Principles of Geospatial Surveying - Allan

Random errors

Errors that vary unpredictably in both magnitude and sign, following statistical distributions (typically normal). Can be reduced by averaging repeated observations. — Allan, Principles of Geospatial Surveying, Ch. 5

Right-handed coordinate system

A 3D coordinate system where, using the right-hand rule, the thumb points along +X, fingers curl toward +Y, and palm faces +Z. Standard convention in surveying and geodesy. — Allan, Principles of Geospatial Surveying, Ch. 3

Polar coordinates in surveying

A system expressing position as distance (radius) and direction (angle) from a reference point and line. In surveying: horizontal distance and bearing/azimuth from the instrument station. — Allan, Principles of Geospatial Surveying, Ch. 3

Cylindrical coordinates

Coordinates (r, θ, z) combining polar coordinates in one plane with height along a perpendicular axis. Useful for tunnels, shafts, and cylindrical structures. — Allan, Principles of Geospatial Surveying, Ch. 3

Geodetic latitude (φ)

The angle between the equatorial plane and the normal (perpendicular) to the reference ellipsoid at a point. Differs from geocentric latitude except at poles and equator. — Allan, Principles of Geospatial Surveying, Ch. 3

Geocentric vs. geodetic latitude difference

Arises because Earth is an oblate spheroid. On a sphere, all surface normals pass through the center. On an ellipsoid, only at poles and equator does the normal pass through the center. — Allan, Principles of Geospatial Surveying, Ch. 3

Conformal transformation

A transformation that preserves angles and shapes of infinitesimally small figures. Scale may vary from point to point but is the same in all directions at any given point. — Allan, Principles of Geospatial Surveying, Ch. 4

Helmert (7-parameter) transformation

Similarity transformation with 7 parameters: 3 translations (TX, TY, TZ), 3 rotations (ωX, ωY, ωZ), and 1 uniform scale factor (s). Preserves shape while allowing translation, rotation, and uniform scaling. — Allan, Principles of Geospatial Surveying, Ch. 4

2D conformal transformation parameters

Four parameters: two translations, one rotation, and one scale factor. Requires minimum 2 common points (each provides 2 equations: E and N). — Allan, Principles of Geospatial Surveying, Ch. 4

Systematic errors

Errors with consistent magnitude and sign under similar conditions. Cannot be reduced by repeated observations—must be eliminated through calibration, corrections, or improved technique. — Allan, Principles of Geospatial Surveying, Ch. 5

Standard deviation (σ)

Measures dispersion of observations about their mean. For normal distribution: ~68% within ±1σ, ~95% within ±2σ, ~99.7% within ±3σ. — Allan, Principles of Geospatial Surveying, Ch. 5

Standard error of the mean

σₘ = σ/√n, where σ is standard deviation of single observation and n is number of observations. Shows why averaging improves precision. — Allan, Principles of Geospatial Surveying, Ch. 5

Weight of an observation

w = σ₀²/σ². Inversely proportional to variance—more precise observations (smaller σ) receive larger weights in least squares adjustment. — Allan, Principles of Geospatial Surveying, Ch. 5

Variance propagation

Computes precision of derived quantities from measured quantities. For Z = f(X,Y): σ²_Z = (∂f/∂X)²σ²_X + (∂f/∂Y)²σ²_Y + 2(∂f/∂X)(∂f/∂Y)σ_XY. — Allan, Principles of Geospatial Surveying, Ch. 5

Principle of least squares

Minimizes Σwᵢvᵢ² (sum of weighted squared residuals). Provides most probable values when errors are normally distributed; yields unique, unbiased estimates. — Allan, Principles of Geospatial Surveying, Ch. 6

Normal equations (least squares)

Formed by differentiating weighted sum of squared residuals with respect to each unknown and setting derivatives to zero. Yields system of linear equations for least squares solution. — Allan, Principles of Geospatial Surveying, Ch. 6

Matrix solution for least squares

X = (AᵀWA)⁻¹AᵀWL, where A is design matrix, W is weight matrix, L is observation vector. N = AᵀWA is normal matrix; Qx = N⁻¹ gives variance-covariance of unknowns. — Allan, Principles of Geospatial Surveying, Ch. 6

Redundancy (degrees of freedom)

r = n - u (observations minus unknowns). Allows error detection and provides basis for statistical testing. At least one redundant observation needed for quality assessment. — Allan, Principles of Geospatial Surveying, Ch. 6

Reference variance (variance factor)

σ̂₀² = vᵀWv / (n - u). Estimated from weighted sum of squared residuals divided by degrees of freedom. Should be close to 1.0 if a priori weights were realistic. — Allan, Principles of Geospatial Surveying, Ch. 6

GPS positioning principle

Trilateration using pseudo-range measurements to satellites. Intersection of range spheres determines position. Minimum 4 satellites needed: 3 for position, 1 for receiver clock error. — Allan, Principles of Geospatial Surveying, Ch. 7

Differential GPS (DGPS)

Eliminates spatially correlated errors (especially ionospheric delay) by differencing observations between reference station and rover experiencing similar conditions. — Allan, Principles of Geospatial Surveying, Ch. 7

Carrier phase vs. code measurement precision

Carrier wavelength (~19 cm for L1) is much shorter than code chip length (~300 m for C/A). Carrier phase measurable to millimeters, enabling centimeter-level positioning. — Allan, Principles of Geospatial Surveying, Ch. 7

Integer ambiguity (GPS)

The unknown number of complete carrier wavelengths between satellite and receiver. Receiver only measures fractional phase. Must be resolved for high-precision carrier phase positioning. — Allan, Principles of Geospatial Surveying, Ch. 7

Resection

Position determination by measuring angles AT the unknown point TO three or more known control points. Instrument occupies unknown point. — Allan, Principles of Geospatial Surveying, Ch. 8

Danger circle (resection)

The circle passing through the three known points and the unknown point. If unknown lies on this circle, no unique solution exists—any point on the circle produces same angles. — Allan, Principles of Geospatial Surveying, Ch. 8

Forward intersection

Position determination by measuring directions FROM two or more known points TOWARD the unknown point. Instruments occupy known points; ideal for inaccessible targets. — Allan, Principles of Geospatial Surveying, Ch. 8

Traverse closure

Checks both angular and distance measurements. Angular closure: measured angles vs. theoretical sum. Linear closure: misclosure in E and N from combined errors. — Allan, Principles of Geospatial Surveying, Ch. 8

Orthometric vs. ellipsoidal height

h = H + N. Ellipsoidal height (h) = orthometric height (H) + geoid undulation (N). Orthometric measured above geoid; ellipsoidal measured above reference ellipsoid. — Allan, Principles of Geospatial Surveying, Ch. 9

Curvature and refraction correction

Combined effect ≈ 0.0675d² meters (d in km). Curvature: +0.0785d². Refraction: -0.011d². Net effect causes level surface to drop away from horizontal sight. — Allan, Principles of Geospatial Surveying, Ch. 9

Trigonometric heighting

ΔH = S × sin(α) + (c + r) - i + t, where S is slope distance, α is vertical angle, c + r is curvature/refraction, i is instrument height, t is target height. — Allan, Principles of Geospatial Surveying, Ch. 9

Reciprocal leveling

Sights taken in both directions across an obstacle. Mean of two elevation differences cancels curvature, refraction, and collimation errors (if conditions constant). — Allan, Principles of Geospatial Surveying, Ch. 9

Degree of curve (chord definition)

Central angle D subtended by a 100-foot chord. Radius R = 50/sin(D/2). Larger degree = sharper curve (smaller radius). Used in North America for railways. — Allan, Principles of Geospatial Surveying, Ch. 11

Transition (spiral) curve

Provides gradual change from infinite radius (tangent) to finite radius (circular curve). Allows smooth introduction of superelevation for vehicle comfort and stability. — Allan, Principles of Geospatial Surveying, Ch. 11

Vertical parabolic curve property

Rate of change of grade is constant throughout the curve. r = (g₂ - g₁)/L, where g₁, g₂ are entrance/exit grades and L is curve length. — Allan, Principles of Geospatial Surveying, Ch. 11

Conformal (orthomorphic) projection

Preserves angles and shapes of small features. Scale same in all directions at any point (though varies from point to point). Examples: Transverse Mercator, Lambert Conformal Conic. — Allan, Principles of Geospatial Surveying, Ch. 10

UTM (Universal Transverse Mercator)

60 zones of 6° longitude each, from 80°S to 84°N. Each zone has own projection centered on central meridian (assigned 500,000 m E). Coordinates in meters. — Allan, Principles of Geospatial Surveying, Ch. 10

Lambert Conformal Conic—standard parallels

Two standard parallels where cone intersects ellipsoid. Scale exact (unity) along these lines, smaller between, larger outside. Ideal for east-west extent regions. — Allan, Principles of Geospatial Surveying, Ch. 10

Grid convergence (γ)

Angle between grid north (parallel to central meridian) and true north (along local meridian). Zero on central meridian; increases with distance from it. — Allan, Principles of Geospatial Surveying, Ch. 10

Deformation monitoring

Determining changes in position/shape over time. Requires repeated observations from stable reference points. Precision depends on expected movement rates. — Allan, Principles of Geospatial Surveying, Ch. 12

As-built survey

Performed after construction to verify actual positions and dimensions against design specifications. Provides records for maintenance and may be required contractually. — Allan, Principles of Geospatial Surveying, Ch. 12

EDM phase comparison principle

Modulated carrier wave compared with returned signal phase. Phase difference relates to fractional wavelength of distance. Multiple frequencies resolve integer ambiguity. — Allan, Principles of Geospatial Surveying, Ch. 13

Automatic level compensator

Suspended prism/mirror using gravity to automatically bring line of sight horizontal when roughly leveled. Eliminates precise leveling requirement, speeds field work. — Allan, Principles of Geospatial Surveying, Ch. 13