Theodolite, Compass & Traverse Surveying | Geomatics Engineering (Surveying) - Civil Engineering (CE) PDF Download

Traverse Surveying

Latitude and Departure

The latitude and departure are the rectangular components of a survey line when resolved along the true north-south and east-west axes respectively. For a line AB of length l and reduced bearing (or azimuth) θ, the components are

L = +l cos θ

D = l sin θ

Latitude: Projection of a line on the north-south direction.

Departure: Projection of a line on the east-west direction.

Latitude and Departure in Various Quadrants

The sign of latitude and departure depends on the quadrant in which the line lies. The algebraic signs for L and D must be applied consistently when summing components for a traverse. Diagrams that show the four quadrants and the corresponding signs of latitude and departure are given below.

Here, l₁, l₂, l₃ and l₄ are the lengths of OA, OB, OC and OD respectively.

Independent Coordinate and Closing Error

When a traverse is computed by resolving each line into latitudes and departures and summing those components, the coordinates of stations are obtained independently in northing and easting. If the algebraic sums of latitudes or departures for a closed traverse are not zero, a closing error exists.

The closing error in latitude and departure may be shown symbolically (see figures or formula images below).

Conditions for a Correct Closed Traverse

• Sum of all internal angles of a closed traverse = (2n - 4) × 90°, where n is the number of sides.
• Sum of all exterior (or deflection) angles = 360°. That is, θ_A + θ_B + θ_C + ... = 360° for a closed polygonal traverse.
• Algebraic sum of latitudes, ∑L = 0, and algebraic sum of departures, ∑D = 0, for a properly closed and computed traverse.

Adjustment (Balancing) of Closing Error

When a closing error is present it must be distributed among the traverse sides. Common methods are described below.

1. Bowditch (Compass) Rule

   The Bowditch method (also called the Compass rule) distributes the total linear error in latitude and departure to each line in proportion to the length of that line. It is based on the practical observation that the error in linear measurement is proportional to √l and the error in angular measurement is proportional to 1/√l, so the dominant source of coordinate error is proportional to length.

   

   Notation used in the formula image:

   • l = length of the line
   • C_L = correction in latitude for that line
   • ∑L = total error in latitude (algebraic)
   • C_D = correction in departure for that line
   • ∑D = total error in departure (algebraic)
   • ∑l = sum of lengths of all traverse lines (absolute)
2. Transit Rule

   The transit rule distributes the total error in latitude and departure to each line in proportion to the absolute value of its latitude and departure respectively.

   Corrections are given by

   C_L = (L / L_T) × ∑L

   C_D = (D / D_T) × ∑D

   Where

   • ∑L = total algebraic error in latitude
   • ∑D = total algebraic error in departure
   • L = latitude of the particular line (algebraic)
   • D = departure of the particular line (algebraic)
   • L_T = sum of absolute values of all latitudes (i.e., without sign)
   • D_T = sum of absolute values of all departures (i.e., without sign)
3. Axis (Method of Meridians or Polar Method)

   The axis method adjusts traverse coordinates by projecting the network onto chosen axes and applying corrections so that the final coordinates meet closure conditions. The formula images below show corrections to a particular length or coordinate produced by axis adjustments.

   
   
   

Triangulation

Definition: Horizontal control for geodetic surveys is often established either by triangulation or by a precise traverse. In triangulation a network of inter-connected triangles is observed: the length of only one line (the base line) is measured directly and the angles of the triangles are measured precisely to obtain the remaining sides by trigonometry.

Criterion of Strength of Figure

The strength of figure is a measure used when designing a triangulation network; it indicates how well the figure will withstand observational errors and still permit required precision in computed sides. Quantitative expressions are used to estimate probable errors in computed sides based on the number of directions observed and the angular precision.

A relation given in some treatments expresses the square of a probable error in the logarithm of a side in terms of observational quantities:

L₂ = (4/3) d² R

Where

• d = probable error of an observed direction (in seconds)
• D = number of directions observed (forward and/or backward)
• δ_A = effect in the sixth place of the logarithm of sine of the distance angle A of each triangle
• δ_B = similar effect for the other distance angle B
• C = number of independent conditions from angles and sides, given by C = (n′ - s′ + 1) + (n - 2s + 3)
• n = total number of lines
• n′ = number of lines observed in both directions
• s = total number of stations
• s′ = number of occupied stations
• (n′ - s′ + 1) = number of angle conditions
• (n - 2s + 3) = number of side conditions

Signals and Towers

A signal is a device or mark erected to define the exact position of an observed station in triangulation. Signals and towers must be designed to be visible at the required sighting distance and to minimise errors of bisection (phase).

• Non-luminous signals: For non-luminous targets (daytime, unlit), approximate practical dimensions are given by
  • Diameter of signal (cm) = 1.3 D to 1.9 D
  • Height of signal (cm) = 13.3 D

  where D is the distance of sight in kilometres.

• Luminous (sun) signals: These are used when sight distances are very large (for example, when distance of sight > 30 km) or when enhanced contrast is required for observation.
• Phase of signals: Phase is the error of bisection which arises when a signal is partly in sunlight and partly in shade; the observer may misidentify the geometrical centre. Phase corrections are applied to observed directions when necessary.

Phase correction when the observation is made on the bright portion is illustrated below. Symbols used in the figure are explained thereafter.

Where

• α = angle which the direction of the sun makes with the line of sight
• r = radius of the signal
• D = distance of sight

When the observation is made on the bright line (edge) of the illuminated part of the signal, the appropriate correction diagram is shown below.

Routine of a Triangulation Survey

The general routine in a triangulation survey normally includes the following operations in order:

1. Reconnaissance (selection of stations and figure)
2. Erection of signals and towers
3. Measurement of base lines
4. Measurement of horizontal angles at stations
5. Astronomical observations for azimuths or direction control
6. Computations (adjustment, reduction and plotting)

Intervisibility and Height of Stations

1. Distance to the visible horizon from a station: If there is no obstruction due to intervening ground, the geometric distance D from a station of height h above datum to the visible horizon is given by

   D = √(2Rh + h²) where R is the mean radius of the Earth.

   For usual heights small compared with the Earth's radius the term h² is negligible and

   D ≈ √(2Rh).

   With h in metres and D in kilometres the practical approximate relation is

   D ≈ 3.57 √h

   which is useful for quick checks of intervisibility (assuming standard refraction conditions).

2. Relative elevations of stations to secure intervisibility: If two stations A and B are separated by a known distance D and there is no intervening obstruction, the minimum required heights of the stations to see each other can be estimated by treating the horizon distances from each station. If h₁ is the known elevation of station A above datum and h2

   is the elevation required at B so that the line of sight is tangent to the Earth at some intervening point, the relation (using practical constants) may be written as:

   h = 0.06728 D² (with D in km gives h in metres)

   Accordingly, if D₁ is the distance from A to the tangency point and D₂ = D - D₁ is the distance from B to that point then

   h₁ = 0.06728 D₁²

   h₂ = 0.06728 D₂²

   
   
   

   Here D₁ and D₂ are in kilometres and h₁, h₂ are in metres. These relations include the customary reduction factor for the combined effect of curvature and average atmospheric refraction used in field practice.

3. Profile of the intervening ground: During reconnaissance the elevations and positions of peaks and ridges in the intervening ground between proposed stations should be determined. Comparing these elevations with the proposed line of sight establishes whether any obstruction exists. The problem can be solved using the horizon relations above or other methods such as the McCaw method described below.

Captain G. T. McCaw's Method

This method provides a practical formula to compute the height of the line of sight at an intervening obstruction point C between two stations A and B. It accounts for the geometry of the problem and the curvature/refraction reductions used in triangulation practice.

Let

• h₁ = height of station A above datum
• h₂ = height of station B above datum
• h = height of the line of sight at the obstruction C
• 2s = distance between the two stations A and B
• (s + x) = distance of obstruction C from A
• (s - x) = distance of obstruction C from B
• ζ = zenith distance or direction parameter used in the relation as appropriate


The formula for the height h of the line of sight at the obstruction is given by the relation illustrated below.



When x, s and R are substituted in miles and h₁, h₂, h are in feet, appropriate numerical factors for curvature and refraction are used. When x, s and R are in kilometres and heights are in metres, the factor commonly used for combined curvature and refraction reduction is approximately 0.06728 (that is, 1 - 2m/2R ≈ 0.06728 under the standard assumptions used in the method).

Notes and practical points:

• Always apply algebraic signs consistently for latitudes and departures when computing closure and applying balancing methods.
• When planning triangulation figures prefer well-shaped figures (wide base, small included angles avoided) to improve geometric strength.
• Signals and tower dimensions must be chosen for visibility and minimal phase error; phase corrections should be applied if part of the signal is shaded.
• Intervisibility checks must include both geometric horizon considerations and the profile of intervening ground; reconnaissance is essential.