GELE Adjustment Computations (Least Squares) — Adjustment of Level Nets and TraversesStudy Notes

Every surveying measurement contains unavoidable random and systematic errors. In a closed level loop, you begin at a known elevation, run a series of level sections, and return to the starting point. Ideally, your computed final elevation equals the starting elevation. In reality, it does not—the difference is the misclosure. For a level loop: • Start at elevation H₀ (known) • Measure level differences over n sections: Δh₁, Δh₂, ..., Δhₙ • Compute final elevation: H_f = H₀ + Σ(Δhᵢ) • Misclosure: e = H_f − H₀ = Σ(Δhᵢ) The misclosure e should be close to zero; if it is not, the survey must be adjusted before use. For a closed traverse: • Start and end at the same point (X₀, Y₀) • Measure n sides with lengths L₁, L₂, ..., Lₙ and bearings • Compute latitude and departure for each side: Latᵢ = Lᵢ cos(Az), Depᵢ = Lᵢ sin(Az) • Sum latitudes and departures: ΣLat and ΣDep • In a perfect traverse: ΣLat = 0 and ΣDep = 0 • Error of closure: EC = √[(ΣLat)² + (ΣDep)²] Under Philippine geodetic practice (RA 4374 and RA 8560), survey closure tolerances must be documented and must meet the standards for the project class. For cadastral surveys under PD 1529 and CA 141, relative precision of at least 1:5000 is commonly required; for engineering surveys, 1:10,000 may be specified.

The distance-proportional adjustment (also called length-proportional adjustment) distributes the total misclosure e across all sections in proportion to the length (or number of setups) of each section. This is the most common method in practice and is justified when errors accumulate uniformly over distance—a reasonable assumption for leveling. **Principle:** If a level loop has a misclosure e distributed over total distance L_total, then each section i of length Lᵢ receives a correction proportional to its contribution to the total: c_i = −e × (Lᵢ / L_total) The negative sign ensures corrections oppose the misclosure direction. The sum of all corrections equals −e, thus removing the entire misclosure. **Procedure:** 1. Measure all level sections (e.g., in kilometres or metres) 2. Compute the sum: L_total = Σ Lᵢ 3. Calculate the misclosure: e = H_f − H₀ (or the algebraic sum of all level differences) 4. Apply correction to each section: c_i = −e × (Lᵢ / L_total) 5. Adjusted elevation difference for section i: Δh_adjusted,i = Δhᵢ + c_i 6. Verify: Σ(Δh_adjusted,i) = 0 (should return exactly to starting elevation) **Alternative: Adjustment by Number of Setups** When distances are not recorded but the number of setups (or backsight–foresight pairs) is known, substitute n_i (setups in section i) for Lᵢ: c_i = −e × (n_i / n_total) This is equally valid if errors are assumed to be independent of distance and depend instead on the number of instrument positions. **Allowable Misclosure (Philippine Standard)** For leveling networks in the Philippines, the allowable misclosure is typically: Allow = ±k√K where K = total distance in kilometres and k = 5 to 10 mm√km (depending on survey class and instrument type; k ≈ 5 mm√km for third-order leveling, k ≈ 10 mm√km for lower-order work). If the computed misclosure exceeds this, the work must be rerun. **Worked Example – Level-Loop Adjustment:** A level loop in a subdivision survey (Quezon City, Metro Manila) starts at benchmark BM-10 (elevation 42.500 m, referenced to MSL via PRS92). The survey comprises four sections: • Section 1: 2.0 km, measured Δh = +1.234 m • Section 2: 3.0 km, measured Δh = −0.456 m • Section 3: 1.5 km, measured Δh = +0.567 m • Section 4: 2.5 km, measured Δh = −1.321 m Total distance: L_total = 2.0 + 3.0 + 1.5 + 2.5 = 9.0 km Sum of level differences: Σ(Δh) = 1.234 − 0.456 + 0.567 − 1.321 = 0.024 m Misclosure: e = +0.024 m (the loop closes 24 mm too high) Allowable misclosure (k = 5 mm√km): Allow = 5 × √9.0 = 5 × 3.0 = 15 mm Actual misclosure: 24 mm > 15 mm threshold—technically marginal and should be investigated; however, for this example, we will proceed with adjustment. Corrections (distance-proportional): • c₁ = −0.024 × (2.0 / 9.0) = −0.024 × 0.2222 = −0.00533 m ≈ −5.3 mm • c₂ = −0.024 × (3.0 / 9.0) = −0.024 × 0.3333 = −0.00800 m ≈ −8.0 mm • c₃ = −0.024 × (1.5 / 9.0) = −0.024 × 0.1667 = −0.00400 m ≈ −4.0 mm • c₄ = −0.024 × (2.5 / 9.0) = −0.024 × 0.2778 = −0.00667 m ≈ −6.7 mm Verify sum: −5.3 − 8.0 − 4.0 − 6.7 = −24.0 mm = −e ✓ Adjusted elevation differences: • Δh₁,adj = 1.234 − 0.00533 = 1.22867 m • Δh₂,adj = −0.456 − 0.00800 = −0.46400 m • Δh₃,adj = 0.567 − 0.00400 = 0.56300 m • Δh₄,adj = −1.321 − 0.00667 = −1.32767 m Final verification: Σ(Δh,adj) = 1.22867 − 0.46400 + 0.56300 − 1.32767 = 0.00000 m ✓ Adjusted elevations at each section endpoint (starting from BM-10 at 42.500 m): • After section 1: 42.500 + 1.22867 = 43.72867 m • After section 2: 43.72867 − 0.46400 = 43.26467 m • After section 3: 43.26467 + 0.56300 = 43.82767 m • After section 4: 43.82767 − 1.32767 = 42.50000 m (returns to BM-10) ✓

A closed traverse is a polygon survey: you start at a known point, measure sides (distances and bearings/azimuths), and return to the starting point. In a perfect traverse, the sum of latitude components equals zero (you go north and south equally) and the sum of departure components equals zero (you go east and west equally). In practice, angular and distance errors cause a closure error. **Key Definitions:** • Latitude (Lat) of a side: Lᵢ cos(Azimuth) • Departure (Dep) of a side: Lᵢ sin(Azimuth) • Error of closure: EC = √[(ΣLat)² + (ΣDep)²] • Perimeter: P = ΣLᵢ (sum of all side lengths) • Relative precision: RP = EC / P (expressed as a ratio, e.g., 1:5000) **The Compass Rule (Bowditch Method):** The compass rule assumes that errors in distance measurement are the dominant source of closure error. It distributes both the latitude and departure errors in proportion to the length of each side: Latitude correction: c_lat,i = −(ΣLat) × (Lᵢ / P) Departure correction: c_dep,i = −(ΣDep) × (Lᵢ / P) These corrections are applied to the measured latitude and departure of each side to produce adjusted values. **Procedure:** 1. For each side i, measure length Lᵢ and bearing/azimuth Azᵢ 2. Compute latitude and departure: Latᵢ = Lᵢ cos(Azᵢ), Depᵢ = Lᵢ sin(Azᵢ) 3. Sum latitudes and departures: ΣLat and ΣDep (should be zero in a perfect survey) 4. Compute error of closure: EC = √[(ΣLat)² + (ΣDep)²] 5. Compute perimeter: P = ΣLᵢ 6. Compute relative precision: RP = EC / P 7. For each side, apply corrections: - c_lat,i = −(ΣLat) × (Lᵢ / P) - c_dep,i = −(ΣDep) × (Lᵢ / P) 8. Adjusted latitude and departure: Lat_adj,i = Latᵢ + c_lat,i; Dep_adj,i = Depᵢ + c_dep,i 9. Compute adjusted coordinates using adjusted latitudes and departures 10. Verify: ΣLat_adj = 0 and ΣDep_adj = 0 **The Transit Rule (Alternative):** When angle measurements are less reliable than distances (e.g., compass-based surveys), the transit rule distributes errors in proportion to the magnitude of the latitude or departure components: c_lat,i = −(ΣLat) × (|Latᵢ| / Σ|Lat|) c_dep,i = −(ΣDep) × (|Depᵢ| / Σ|Dep|) The transit rule is less common in modern practice but may be specified for older or lower-precision surveys. **Philippine Cadastral and Engineering Standards:** Under RA 4374 and RA 8560, and for land registration under PD 1529 and CA 141, the required relative precision depends on survey class: • Cadastral surveys (PD 1529): minimum 1:5000 relative precision • Engineering surveys: 1:10,000 or better for infrastructure projects • First-order control: 1:100,000 or better If a traverse does not meet the required standard, it must be rerun or adjusted traverses must be approved by an authorized geodetic engineer. **Worked Example – Traverse Adjustment (Compass Rule):** A closed traverse of a agricultural parcel (Laguna province, part of PPCS/UTM zone) contains four sides: | Side | Length (m) | Azimuth | |------|------------|----------| | AB | 325.46 | 45° 30' | | BC | 412.80 | 135° 20' | | CD | 289.35 | 225° 45' | | DA | 198.50 | 315° 15' | Step 1: Compute latitude and departure for each side (using decimal degrees and sine/cosine): Azimuth (decimal): AB = 45.5°, BC = 135.333°, CD = 225.75°, DA = 315.25° AB: Lat = 325.46 × cos(45.5°) = 325.46 × 0.70537 = 229.58 m Dep = 325.46 × sin(45.5°) = 325.46 × 0.70883 = 230.75 m BC: Lat = 412.80 × cos(135.333°) = 412.80 × (−0.71844) = −296.41 m Dep = 412.80 × sin(135.333°) = 412.80 × 0.69554 = 287.10 m CD: Lat = 289.35 × cos(225.75°) = 289.35 × (−0.70737) = −204.83 m Dep = 289.35 × sin(225.75°) = 289.35 × (−0.70675) = −204.56 m DA: Lat = 198.50 × cos(315.25°) = 198.50 × 0.70507 = 139.95 m Dep = 198.50 × sin(315.25°) = 198.50 × (−0.70881) = −140.65 m Step 2: Sum latitudes and departures: ΣLat = 229.58 − 296.41 − 204.83 + 139.95 = −131.71 m (should be ~0) ΣDep = 230.75 + 287.10 − 204.56 − 140.65 = 172.64 m (should be ~0) (Note: These are large errors; the azimuth values given here produce this for teaching purposes.) Step 3: Error of closure and perimeter: EC = √[(−131.71)² + (172.64)²] = √[17347.6 + 29804.0] = √47151.6 ≈ 217.2 m P = 325.46 + 412.80 + 289.35 + 198.50 = 1226.11 m Relative precision = 217.2 / 1226.11 ≈ 1:5.64 (this is very poor; 1:5000 is typical requirement) (For realistic data, EC would be much smaller. This example uses exaggerated errors for clarity.) Step 4: Apply compass-rule corrections: For side AB (L = 325.46 m): c_lat,AB = −(−131.71) × (325.46 / 1226.11) = 131.71 × 0.2655 = +34.96 m c_dep,AB = −(172.64) × (325.46 / 1226.11) = −172.64 × 0.2655 = −45.86 m For side BC (L = 412.80 m): c_lat,BC = 131.71 × (412.80 / 1226.11) = 131.71 × 0.3367 = 44.35 m c_dep,BC = −172.64 × 0.3367 = −58.15 m For side CD (L = 289.35 m): c_lat,CD = 131.71 × (289.35 / 1226.11) = 131.71 × 0.2361 = 31.09 m c_dep,CD = −172.64 × 0.2361 = −40.74 m For side DA (L = 198.50 m): c_lat,DA = 131.71 × (198.50 / 1226.11) = 131.71 × 0.1619 = 21.33 m c_dep,DA = −172.64 × 0.1619 = −27.95 m Verify correction sums: Σc_lat = 34.96 + 44.35 + 31.09 + 21.33 = 131.73 ≈ 131.71 ✓ Σc_dep = −45.86 − 58.15 − 40.74 − 27.95 = −172.70 ≈ −172.64 ✓ Step 5: Adjusted latitudes and departures: AB_adj: Lat = 229.58 + 34.96 = 264.54 m; Dep = 230.75 − 45.86 = 184.89 m BC_adj: Lat = −296.41 + 44.35 = −252.06 m; Dep = 287.10 − 58.15 = 228.95 m CD_adj: Lat = −204.83 + 31.09 = −173.74 m; Dep = −204.56 − 40.74 = −245.30 m DA_adj: Lat = 139.95 + 21.33 = 161.28 m; Dep = −140.65 − 27.95 = −168.60 m Final verification: ΣLat_adj = 264.54 − 252.06 − 173.74 + 161.28 = 0.02 ≈ 0 ✓ (rounding) ΣDep_adj = 184.89 + 228.95 − 245.30 − 168.60 = −0.06 ≈ 0 ✓ The adjusted traverse now closes (within rounding error) and is ready for coordinate computation and property registration under PD 1529/CA 141.

The choice between adjustment methods depends on the survey type, the nature of errors, and the precision standards being applied. **Level-Loop Adjustment (Distance-Proportional):** • Use when: Running a closed leveling loop or traverse using only elevation measurements • Assumption: Errors accumulate uniformly with distance (or setups) • Advantage: Simple to compute; suitable for moderate-precision leveling • Standard: RA 4374/RA 8560 leveling work; typical for engineering and land-surveying applications • Tolerance: Allow = ±k√K (k ≈ 5 mm√km for third-order) • When to reject: If misclosure exceeds tolerance, the entire loop must be releveled **Compass Rule (Bowditch) for Traverses:** • Use when: Closing a polygon traverse where distance measurements are reliable and reasonably precise • Assumption: Angular errors are small (well-trained crew, quality transit/theodolite); distance errors dominate • Advantage: Distributes errors fairly; standard in modern practice • Standard: Default method in Philippine cadastral surveys (RA 4374, PD 1529) • Requirement: Relative precision typically 1:5000 (cadastral), 1:10,000 (engineering) • When to apply: After all field measurements are complete; before computing final coordinates **Transit Rule for Traverses:** • Use when: Angular measurements are less reliable than distances (e.g., compass or low-grade transit used) • Assumption: Distance errors are small; angle errors significant • Advantage: Distributes error based on latitude/departure magnitude, not length • Standard: Less common in modern Philippine practice; may be required for lower-order surveys or older data • When to apply: Only if compass rule would be inappropriate (documented reason required) **Precision and Acceptability:** Before adjustment, verify that closure is acceptable: - Cadastral surveys (PD 1529, CA 141): minimum 1:5000 - Engineering surveys: minimum 1:10,000 - First-order control networks: minimum 1:100,000 If closure is borderline or fails, field work must be revisited; adjustment alone cannot correct bad data. **Integration with Philippine Standards:** Under RA 4374 (Updating the Magna Carta for Land Surveyors), all adjustment must be transparently documented. A Professional Geodetic Engineer's stamp certifies that the work meets: 1. RA 8560 (National Mapping and Resource Information Authority oversight) 2. PD 1529 (Land Registration Decree, PD 1529) 3. CA 141 (Public Land Act); for government lands 4. PPCS/UTM grid requirements for coordinate reporting PRS92 (Philippine Reference System 1992, referenced to WGS84) is the mandatory datum for modern surveys; older data in PRS27 or local datums must be transformed using official parameters provided by NAMRIA. **Least Squares vs Proportion Methods:** This chapter focuses on proportional adjustment (simple and intuitive). Full least-squares adjustment is covered in the next chapter. Proportional methods are: - Faster to hand-compute - Suitable for single, isolated networks - Adequate for routine engineering and cadastral work Least-squares adjustment is: - More rigorous, using error theory - Essential for large, multi-station networks - Preferred for first-order control networks - Provides statistical confidence measures (standard errors, covariance matrices)

After adjustment is complete, verification and documentation are essential for professional credibility and compliance with Philippine standards. **Verification Steps:** 1. **Level-Loop Adjustment Verification:** - Check that Σ(corrections) = −e (the original misclosure) - Verify that Σ(adjusted elevation differences) = 0 - Spot-check a few adjusted elevations by hand - Confirm that allowable tolerance was met before adjustment 2. **Traverse Adjustment Verification:** - Check that Σ(c_lat) = −(ΣLat) and Σ(c_dep) = −(ΣDep) - Verify that Σ(Lat_adj) = 0 and Σ(Dep_adj) = 0 (within rounding) - Recompute coordinates from adjusted latitudes/departures - Confirm closure at starting point (round-trip coordinates agree) - Check relative precision: RP = EC / P should meet project standard 3. **Computational Accuracy:** - Use a calculator or spreadsheet and verify calculations independently - Check for sign errors (correction sign must oppose misclosure) - Verify all trigonometric values (sin/cos) with separate references - Review decimal places and rounding rules (typical: 0.01 m for traverse, 0.001 m for leveling) **Quality Control Checks:** - **Proportionality Check:** Corrections should scale with distance/component size. If one side gets an unusually large or small fraction, investigate. - **Physical Reasonableness:** Adjusted elevations should follow topography. If an adjustment reverses a slope, review data. - **Comparison with Field Notes:** The adjustment magnitude should match the field misclosure; if not, a computational error occurred. - **Independent Verification:** Have a second surveyor or technician verify calculations without seeing the first result. **Documentation Requirements (RA 4374, RA 8560, PD 1529):** Every adjusted survey must include: 1. **Field Data Sheet:** Uncorrected measurements, bearings, distances, elevations 2. **Misclosure Calculation:** Show ΣLat, ΣDep, EC, or level misclosure e; include tolerance computation 3. **Adjustment Methodology:** State which rule applied (compass, transit, distance-proportional) and justification 4. **Correction Table:** List each side/section with its correction and adjusted values 5. **Verification:** Show that closures are satisfied (Σ(c) = −e, Σ(adj) = 0) 6. **Final Coordinates or Elevations:** Report adjusted values to project precision (typically 0.01 m) 7. **Professional Statement:** Certified by a PRC-licensed Geodetic Engineer; must include: - Date of survey and adjustment - Instruments used (theodolite type, leveling staff, tape type) - Weather conditions and known corrections (temperature, tension) - Datum reference (PRS92/WGS84, PPCS/UTM grid) - Compliance statement with RA 4374, RA 8560, PD 1529 or CA 141 - Standard error estimate or confidence statement **Reporting Format Example (Philippine Convention):** --- **ADJUSTMENT COMPUTATION REPORT** **Project:** Cadastral Survey, Barangay San Isidro, Laguna **Date:** 15 August 2024 **Licensed Geodetic Engineer:** Maria Santos, PRC-GE #1234 (RA 4374) **TRAVERSE ADJUSTMENT (COMPASS RULE)** | Side | Length (m) | Azimuth | Lat (m) | Dep (m) | c_lat (m) | c_dep (m) | Lat_adj (m) | Dep_adj (m) | |------|-----------|---------|---------|---------|----------|----------|-------------|------------| | AB | 287.52 | 45°30' | 203.34 | 203.78 | −0.042 | +0.051 | 203.298 | 203.831 | | BC | 352.18 | 135°18' | −250.56 | 248.43 | −0.051 | +0.062 | −250.611 | 248.492 | | CD | 298.34 | 225°12' | −211.45 | −211.23 | −0.044 | +0.053 | −211.494 | −211.177 | | DA | 213.67 | 315°42' | 151.23 | −150.98 | −0.031 | +0.038 | 151.199 | −150.942 | | **SUM** | **1151.71** | | **−107.44** | **90.00** | **−0.168** | **+0.204** | **0.000** | **0.000** | **Error of Closure:** EC = √[(−0.107)² + (0.090)²] = 0.140 m **Relative Precision:** 0.140 / 1151.71 = 1:8227 **Standard:** PD 1529 (Cadastral, minimum 1:5000) — ✓ **PASSES** **Adjustment Method:** Compass Rule (Bowditch), assuming distance errors dominate **Verification:** ✓ Σ(c_lat) = −0.168 = −(ΣLat); ✓ Σ(c_dep) = +0.204 = −(ΣDep); ✓ Σ(Lat_adj) ≈ 0; ✓ Σ(Dep_adj) ≈ 0 **Datum:** PRS92 (WGS84 equivalent), PPCS/UTM Zone 51N **Instruments:** Leica TPS1200 total station (±5 mm ± 5 ppm), calibrated 1 August 2024 **Conditions:** Clear weather, 28–32°C, no wind **Corrections Applied:** None (total station compensates internally) **Certification:** This survey was executed and adjusted in accordance with: - RA 4374 (Magna Carta for Land Surveyors) - RA 8560 (NAMRIA oversight) - PD 1529 (Land Registration Decree) - PRC Geodetic Engineer Licensure standards The adjusted coordinates are ready for submission to the Land Registration Authority under CA 141. --- (This format is typical for Philippine cadastral submissions; adapt as required by local LRA guidelines.)