6.2 Geodesy, Earth Models, and Coordinate Systems

Three Earth Surfaces

Geodesy is the science of measuring the size and shape of the earth, its gravity field, and how points are positioned on it. The FS exam tests the concepts, not advanced derivations, so you must keep three surfaces clearly separated:

1. The physical (topographic) surface - the actual rugged ground where measurements are made.
2. The geoid - an equipotential surface of the earth's gravity field that best fits global mean sea level. It is smooth but undulating because gravity varies with mass distribution. The geoid is the surface a still ocean would form, and it is the reference for orthometric heights (elevations) because water flows along it.
3. The reference ellipsoid - a smooth mathematical figure (an oblate spheroid) defined by a semi-major axis a and a flattening f. It approximates the geoid well enough for computation. GRS80 (a = 6,378,137 m, 1/f = 298.257222101) is the ellipsoid used by NAD 83 and is essentially identical to WGS 84.

The key insight: the geoid is a gravity surface, the ellipsoid is a geometric surface, and they do not coincide. Their separation at a point is the geoid undulation N.

Heights and Geoid Undulation

Because there are two reference surfaces, there are two kinds of height the FS exam contrasts constantly:

• Ellipsoid (geodetic) height, h - the distance from the ellipsoid to the point, measured along the ellipsoid normal. GNSS directly produces h. It has no physical meaning for drainage because it ignores gravity.
• Orthometric height, H - the distance from the geoid to the point along the curved plumb line. This is the elevation surveyors and engineers use; water flows downhill in H, not in h.

They are connected by the geoid undulation N through the relation h = H + N, equivalently N = h - H and H = h - N. In the conterminous US, N is negative (the geoid lies below the GRS80 ellipsoid), roughly -8 m to -53 m. Globally N varies about plus or minus 100 m. To convert a GNSS-derived ellipsoid height to a usable elevation, you apply a geoid model (in the US, GEOID18 for NAD 83(2011)/NAVD 88 work).

Three Ways to State a Position

The same point can be described three equivalent ways, and FS questions move between them:

• Geocentric / Earth-Centered, Earth-Fixed (ECEF): Cartesian X, Y, Z with the origin at the earth's center of mass, the Z-axis through the reference pole, and the X-axis through the prime meridian at the equator. GNSS solves in ECEF.
• Geodetic coordinates: latitude (phi), longitude (lambda), and ellipsoid height (h), referenced to a specific ellipsoid and datum. Latitude is measured along the ellipsoid normal, not toward the geocenter, so geodetic latitude differs slightly from geocentric latitude.
• Projected (plane) coordinates: northing and easting (e.g., State Plane or UTM) on a flat grid after a map projection.

Converting ECEF to geodetic is a standard closed or iterative computation; converting geodetic to projected applies a map-projection formula. The exam expects you to know which description a tool produces and that a position is incomplete without its datum and height type.

Ellipsoid Parameters and GNSS Geoid Models

A reference ellipsoid is fully defined by two parameters - the semi-major axis a and a measure of polar flattening, usually the flattening f or the first eccentricity e (with e-squared = 2f - f-squared). For GRS80, a = 6,378,137 m and the inverse flattening 1/f = 298.257222101; WGS 84 shares the same a and a flattening different only in the last digits, so for survey purposes NAD 83 (GRS80) and WGS 84 ellipsoids are treated as geometrically equivalent even though their datum realizations differ at the meter level.

Geodesy reaches surveyors most directly through GNSS heighting. Because a receiver yields ellipsoid height h, every elevation derived from GNSS depends on a geoid model to supply N. The current US model for NAD 83(2011)/NAVD 88 work is GEOID18; the modernized NSRS pairs its new frames with GEOID2022. A geoid model is a grid of undulation values; the software interpolates N at the point and computes H = h - N. The accuracy of the resulting elevation is therefore limited by the accuracy of the geoid model, which is why high-precision vertical work still relies on differential leveling tied to published benchmarks.

Keeping ellipsoid, geoid, datum realization, and height type distinct is exactly the conceptual discipline the FS geodesy questions reward.