GELE Surveying (Geomatics) — Traverse and Omitted MeasurementsRevision Notes

Formulas

Exam Tips

• Board exams frequently give bearings in the N–S notation. Practice converting: e.g., S 60° W = azimuth 240°.
• Set up a table: Line | Length | Bearing | Lat (+/−) | Dep (+/−) — this organized format prevents sign errors.
• Always verify: ΣN Lat vs. ΣS Lat and ΣE Dep vs. ΣW Dep before computing error of closure.
• If azimuth is given, use Lat = L cos(Az) and Dep = L sin(Az) directly — no sign assignment needed.

Key Points

• Every traverse line is resolved into two rectangular components: Latitude (north–south) and Departure (east–west).
• Latitude = L cos θ, where θ is the bearing angle measured from north or south toward east or west.
• Departure = L sin θ, where θ is the same bearing angle.
• Sign of Latitude: N-bearing lines → +Lat; S-bearing lines → −Lat.
• Sign of Departure: E-bearing lines → +Dep; W-bearing lines → −Dep.
• When using whole-circle azimuths (0°–360°): Lat = L cos(Az), Dep = L sin(Az) — signs come out automatically from trigonometry.
• For a perfectly closed traverse: ΣLat = 0 and ΣDep = 0.
• A quick sanity check: the algebraic sum of all north latitudes must equal the algebraic sum of all south latitudes (and similarly for east/west departures).

Definitions

Section Title

1. Latitudes and Departures

Common Mistakes

• Forgetting to assign the correct sign (N/S for Lat, E/W for Dep) — a single sign error makes the entire closure computation wrong.
• Confusing the angle used: bearing angle θ is always acute (0°–90°); azimuth can be 0°–360°.
• Using sin for latitude and cos for departure — the formulas are Lat = L cos θ and Dep = L sin θ, NOT the reverse.
• Mixing bearings and azimuths in the same table without converting first.
• Rounding intermediate cos/sin values — carry at least 4 decimal places until the final answer.

Formulas

Exam Tips

• Board exam problems often give ΣLat and ΣDep directly — immediately apply the Pythagorean theorem for EC.
• Remember: a 3-4-5 Pythagorean triple often appears in board exams (e.g., ΣLat = 0.30, ΣDep = 0.40 → EC = 0.50).
• RP denominator n = Perimeter / EC — compute this by division, then round down to the nearest 100 or 500 for reporting.
• Know typical precision benchmarks: 1:3 000 (ordinary), 1:5 000 (engineering), 1:10 000 (precise).

Key Points

• In practice, ΣLat ≠ 0 and ΣDep ≠ 0 due to measurement errors; the residuals are called the latitude misclosure (eL) and departure misclosure (eD).
• Error of Closure (EC) is the linear distance from the computed end point back to the true starting point of a closed traverse.
• EC = √[(ΣLat)² + (ΣDep)²].
• Relative Precision (RP) = EC / Perimeter, expressed as the ratio 1 : n (e.g., 1 : 5 000). A smaller numerator (larger n) means higher accuracy.
• Typical acceptable precision: 1:3 000 for ordinary surveys; 1:5 000 to 1:10 000 for engineering surveys; 1:10 000+ for precise control surveys.
• The direction of the error is: bearing of closing error = arctan(ΣDep / ΣLat) — useful for quality checks.

Definitions

Section Title

2. Error of Closure and Relative Precision

Common Mistakes

• Reporting RP as a decimal (e.g., 0.0005) instead of the ratio form 1:2 000 — always express as 1:n.
• Forgetting to take the absolute value before squaring — signs inside the square root are automatically eliminated by squaring, but keeping track of signs avoids computational confusion.
• Using only one component (ΣLat or ΣDep) as the error instead of the Pythagorean combination.
• Confusing EC (a length, in metres) with RP (a dimensionless ratio).

Formulas

Exam Tips

• In board problems, Bowditch Rule is almost always the intended method unless the problem explicitly states 'Transit Rule.'
• Set up a balancing table: Line | L | Lat | Dep | C_Lat | C_Dep | Adj Lat | Adj Dep — this systematic approach earns full partial credit.
• After computing all corrections, sum them: ΣC_Lat must equal −ΣLat and ΣC_Dep must equal −ΣDep. Use this as a check before proceeding.
• For coordinate computation after balancing: N_B = N_A + Adj Lat(AB), E_B = E_A + Adj Dep(AB). Start from the known station and work around.

Key Points

• Balancing distributes the misclosure (eL and eD) back into the individual line latitudes and departures so that ΣLat = 0 and ΣDep = 0 after adjustment.
• Two principal rules: (1) Bowditch (Compass) Rule — corrections proportional to line length; (2) Transit Rule — corrections proportional to the absolute value of the latitude or departure of each line.
• Bowditch Rule is preferred when linear measurements and angular measurements have equal reliability (most common in Philippine board problems).
• Transit Rule is preferred when angular accuracy is higher than linear accuracy (theodolite traverses with rough taping).
• The correction always has the opposite sign of the misclosure.
• After applying corrections, compute adjusted latitudes and departures, then verify ΣLat = ΣDep = 0.
• Adjusted coordinates (Northing, Easting) of each station are computed by successive addition of adjusted latitudes and departures from a known starting point.

Definitions

Section Title

3. Traverse Balancing (Adjustment)

Common Mistakes

• Applying the correction with the SAME sign as the misclosure instead of the OPPOSITE sign — always negate the misclosure when computing corrections.
• Forgetting that the sum of all individual corrections must exactly equal the total misclosure (use a check column).
• Rounding each correction independently — small rounding errors can cause the final adjusted sums to differ slightly from zero; assign the residual rounding error to the longest line.
• Confusing which rule to use: Bowditch → proportional to length; Transit → proportional to lat/dep magnitude.

Formulas

Exam Tips

• Organize the known data into a Lat/Dep table first, sum the known values, then apply the closure conditions to isolate the unknowns.
• When two lines are missing, check: if one missing line's bearing is known, substitute Lat = L cos θ and Dep = L sin θ into the closure equations to get two equations in (at most) two unknowns.
• The quadrant check is mandatory: write it out explicitly to avoid sign errors in the final bearing.
• Board problems often test a 4- or 5-sided traverse with one missing side — practice this type until it is routine (target: solve in under 5 minutes).

Key Points

• If one or two measurements (length and/or bearing) of a traverse line are missing, they can be computed from the closure conditions ΣLat = 0 and ΣDep = 0.
• CASE 1 — One side completely omitted (length AND bearing unknown): Use the two closure equations to find the latitude and departure of the missing line, then back-calculate length and bearing.
• CASE 2 — One side's length OR bearing unknown: Substitute the known partial data into the two closure equations and solve the single unknown.
• CASE 3 — Two sides with partial information missing (e.g., two bearings or two lengths): Set up simultaneous equations from ΣLat = 0 and ΣDep = 0; two unknowns, two equations.
• The missing line's length: L_missing = √(Lat_missing² + Dep_missing²).
• The missing line's bearing angle: θ = arctan(|Dep_missing| / |Lat_missing|), then assign the quadrant based on the signs of Lat_missing and Dep_missing.
• This topic is one of the highest-value items in PRC board exam traverses — expect at least one problem per examination.

Definitions

Section Title

4. Omitted Measurements

Common Mistakes

• Forgetting to include the missing line's contribution in ΣLat = 0 — the equation is: Σ(known Lats) + Lat_missing = 0.
• Assigning the wrong quadrant to the bearing — always check the SIGNS of Lat_missing and Dep_missing: (+Lat, +Dep) = NE; (−Lat, +Dep) = SE; (−Lat, −Dep) = SW; (+Lat, −Dep) = NW.
• For Case 3 (two unknowns on different lines), forgetting to set up two simultaneous equations — one from ΣLat = 0 and one from ΣDep = 0.
• Computing the arctan in degrees but the calculator is set to radians — always verify calculator mode.

Formulas

Exam Tips

• Board exams often combine traverse balancing + area computation in a single problem — master both as a sequence.
• For a quick area estimate, use the Coordinate Method on a calculator if coordinates are already computed.
• Always convert final area to hectares if the problem asks for land area (Philippine context: lots are reported in m² or ha).

Key Points

• After balancing, the area enclosed by a traverse is computed using the Double Meridian Distance (DMD) method or the coordinate method.
• DMD of first line = its departure; DMD of each succeeding line = DMD of preceding line + Dep of preceding line + Dep of current line; DMD of last line = −(its departure), as a check.
• Double Area (2A) = Σ(DMD × Adjusted Latitude); Area = |2A| / 2.
• Alternatively, Coordinate Method: 2A = |Σ(N_i × E_{i+1} − N_{i+1} × E_i)|, which is mathematically equivalent.
• Area results should be in m² or converted to hectares (1 ha = 10 000 m²).

Definitions

Section Title

5. Coordinate (DMD) Method for Area — Bonus Exam Topic

Common Mistakes

• Forgetting to use ADJUSTED (balanced) latitudes and departures in the DMD computation — raw computed values give incorrect areas.
• Not verifying that the last DMD = −(last departure) — this is the built-in check for DMD computation.
• Sign errors in the DMD × Lat products — keep the signs and sum algebraically; take the absolute value only at the end.