The Geoid, Gravity and Heights - Cheat Sheet
Your 30-minute revision companion for the geoid, three height systems, gravity, and their critical interplay in modern GNSS surveying. Every formula, definition, and pitfall you need to ace board-exam questions on ellipsoidal vs orthometric heights.
The Three Height Systems (Absolute Priority)
Formulas
h = H + N
Meaning
h = ellipsoidal height (GNSS-derived); H = orthometric height (spirit-levelled elevation); N = geoid undulation (height of geoid above ellipsoid)
Watch out
Sign of N: NEGATIVE where geoid is BELOW ellipsoid (common in SE Asia, including Philippines). Double negative in H = h − N trips students — write it out: H = h − (−30) = h + 30.
When to use
Every time you need to convert between GNSS heights and ground elevations; the fundamental relationship in modern surveying.
H = h − N
Meaning
Rearranged form: orthometric height equals ellipsoidal height minus geoid undulation.
Watch out
Apply sign of N correctly. If N = −30 m (geoid 30 m below ellipsoid), then H = h − (−30) = h + 30. Most students forget the sign convention.
When to use
Converting from GNSS (gives h) to elevation (need H); the standard board-exam transformation.
N = h − H
Meaning
Geoid undulation derived from GNSS ellipsoidal height and known orthometric height.
Watch out
Result can be positive or negative. In Philippines (PRS92 context), typical N values range ≈ −30 to +40 m. Negative N is normal for much of the archipelago.
When to use
Determining geoid undulation at a point when both h (GNSS) and H (levelling) are known.
Common values
Value
9.78 m/s²
Symbol
g₀
Quantity
Standard gravity (normal value at equator, sea level)
Value
9.83 m/s²
Symbol
g_pole
Quantity
Standard gravity at pole (latitude 90°, sea level)
Value
−0.0003086 m/s² per metre (or ~−3.086 mGal/100 m)
Symbol
∂g/∂h
Quantity
Free-air gravity gradient (change per metre altitude)
Value
−30 to +40 m (varies regionally; negative dominant in Luzon and Visayas)
Symbol
N
Quantity
Typical geoid undulation in Philippines
Value
6,378,137 m
Symbol
a
Quantity
WGS84 ellipsoid semi-major axis
Value
6,356,752.3 m
Symbol
b
Quantity
WGS84 ellipsoid semi-minor axis
Important facts
- GNSS gives h (ellipsoidal) directly — NOT elevation. Always apply a geoid model (N) to get H.
- h = H + N is the fundamental equation taught in geodesy; every surveyor must know this.
- Sign of N: Negative N is common in Philippines (geoid below WGS84 ellipsoid in much of the country).
- WGS84 ellipsoid is the global reference; PRS92 (Philippine Reference System) is a local realization centred on Philippines.
- Geoid undulation is caused by density variations in Earth's crust and mantle (mass anomalies).
- Orthometric heights are what engineering projects use (roads, dams, building elevations).
- Spirit levelling is the traditional method to establish orthometric heights; far more accurate than raw GNSS for local engineering work.
- Geoid models (e.g., EGM2008, local grid models) are essential; without N, cannot convert h to H.
- Gravity variations with latitude are ~0.5 %; elevation changes cause free-air gravity gradient of ~0.0003 m/s² per metre.
- Bouguer anomaly accounts for mass density; measured gravity minus normal gravity (at sea level) plus topographic correction.
Key definitions
Geoid
Example
Mean sea level surface (if oceans were global, undisturbed); undulates ±100 m from ellipsoid due to mass anomalies.
The equipotential surface of Earth's gravity field that best represents global mean sea level; the reference for orthometric heights.
Ellipsoidal Height (h)
Example
GNSS receiver displays h = 52.30 m; this is NOT the same as ground elevation.
Perpendicular distance from WGS84 ellipsoid surface to point; directly measured by GNSS (GPS/GNSS receivers).
Orthometric Height (H)
Example
Benchmark surveyed with levelling staff reads H = 112.20 m above mean sea level (true elevation for roads, buildings).
Distance measured along local gravity direction from geoid (MSL) to point; obtained by spirit levelling; the practical 'elevation' used in engineering.
Geoid Undulation (N)
Example
In Metro Manila area, N ≈ −30 m (geoid 30 m below WGS84 ellipsoid); in some Mindanao regions, N ≈ +10 m.
Height of geoid above ellipsoid; positive where geoid is above ellipsoid, negative where below.
Gravity (g)
Example
g ≈ 9.78 m/s² at equator; g ≈ 9.83 m/s² at poles; decreases with altitude (free-air effect).
Acceleration due to Earth's gravitational field; varies with latitude, elevation, and local density anomalies.
Geopotential Number (C)
Example
Used for precise levelling networks in PRS92; accounts for variation of g with latitude and altitude.
Integral of gravity along a level path from geoid to point; provides rigorous (dynamic) height independent of gravity variations.
Diagrams to know
- Three-surface diagram: ellipsoid, geoid, terrain; showing h, H, N on a vertical section.
- Sign convention for N: geoid above vs below ellipsoid (positive vs negative N).
- Gravity variation curve vs latitude (9.78 to 9.83 m/s²).
Gravity and Its Variation
Formulas
g(φ) = 9.78 + 0.0052 sin²(φ) + 0.0000058 sin²(2φ) m/s²
Meaning
International Gravity Formula (1967); g is normal gravity at sea level for latitude φ (φ in degrees or radians).
Watch out
This gives NORMAL gravity (theoretical, on ellipsoid). Measured gravity ≠ g(φ); difference is the Bouguer anomaly or free-air anomaly.
When to use
Computing normal gravity at a benchmark to then determine gravity anomaly or apply geopotential corrections.
g_measured = g_normal + ΔgFree-Air + ΔgBouguer + ΔgTerrain
Meaning
Measured gravity = normal gravity + free-air correction + Bouguer correction + terrain correction.
Watch out
All corrections must have consistent signs and units. Free-air is positive with elevation; Bouguer (negative mass) reduces gravity below.
When to use
Gravity surveys and gravimetry to isolate local mass anomalies; critical for geoid determination.
ΔgFree-Air = −(∂g/∂h) × h ≈ −0.0003086 h (mGal)
Meaning
Free-air correction for elevation h (in metres); gravity decreases with altitude.
Watch out
Sign: negative because gravity decreases upward. If point is above sea level, ΔgFree-Air is negative (gravity reduced).
When to use
Converting gravity measured at elevation h to equivalent value at sea level.
Common values
Value
−0.3086 mGal/metre
Symbol
Γ_FA
Quantity
Free-air gravity gradient
Value
−0.1119 mGal/metre
Symbol
Γ_B
Quantity
Bouguer gravity gradient (for crustal density ~2.67 g/cm³)
Value
6,371 km
Symbol
R
Quantity
Earth's mean radius
Important facts
- Gravity increases from equator to poles (~0.5 %) due to Earth's oblateness and centrifugal effect.
- Free-air correction (−0.0003086 m/s² per metre) is the dominant effect for elevation changes.
- Bouguer correction removes assumed mass slab; Bouguer anomaly reveals true subsurface density structure.
- Geopotential numbers provide rigorous heights for high-precision levelling networks (e.g., PRS92).
- Gravity observations are essential to define the geoid; satellite and terrestrial gravity data feed into geoid models.
- Isostatic compensation complicates gravity: tall mountains have lighter roots, reducing net gravity anomaly.
- In engineering surveying, precise levelling + gravity observations give orthometric heights; GNSS alone cannot.
- Gravity anomalies can exceed ±100 mGal in tectonically active regions (Philippines is highly anomalous due to subduction).
Key definitions
Normal Gravity
Example
At latitude 14°N (Luzon), g ≈ 9.794 m/s²; at 10°S (Mindanao), g ≈ 9.782 m/s².
Theoretical gravity on WGS84 ellipsoid at sea level for a given latitude; depends only on latitude and ellipsoid parameters.
Measured Gravity
Example
A gravimeter reading 978.5 mGal at a point in Metro Manila includes effects of crustal density anomalies and elevation.
Actual gravity observed at a station using gravimeter; differs from normal gravity due to local mass variations and elevation.
Free-Air Gravity Anomaly
Example
Positive free-air anomaly over oceanic ridge (dense mantle rock); negative over continental plateau.
Difference between measured gravity (corrected for elevation only) and normal gravity; indicates mass anomaly without topographic effect.
Bouguer Anomaly
Example
Used in mineral exploration and crustal studies to map density variations; negative Bouguer anomaly indicates deficit of dense material.
Difference between measured gravity (free-air corrected and topographic mass removed) and normal gravity; reveals subsurface mass.
Geopotential Number (C)
Example
C = ∫g dH from geoid upward; gives rigorous height independent of latitude-dependent gravity variations.
Integral of gravity from geoid to point along a plumb line; equals W₀ − W, where W is potential at point.
Diagrams to know
- Gravity variation with latitude (sine curve from 9.78 to 9.83 m/s²).
- Free-air correction vs elevation (linear decrease with altitude).
- Components of measured gravity: normal + anomalies.
Geoid Models and Their Use
Common values
Value
±0.1 to 0.15 m (1σ)
Symbol
σ_EGM
Quantity
Typical uncertainty in global geoid models (EGM2008)
Value
±0.05 to 0.1 m (in well-surveyed areas)
Symbol
σ_local
Quantity
Typical uncertainty in local geoid models (PRS92)
Value
±100 m (±30 m in SE Asia / Philippines region)
Symbol
N_range
Quantity
Maximum geoid undulation range globally
Important facts
- No single geoid model is universal; accuracy depends on local gravity data coverage and quality.
- Global models (EGM2008) have ~0.1 m uncertainty; local models can achieve 0.05 m with dense gravity surveys.
- Geoid undulation varies smoothly over 10s of kilometres; do not extrapolate N beyond survey area.
- Satellite altimetry (TOPEX, Jason, Sentinel-3) provides mean sea surface; geoid model = mean sea surface − sea surface topography.
- PRS92 geoid model is legally required for all official Philippine surveys (RA 4374, RA 8560).
- Geoid models are regularly updated as satellite gravity data (GRACE, GOCE) improve.
- Use the most recent geoid model available for your survey region; older models may have 0.2–0.5 m errors.
Key definitions
Geoid Model
Example
EGM2008 (global, 1 arcminute grid); PRS92 local geoid model (used for Philippine surveys).
Grid or function describing geoid undulation (N) as function of latitude and longitude; derived from gravity data and satellite measurements.
EGM2008
Example
For a point at 14.5°N, 121.0°E (Metro Manila), EGM2008 gives N ≈ −30.5 m.
Earth Gravitational Model 2008; global geoid model with ~0.1 m accuracy; used worldwide before local models available.
PRS92 Geoid Model
Example
More accurate than EGM2008 for Philippine territory; accuracy ≈ 0.05 to 0.1 m in well-surveyed areas.
Philippine Reference System geoid model; local refinement of global models for improved accuracy in the Philippines.
Diagrams to know
- Geoid map of Philippines showing N variations (−40 to +20 m across archipelago).
- Schematic: satellite, geoid model grid, user interpolating N for survey point.
Practical Conversion: GNSS to Elevation (Board-Exam Workflow)
Formulas
H = h − N
Meaning
Step-by-step: (1) Get h from GNSS; (2) Look up N from geoid model at survey point; (3) Calculate orthometric height H.
Watch out
Sign of N is critical. Write it explicitly: if N = −30 m, then H = h − (−30) = h + 30 m. Double-check sign every time.
When to use
Every practical GNSS survey where elevation is needed (roads, dams, building foundations, levelling networks).
Important facts
- GNSS is faster and more accurate than spirit levelling for gross elevation; but requires geoid model for final result.
- Spirit levelling + gravity is still the rigorous method for high-precision orthometric heights (±0.01 m).
- For engineering projects, combine GNSS + geoid model with spot levelling checks at control points.
- Geoid undulation varies smoothly; interpolate N from grid models using bilinear or bicubic methods.
- Always document which geoid model and version used in survey report (required by RA 8560).
- Uncertainty in final H = √(σ_h² + σ_N²); if GNSS σ_h = 0.05 m and geoid σ_N = 0.1 m, then σ_H ≈ 0.11 m.
- For critical surveys, run independent spirit levelling verification along key routes.
Key definitions
GNSS-to-Elevation Workflow
Example
GNSS gives 52.30 m; geoid model (EGM2008/PRS92) gives N = −30.10 m; elevation H = 52.30 − (−30.10) = 82.40 m.
Process: GNSS measurement → ellipsoidal height h; apply geoid model → geoid undulation N; compute H = h − N; result is usable elevation.
Diagrams to know
- Flowchart: GNSS receiver → h; geoid model lookup → N; calculator H = h − N → elevation.
- Three-surface cross-section showing where h, H, N are measured.
Heights in Philippine Context (PRS92 & Legal References)
Important facts
- All Philippine surveys use PRS92 by law (RA 8560); international projects may also reference WGS84 globally.
- PRS92 uses Philippine local geoid model; must apply for accurate h-to-H conversion in Philippines.
- Orthometric heights in Philippines are referenced to mean sea level (MSL) at Tidal Station, Port of Manila (historical datum).
- Geodetic Engineer stamp required on all official surveys; uses PRS92 ellipsoidal coordinates + orthometric heights.
- Survey reports must document: GNSS ellipsoidal heights (h), geoid model used (PRS92 version and date), computed orthometric heights (H), accuracy (σ).
- Cadastral surveys (land title) must tie to national coordinate system; elevations used for site description, not legal boundary.
- Land title registration under Torrens System (RA 496, RA 8560) requires accurate cadastral survey; professional liability high.
Key definitions
PRS92 (Philippine Reference System)
Example
All official maps, surveys, and engineering projects in Philippines must reference PRS92 (RA 8560).
National reference system for the Philippines; uses WGS84 ellipsoid centred at station Libmanan (Camarines Sur).
RA 8560 (Geodetic Engineering Law)
Example
Any cadastral, topographic, or engineering survey must be done and signed off by licensed Geodetic Engineer using PRS92.
Republic Act 8560 (1998); mandates all surveying and mapping in Philippines use PRS92; requires Geodetic Engineer licensure.
RA 4374 (Geotechnical and Geological Sciences Law)
Example
Combined with RA 8560, these define professional standards for all land surveying and mapping in Philippines.
Companion law regulating geological/geotechnical professions; works alongside Geodetic Engineering law.
PD 1529 (Forerunner decree establishing Geodetic Engineer Board Exam)
Example
PRC Geodetic Engineer Licensure Exam is administered under this decree (now superseded by RA 8560, but PD 1529 still referenced).
Presidential Decree 1529 (pre-1998); established the professional board examination and licensure for Geodetic Engineers.
CA 141 (Public Land Act provisions on surveys)
Example
Lot surveys must follow CA 141 procedures; cadastral surveys tied to PRS92 using ellipsoidal coordinates converted to elevations.
Commonwealth Act 141; historical law defining land survey and title registration; still used for cadastral survey standards.
Diagrams to know
- Philippine archipelago with PRS92 zone coverage and typical N values by region.
Must remember
- The fundamental equation: h = H + N (ellipsoidal = orthometric + geoid undulation). Rearranged: H = h − N converts GNSS to elevation.
- GNSS gives h (ellipsoidal height), NOT elevation H. A geoid model (N) is ALWAYS required to convert h to H. Without N, you cannot determine true elevation.
- Sign convention for N: Negative N (common in Philippines) means geoid is BELOW the ellipsoid. In H = h − N, be careful with double negatives: H = h − (−30) = h + 30.
- Gravity increases from equator (9.78 m/s²) to poles (9.83 m/s²) due to Earth's oblateness and centrifugal effect; this variation defines the normal gravity baseline.
- Free-air gravity gradient is −0.0003086 m/s² per metre; this is the largest gravity correction for elevation changes in surveying.
- Geoid undulation (N) in Philippines ranges typically −30 to +40 m and varies smoothly; interpolate from grid models (PRS92 or EGM2008) — never extrapolate beyond survey area.
- Spirit levelling + gravity observations provide rigorous orthometric heights (±0.01 m accuracy); GNSS + geoid model are faster but less accurate (±0.1 m) for engineering projects.
- RA 8560 (Philippine law) mandates all surveys use PRS92 reference system; Geodetic Engineer licensure required for official cadastral, topographic, or engineering surveys.
- Orthometric heights (H) are what engineers actually use for elevation control, design grades, and project elevations; these reference the geoid (mean sea level), not the ellipsoid.
- Board-exam pitfall #1: confusing h and H, forgetting sign of N, or not applying a geoid model. Always write out: 'GNSS h = ___ m; PRS92 geoid N = ___ m; Elevation H = h − N = ___ m.'
Last minute tips
- If an exam question says 'GNSS gave 52 m, find elevation,' immediately ask yourself: 'Is this asking for h (ellipsoidal) or H (orthometric)?' The answer is always H. Set up H = h − N and look for N in the problem or geoid model table provided.
- Sign trick for N: If geoid is 'below' ellipsoid, N is negative (Philippines). When you compute H = h − N, the double negative flips the sign. Example: h = 52 m, N = −30 m ⇒ H = 52 − (−30) = 82 m. Do NOT write H = 52 − 30 = 22 m (wrong sign error).
- Gravity varies smoothly with latitude; if a problem gives gravity values at two latitudes, use proportional reasoning. Memorize: g ≈ 9.78 m/s² (equator), 9.83 m/s² (poles), free-air gradient ≈ −0.31 mGal/metre.
- Geoid model lookup: PRS92 is standard for Philippines; EGM2008 is global backup. Problem will either provide N directly, give you a grid table to interpolate from, or ask you to state 'Cannot determine H without geoid model N.' Never assume N = 0.
- Legal/professional note: Any answer referencing Philippine surveys must cite PRS92, not generic WGS84 ellipsoid. If problem says 'Philippines survey,' always include 'Using PRS92 geoid model…' in your answer for full exam credit.
Comparison tables
Three Height Systems — Comparison
| Height Type | Symbol | Reference Surface | How Measured | Used For | Typical Accuracy | |
|---|---|---|---|---|---|---|
| Ellipsoidal Height | h | WGS84 ellipsoid | GNSS receiver (direct output) | Geometric; not engineering elevation | ±0.05 m (RTK-GNSS) | |
| Orthometric Height | H | Geoid (MSL) | Spirit levelling + gravity, or GNSS + geoid model | Engineering projects, elevations, benchmarks | ±0.01 m (spirit levelling); ±0.1 m (GNSS + geoid) | |
| Geoid Undulation | N | Ellipsoid surface | Derived from gravity data / geoid model | Lookup from grid model (EGM2008, PRS92) | Conversion tool: H = h − N | ±0.05 to 0.15 m (model dependent) |
Sign of N — Critical Decision Tree
| Condition | Sign of N | Meaning | Example Region | Calculation: H = h − N |
|---|---|---|---|---|
| Geoid BELOW ellipsoid | N < 0 (negative) | Geoid surface is below WGS84 ellipsoid | Much of Philippines (Luzon, Visayas), N ≈ −30 m | H = h − (−30) = h + 30; elevation H is higher than h |
| Geoid ABOVE ellipsoid | N > 0 (positive) | Geoid surface is above WGS84 ellipsoid | Some Pacific islands, parts of Indonesia, N ≈ +20 m | H = h − (+20) = h − 20; elevation H is lower than h |
| Geoid coincides with ellipsoid | N ≈ 0 | Hypothetical perfect match (rare globally) | None (used only for teaching) | H ≈ h; no correction needed |
Gravity Variations — Magnitude & Significance
| Effect | Magnitude | Cause | Direction of Change | Correction Type |
|---|---|---|---|---|
| Latitude effect on g | ≈ 0.5 % (9.78 to 9.83 m/s²) | Earth's oblateness + centrifugal effect | Increases from equator to poles | Normal gravity formula (table or model) |
| Elevation (free-air) | −0.0003086 m/s² per metre | Distance from Earth's center | Gravity decreases with altitude | Free-air correction (ΔgFA = −0.0003086 h) |
| Mass density (Bouguer) | ±0.1 to 0.2 m/s² or more | Local crustal/mantle density anomalies | Positive over dense bodies (mafic rock); negative over light bodies | Bouguer correction + terrain correction |
| Terrain effect (local topography) | ±10 to 100 mGal (varies with slope) | Proximity to high/low topography | High ground pulls gravity upward (positive); deep basins reduce it | Terrain correction (computed from DEM) |
Common Board-Exam Mistakes & How to Avoid Them
| Mistake | Why It Happens | Correct Approach | Example Correction |
|---|---|---|---|
| Using GNSS height h as elevation | Students forget GNSS gives h, not H | Always apply geoid model: H = h − N | GNSS h = 52.30 m, N = −30.10 m ⇒ H = 82.40 m (not 52.30 m) |
| Wrong sign on N (positive/negative) | Forgetting that N < 0 in Philippines | Check geoid map or model output; write N explicitly | If N = −30 m (geoid below), H = h − (−30) = h + 30 (double negative) |
| Confusing orthometric (H) with ellipsoidal (h) | Similar notation; different meanings | Remember: H from spirit levelling/gravity; h from GNSS satellite | Engineering uses H; GNSS outputs h; never mix them |
| Forgetting geoid model entirely | Students skip the lookup step in workflow | Geoid model is mandatory; H cannot be computed without N | Always state: 'Using PRS92 geoid model, N = ... m at this point' |
| Ignoring uncertainty (σ) in geoid model | Reporting H without error bounds | Compute total error: σ_H = √(σ_h² + σ_N²) | h = 52.30 ± 0.05 m; N = −30.10 ± 0.10 m ⇒ H = 82.40 ± 0.11 m |
Diagrams
The Geoid, Gravity and Heights — Concept Map
Mind map showing how ellipsoidal height (h from GNSS), orthometric height (H from levelling), and geoid undulation (N from model) are related. Emphasizes h = H + N and practical Philippine requirements.
GNSS-to-Elevation Conversion Workflow
Step-by-step flowchart showing the workflow to convert GNSS ellipsoidal height (h) to usable engineering elevation (H) using a geoid model. Highlights the critical sign decision for N.
Height System State Transitions
State diagram showing the transformation from GNSS measurement (ellipsoidal height h) through geoid model lookup (N) to final orthometric height (H) for engineering use.
Sources of Uncertainty in h→H Conversion
Pie chart showing typical contributions to total uncertainty in converting GNSS heights to elevation. Geoid model uncertainty dominates; emphasizes why geoid accuracy is critical.