CELE Surveying (Geomatics) — Traverse and Omitted MeasurementsStudy Notes

A traverse is a series of connected survey lines whose lengths and directions (bearings or azimuths) are measured in the field. The positions of all points are then calculated relative to a starting point and known direction.

There are two main types of traverses:

Closed Traverse: Forms a closed loop (polygon), returning to the starting point. Used for boundary surveys, area calculations, and control networks. In a closed traverse, the algebraic sum of latitudes and departures should be zero if measurements were perfect.

Open Traverse: Does not close on a known point. Used for routes (roads, pipelines, canals) and reconnaissance. Less reliable because closure cannot be checked directly.

For each line segment in a traverse, we measure:

• Length (L): the distance along the line, in metres
• Bearing or Azimuth (θ): the direction measured from north (or south), typically in degrees, minutes, and seconds

These measurements are then converted into rectangular coordinates using latitude and departure components.

Latitude and departure are the north–south and east–west rectangular components of a traverse line. They are computed from the measured length and bearing/azimuth.

Definitions and Sign Conventions:

Latitude (L) = north–south component

• Positive (+) if north
• Negative (−) if south

Departure (D) = east–west component

• Positive (+) if east
• Negative (−) if west

Standard Formulas:

Latitude = L × cos(bearing angle)
Departure = L × sin(bearing angle)

Where L is the measured length and the bearing angle is the direction from north.

Sign Convention by Quadrant (using bearing notation N θ E, S θ E, S θ W, N θ W):

1. N θ E (First quadrant): Latitude +, Departure +
2. S θ E (Second quadrant): Latitude −, Departure +
3. S θ W (Third quadrant): Latitude −, Departure −
4. N θ W (Fourth quadrant): Latitude +, Departure −

Alternatively, using Azimuth (0° to 360° from north, clockwise):

Latitude = L × cos(azimuth)
Departure = L × sin(azimuth)

The trigonometric functions automatically apply the correct signs based on the quadrant.

Computational Notes for the PRC Exam:

• Always include the sign (+ or −) with each component
• Keep sufficient decimal precision (typically 0.01 m or 0.001 m) during calculations
• Sum all latitudes and departures at the end to check closure
• For closed traverses with perfect measurements: ΣLat = 0 and ΣDep = 0

In a perfect closed traverse, the sum of all latitudes should equal zero, and the sum of all departures should equal zero. In practice, due to measurement errors (instrument limitations, human mistakes, environmental factors), these sums are not exactly zero. This discrepancy is called the error of closure.

Error of Closure (Linear Closure Error):

EC = √[(ΣLat)² + (ΣDep)²]

Where:

• ΣLat = algebraic sum of all latitudes
• ΣDep = algebraic sum of all departures

The error of closure is the straight-line distance between the theoretical closing point and the actual closing point.

Relative Precision (Accuracy Ratio):

Relative Precision = EC / Perimeter

Expressed as a fraction 1/n, where n is calculated as:

n = Perimeter / EC

Interpretation of Relative Precision:

• Common standard: 1/5000 (reasonable for most surveying work)
• 1/10000 or better: high precision
• 1/2000 or worse: lower precision, may require remeasurement

For the PRC exam, you will often be asked to:

1. Calculate the error of closure
2. Express relative precision as 1/n form
3. Judge whether the traverse meets acceptable standards

Acceptable Closure Standards (varies by jurisdiction and traverse type):

• Property boundary surveys: typically 1/5000 or better
• Route surveys (less critical): may accept 1/2000 to 1/3000
• High-precision control networks: 1/10000 or better

Once a closed traverse is found to be acceptable (meets precision standards), the error of closure must be distributed back into the measured values so that the final coordinates satisfy ΣLat = 0 and ΣDep = 0. This process is called balancing or adjustment.

There are two main balancing methods used in surveying, each based on different assumptions about the source of error:

Method 1: Compass Rule (Bowditch Method)

Assumption: Errors in both distance and angle measurement are present and are proportional to the distance measured.

Correction to latitude (line i) = −(ΣLat) × (Li / ΣL)
Correction to departure (line i) = −(ΣDep) × (Li / ΣL)

Where:

• Li = length of line i
• ΣL = total perimeter
• ΣLat, ΣDep = misclosures

Method 2: Transit Rule

Assumption: Angle measurements are precise; errors are primarily in distance (or in direction's influence on position).

Where:

• Lati, Depi = the latitude and departure of line i
• Σ|Lat|, Σ|Dep| = sums of absolute values

Which Method to Use?

For the PRC exam, unless specifically stated:

• Compass (Bowditch) Rule is the most commonly applied method
• Used when both angle and distance errors are expected
• Often the default for general boundary surveys

Transit Rule is used when:

• Angles are known to be measured with high precision (e.g., instrument with electronic angle measurement)
• Distances are measured by tape or less precise methods

Steps to Balance a Traverse (Compass Rule):

1. Calculate ΣLat and ΣDep (misclosures)
2. For each line i, calculate: LatCorrection(i) = −(ΣLat) × (Li / ΣL)
3. For each line i, calculate: DepCorrection(i) = −(ΣDep) × (Li / ΣL)
4. Add corrections to original latitudes and departures
5. Verify: adjusted ΣLat and adjusted ΣDep should both equal zero (within rounding)
6. Calculate adjusted coordinates

Important Notes:

• Corrections are opposite the sign of the misclosure
• Longer lines receive larger corrections (proportional)
• The method is empirical and assumes errors are random
• All coordinates are adjusted; no original measurement is considered error-free

In some traverses, one or two measurements are missing (e.g., a line's length and bearing could not be measured directly, or only one is missing). The missing value(s) can be computed using the closure condition: for a closed traverse, ΣLat = 0 and ΣDep = 0.

Case 1: Omitted Length (Missing Distance, Bearing Known)

If the bearing of the closing line is known but its length is unknown:

Since ΣLat = 0 and ΣDep = 0 must be satisfied:

Latitude of closing line = −(ΣLat of all other lines)
Departure of closing line = −(ΣDep of all other lines)

Then, the missing length:

L_missing = √(Lat² + Dep²)

Where Lat and Dep are the computed closing line components.

Case 2: Omitted Bearing (Missing Direction, Length Known)

If the length is known but the bearing is unknown:

Latitude of closing line = −(ΣLat of all other lines)
Departure of closing line = −(ΣDep of all other lines)

The bearing of the closing line:

Bearing = arctan(Dep / Lat) [with attention to quadrant]

Case 3: Omitted Length and Bearing (Both Unknown)

This is the most common omitted measurement scenario. Both the closing line's length and bearing are missing. Using the same closure conditions:

Lat_closing = −(ΣLat of all known lines)
Dep_closing = −(ΣDep of all known lines)

Then:

L_missing = √(Lat_closing² + Dep_closing²)
Bearing_missing = arctan(Dep_closing / Lat_closing)

Determining Quadrant from Latitude and Departure:

The quadrant (and thus the correct bearing notation N/S–E/W) is determined by the signs:

PRC Exam Tips:

• Omitted measurements are common on the exam
• Always check closure with the computed missing values
• Express final bearings in standard notation (N/S–E/W) unless azimuth is specifically requested
• Include units (metres, degrees, minutes, seconds) in final answers

Once latitudes and departures are balanced (or if the traverse is open and no balancing is needed), coordinates for each traverse point are calculated using a reference starting point and direction.

Coordinate Calculation Method:

Assuming a starting point A with coordinates (Easting_A, Northing_A), the coordinates of subsequent points are calculated by accumulating latitudes and departures:

For point B (end of line AB):
Northing(B) = Northing(A) + Latitude(AB)
Easting(B) = Easting(A) + Departure(AB)

For point C (end of line BC):
Northing(C) = Northing(B) + Latitude(BC) = Northing(A) + Latitude(AB) + Latitude(BC)
Easting(C) = Easting(B) + Departure(BC) = Easting(A) + Departure(AB) + Departure(BC)

Common Starting Coordinates:

• Often taken as (1000.00 m E, 1000.00 m N) for computational convenience
• In UTM, actual zone coordinates are used
• Philippine coordinate systems may reference PCSM (Philippine Coordinate System Manila) or UTM zones

Area Calculation (Double Meridian Distance Method or Shoelace Formula):

Once coordinates are established, area can be calculated using the Shoelace Formula:

Area = ½ |Σ(E_i × N_{i+1} − E_{i+1} × N_i)|

Where (E_i, N_i) are consecutive point coordinates.

Practical Applications in Philippine Surveying:

1. Boundary Surveys (RA 544 – The Cadastral Law): Establish property boundaries for land titles
2. Route Surveys: Roads, utility lines, canals
3. Subdivision Surveys: Divide parcels into smaller lots
4. Control Networks: Establish reference points for other surveying work
5. Construction Stakeout: Set out building corners and alignments

This section highlights typical errors made by candidates in PRC surveying exams and strategies to avoid them.

Common Mistakes:

1. Bearing vs. Azimuth Confusion

   • Mistake: Using bearing angle directly as azimuth, or vice versa

   • Fix: Clearly identify which system is given; convert if necessary. Remember:

     • Bearing: given as N/S–E/W (e.g., N 35° E)
     • Azimuth: 0° to 360° clockwise from north (e.g., 035°)

2. Sign Errors in Latitude/Departure

   • Mistake: Not applying the correct signs based on quadrant
   • Fix: Always use the north/south and east/west labels from the bearing to confirm signs

3. Forgetting to Sum Components

   • Mistake: Calculating individual lines correctly but forgetting to sum for closure check
   • Fix: Always compute ΣLat and ΣDep before proceeding

4. Misapplying Correction Signs

   • Mistake: Adding the misclosure instead of subtracting (or vice versa)
   • Fix: Corrections are opposite the sign of misclosure. If ΣLat = −0.6, add +0.6 m

5. Mixing Units

   • Mistake: Combining bearings in degrees with other units, or inconsistent distance units
   • Fix: Keep all lengths in metres, all angles in consistent units (degrees or degrees-minutes-seconds)

6. Overlooking Decimal Precision

   • Mistake: Rounding intermediate values, causing loss of accuracy in final answers
   • Fix: Carry extra decimal places (at least 0.01 m) through calculations; round only at the end

7. Incorrect Relative Precision Fraction

   • Mistake: Writing EC/Perimeter as a decimal instead of as a fraction 1/n
   • Fix: Always express as 1/n. For example, 0.0002 = 1/5000

8. Transit vs. Compass Rule Confusion

   • Mistake: Applying the wrong correction method, or mixing both in the same problem
   • Fix: Unless explicitly stated, use Compass (Bowditch) Rule. Ensure you divide by the correct denominator (ΣL or Σ|Lat| or Σ|Dep|)

9. Quadrant Determination Errors

   • Mistake: Computing arctan(Dep/Lat) but placing the angle in the wrong quadrant
   • Fix: Always check the signs of latitude and departure to determine the correct N/S and E/W designation

10. Not Verifying Omitted Measurements

    • Mistake: Computing the missing length/bearing but not checking that closure is satisfied
    • Fix: After solving, substitute back into ΣLat = 0 and ΣDep = 0 to confirm

PRC Exam Strategy:

1. Read the Problem Carefully

   • Identify what is given (lengths, bearings/azimuths, coordinates)
   • Identify what is asked (latitude/departure, closure error, missing values, etc.)
   • Note any specific instructions (method of balancing, unit preferences)

2. Set Up the Computation

   • For latitude/departure: prepare a table with columns for Line, Length, Bearing, Latitude, Departure
   • Label bearing clearly (N/S and E/W) to avoid sign mistakes

3. Double-Check Calculations

   • Verify each latitude: L × cos(angle)
   • Verify each departure: L × sin(angle)
   • Use calculator carefully; recompute critical values

4. Apply Closure Check

   • Always sum latitudes and departures
   • Compare against acceptable precision standards
   • If unacceptable, indicate that remeasurement is required

5. Show Work Clearly

   • Points are often awarded for methodology, not just the final answer
   • Intermediate steps and units are important
   • Label all corrections and corrected values

6. Final Verification

   • For balanced traverses, confirm adjusted ΣLat and ΣDep = 0
   • For omitted measurements, confirm closure is satisfied
   • For coordinates, check that the traverse closes (returns to starting point)

7. Express Answers in Standard Form

   • Bearings: N/S–E/W notation with degrees and minutes
   • Distances: metres with appropriate decimal places (typically 0.01 m)
   • Coordinates: both easting and northing, clearly labeled
   • Relative precision: as a fraction 1/n (e.g., 1/5000, not 0.0002)