Theodolite Traversing

Traversing is a fundamental surveying technique used to establish a control network of points by measuring angles and distances between them. A theodolite is a precision instrument used for accurately measuring horizontal and vertical angles. In traversing, a series of connected straight lines are run between survey stations, forming a framework that covers the entire area to be surveyed. The process involves setting up the theodolite over each station, measuring the horizontal angles between adjacent lines of the traverse, and measuring the distances between stations.

Types of Traversing

Traversing can be broadly classified into two types based on the angles measured:

• Open Traversing: This type of traverse starts from a point of known coordinates and ends at a point of unknown coordinates, with no closure. It is generally not recommended for precise work due to the accumulation of errors.
• Closed Traversing: This traverse starts from a point of known coordinates, proceeds through a series of stations, and eventually returns to the starting point or to another point of known coordinates. The closure of the traverse provides a check on the accuracy of the measurements.

Theodolite Setup for Traversing

Accurate setup of the theodolite is crucial for precise measurements. The process involves:

1. Centering: The theodolite is positioned directly over the survey station mark using a plumb bob or optical plummet.
2. Levelling: The instrument is made level using the levelling screws and bubble indicators. This ensures that the horizontal circle is truly horizontal.
3. Setting Horizontal Circle: The lower clamp and tangent screw are used to set the vernier to zero or to a specific reading for the first observation.
4. Clamping and Tangenting: The upper clamp and tangent screw are used for fine adjustments when sighting the target.

Measurements in Theodolite Traversing

The primary measurements in theodolite traversing are horizontal angles and distances.

• Horizontal Angles: These are measured between successive lines of the traverse. The theodolite is set up over a station, and angles are read to the backsight station and then to the foresight station. Angles can be measured either as interior angles (inside the traverse) or exterior angles (outside the traverse), or as azimuths (angles measured clockwise from a reference meridian).
• Distances: Distances between stations are measured using methods like chaining, stadia methods (using the theodolite's telescope), or electronic distance measurement (EDM) instruments.

Checks in Traversing

Several checks are performed during and after the traverse measurements to ensure accuracy:

• Sum of Interior Angles: For a closed traverse with 'n' sides, the sum of interior angles should be (2n - 4) × 90 degrees. Any significant deviation indicates errors.
• Sum of Exterior Angles: The sum of exterior angles should be (2n + 4) × 90 degrees.
• Sum of Deflection Angles: The algebraic sum of deflection angles (the angle between the preceding line and the succeeding line) should be equal to 360 degrees.
• Bowditch's Rule and Transit Rule: These are methods used for adjusting the errors in a closed traverse by distributing the misclosure proportionally to the measured distances and angles.

Adjustment of Theodolite

A theodolite, like any precision instrument, requires periodic adjustments to ensure its readings are accurate. These adjustments are made to correct for wear and tear, damage, or improper handling. The goal is to bring the instrument into a state where its fundamental lines and planes are in their correct geometric relationship with each other.

Fundamental Lines and Planes

Understanding these is key to adjustments:

• Line of Sight: The optical axis of the telescope.
• Horizontal Line of Sight: When the line of sight is perpendicular to the horizontal axis.
• Vertical Axis: The axis around which the instrument rotates horizontally.
• Horizontal Axis (Trunnion Axis): The axis around which the telescope rotates vertically.
• Vertical Axis of the Instrument: The line passing through the centre of the upper and lower pivots.
• Horizontal Axis of the Instrument: The axis around which the telescope rotates.
• Collimation Line: The line of sight.

Essential Adjustments

There are two main categories of adjustments:

1. Temporary Adjustments: These are made every time the instrument is set up at a station. They include:
   • Centering
   • Levelling
   • Setting the vernier to zero (or a specific reading)
2. Permanent Adjustments: These are made when the instrument is found to be out of adjustment and are intended to correct the relationships between the fundamental lines. The primary permanent adjustments are:
   • Adjustment of Plate Bubbles (Levelling Head Adjustment): To make the vertical axis truly vertical when the plate bubbles indicate the instrument is level.
   • Adjustment of the Horizontal Axis (Trunnion Axis) to be Perpendicular to the Vertical Axis: This ensures that when the telescope is plunged, it sweeps a vertical plane.
   • Adjustment of the Line of Sight (Collimation) to be Perpendicular to the Horizontal Axis: This ensures that the line of sight is truly horizontal when the telescope bubble is in the centre and the instrument is level.

Procedure for Permanent Adjustments

1. Adjustment of Plate Bubbles

Objective: To make the plate bubbles perpendicular to the vertical axis.

1. Set up the theodolite and level it carefully using the plate bubbles.
2. Rotate the instrument through 180 degrees (plunge the telescope).
3. If the bubble remains central, the plate bubble is in adjustment. If not, use the levelling screws to bring the bubble halfway back to the centre.
4. Now, use the small capstan screws on the plate bubble mount to bring the bubble exactly to the centre.
5. Repeat the process by rotating through 90 degrees and then 180 degrees.

2. Adjustment of the Horizontal Axis (Trunnion Axis)

Objective: To make the horizontal axis perpendicular to the vertical axis.

1. Set up the theodolite and level it.
2. Sight a high point (e.g., the top of a distant pole) with the telescope in the face left position. Clamp the instrument and read the vertical circle.
3. Plunge the telescope and sight the same point again. Clamp and read the vertical circle.
4. If the readings are the same (or differ by 180 degrees), the horizontal axis is perpendicular to the vertical axis.
5. If there is a difference, the horizontal axis is out of adjustment. Let the difference in vertical circle reading be 'd'. The error is d/2.
6. To adjust, loosen the horizontal axis pivot screws and adjust the horizontal axis until the vertical circle reading is correct (i.e., the difference is 180 degrees). This is a delicate adjustment and may require a surveyor's skilled hand.
7. Re-check by sighting a low point and repeating the process.

3. Adjustment of the Line of Sight (Collimation)

Objective: To make the line of sight perpendicular to the horizontal axis (i.e., the telescope bubble is parallel to the line of sight).

This is often done using the two-peg test or the axis-method.

Axis-Method:

1. Set up the theodolite midway between two points (A and B), about 20-30 meters apart.
2. Level the instrument. Set the telescope bubble to be central.
3. Sight point A with the telescope, clamp, and measure the horizontal distance. Let this reading be R1.
4. Plunge the telescope and sight point A again. Clamp and measure the horizontal distance. Let this reading be R2.
5. If R1 = R2, the line of sight is perpendicular to the horizontal axis.
6. If R1 ≠ R2, the line of sight is out of adjustment. Use the collimation screws at the object end of the telescope to adjust the line of sight until the reading is the mean of R1 and R2.
7. Now, move the theodolite to a position close to point A (e.g., about 1-2 meters away).
8. Level the instrument. Set the telescope bubble to be central.
9. Sight point B with the telescope, clamp, and measure the horizontal distance. Let this reading be R3.
10. Plunge the telescope and sight point B again. Clamp and measure the horizontal distance. Let this reading be R4.
11. If R3 = R4, the line of sight is perpendicular to the horizontal axis.
12. If R3 ≠ R4, the line of sight is out of adjustment. Use the collimation screws to adjust the line of sight until the reading is the mean of R3 and R4.
13. Repeat the process until consistent results are obtained.

Levelling

Levelling is the process of determining the relative vertical positions of points on the Earth's surface. It establishes the elevation or height of points above a reference datum, commonly mean sea level. This information is crucial for designing infrastructure like roads, canals, buildings, and for earthwork calculations.

Basic Principles of Levelling

Levelling relies on the principle of establishing a horizontal line of sight and observing the vertical distances to points.

• Horizontal Line of Sight: Achieved using a levelling instrument (like a dumpy level, automatic level, or theodolite with a horizontal telescope) which provides a stable horizontal line of sight.
• Staff Reading: A graduated staff (levelling staff) is held vertically at the point whose elevation is to be determined. The instrument operator sights the staff and reads the value where the horizontal line of sight intersects the staff.
• Vertical Distance: The difference between the instrument's height of collimation (height of the horizontal line of sight above the datum) and the staff reading at a point gives the elevation of that point.

Types of Levelling

Levelling can be performed using various methods depending on the required accuracy and terrain:

• Direct Levelling (Differential Levelling): This is the most common and accurate method. It involves setting up the levelling instrument at intermediate points between the starting and ending points. It is used to determine the difference in elevation between two points that are too far apart or too high to be taken in a single setup.
• Trigonometric Levelling: Uses vertical angle and horizontal distance measurements to calculate elevation differences. It's faster but less accurate than direct levelling, suitable for rough surveys or long distances.
• Barometric Levelling: Uses atmospheric pressure variations measured by a barometer to determine elevation differences. It's quick but very approximate, suitable for reconnaissance surveys.
• GPS Levelling: Utilizes Global Positioning System (GPS) receivers to determine elevations. Accuracy depends on the GPS equipment and signal quality.

Procedure for Differential Levelling

Differential levelling involves a series of instrument setups and staff readings.

1. Setup: Place the levelling instrument on a stable tripod at an intermediate point where it can clearly see both the backsight and foresight stations.
2. Backsight Reading (BS): Clamp the levelling staff vertically on a point of known elevation (or the starting point). The instrument operator sights the staff and records the reading where the horizontal line of sight intersects the staff. This is the backsight reading.
3. Foresight Reading (FS): Move the levelling staff to the next point whose elevation needs to be determined. The instrument operator sights this staff and records the reading. This is the foresight reading.
4. Calculating Elevation: The elevation of the new point is calculated as:

   Elevation of Foresight Point = Elevation of Backsight Point + Backsight Reading - Foresight Reading

5. Shifting the Instrument: If the distance to the next point is too far, or if the terrain is difficult, the instrument is moved forward to a new setup position. A new backsight is taken on the last point (now acting as a backsight station), and a new foresight is taken on the subsequent point.
6. Checks: The accuracy is checked by ensuring that the sum of all backsight readings equals the sum of all foresight readings (assuming the last point's elevation is known, or by comparing the calculated elevation with a known elevation). Alternatively, the sum of BS minus the sum of FS should equal the total difference in elevation between the start and end points.

Types of Levelling Staves

• Slattern Staff: A plain staff with markings, used for short distances.
• Folding Staff: A staff that can be folded, convenient for transport.
• Invar Staff: Made of Invar metal (an alloy of iron and nickel), which has a very low coefficient of thermal expansion. Used for high-precision levelling to minimize errors due to temperature changes.

Contouring

Contouring is the process of drawing lines on a map that connect points of equal elevation. These lines are called contour lines or contours. Contour maps are essential for visualizing the shape and elevation of the terrain, providing vital information for planning and design in various engineering and geographical applications.

Characteristics of Contour Lines

Understanding these characteristics is key to interpreting contour maps:

• Equal Elevation: Every point on a single contour line has the same elevation.
• Contour Interval (CI): The constant vertical distance between successive contour lines. The CI is usually specified on the map and depends on the scale of the map and the nature of the terrain (e.g., smaller CI for flat terrain, larger CI for steep terrain).
• Contour Gradient: The slope of the ground is represented by the horizontal distance between contour lines. Closely spaced contours indicate steep slopes, while widely spaced contours indicate gentle slopes.
• Uniform Slope: Contours are evenly spaced.
• Convex Slope: Contours are widely spaced at the top and close together at the bottom.
• Concave Slope: Contours are close together at the top and widely spaced at the bottom.
• Hill or Elevation: Closed contours with decreasing elevations inwards represent a hill.
• Depression or Hollow: Closed contours with increasing elevations inwards represent a depression. Small depressions within a higher contour may be marked with hachure lines (short lines pointing inwards).
• Ridge: 'V' shaped contours pointing downhill.
• Valley: 'V' shaped contours pointing uphill.
• Separation of Contours: Contour lines never cross each other, except in the rare case of an overhanging cliff, where they might appear to merge.
• Continuity: Contour lines are continuous. They either close on themselves within the map or extend to the edge of the map.

Methods of Contouring

There are two primary methods for obtaining data for contouring:

1. Direct Method: This involves directly locating points of equal elevation in the field using levelling instruments.
   • Using a Level and Staff: The surveyor establishes a grid of points over the area and determines the elevation of each grid intersection using differential levelling. These elevations are then plotted on a plan, and contour lines are interpolated by connecting points of equal elevation.
   • Using a Hand Level or Abney Level: For less accurate work, a hand level can be used to walk along lines of constant elevation.
2. Indirect Method: This method involves surveying the area using other methods (like traversing or tacheometry) to determine the horizontal positions (X, Y coordinates) and elevations (Z values) of a sufficient number of points. These points are then plotted on a plan, and contour lines are interpolated.
   • Grid Method: Points are established at regular intervals (forming a grid), and their elevations are determined. Contour lines are then drawn by interpolation.
   • Tacheometric Method: Tacheometry directly provides both horizontal distance and difference in elevation from instrument stations, making it efficient for contouring.
   • Photogrammetry: Using aerial photographs to create stereoscopic models from which elevations and contours can be derived.

Interpolation of Contours

Interpolation is the process of estimating elevations or determining the position of contour lines between surveyed points whose elevations are known.

• Graphical Interpolation: This is the most common method.
  • On a Plain Sheet: Points of known elevation are plotted on a map. To find the position of a contour line (e.g., 100m), locate points on lines connecting known elevations that represent 100m. For example, if point A is 95m and point B is 105m, the 100m contour line will intersect the line AB at its midpoint.
  • Using Profile Paper: A profile of the ground is drawn on profile paper, and contour lines are plotted directly on it.
  • Using a Contour Template: A special ruler with calibrated scales can be used to speed up interpolation.
• Arithmetical Interpolation: This involves calculating the exact position of a contour line between two points using proportional division. If point P has elevation E_P and point Q has elevation E_Q, and the desired contour elevation is E_C, the distance 'd' along PQ where the contour line lies can be calculated:

  d = Distance PQ × (E_C - E_P) / (E_Q - E_P)

Uses of Contouring

• Engineering Projects: Planning roads, railways, canals, dams, and pipelines by visualizing terrain and calculating earthwork volumes.
• Military Operations: Understanding terrain for tactical advantage.
• Hydrology: Studying water flow and drainage patterns.
• Agriculture: Planning irrigation and drainage systems.
• Geology: Mapping geological formations.

Curvature and Refraction Corrections

When surveying over long distances, the curvature of the Earth and atmospheric refraction can significantly affect the accuracy of measurements, particularly in levelling. Corrections must be applied to account for these effects.

Effect of Earth's Curvature

The Earth is not flat; it is approximately spherical. When using a level instrument, the line of sight is straight, but the true horizontal level at the instrument's position curves away from the Earth's surface. This causes the observed staff reading to be lower than the true reading, meaning the calculated elevation will be higher than it should be.

The error due to curvature (c_c) is approximately proportional to the square of the distance (d).

For a distance 'd' in kilometers, the curvature correction is approximately:

c_c = 0.0785 d² meters

Where 'd' is the distance in kilometers.

If 'd' is in meters, the formula is:

c_c = d² / (2 * R)

Where R is the radius of the Earth (approximately 6371 km).

This correction is always subtractive from the observed staff reading because the observed reading is less than the true reading.

Effect of Atmospheric Refraction

Atmospheric refraction is the bending of light rays as they pass through layers of air with different densities, caused by variations in temperature and pressure. Near the Earth's surface, the air is generally denser, causing light rays to bend downwards towards the Earth. In levelling, this bending makes the distant object appear slightly higher than it actually is. Therefore, the observed staff reading is higher than it would be if the line of sight were perfectly straight.

The error due to refraction (c_r) is also approximately proportional to the square of the distance.

For a distance 'd' in kilometers, the refraction correction is approximately:

c_r = 0.0112 d² meters

Where 'd' is the distance in kilometers.

This correction is always additive to the observed staff reading because the observed reading is more than the true reading.

Combined Correction for Curvature and Refraction

In most practical levelling operations, the effects of curvature and refraction occur simultaneously. The bending of the light ray due to refraction partially counteracts the effect of Earth's curvature. The combined effect is usually taken as the difference between the two corrections.

Combined correction (c_cr) = c_c - c_r

For a distance 'd' in kilometers:

c_cr = 0.0785 d² - 0.0112 d² = 0.0673 d² meters

This combined correction is always subtractive from the observed staff reading, as the effect of curvature is greater than that of refraction. This correction applies to the foresight reading.

Application in Levelling

These corrections are most significant in:

• Long Sight Distances: When the backsight and foresight distances are large and unequal.
• High Precision Levelling: Where very accurate elevation differences are required.

In standard differential levelling, the instrument is typically set up midway between the backsight and foresight stations to equalize the distances. When BS and FS distances are equal (d_bs = d_fs), the errors due to curvature and refraction at both points are equal and opposite, thus cancelling each other out. Therefore, no correction is needed.

However, if the distances are unequal, the difference between the BS and FS corrections must be applied.

Let d_bs be the distance to the backsight and d_fs be the distance to the foresight.

Combined correction for BS = 0.0673 d_bs² (subtractive from observed BS)

Combined correction for FS = 0.0673 d_fs² (subtractive from observed FS)

The net correction to be applied to the calculation of elevation is:

Net Correction = (Correction for BS) - (Correction for FS)

= (0.0673 d_bs²) - (0.0673 d_fs²)

= 0.0673 (d_bs² - d_fs²) meters

If d_fs > d_bs, the net correction is negative, meaning the calculated elevation of the foresight point will be lower than it should be, and a negative correction (subtraction) is needed.

If d_bs > d_fs, the net correction is positive, meaning the calculated elevation of the foresight point will be higher than it should be, and a positive correction (addition) is needed.