Positioning Terminology: Error, Reference Surfaces, and Coordinate Systems

Positional Error & Uncertainty

Definition:
Positional error is the measurable difference between a point’s observed (measured) position and its actual or reference position, typically expressed as a linear distance. Uncertainty refers to the estimated interval within which the true position lies, given the measurement process’s limitations. Both are fundamental for evaluating spatial data’s reliability and suitability for use.

Usage:
In aviation, surveying, and geodetic applications, positional error and uncertainty must be rigorously assessed. For example, FAA Advisory Circular 150/5300-18C and ICAO standards require that critical features such as runway thresholds be measured with uncertainties below specific thresholds (often just a few centimeters). These values are determined through statistical analysis, commonly using the 95% confidence level (2σ), and are crucial for navigation, obstacle clearance, and engineering design.

Sources of Error:

• Instrument precision and calibration
• Operator skill and procedure
• Environmental factors (e.g., atmospheric effects, multipath in GPS)
• Geodetic model or datum inconsistencies
• Random (noise) and systematic (bias) errors

Expression and Standards:
Uncertainty is commonly expressed as a radius (e.g., Circular Error Probable, CEP) or an error ellipse around the measured point. Methodologies for quantifying and reporting uncertainty are defined by standards such as the Federal Geographic Data Committee (FGDC) and the National Standard for Spatial Data Accuracy (NSSDA). The Root Mean Square Error (RMSE) is a fundamental metric, often scaled by 1.7308 to yield a 95% confidence interval for horizontal positions.

Illustrative Example:
A GPS survey of a runway end marker yields an RMSE of 0.015 m. The 95% confidence positional uncertainty is ±0.026 m (0.015 m × 1.7308). If the standard requires ≤0.03 m, the result is compliant.

Relevant Standards:

• FAA AC 150/5300-18C
• ICAO Annex 14, ICAO Doc 9674
• FGDC, NSSDA

Reference Surface

Definition:
A reference surface is a mathematically or physically defined surface to which positions are referenced for measurement, mapping, and navigation. The most common are the ellipsoid, geoid, and local sphere.

Usage:
Reference surfaces underpin all coordinate systems and datums. The ellipsoid is standard for global and national horizontal mapping; the geoid is used for vertical datums (heights referenced to mean sea level). For aeronautical data, ICAO and FAA require referencing to globally recognized surfaces—typically the WGS84 ellipsoid for horizontal and a defined geoid for elevations.

Types:

• Ellipsoid: Smooth, regular surface approximating the Earth’s shape for latitude/longitude.
• Geoid: Irregular, gravity-based surface matching mean sea level; used for elevations.
• Local Sphere: Simplified sphere for small-area surveys where ellipsoid/geoid differences are negligible.

Example:
A runway end is referenced by latitude, longitude, and ellipsoid height (WGS84), plus orthometric height (NAVD88) over the geoid.

Standards:

• ICAO WGS 84 Implementation Manual
• FAA AC 150/5300-18C

Ellipsoid

Definition:
An ellipsoid (or spheroid) is a mathematically defined, smooth, closed surface generated by rotating an ellipse about its minor axis. It approximates the Earth’s mean sea level and is simple enough for computational use.

Parameters:

• Semi-major axis (a)
• Semi-minor axis (b)
• Flattening (f = (a-b)/a)
• First eccentricity (e)

Common Models:

• WGS84: Global standard (a = 6,378,137.0 m; f = 1/298.257223563)
• GRS80: NAD83 (North America); nearly identical to WGS84

Usage:
The ellipsoid is the reference for geodetic coordinate systems. All GPS and aeronautical data use the WGS84 ellipsoid, ensuring global compatibility.

Example:
A survey station’s coordinates (latitude, longitude, ellipsoid height) referenced to WGS84 can be seamlessly used with GNSS data worldwide.

Geoid

Definition:
The geoid is the equipotential surface of Earth’s gravity field that best fits global mean sea level, including under continents. Unlike the ellipsoid, the geoid is irregular, reflecting local gravity variations.

Usage:
The geoid is the reference for orthometric heights (elevations above mean sea level). Vertical datums like NAVD88 (U.S.) or EGM96 (global) are essentially geoid models. The geoid is essential for converting GPS-derived ellipsoid heights to usable elevations for engineering and aviation.

Properties:

• Matches mean sea level, varies locally up to ±100 m from the ellipsoid
• Determined using satellite altimetry, gravimetric surveys, and leveling

Example:
A runway threshold’s elevation is 57.6 m above the geoid (NAVD88), but the GPS ellipsoid height is 65.2 m. The geoid undulation is -7.6 m.

Standards:
ICAO and FAA require specifying the geoid model (e.g., GEOID12B, EGM96) for all aeronautical height data.

Local Sphere

Definition:
A local sphere is a spherical surface used for small-area surveys (typically <100 km radius), with a radius chosen to fit the local curvature of the ellipsoid.

Usage:
Used in small engineering or mapping projects where sub-centimeter accuracy is not required. For large areas, ellipsoid or geoid referencing is preferred.

Example:
A small airport layout may use a local sphere radius of 6,378,000 m for preliminary work, then convert to ellipsoidal coordinates for regulatory compliance.

Datum (Horizontal, Vertical, Geodetic)

Definition:
A datum is a set of reference parameters specifying the origin, orientation, and scale of a coordinate system, typically tied to a reference surface and control points.

Types:

• Horizontal Datum: Defines latitude and longitude on an ellipsoid (e.g., WGS84, NAD83)
• Vertical Datum: Defines “zero” for elevations, usually the geoid (e.g., NAVD88, EGM96)
• Geodetic Datum: Integrates both horizontal and vertical components

Usage:
All spatial data must specify the datum. Coordinates referenced to different datums can differ by tens or hundreds of meters. Modern datums use satellite and gravity data for high precision.

Example:
A runway end is reported as 33°55'48.2"N, 118°24'28.9"W, height 28.3 m (NAD83 (2011) geodetic datum, NAVD88 vertical datum).

Datum Transformation

Definition:
Datum transformation mathematically converts coordinates between datums, accounting for differences in origin, scale, orientation, and ellipsoid parameters.

Methods:

• Three-parameter transformation: Translation only
• Seven-parameter (Helmert) transformation: Translation, rotation, scale
• Grid-based transformation: Uses empirical grids for local corrections

Usage:
Essential for integrating data from sources using different datums. ICAO mandates WGS84 for aviation; FAA requires documentation for any data not originally in WGS84.

Example:
A position in NAD27 is transformed to WGS84 using a seven-parameter transformation for GNSS navigation.

Coordinate System

Definition:
A coordinate system is a framework for specifying point positions using numerical values (coordinates), based on a defined origin, axes, and units, referenced to a surface or datum.

Types:

• Geodetic Coordinate System: Latitude, longitude, ellipsoidal height
• Geocentric Coordinate System: Cartesian X, Y, Z from Earth’s center
• Local (Project) Coordinate System: Rectangular grid aligned to a local origin

Example:
A runway centerline is mapped in geodetic coordinates (WGS84) then transformed to a local engineering grid.

Key Consideration:
Always specify both the coordinate system and datum/reference surface. Omission can cause significant misplacement, especially when combining data from different systems.

Geodetic Coordinate System

Definition:
A geodetic coordinate system is a 3D curvilinear system based on an ellipsoid, defined by latitude (φ), longitude (λ), and ellipsoidal height (h).

Usage:
Standard for GPS, geodetic surveying, and aeronautics. Required by ICAO and FAA for all aeronautical positions.

Example:
Navigation fix: 51°28'40.12"N, 0°27'41.21"W, height 45.0 m (WGS84).

Advantages:

• Global applicability
• Direct GNSS compatibility
• Supports data integration across regions

Geocentric Coordinate System

Definition:
A geocentric coordinate system is a 3D Cartesian system with the origin at the Earth’s center of mass.

• X-axis: Intersects the equator at the prime meridian
• Y-axis: 90° east of X-axis
• Z-axis: Earth’s mean rotation axis (north pole)

Usage:
Essential for satellite geodesy, GNSS, and datum transformations.

Example:
A GPS satellite’s position: X = 1,567,890 m, Y = 4,567,890 m, Z = 6,789,012 m (WGS84 geocentric system).

Local Coordinate System

Definition:
A local coordinate system is a 2D or 3D Cartesian grid for a specific project, with its own origin, orientation, and scale.

Usage:
Common for engineering, construction, and mapping within limited areas. Simplifies calculations and reduces distortion versus global systems.

Example:
A construction site uses a local grid with (0,0,0) at the southwest corner, all elements referenced in meters north, east, and elevation above a site benchmark.

For more details on standards and implementation, see FAA AC 150/5300-18C, ICAO Annexes, and FGDC/NSSDA publications, or contact our geodesy experts for a consultation.