GELE Geodesy — The Geoid, Gravity and HeightsSummary

A mathematical surface of revolution (oblate spheroid) defined by two parameters: semi-major axis (a) and flattening (f). WGS84 specifies a = 6,378,137 m and f = 1/298.257223563. The ellipsoid is smooth, featureless, and serves as the reference for GNSS coordinate systems. It does not align with actual sea level; hence the need for the geoid.

The equipotential surface of Earth's gravity field that best approximates global mean sea level (MSL). It undulates relative to the ellipsoid by ±100 m globally and ±60 m regionally (Philippines: roughly –15 to +40 m relative to WGS84 ellipsoid). The undulation N is caused by lateral variations in crustal density, mantle structure, and isostatic imbalance. The geoid is physically meaningful: a perfectly still ocean (ignoring currents, wind, pressure) would conform to the geoid.

The perpendicular distance from a point on the Earth's surface to the reference ellipsoid. Provided directly by GNSS receivers (precise to ±10–20 mm for modern static observations). Measured along the ellipsoid normal, not vertically. Independent of gravity. Example: a GNSS receiver at a benchmark reads h = 52.30 m, meaning the point is 52.30 m above the WGS84 ellipsoid surface along the normal direction.

The height above the geoid, measured along the local vertical (direction of gravity). Obtained from spirit leveling and is the practical elevation used in engineering, drainage, flood management, and utility design. Rigorously defined via geopotential numbers (W) and mean gravity along the plumb line. Example: a benchmarked point at H = 82.40 m is 82.40 m above mean sea level in the gravitational sense. In the Philippines, orthometric heights are referenced to the Philippine Geodetic Datum (referenced to Corregidor tide gauge, now datum for PRS92).

The vertical distance from the ellipsoid to the geoid, measured along the ellipsoid normal. Positive where geoid is above the ellipsoid; negative where below. Varies spatially due to Earth's internal mass distribution. Example: if N = –30.10 m at a point, the geoid is 30.10 m below the WGS84 ellipsoid. For the Philippines, N ranges from approximately –15 m (parts of Mindanao) to +40 m (parts of Luzon), reflecting variations in crustal thickness and isostatic adjustment.

Rearranged as H = h – N. This is the core formula converting GNSS ellipsoidal height (h) to orthometric height (H) given the geoid undulation (N). Sign convention: if N is negative (geoid below ellipsoid), subtracting it increases H; if N is positive (geoid above ellipsoid), subtracting it decreases H. Example: h = 52.30 m, N = –30.10 m → H = 52.30 – (–30.10) = 82.40 m.

Gravitational acceleration (g) is the downward acceleration due to Earth's gravity field, measured in m/s² (≈ 9.78 at the equator, ≈ 9.83 at the poles due to Earth's oblate shape and centrifugal effects). Varies with: (1) latitude (centrifugal effect, flattening); (2) elevation (free-air correction, ≈ 0.3086 mGal/m); (3) local terrain mass (Bouguer anomaly). Gravity measurements define the geoid via the fundamental relation between potential and gravity. Leveling combined with gravity observations yields rigorous orthometric heights.

The integral of gravity along a level surface from the geoid to the point: C = W₀ – W, where W is the potential at the point and W₀ is the potential of the geoid. Measured in m²/s² (or 'GPU' units, 1 GPU = 1 m²/s²). Geopotential numbers are what spirit leveling + gravity observations physically measure. They define rigorous (dynamic or orthometric) heights via H_o = C / g̅, where g̅ is the mean gravity between geoid and point. Invariant—independent of the path of leveling, even if the leveling circuit is not closed.

Dynamic heights use a constant reference gravity (g₀ = 9.78031846 m/s² at 45° latitude) to convert geopotential numbers: H_d = C / g₀. Orthometric heights use the mean gravity along the plumb line: H = C / g̅, and require gravity measurements. For rough engineering, they differ by < 0.5 % over typical height ranges; for precise geodetic networks, the distinction matters. The Philippine national leveling network referenced orthometric heights historically; modern adjustments may incorporate geopotential approaches.

Global models (e.g., EGM2008: ±0.5 m accuracy globally; regional models (e.g., PGM09 for the Philippines, refined with airborne gravity data: ±0.1–0.3 m accuracy) provide N values at any point via interpolation. The Philippines uses PGM09 (Philippine Geoid Model 2009) for GNSS-to-elevation conversions in cadastral surveying and engineering. Users select a model appropriate to accuracy requirements and geographic extent. Modern software (Trimble TerraSync, Leica GeoOffice) integrates geoid models for real-time GNSS-to-elevation conversion.

A surveyor establishes a benchmark using a high-precision GNSS receiver in Calabarzon, Philippines. Raw GNSS gives h = 156.42 m (WGS84 ellipsoidal height). Using PGM09, the local geoid undulation is N = 25.65 m. Calculate the orthometric height: H = 156.42 – 25.65 = 130.77 m above Philippine MSL. This value is then used for engineering design (road grades, water supply lines, drainage networks). Without the geoid model, the surveyor would incorrectly assume the elevation is 156.42 m—an error of 25.65 m that would lead to design failure.

A national leveling network spanning 500 km has been adjusted using classical methods (least-squares, assuming infinite gravity). Geopotential numbers (C values) are computed at all benchmarks via leveling + gravity data. Simultaneously, a GNSS survey of the same benchmarks provides ellipsoidal heights (h). Using the official geoid model (PGM09), orthometric heights are derived: H_gnss = h – N. These are compared with the leveled orthometric heights: H_leveled. Discrepancies > ±50 mm indicate systematic errors in the leveling adjustment, crustal deformation, or geoid model inadequacy. This cross-validation strengthens confidence in both the leveling network and the geoid model.

Engineers designing a major flood-control project (e.g., Marikina River dike raise) must know the elevation of the terrain and predicted flood levels (referred to MSL, i.e., orthometric heights). GNSS rapid surveys of 50 cross-section points yield ellipsoidal heights (h). Using PGM09, these are converted to orthometric heights (H = h – N). Design flood level is given as H = 24.50 m (MSL). Dike crest is set at H = 27.00 m (freeboard). Without correct geoid undulation, if the surveyor mistakenly uses h instead of H, the dike could be ±25 m off—catastrophic failure in a flood event.

Under RA 4374 (Cadastral Act) and RA 8560 (Property Registration Act), surveyed boundaries and areas are recorded in official cadastral maps. For sloping terrain, the original survey was conducted with orthometric heights (H) because leveling was the primary method. When a modern resurvey uses GNSS, heights must be converted from ellipsoidal (h) back to orthometric (H) for consistency with the legal record. Using the current geoid model ensures that the resurvey is compatible with the official cadastral datum and that area calculations (projected onto a standard elevation, e.g., H = 0) are legally defensible.

A contractor building a shopping mall in Metro Manila uses GNSS to set out building corners and slab elevations. Architectural plans specify floor levels in orthometric heights (e.g., basement at H = –3.20 m, ground floor at H = 0.00 m, roof at H = 42.50 m). The builder's GNSS instrument provides ellipsoidal heights (h). Without converting via the geoid model (N ≈ 28 m in Metro Manila), the builder would miss levels by ~28 m vertically—impossible. Correct procedure: measure h for each point, subtract N to get H, then compare against the architectural plan. This is a daily task on major construction projects.

A surveying company publishes a geoid undulation map (N in meters) covering the Visayas. The map shows N ranging from –8 m in parts of Panay to +35 m in parts of Negros. A surveyor interpreting this map must understand: (1) where N is most negative (Panay, geoid is lowest relative to ellipsoid, meaning ellipsoidal heights are largest relative to orthometric heights), (2) where N is most positive (Negros, geoid is highest, ellipsoidal heights are smallest relative to orthometric heights), (3) the spatial gradient of N (e.g., N changes by ~40 m over 150 km, or ~0.27 m/km on average). For a survey across this region, ignoring spatial variation of N leads to height errors that accumulate.

A surveyor uses a modern GNSS receiver (Trimble R12i) that can output both WGS84 ellipsoidal height (h) and orthometric height (H) if a geoid model is loaded. If the loaded geoid model is outdated (e.g., EGM96, which is 0.5–1 m less accurate than PGM09 for the Philippines), the displayed H is biased. The surveyor may not realize the problem until comparing the GNSS results with leveled benchmarks and finding systematic offsets. Proper practice: verify that the receiver is using the correct, current geoid model (PGM09) for Philippine surveys, and understand that the receiver's H is only as accurate as its embedded geoid model.

A precision agriculture company uses GNSS-based yield mapping for rice paddies in Nueva Ecija. Yields vary with micro-topography; subtle height differences (few centimeters) affect water retention and thus crop health. The company surveys field-elevation points with high-precision GNSS (cm-level) and uses PGM09 to convert to orthometric heights (H). A digital elevation model (DEM) is constructed showing H at each pixel. This DEM reveals drainage patterns and informs irrigation/subsidence management. Without proper height conversion, the DEM is useless for terrain analysis.