CELE Surveying (Geomatics) — Advanced and Geodetic SurveyingStudy Notes

Triangulation and trilateration are complementary techniques for extending control networks over large distances. **Triangulation** involves measuring the angles of interconnected triangles from a known baseline and calculating side lengths using the law of sines. This was the historical backbone of national control surveys. A baseline of known length is established (often 0.5–2 km), and from its endpoints, angles to distant points are measured using theodolites. The network grows outward, with each new side becoming the base for the next triangle. **Trilateration** reverses the process: distances are measured directly (now practical with Electronic Distance Measurement devices), and angles are computed. Modern surveys combine both methods with GNSS positioning for maximum accuracy and redundancy. **Law of Sines Application:** In triangle ABC with sides a, b, c opposite angles A, B, C: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ For triangulation, if baseline AB = c is known, and angles A and B are measured, then: $$BC = a = c \cdot \frac{\sin A}{\sin C}$$ $$AC = b = c \cdot \frac{\sin B}{\sin C}$$ **Example 1.1 — Triangulation Chain** A baseline AB = 1500 m is established. From point A, the angle to point C is 42°. From point B, the angle to point C is 35°. Find the distances AC and BC. *Solution:* Angle C = 180° − 42° − 35° = 103° Using the law of sines: $$AC = AB \cdot \frac{\sin B}{\sin C} = 1500 \cdot \frac{\sin 35°}{\sin 103°} = 1500 \cdot \frac{0.57358}{0.97437} = 882.4 \text{ m}$$ $$BC = AB \cdot \frac{\sin A}{\sin C} = 1500 \cdot \frac{\sin 42°}{\sin 103°} = 1500 \cdot \frac{0.66913}{0.97437} = 1031.0 \text{ m}$$ **Triangulation vs Trilateration:** - **Triangulation:** Angle-based; traditional method; dependent on theodolite precision; network grows systematically. - **Trilateration:** Distance-based; modern EDM/GNSS; less dependent on atmospheric refraction; faster for large areas. - **Combined approach:** Modern surveys use triangulation-trilateration networks ("trilaterometry") with GNSS constraints for optimal control.

Stadia measurement is a rapid, indirect method of determining horizontal and vertical distances using a telescope fitted with stadia hairs (upper, middle, and lower horizontal crosshairs) and a graduated rod. The intercept s (in metres) between upper and lower stadia hairs is read on the rod, and the distance is computed using the stadia equation. **Horizontal Stadia Distance:** For a horizontal sight (vertical angle α = 0°): $$D = K \cdot s + C$$ where: - D = horizontal distance (m) - K = stadia interval factor (typically 100, sometimes 101 or other values depending on the instrument) - s = stadia intercept (m) = reading on upper hair − reading on lower hair - C = additive constant (m), typically 0–0.3 m for internal-focusing telescopes, larger for older external-focusing instruments **Inclined Sights:** When the line of sight makes a vertical angle α (positive above horizontal, negative below): $$D_H = K \cdot s \cdot \cos^2 \alpha$$ $$V = \frac{1}{2} K \cdot s \cdot \sin 2\alpha$$ where: - D_H = horizontal distance component (m) - V = vertical distance component (m); positive when C is above the instrument height - sin(2α) = 2sin(α)cos(α) **Instrument Calibration:** Before field use, K and C are determined by measuring known baseline distances at various intercepts and solving: $$D = K \cdot s + C$$ For modern internal-focusing telescopes, C ≈ 0 and K ≈ 100 (sometimes written as 1:100 or 100× magnification). **Example 2.1 — Horizontal Stadia** A stadia intercept of s = 0.85 m is read on a horizontal sight with K = 100 and C = 0. Find the distance. *Solution:* $$D = K \cdot s + C = 100 \times 0.85 + 0 = 85 \text{ m}$$ **Example 2.2 — Inclined Stadia (above horizontal)** The same intercept (s = 0.85 m) is read on a sight inclined at angle α = +5°. Find the horizontal distance and vertical component. *Solution:* Horizontal distance: $$D_H = K \cdot s \cdot \cos^2 \alpha = 100 \times 0.85 \times \cos^2(5°)$$ $$\cos(5°) = 0.99619, \quad \cos^2(5°) = 0.99240$$ $$D_H = 85 \times 0.99240 = 84.35 \text{ m}$$ Vertical component: $$V = \frac{1}{2} K \cdot s \cdot \sin 2\alpha = \frac{1}{2} \times 100 \times 0.85 \times \sin(10°)$$ $$\sin(10°) = 0.17365$$ $$V = 42.5 \times 0.17365 = 7.38 \text{ m}$$ **Example 2.3 — Inclined Stadia (below horizontal)** An inclined sight at α = −3° has intercept s = 1.20 m, K = 100, C = 0. Find D_H and V. *Solution:* $$\cos(3°) = 0.99863, \quad \cos^2(3°) = 0.99727$$ $$D_H = 100 \times 1.20 \times 0.99727 = 119.67 \text{ m}$$ $$\sin(−6°) = −0.10453$$ $$V = \frac{1}{2} \times 100 \times 1.20 \times (−0.10453) = −6.27 \text{ m}$$ (Negative V indicates the rod is below the instrument height.) **Common Pitfalls in Stadia Calculations:** 1. **Using cos(α) instead of cos²(α)** for horizontal distance — remember the correction factor is the square of the cosine. 2. **Forgetting the factor of 1/2** in the vertical component formula. 3. **Using sin(α) instead of sin(2α)** — the double angle appears in the vertical formula. 4. **Confusing the sign of α** — upward sights are positive, downward are negative. 5. **Ignoring the additive constant C** — modern instruments have C ≈ 0, but always verify with calibration.

Surveying methods must account for the physical shape of the Earth to maintain accuracy over large distances. The choice between plane and geodetic approaches depends on the survey extent and required precision. **Plane Surveying:** - Assumes the Earth is flat (horizontal plane). - Treats coordinates as rectangular (E, N) rather than ellipsoidal (latitude, longitude). - Suitable for areas up to approximately 5–10 km² (roughly a 3 km radius) depending on precision requirements. - All calculations use plane trigonometry (Euclidean geometry). - Convergence of meridians and Earth's curvature are negligible. - Faster computations; simpler field procedures. **Geodetic Surveying:** - Accounts for Earth's curvature (ellipsoidal shape). - Uses ellipsoidal coordinates (latitude φ, longitude λ) on a reference ellipsoid (e.g., WGS 84). - Necessary for large areas (>10 km²), national control networks, and long baselines (>10 km). - Incorporates atmospheric refraction corrections. - Requires spherical (or ellipsoidal) trigonometry. - Projections (e.g., UTM, Transverse Mercator) convert ellipsoidal to plane coordinates for detailed mapping. - More computationally intensive but maintains accuracy over continental scales. **Curvature and Refraction Effects:** When observing from a height h₁ to a height h₂ over a sight distance D (in kilometres), Earth's curvature causes an apparent vertical displacement. The combined curvature-refraction correction is: $$h_{cr} = 0.0675 \cdot D_{km}^2 \text{ (metres)}$$ This formula assumes: - D_km is the slant distance converted to kilometres. - Standard atmospheric conditions (refraction coefficient ≈ 0.13). - The correction is subtractive from vertical angles (curvature makes distant points appear lower than they are). For practical purposes: - At D = 1 km: h_cr ≈ 0.07 m (negligible for plane surveys) - At D = 5 km: h_cr ≈ 1.69 m (significant; geodetic survey required) - At D = 10 km: h_cr ≈ 6.75 m (critical for precision levelling) **Example 3.1 — Curvature-Refraction Correction** A geodetic sight spans 4 km horizontally. Calculate the combined curvature-refraction correction. *Solution:* $$h_{cr} = 0.0675 \times 4^2 = 0.0675 \times 16 = 1.08 \text{ m}$$ This means a point 4 km away appears approximately 1.08 m lower due to Earth's curvature and atmospheric refraction. **Example 3.2 — Deciding Between Plane and Geodetic Methods** A survey covers an area of 15 km × 20 km. Should plane or geodetic methods be used? *Solution:* Area = 300 km² >> 10 km²; diagonal ≈ 25 km. Earth's curvature effect at 25 km: $$h_{cr} = 0.0675 \times 25^2 = 0.0675 \times 625 = 42.2 \text{ m}$$ This is substantial. **Geodetic surveying is required.** Measurements must reference an ellipsoid, and results must account for convergence of meridians and projection distortions. **Conversion from Geodetic to Plane Coordinates:** In practice, geodetic surveys establish control points in ellipsoidal coordinates (φ, λ), which are then projected onto a plane coordinate system (Easting, Northing) using a map projection (e.g., UTM). The surveyor calculates details in the plane system but ensures the geodetic framework is correct at the coarse level. **Philippine Context — NGCP (National Geodetic Control Points):** The Philippines uses the Philippine Reference System 1992 (PRS92) based on WGS 84. The National Geodetic Control Network, maintained by the National Mapping and Resource Information Authority (NAMRIA), provides bench marks and control points nationwide. Surveyors tie local surveys to NGCP bench marks using geodetic methods to ensure consistency and legal validity (per RA 544, Geomatics Engineer License Law).

Over large distances, Earth's curvature necessitates spherical (or ellipsoidal) trigonometry instead of plane trigonometry. On a sphere (or ellipsoid), triangles formed by great circles (geodesics) obey different rules than plane triangles. **Spherical Triangle Fundamentals:** A spherical triangle is formed by three great circles (geodesics) on a sphere's surface. Its sides (a, b, c) are measured as angles (in radians or degrees) subtended at the Earth's centre, and its angles (A, B, C) are those between the tangent planes at vertices. **Spherical Law of Sines:** For a spherical triangle: $$\frac{\sin a}{\sin A} = \frac{\sin b}{\sin B} = \frac{\sin c}{\sin C}$$ This is analogous to the plane law of sines but applies to the surface of a sphere. **Spherical Law of Cosines (for sides):** $$\cos c = \cos a \cos b + \sin a \sin b \cos C$$ and for angles: $$\cos C = -\cos A \cos B + \sin A \sin B \cos c$$ **Small-Angle Approximation:** For triangulation networks where sides are small relative to Earth's radius (R ≈ 6,371 km), spherical effects are minimal. The approximation uses: $$\text{Spherical excess} = E \approx \frac{A_\text{area}}{R^2}$$ where A_area is the triangle's area in km². For triangles spanning <10 km per side on Earth's surface, E < 1 arcsecond, and plane trigonometry is acceptable. For national control networks or intercontinental geodesy, the spherical excess must be computed and applied. **Example 4.1 — Spherical Triangle (simplified)** Consider a geodetic network spanning a region where plane assumptions break down. A triangulation network on a sphere with baseline a = 0.01 radians (about 63.7 km at Earth's equator), angle A = 45°, angle B = 55°. Find angle C and side b. *Solution:* For a small-angle approximation (where spherical effects are minimal but not negligible), we first compute the spherical excess: $$E \approx \frac{\text{area}}{R^2}$$ However, for this problem, we use the spherical law of sines directly. Angle C = 180° − 45° − 55° = 80° (plane assumption for now). Using spherical law of sines: $$\frac{\sin b}{\sin B} = \frac{\sin a}{\sin A}$$ $$\sin b = \sin a \cdot \frac{\sin B}{\sin A} = \sin(0.01) \times \frac{\sin 55°}{\sin 45°}$$ $$\sin b ≈ 0.00999976 \times \frac{0.81915}{0.70711} ≈ 0.011596$$ $$b ≈ 0.01160 \text{ radians} ≈ 73.9 \text{ km}$$ **Practical Application in the Philippines:** For a national triangulation network, geodetic methods account for: 1. The ellipsoidal shape (WGS 84 or PRS92). 2. Convergence of meridians (varies by latitude; significant in north–south networks). 3. Spherical excess and curvature corrections. 4. Projection distortions when converting to plane coordinates (UTM for the Philippines uses zones, each ±3° wide in longitude). Modern software (GNSS, Total Stations with geodetic modules) handles these computations automatically, but understanding the underlying spherical geometry is essential for error assessment and field adjustments.

Contemporary surveys integrate traditional methods (triangulation, stadia, theodolites) with modern technologies (EDM, total stations, GNSS) to achieve high accuracy over large areas efficiently. **Electronic Distance Measurement (EDM):** - Measures distances by timing the travel of electromagnetic waves (infrared or microwave) to a reflective prism. - Precision: ±(5 mm + 5 ppm × distance) typical; sub-centimetre for geodetic-grade instruments. - Eliminates stadia's dependence on rod intercept and atmosphere. - Works in low light; not limited to daylight like stadia. - Modern EDMs integrated into total stations. **Total Stations:** - Combine theodolite (angle measurement), EDM (distance), electronic data logger, and computation module. - Measure horizontal angles, vertical angles, and distances simultaneously. - Real-time computation of coordinates (E, N, elevation). - Integral components: horizontal circle (0° to 360°), vertical circle (−180° to +180°), collimation optics, EDM emitter. - Accuracy: ±(1–2 arcseconds) angular; ±(5 mm + 5 ppm) distance; ±1 cm elevation for 100 m sights. **GNSS (Global Navigation Satellite System):** - GPS (USA), GLONASS (Russia), Galileo (EU), BeiDou (China), and regional systems. - Determines 3D position (latitude, longitude, ellipsoidal height) directly from satellite signals. - Accuracy: ±10 m (single-point, civilian); ±1 cm (differential GPS/dGPS); ±5 mm (Real-Time Kinematic/RTK). - Requires clear sky visibility; less accurate in dense urban or forested areas. - Independent of line-of-sight constraints; can operate at night and in poor visibility. - Integration: GNSS provides absolute coordinates; total stations and levels provide relative detail and tie to local reference frames. **Hybrid Approaches:** Modern practice combines: 1. **GNSS for primary control:** Establish national/regional control points using multi-constellation satellite receivers (GPS/GLONASS/Galileo) in RTK mode. 2. **Total stations for detailed survey:** From GNSS control points, use total stations to map details (buildings, roads, utilities) with centimetre accuracy. 3. **Stadia for rapid reconnaissance:** Use stadia for rough distance checks or areas where EDM/GNSS are impractical. 4. **Levels for elevation control:** Establish vertical datums via differential levelling or RTK ellipsoidal heights, verified with bench marks. **Example 5.1 — Integrated Survey Workflow** A 20 km × 15 km area in the Philippines requires cadastral mapping. Procedure: 1. **GNSS reconnaissance (1–2 days):** Establish 5–6 GNSS control points distributed across the area using RTK-GNSS receivers. Accuracies: ±2 cm horizontal, ±3 cm vertical (PRS92 datum). 2. **Total station infill (5–10 days):** From GNSS controls, set up total stations at strategic points. Map 200–300 boundary monuments, building corners, and infrastructure. Precisions: ±5 cm per 500 m baseline. 3. **Levelling or ellipsoidal heights (2–3 days):** Establish vertical control using differential levelling (bench marks to ±5 mm per km) or verification of GNSS ellipsoidal heights. 4. **Data processing (2–3 days):** Adjust all observations in a least-squares network to reconcile GNSS, total station, and level data. Produce final coordinates (E, N, elevation) in PRS92. **Quality Assurance:** - Close loops: Every total station traverse must return to its starting point; angular and linear closures checked against tolerances (e.g., ±30 arcsec per traverse for cadastral surveys). - Redundancy: Multiple sights to each point from different stations; overdetermined observations detected and refined. - Datum consistency: All observations refer to PRS92 ellipsoid and Philippine UTM Zone 51N or 52N projection. **Philippine Context — RA 544 Requirements:** Under RA 544 (Geomatics Engineer License Law), surveys affecting property rights must be conducted by licensed Geomatics Engineers using approved methods and control points. Modern integrated surveys combining GNSS, total stations, and levels satisfy these requirements when properly adjusted and reported to the Land Registration Authority (LRA) or Bureau of Lands (BL).

Accurate surveying requires systematic understanding of error sources and application of appropriate corrections. Errors in triangulation, stadia, and geodetic work stem from instrumental, environmental, and procedural sources. **Instrumental Errors:** 1. **Theodolite collimation error (c):** Horizontal axis not perpendicular to vertical axis. Correction: - Measure angle with telescope normal (direct) and reversed (indirect); average eliminates c. 2. **Levels (bubble tubes) not calibrated:** Sensitivity of ±2 arcsec per mm deviation is assumed; if different, apply correction factor or re-calibrate. 3. **EDM zero error and scale factor:** Every EDM has a small constant offset (±1–5 cm) and a scale error (±2–5 ppm). Determined by measurement over a calibrated baseline; applied systematically to all distances. 4. **Stadia intercept parallax:** Reading the rod through an unfocused eyepiece introduces parallax error. **Solution:** Focus eyepiece on rod, not on crosshairs, to eliminate parallax. **Environmental Errors:** 1. **Atmospheric refraction (EDM and levelling):** Temperature, pressure, and humidity affect light propagation. - For levelling: Use short sights (≤60 m); balance back-sights and foresights. - For EDM: Apply temperature and pressure corrections to compute the refractive index; modern instruments auto-correct. 2. **Temperature effect on levelling staff/rod:** Wooden rods expand/contract; affect scale. Use metal invar staffs for precision levelling; apply temperature correction: $$\Delta L = \alpha L \Delta T$$ where α ≈ 1.2 × 10⁻⁵/°C for wood, 1 × 10⁻⁶/°C for metal. 3. **Earth's curvature and refraction:** Combined correction h_cr = 0.0675 D² applied to vertical angles and heights over distances >1 km. **Procedural Errors:** 1. **Systematic centering errors:** Theodolite/GNSS antenna not truly centred over station; applies a systematic offset to all observations. **Solution:** Measure offsets (e.g., rod-to-theodolite horizontal distance) and apply corrections in processing. 2. **Closure and misclosure:** In a traverse, the computed closure (difference between starting and ending coordinates) reveals accumulated error. For a closed polygon: - **Linear misclosure:** Δ E, Δ N discrepancies. - **Angular misclosure:** Sum of interior angles ≠ (n−2)180° for n sides. Each should be ≤√(n) × allowable angular tolerance (e.g., ±30 arcsec for cadastral work). - **Relative misclosure:** (linear misclosure)/(perimeter) ≈ 1/5000 acceptable for most surveys; 1/10000 for precise work. 3. **Blunders (mistakes):** Misread rod, wrong angle reversal, station misidentification. Detected by redundancy (remeasuring lines, angles) and statistical outlier tests (e.g., 3-sigma rejection in least-squares). **Common Exam Pitfalls:** 1. **Forgetting cos²(α) in stadia inclined sights** — use cos²(α), not cos(α) or tan(α). 2. **sin(2α) in vertical stadia component** — double angle, not single. 3. **Curvature correction formula h_cr = 0.0675 D² in km** — note the units: D in kilometres, h_cr in metres. 4. **Triangulation vs trilateration** — which one measures angles (triangulation) vs distances (trilateration). 5. **Plane vs geodetic decision** — plane for <10 km²; geodetic for larger areas and long baselines. 6. **Sign convention for vertical angles** — positive above horizontal, negative below. 7. **Spherical vs plane trigonometry** — when to apply spherical law of sines/cosines. **Example 6.1 — Error Correction in Levelling** A level circuit (closed loop) has measured elevations: - Forward loop (A → B → C → D → A): closure error = +12 mm over 1.2 km. - Acceptable tolerance for engineering survey: ±20 mm per √km = ±20 × √1.2 ≈ ±22 mm. - **Result:** Closure is acceptable; distribute error proportionally to distances traveled. If distance A→B = 300 m, B→C = 250 m, C→D = 400 m, D→A = 250 m (total 1200 m): Error per segment = 12 mm × (segment distance / total distance) - A→B correction: 12 × (300/1200) = 3.0 mm (subtract from all elevations computed via A→B forward) - B→C correction: 12 × (250/1200) = 2.5 mm - C→D correction: 12 × (400/1200) = 4.0 mm - D→A correction: 12 × (250/1200) = 2.5 mm **Example 6.2 — Traverse Closure and Linear Misclosure** A 4-sided traverse (rectangular perimeter, approximately 800 m × 600 m) measured with total station gives: - Sum of interior angles: (360°02′15″) vs theoretical (360°00′00″) for a quadrilateral. - Angular misclosure = 2′15″ = 135 arcsec. - Allowable for cadastral work: ±30 arcsec × √4 = ±60 arcsec. - **Result:** Angular misclosure exceeds tolerance; remeasure angles or recheck instrument calibration. Linear closure: - Computed closing coordinates: ΔE = −0.15 m, ΔN = +0.08 m. - Linear misclosure = √(0.15² + 0.08²) = 0.17 m. - Relative misclosure = 0.17 m / 2800 m perimeter ≈ 1/16,500 (acceptable for most work, though tight). **Example 6.3 — Stadia Accuracy and Precision** A stadia distance is measured 5 times, intercepts: 0.850, 0.851, 0.848, 0.852, 0.850 m. K = 100, C = 0. Distances: 85.0, 85.1, 84.8, 85.2, 85.0 m. Mean distance: 85.02 m. Standard deviation: σ ≈ 0.15 m. Relative precision: 0.15 / 85.02 ≈ 1/567 (acceptable for reconnaissance; marginal for engineering surveys). Comparison: Modern EDM achieves 1/10,000 or better, so stadia is reserved for rapid field checks, not primary measurements.