds through points Q and R. The following data is recorded:
PQ: Distance = 120 m, Bearing = 30°
QR: Distance = 150 m, Bearing = 75°
Calculate the latitude and departure of each leg.

Step 1: Calculate latitude and departure for PQ:

\( L_{PQ} = 120 \times \cos 30^\circ = 120 \times 0.866 = 103.92 \) m

\( P_{PQ} = 120 \times \sin 30^\circ = 120 \times 0.5 = 60 \) m

Step 2: Calculate latitude and departure for QR:

\( L_{QR} = 150 \times \cos 75^\circ = 150 \times 0.2588 = 38.82 \) m

\( P_{QR} = 150 \times \sin 75^\circ = 150 \times 0.9659 = 144.89 \) m

Answer:

PQ: Latitude = 103.92 m, Departure = 60 m
QR: Latitude = 38.82 m, Departure = 144.89 m

Example 5: Computing Area Using Coordinates from Compass Survey Medium

The coordinates of a closed traverse ABCD are:
A(0,0), B(70,70), C(10,130), D(-60,60). Calculate the area enclosed by the traverse using the shoelace formula.

Step 1: List coordinates in order, repeating the first point at the end:

A(0,0), B(70,70), C(10,130), D(-60,60), A(0,0)

Step 2: Apply shoelace formula:

\[ \text{Area} = \frac{1}{2} \left| (x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1) - (y_1 x_2 + y_2 x_3 + y_3 x_4 + y_4 x_1) \right| \]

Calculate sums:

\( S_1 = 0 \times 70 + 70 \times 130 + 10 \times 60 + (-60) \times 0 = 0 + 9100 + 600 + 0 = 9700 \)
\( S_2 = 0 \times 70 + 70 \times 10 + 130 \times (-60) + 60 \times 0 = 0 + 700 - 7800 + 0 = -7100 \)

Step 3: Calculate area:

\[ \text{Area} = \frac{1}{2} |9700 - (-7100)| = \frac{1}{2} \times 16800 = 8400 \text{ m}^2 \]

Answer: The area enclosed by the traverse is 8400 square meters.

Tips & Tricks

Tip: Always check fore and back bearings for consistency.

When to use: While recording bearings in the field to detect errors early.

Tip: Use Bowditch's rule for error distribution in closed traverses.

When to use: When there is a small closing error in traverse survey.

Tip: Mark local attraction by comparing bearings at different stations.

When to use: When compass readings seem inconsistent or abnormal.

Tip: Memorize the formula for latitude and departure as \( D \cos \theta \) and \( D \sin \theta \).

When to use: For quick calculation during exams.

Tip: Use a consistent unit system (metric) throughout calculations.

When to use: To avoid conversion errors in problem-solving.

Common Mistakes to Avoid

❌ Confusing fore bearing and back bearing values

✓ Remember back bearing = fore bearing ± 180°, adjusted for quadrant

Why: Students often forget to add or subtract 180° or ignore quadrant rules.

❌ Ignoring local attraction effects

✓ Check bearings at multiple stations and correct or avoid affected points

Why: Local magnetic disturbances cause incorrect readings if unaccounted.

❌ Incorrect sign usage in latitude and departure calculations

✓ Assign positive or negative signs based on direction (N/S, E/W)

Why: Misinterpretation of direction leads to wrong coordinate plotting.

❌ Not closing the traverse properly or ignoring closing error

✓ Always calculate closing error and apply Bowditch's rule if needed

Why: Neglecting closure leads to inaccurate survey results.

❌ Mixing units or inconsistent use of metric system

✓ Use meters and degrees consistently throughout calculations

Why: Unit inconsistency causes calculation errors.

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Compass Surveying

Introduction to Compass Surveying

Magnetic Bearings and Compass Use

Traverse Surveying Using Compass

Calculations in Compass Surveying

Error Handling in Compass Surveying

Formula Bank

Tips & Tricks

Common Mistakes to Avoid

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