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Surveying principles, leveling, traversing, instruments and field calculation shortcuts.
| Concept | Definition | Key Detail |
|---|---|---|
| Surveying | Art of determining relative positions of points on/below/above Earth | First step in any civil engineering project |
| Plane Surveying | Earth surface treated as flat (spherical curvature neglected) | Valid for areas < 250 km² |
| Geodetic Surveying | Earth curvature accounted for | Large areas, national surveys |
| Primary Control | Establishing major points (triangulation, traverse) | High precision, covers large area |
| Secondary Control | Filling in details between primary points | Lower precision, smaller area |
| Horizontal Control | Latitudes and departures (position in plan) | Traversing, triangulation, trilateration |
| Vertical Control | Elevations (benchmarks, leveling) | Leveling, trigonometric leveling |
| Type | Length | Use |
|---|---|---|
| Metric Chain | 20 m, 30 m | General surveying in India |
| Engineer's Chain | 100 ft (100 links × 1 ft) | Imperial system surveys |
| Gunter's Chain | 66 ft (100 links × 0.66 ft) | Land surveys (1 chain = 1 acre width) |
| Revenue Chain | 33 ft | Revenue/land records |
| Steel Tape | 10 m, 15 m, 20 m, 30 m | Precise linear measurements |
| Invar Tape | Various | Least thermal expansion (for high precision) |
| Correction | Formula | When Applied |
|---|---|---|
| Temperature | Ct = αL(Tm − Ts) | Actual temperature ≠ standard temp |
| Pull/Tension | Cp = (P − Ps)L/(AE) | Actual pull ≠ standard pull |
| Sag | Cs = −w²L³/(24P²) | Tape hangs under its own weight |
| Slope | Ch = L(1 − cosθ) ≈ h²/(2L) | Measuring along slope |
| Standardization | Ck = (L − Ls)/Ls × L | Actual tape ≠ nominal length |
| Combined | C_total = Σ corrections | Sum of all applicable corrections |
┌─────────────────────────────────────────────────────────────────┐
│ CHAIN SURVEY — PRINCIPLES & TOOLS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ PRINCIPLE OF SURVEYING: │
│ Work from WHOLE to PART │
│ • Establish control points (framework) first │
│ • Then fill in details │
│ │
│ CHAIN LINE: Main survey line (direction of chain measurement) │
│ BASE LINE: Longest & most accurately measured chain line │
│ CHECK LINE: Measurement to verify accuracy of work │
│ TIE LINE: Connects survey stations to nearby features │
│ OFFSET: Lateral measurement from chain line to detail │
│ │
│ OFFSET TYPES: │
│ Perpendicular offset (preferred) — shortest distance │
│ Oblique offset — measured at angle to chain line │
│ │
│ SURVEY STATIONS: │
│ Main station: At end of main chain line │
│ Subsidiary station: On main line for convenience │
│ Tie station: Point connected by tie line │
│ │
│ CORRECTED DISTANCE: │
│ L_true = L_measured + Ct + Cp + Cs + Ck + Ch │
│ (signs of corrections may be + or − as applicable) │
└─────────────────────────────────────────────────────────────────┘| Type | Description | Accuracy |
|---|---|---|
| Simple Leveling | One setup of instrument | Low (±100 mm/km) |
| Differential Leveling | Multiple setups to cover long distances | Medium (±10 mm/km) |
| Reciprocal Leveling | Leveling across rivers/wide gaps | High (accounts for curvature & refraction) |
| Trigonometric Leveling | Using vertical angles & distances | Variable (depends on distance) |
| Barometric Leveling | Using atmospheric pressure | Low (±1-3 m) |
| Stadia Leveling | Using tacheometric principle | Medium (±150 mm/km) |
| Profile Leveling | Along a defined line (road, canal) | Medium |
| Cross-Section Leveling | Perpendicular to centerline | Medium |
| Instrument | Key Feature | Use |
|---|---|---|
| Dumpy Level | Rigid telescope, precise | General leveling work |
| Tilting Level | Telescope can tilt slightly | Precise work, easy setup |
| Auto Level | Self-leveling compensator | Most common modern instrument |
| Digital Level | Bar-coded staff + electronic read | Highest precision, automated |
| Laser Level | Laser beam reference plane | Construction layout, grading |
┌─────────────────────────────────────────────────────────────────┐
│ LEVELING — KEY FORMULAS & CONCEPTS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ FUNDAMENTAL EQUATION: │
│ Height of Instrument (HI) = RL of BM + BS (Backsight) │
│ RL of point = HI − FS (Foresight) │
│ RL of point = HI − IS (Intermediate Sight) │
│ │
│ BOOKING METHODS: │
│ ───────────────── │
│ Height of Instrument Method: │
│ HI = RL(BM) + BS │
│ RL(next) = HI − FS │
│ Check: Last RL = ΣBS − ΣFS + RL(first) │
│ │
│ Rise and Fall Method: │
│ Rise = BS − FS (if +ve, elevation increases) │
│ Fall = FS − BS (if +ve, elevation decreases) │
│ RL(n) = RL(n-1) + Rise − Fall │
│ Check: ΣBS − ΣFS = ΣRise − ΣFall = Last RL − First RL │
│ │
│ CORRECTIONS IN LEVELING: │
│ ───────────────────────── │
│ Curvature Correction: │
│ Cc = 0.0785 × d² (meters, where d is in km) │
│ (Earth curvature causes staff reading to be too HIGH) │
│ │
│ Refraction Correction: │
│ Cr = 0.0112 × d² (meters, where d is in km) │
│ (Atmospheric refraction causes reading to be too LOW) │
│ │
│ Combined Correction: │
│ Cc − Cr = 0.0673 × d² (combined effect reduces reading) │
│ │
│ RECIPROCAL LEVELING (across river): │
│ True difference = (h_AB − h_BA) / 2 │
│ where h_AB and h_BA are differences from both banks │
└─────────────────────────────────────────────────────────────────┘┌─────────────────────────────────────────────────────────────────┐
│ LEVELING — HEIGHT OF INSTRUMENT METHOD │
├─────────────────────────────────────────────────────────────────┤
│ │
│ BM RL = 100.00 m │
│ │
│ ┌──────┬──────┬──────┬──────────┬────────┐ │
│ │ Sta │ BS │ IS │ FS │ RL │ │
│ ├──────┼──────┼──────┼──────────┼────────┤ │
│ │ BM │ 1.625│ │ │ 100.000│ │
│ │ 1 │ │ 2.150│ │ 99.475│ │
│ │ 2 │ │ │ 1.830 │ 99.795│ ← HI: 101.625 │
│ │ 3 │ 2.465│ │ │ │ ← New setup │
│ │ 4 │ │ 1.920│ │ 100.340│ │
│ │ 5 │ │ │ 2.110 │ 100.150│ │
│ │ BM │ │ │ 1.765 │ 100.495│ ← Close │
│ └──────┴──────┴──────┴──────────┴────────┘ │
│ │
│ Check: ΣBS = 1.625 + 2.465 = 4.090 │
│ ΣFS = 1.830 + 2.110 + 1.765 = 5.705 │
│ ΣBS − ΣFS = −1.615 │
│ Last RL − First RL = 100.495 − 100.000 = +0.495 │
│ MISMATCH → Error in booking (recheck!) │
│ │
│ CORRECT CLOSED LOOP: ΣBS = ΣFS (should be equal) │
│ Misclosure = ΣBS − ΣFS = 0 for perfect closure │
│ Permissible = ±K√n where K = 0.025m to 0.1m, n = #setups │
└─────────────────────────────────────────────────────────────────┘±25√K mm where K = distance in km. For precise leveling, use ±4√K mm. Distribute misclosure proportionally to distances or number of setups.| Type | Least Count | Accuracy | Use |
|---|---|---|---|
| Vernier Theodolite | 20″ or 30″ | ±1′ | General survey, traverse |
| Precise Theodolite | 1″ or 0.1″ | ±1″ | Triangulation, geodetic work |
| Electronic/Digital | 1″ (display) | ±1″ | Modern surveys, automated |
| Total Station | 1″ + EDM | ±(1mm + 1ppm) | All-in-one: angles + distance |
| Term | Definition |
|---|---|
| Transit | Telescope can transit (rotate 180° in vertical plane) |
| Face Left (FL) | Vertical circle on left of observer |
| Face Right (FR) | Vertical circle on right of observer |
| Swinging | Rotating telescope in horizontal plane |
| Transiting | Rotating telescope 180° in vertical plane |
| Plunging | Reversing the telescope direction |
| Repetition | Measuring angle multiple times for precision |
| Reiteration | Measuring angles around horizon in sets |
┌─────────────────────────────────────────────────────────────────┐
│ THEODOLITE — MEASUREMENT METHODS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ HORIZONTAL ANGLE MEASUREMENT: │
│ ──────────────────────────── │
│ Method of Repetition (for precision): │
│ 1. Set up and level at station O │
│ 2. With Face Left: measure angle AOB (initial reading) │
│ 3. Release upper plate, swing to A again │
│ 4. Lock lower plate, measure AOB again (angles accumulate) │
│ 5. Repeat n times │
│ 6. Divide accumulated reading by n = average angle │
│ 7. Transit to Face Right, repeat │
│ 8. Average of FL and FR = final angle │
│ │
│ Method of Reiteration (multiple angles from one station): │
│ 1. Measure angle 1-2 (first angle) │
│ 2. Measure angle 1-3 (cumulative) │
│ 3. Measure angle 1-4 (cumulative) │
│ 4. Close on 1 to check closure error │
│ │
│ VERTICAL ANGLE: │
│ Vertical Angle = Zenith Angle − 90° (or 270° − Zenith Angle) │
│ Angle of Elevation: +ve (looking up) │
│ Angle of Depression: −ve (looking down) │
│ │
│ FACE LEFT & FACE RIGHT AVERAGE: │
│ Eliminates: │
│ • Collimation error (line of sight not ⊥ to horizontal axis) │
│ • Horizontal axis not ⊥ to vertical axis │
│ • Plate bubble errors │
│ │
│ TRIGONOMETRIC LEVELING: │
│ Δh = D × tan(α) + h_i − h_t │
│ where: │
│ D = horizontal distance │
│ α = vertical angle (elevation/depression) │
│ h_i = height of instrument │
│ h_t = target height / staff reading │
│ │
│ For long distances: │
│ Δh = D × tan(α) + h_i − h_t + (1 − 2m) × 0.0673 × d² │
│ where m = coefficient of refraction (≈ 0.07 for standard) │
└─────────────────────────────────────────────────────────────────┘±(a + b×D) where a = constant error (mm) and b = scale error (ppm).| Type | Description | Check Available |
|---|---|---|
| Open Traverse | Start and end at different known points | No inherent check; needs external check |
| Closed Traverse (Loop) | Starts and ends at same point | Σ angles = (n−2)×180°; Σ lat/deps = 0 |
| Closed Traverse (Link) | Connects two known points | Known azimuth + coordinates at both ends |
| Step | Operation | Formula / Check |
|---|---|---|
| 1 | Angular Check (Closed) | Σ observed angles = (n−2)×180° (for polygon) |
| 2 | Angular Misclosure | e = Σ obs − Σ theoretical |
| 3 | Balance Angles | Correction = e/n distributed equally |
| 4 | Compute Azimuths | Az = prev Az + 180° − interior angle (clockwise traverse) |
| 5 | Compute Lat/Dep | L = D×cos(Az); Δ = D×sin(Az) |
| 6 | Check ΣLat, ΣDep | ΣL = X_final − X_initial; ΣD = Y_final − Y_initial |
| 7 | Balance (Bowditch) | CL = −e_L × (D/ΣD); CD = −e_D × (D/ΣD) |
| 8 | Compute Coordinates | X = X_prev + L_bal; Y = Y_prev + D_bal |
┌─────────────────────────────────────────────────────────────────┐
│ TRAVERSE — COMPUTATION EXAMPLE │
├─────────────────────────────────────────────────────────────────┤
│ │
│ CLOSED TRAVERSE (4-sided polygon, clockwise): │
│ Known: Station A coordinates (1000, 1000), Azimuth AB = 45° │
│ │
│ ┌────────┬────────┬──────────┬───────────┬───────────┐ │
│ │ Line │ Length │ Bearing │ Latitude │ Departure │ │
│ ├────────┼────────┼──────────┼───────────┼───────────┤ │
│ │ AB │ 200 │ N 45° E │ +141.42 │ +141.42 │ │
│ │ BC │ 150 │ S 60° E │ -75.00 │ +129.90 │ │
│ │ CD │ 180 │ S 20° W │ -169.14 │ -61.56 │ │
│ │ DA │ 170 │ N 70° W │ +58.14 │ -159.72 │ │
│ └────────┴────────┴──────────┴───────────┴───────────┘ │
│ │
│ CHECK: │
│ ΣL = 141.42 − 75.00 − 169.14 + 58.14 = −44.58 │
│ ΣD = 141.42 + 129.90 − 61.56 − 159.72 = +50.04 │
│ Should be 0, 0 for closed loop → misclosure exists │
│ │
│ BOWDITCH ADJUSTMENT: │
│ Correction = −e × (Di / ΣD) │
│ e_L = −44.58 (total latitude error) │
│ e_D = +50.04 (total departure error) │
│ ΣD = 200 + 150 + 180 + 170 = 700 m │
│ │
│ For line AB: │
│ CL = +44.58 × (200/700) = +12.74 │
│ CD = −50.04 × (200/700) = −14.30 │
│ │
│ ADJUSTED COORDINATES: │
│ A: (1000.00, 1000.00) │
│ B: (1000+141.42+12.74, 1000+141.42−14.30) = (1154.16, 1127.12)│
│ C: ... (continue similarly) │
│ A (check): should return to (1000, 1000) │
│ │
│ AREA BY COORDINATES: │
│ Area = ½ |Σ(Xi × Yi+1 − Xi+1 × Yi)| │
│ Apply DMD (Double Meridian Distance) method or coordinates │
└─────────────────────────────────────────────────────────────────┘Area = ½|Σ(x_i·y_{i+1} − x_{i+1}·y_i)|.┌─────────────────────────────────────────────────────────────────┐
│ TACHEOMETRIC SURVEYING │
├─────────────────────────────────────────────────────────────────┤
│ │
│ STADIA METHOD (most common): │
│ ───────────────────────────── │
│ Principle: Telescope has stadia wires (upper, middle, lower) │
│ Staff intercept s = difference between top and bottom readings │
│ │
│ HORIZONTAL SIGHT (Level line of sight): │
│ ──────────────────────────────────── │
│ D = K × s + C │
│ │
│ Where: │
│ D = horizontal distance from instrument to staff │
│ K = multiplying constant (usually 100) │
│ s = staff intercept (top − bottom reading) │
│ C = additive constant (usually 0 for modern instruments) │
│ │
│ INCLINED SIGHT (line of sight at angle θ): │
│ ────────────────────────────────────── │
│ D = K × s × cos²θ + C × cosθ (horizontal distance) │
│ V = K × s × sinθ × cosθ + C × sinθ (vertical component) │
│ RL = HI + V − r │
│ │
│ Where: │
│ θ = vertical angle │
│ V = vertical distance (+ for elevation, − for depression) │
│ HI = height of instrument above station │
│ r = middle hair reading on staff │
│ │
│ TANGENTIAL METHOD: │
│ ───────────────── │
│ Uses two observations at different vertical angles: │
│ D = (s₁ − s₂) / (tanθ₂ − tanθ₁) │
│ (s₁, s₂ are staff readings; θ₁, θ₂ are vertical angles) │
│ │
│ SUBTENSE BAR METHOD: │
│ ──────────────────── │
│ A bar of known length b is placed horizontally: │
│ D = (b/2) / tan(α/2) │
│ where α = subtended angle at theodolite │
│ Approximation: D ≈ b / α (for small α in radians) │
│ │
│ STADIA CONSTANTS DETERMINATION: │
│ Set up at known distances D1, D2 and measure intercepts s1, s2│
│ From D = Ks + C: │
│ K = (D1 − D2)/(s1 − s2) │
│ C = D1 − K × s1 │
└─────────────────────────────────────────────────────────────────┘| Constant | Typical Value | Notes |
|---|---|---|
| K (Multiplying) | 100 | f/i ratio for external focusing telescope |
| C (Additive) | 0 | f + d for internal focusing telescope |
| C (Additive) | 0.3 − 0.5 m | f + d for external focusing telescope |
| Application | Method | Advantage |
|---|---|---|
| Contouring | Stadia leveling | Fast for hilly terrain |
| Topographic Mapping | Stadia + theodolite | Simultaneous horizontal position + elevation |
| Hydrographic Survey | Subtense bar on boat | Distance from shore to boat |
| Bridge Site Survey | Stadia traversing | Rapid data collection |
| Mine Surveying | Tangential method | Useful underground |
D = Ks·cos²θ for inclined sights. The cos²θ factor is critical — many exam errors come from forgetting it. The staff should be held vertical for accurate stadia readings.┌─────────────────────────────────────────────────────────────────┐
│ HORIZONTAL CURVES — CIRCULAR │
├─────────────────────────────────────────────────────────────────┤
│ │
│ NOTATION: │
│ R = radius of curve T = tangent length │
│ Δ = deflection angle L = length of curve │
│ C = chord length LC = long chord │
│ E = external distance M = mid-ordinate │
│ PC = Point of Curvature PT = Point of Tangency │
│ PI = Point of Intersection O = center of curve │
│ │
│ FUNDAMENTAL RELATIONSHIPS: │
│ T = R × tan(Δ/2) Tangent length │
│ L = (π × R × Δ) / 180 Curve length (arc) │
│ C = 2R × sin(Δ/2) Long chord │
│ E = R × [sec(Δ/2) − 1] External distance │
│ M = R × [1 − cos(Δ/2)] Mid-ordinate │
│ LC = 2R × sin(Δ/2) Long chord (same as C) │
│ │
│ DEGREE OF CURVE (D): │
│ By arc definition: D = (L × 360) / (2πR) │
│ By chord definition: D = 2 × arcsin(20/R) × (180/π) │
│ (based on 100 ft / 30 m standard chord) │
│ Relation: D = 1718.9/R (arc definition, D in degrees) │
│ │
│ COMPOUND CURVE: │
│ Two or more curves of different radii, same direction │
│ Common tangent at point of compound curvature (PCC) │
│ │
│ REVERSE CURVE: │
│ Two curves of same/different radii, opposite directions │
│ Common tangent at point of reverse curvature (PRC) │
│ │
│ TRANSITION CURVE: │
│ Type: Spiral (clothoid), cubic parabola, Bernoulli's lemniscate│
│ Purpose: Gradual introduction of centrifugal force │
│ Length: Ls = V³ / (C × R) │
│ V = design speed (m/s) │
│ C = rate of change of centrifugal acceleration (0.3-1.0) │
│ R = radius of circular curve │
│ Shift: S = Ls² / (24R) │
│ Superelevation: e = V² / (gR) + f (f = lateral friction) │
└─────────────────────────────────────────────────────────────────┘┌─────────────────────────────────────────────────────────────────┐
│ VERTICAL CURVES │
├─────────────────────────────────────────────────────────────────┤
│ │
│ TYPES: │
│ Summit (Crest) Curve: Convex up (g1 > 0, g2 < 0) │
│ Valley (Sag) Curve: Concave up (g1 < 0, g2 > 0) │
│ │
│ NOTATION: │
│ g1 = grade of incoming tangent (%) │
│ g2 = grade of outgoing tangent (%) │
│ L = length of curve (horizontal) │
│ BVC = Beginning of Vertical Curve │
│ EVC = End of Vertical Curve │
│ PVI = Point of Vertical Intersection │
│ │
│ PARABOLIC CURVE EQUATIONS: │
│ Elevation at any point x from BVC: │
│ y = y_BVC + g1 × x + (g2 − g1) × x² / (2L) │
│ │
│ Tangent offset at PVI (middle): │
│ E = (g2 − g1) × L / 8 │
│ (always positive for summit, negative for valley) │
│ │
│ Tangent offset at distance x from BVC: │
│ e = (g2 − g1) × x² / (2L) │
│ │
│ HIGH/LOW POINT ON VERTICAL CURVE: │
│ x = −g1 × L / (g2 − g1) (distance from BVC) │
│ │
│ If 0 < x < L → high/low point exists on curve │
│ If x < 0 or x > L → tangent grades are both increasing │
│ or both decreasing (no extreme point on curve) │
│ │
│ MINIMUM LENGTH CRITERIA: │
│ 1. Sight distance: L ≥ S²(A) / (200(√h1 + √h2)²) │
│ where S = sight distance, A = |g1−g2| in %, h = eye ht │
│ 2. Comfort: L ≥ (V² × A) / 395 │
│ where V = speed (km/h), A = algebraic difference in % │
│ 3. Drainage: Minimum 0.5% grade at lowest point │
│ │
│ EXAMPLE: │
│ g1 = +3%, g2 = −2%, L = 200 m, y_BVC = 100 m │
│ A = |3 − (−2)| = 5% │
│ E = 5 × 200 / 8 = 125 → y_PVI = 100 + 3×100 − 125 = 275 │
│ High point: x = −3×200/(−2−3) = −600/−5 = 120 m from BVC │
│ y at x=120: 100 + 3(120) + (−5)(120)²/400 = 100+360−180=280 │
│ Max elevation = 280.00 m │
└─────────────────────────────────────────────────────────────────┘r = (g2−g1)/L. In highway design, minimum curve length for comfort is L ≥ V²A/395 where V is in km/h and A = |g1−g2|%.| Concept | Description |
|---|---|
| Segments | Space (satellites), Control (ground stations), User (receivers) |
| Satellites | 24+ NAVSTAR satellites in 6 orbital planes, altitude ~20,200 km |
| Signals | L1 (1575.42 MHz, C/A + P code), L2 (1227.60 MHz, P code) |
| C/A Code | Coarse/Acquisition — civilian, ~10 m accuracy |
| P Code | Precise — military, ~1-3 m accuracy |
| Minimum Satellites | 4 needed (3 for position, 1 for time correction) |
| Trilateration | Position found by intersecting 4 spheres from 4 satellites |
| Error Source | Magnitude | Mitigation |
|---|---|---|
| Ionospheric Delay | 5-15 m | Dual-frequency receivers, models |
| Tropospheric Delay | 2-30 m | Mathematical modeling |
| Satellite Clock | 1-2 m | Ground control corrections |
| Ephemeris Error | 1-2 m | Updated ephemeris data |
| Multipath | 0.5-5 m | Antenna design, site selection |
| Receiver Noise | 0.3-3 m | Better receiver design |
| Component | Description | Examples |
|---|---|---|
| Hardware | Computers, servers, GPS receivers, plotters | Workstations, field tablets |
| Software | ArcGIS, QGIS, GRASS GIS, AutoCAD Map 3D | Desktop, web, mobile GIS |
| Data | Spatial + attribute data | Shapefiles, GeoJSON, KML, GeoTIFF |
| People | GIS analysts, surveyors, planners | Domain experts, technicians |
| Methods | Analysis procedures, models, workflows | Overlay, buffer, network analysis |
±5mm + 0.5ppm.| Error Type | Cause | Characteristic | Example |
|---|---|---|---|
| Systematic (Cumulative) | Instrument/environmental flaws | Same sign, accumulates | Tape too short, collimation error |
| Random (Accidental) | Human limitations, environmental | ±ve and −ve equally likely | Reading estimation, pointing error |
| Gross (Blunder) | Mistakes, carelessness | Large, usually detectable | Misreading 6 as 9, wrong station ID |
| Concept | Definition | Target Analogy |
|---|---|---|
| Accuracy | Closeness to true value | Near bullseye |
| Precision | Repeatability of measurements | Tight grouping (may be off-center) |
| High Accuracy + Precision | Near true value & repeatable | Tight group at bullseye |
| High Precision, Low Accuracy | Consistent but biased | Tight group away from bullseye |
┌─────────────────────────────────────────────────────────────────┐
│ ERROR ANALYSIS — STATISTICAL CONCEPTS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ MOST PROBABLE VALUE (MPV): │
│ MPV = Σ(x_i) / n = Arithmetic mean │
│ │
│ RESIDUAL: v_i = MPV − x_i │
│ Property: Σ(v_i) = 0 │
│ │
│ STANDARD DEVIATION (σ): │
│ σ = √(Σv²/(n−1)) [for sample] │
│ σ = √(Σv²/n) [for population] │
│ │
│ STANDARD ERROR OF MEAN: │
│ σ_m = σ / √n │
│ (Mean of n measurements is √n times more precise) │
│ │
│ PROBABLE ERROR (e): │
│ e = 0.6745 × σ │
│ Probability that error exceeds e = 50% │
│ │
│ 90% ERROR: 1.6449 × σ │
│ 95% ERROR: 1.9599 × σ │
│ 99% ERROR: 2.5758 × σ │
│ │
│ WEIGHTS IN ADJUSTMENT: │
│ w_i ∝ 1/σ²_i (weight inversely proportional to variance) │
│ Weighted mean = Σ(w_i × x_i) / Σ(w_i) │
│ │
│ ERROR PROPAGATION (for independent measurements): │
│ If u = f(x, y, z): │
│ σ_u² = (∂f/∂x)²σ_x² + (∂f/∂y)²σ_y² + (∂f/∂z)²σ_z² │
│ │
│ LEAST SQUARES ADJUSTMENT: │
│ Principle: Σ(w_i × v_i²) = MINIMUM │
│ Normal equations: AᵀWA × δ = AᵀW × l │
│ (solved for corrections δ) │
└─────────────────────────────────────────────────────────────────┘