GELE Geodesy — The Geoid, Gravity and HeightsDetailed Explanation
A detailed, step-by-step explanation of The Geoid, Gravity and Heights for GELE aspirants. This page goes deeper than the summary and study notes, walking through the reasoning behind each concept so you understand why Professional Regulation Commission (PRC) — Board of Geodetic Engineering tests it the way it does in the GELE Geodesy subtest.
Exam context
For the Geodetic Engineer Licensure Examination, Professional Regulation Commission (PRC) — Board of Geodetic Engineering tests Geodesy under a "Core" label, with The Geoid, Gravity and Heights in the 5th slot across 6 chapters. GELE candidates must clear the 70% weighted average, no sub-test below 50% cut on the 2026 paper, which draws about a meaningful share of Geodesy questions. Date to watch: September 2026.
The Geoid, Gravity and Heights - Detailed Explanation
In geodesy, three distinct surfaces define how we measure and report positions on and above the Earth: the mathematical ellipsoid, the gravity-defined geoid, and the physical terrain. Understanding the relationships among these surfaces — and the three corresponding height types — is fundamental to every GNSS survey, engineering project, and cadastral work in the Philippines. The core equation h = H + N (ellipsoidal height = orthometric height + geoid undulation) is arguably the single most tested formula in the geodesy portion of the PRC Geodetic Engineer Licensure Examination. This chapter builds mastery of that relationship, the nature of gravity, and the practical implications for converting GNSS-derived ellipsoidal heights into the orthometric (mean-sea-level) elevations that Philippine engineering and law require.
Concepts
The Three Reference Surfaces of Geodesy
Geodesy recognizes three distinct reference surfaces, each serving a different purpose: 1. THE ELLIPSOID (Mathematical Surface): A smooth, oblate spheroid defined by semi-major axis a and flattening f. WGS84 (used by GPS/GNSS) and PRS92 (Philippine Reference System of 1992, the national datum of the Philippines) are both GRS80-derived ellipsoids. The ellipsoid has no physical meaning — it is a purely mathematical best-fit to the shape of the Earth. Ellipsoidal heights h are measured along the ellipsoidal normal from the ellipsoid surface to the point. 2. THE GEOID (Gravity Surface): The equipotential surface of the Earth's gravity field that most closely coincides with global mean sea level (MSL). It is a lumpy, irregular surface — it undulates ±100 m relative to the ellipsoid — because the Earth's internal mass distribution is non-uniform. The geoid is the physical reference for orthometric heights H (the 'heights above mean sea level' used in all Philippine engineering works, topographic maps, and legal descriptions under PD 1529 and CA 141). 3. THE TERRAIN (Physical Surface): The actual, irregular surface of the Earth where measurements are made. Points on the terrain have coordinates in all three systems simultaneously. The geoid undulation N is the separation between the geoid and the ellipsoid at a given point, measured along the ellipsoidal normal. N is positive when the geoid is above the ellipsoid and negative when below. In the Philippines, N generally ranges from about +10 m to +22 m (the geoid is above the WGS84 ellipsoid), meaning GNSS ellipsoidal heights h are numerically smaller than the corresponding orthometric heights H over most of the archipelago.
Examples
Since N is positive here (geoid is above the ellipsoid), the orthometric height H is numerically smaller than the ellipsoidal height h. This is the common situation in the Philippines. The benchmark is at 53.12 m above Philippine Mean Sea Level.
Scenario
A geodetic engineer surveys a benchmark in Quezon City. The GNSS receiver reports an ellipsoidal height h = 68.42 m (WGS84). A Philippine geoid model gives N = +15.30 m at that location. What is the orthometric height H of the benchmark?
Solution
H = h − N = 68.42 − 15.30 = 53.12 m above MSL
GNSS gives ellipsoidal height h, not orthometric height H. Without applying a geoid model, a surveyor can seriously misinterpret the true elevation of a point relative to sea level.
Scenario
Explain why a GNSS-derived height of h = 5.00 m does not mean a point is only 5 m above sea level in Metro Manila.
Solution
H = h − N. If N ≈ +15 m in Metro Manila, then H = 5.00 − 15.00 = −10.00 m — the point is actually 10 m BELOW mean sea level. This is not unusual for reclaimed areas or low-lying barangays in coastal Metro Manila.
Applications
- Converting GNSS ellipsoidal heights to mean-sea-level elevations for engineering design.
- Establishing vertical control benchmarks consistent with NAMRIA's national leveling network.
- Flood risk and storm surge mapping using accurate orthometric heights.
- PD 1529 (Property Registration Decree) and CA 141 (Public Land Act) surveys require correct elevation data on cadastral plans.
- Design of drainage, irrigation, and road infrastructure that depends on accurate MSL-referenced elevations.
Misconceptions
- MISCONCEPTION: 'The ellipsoid and geoid are the same surface.' CORRECTION: The ellipsoid is a smooth mathematical shape; the geoid is a gravity-defined physical surface. They differ by up to ±100 m globally.
- MISCONCEPTION: 'A GNSS height is directly usable as elevation.' CORRECTION: GNSS gives ellipsoidal height h; the orthometric height H requires subtracting the geoid undulation N.
- MISCONCEPTION: 'N is always positive.' CORRECTION: N is negative (up to −107 m) in areas like the Indian Ocean region where the geoid is below the ellipsoid. In the Philippines N is generally positive.
Related Concepts
- Ellipsoidal height h
- Orthometric height H
- Geoid undulation N
- WGS84 and PRS92 reference ellipsoids
- NAMRIA geoid model for the Philippines
- Mean Sea Level (MSL)
Common Exam Questions
Example
Which surface is defined by the Earth's gravity field and approximates mean sea level? Answer: The geoid.
Approach
Distinguish which surface (ellipsoid, geoid, terrain) a given height refers to.
Question Type
Conceptual identification
Example
Geoid undulation N is the height of the geoid above the ellipsoid; positive when geoid is above, negative when below.
Approach
State the definition of geoid undulation N and its sign convention.
Question Type
Definition recall
Key Points To Remember
- Ellipsoid: smooth, mathematical, basis for GNSS — defined by WGS84 (global) and PRS92 (Philippine).
- Geoid: lumpy, physical, gravity-defined equipotential surface ≈ mean sea level — reference for orthometric heights.
- Terrain: the actual ground surface where measurements are taken.
- Geoid undulation N = separation of geoid above (+) or below (−) the ellipsoid.
- In most of the Philippines, N is positive (+10 m to +22 m), so h < H at most sites.
- PRS92 is the official horizontal datum of the Philippines; NAMRIA maintains geoid models for Philippine territory.
- All legally required elevations (PD 1529 surveys, CA 141 public land surveys) are orthometric heights H.
The Fundamental Height Equation: h = H + N
The most important equation in this chapter — and one of the most tested in the PRC board exam — relates the three height quantities: h = H + N Where: h = ellipsoidal height: vertical distance from the ellipsoid surface to the point, measured along the ellipsoidal normal. GNSS/GPS receivers output this directly. H = orthometric height: vertical distance from the geoid to the point, measured along the plumb line (direction of gravity). This is the practical 'elevation above mean sea level' used in all engineering and legal work. N = geoid undulation (also called geoid height or geoid separation): vertical distance from the ellipsoid to the geoid, measured along the ellipsoidal normal. DERIVED FORMS: H = h − N (most commonly used: convert GNSS height to elevation) N = h − H (compute undulation from dual GNSS + leveling) SIGN CONVENTION — THE MOST COMMON SOURCE OF ERRORS: • If N > 0: geoid is above ellipsoid → H = h − N gives H < h • If N < 0: geoid is below ellipsoid → H = h − N gives H > h (e.g., H = 52.30 − (−30.10) = 82.40 m) PHYSICAL MEANING: Think of h as the 'GPS reading', H as the 'surveyor's elevation', and N as the 'correction' that bridges the two. Without N, GPS heights have no practical engineering use.
Examples
N is negative here, meaning the geoid is below the ellipsoid at this point. Subtracting a negative number increases H above h. The double negative is the most common arithmetic mistake in board exams — always write out the substitution fully.
Scenario
BOARD-TYPE PROBLEM: A GNSS survey gives h = 52.30 m. The geoid model gives N = −30.10 m. Find the orthometric height H.
Solution
H = h − N = 52.30 − (−30.10) = 52.30 + 30.10 = 82.40 m
N is positive, meaning the geoid is 12.30 m above the ellipsoid at this benchmark. This is a typical value in many parts of the Philippines.
Scenario
BOARD-TYPE PROBLEM: A benchmark has a levelled orthometric height H = 112.20 m and a GNSS ellipsoidal height h = 124.50 m. Find the geoid undulation N.
Solution
N = h − H = 124.50 − 112.20 = +12.30 m
Here we use the forward form h = H + N. This checks that the GNSS measurement is consistent with the known orthometric height and geoid undulation.
Scenario
BOARD-TYPE PROBLEM: N = +42.30 m and H = 155.70 m at a triangulation station. A GNSS survey is done at the same station. What ellipsoidal height h should the GNSS receiver report?
Solution
h = H + N = 155.70 + 42.30 = 198.00 m
(1) Negative N increases H above h — a classic board-exam trap. (2) The large positive N (42.2 m) is atypical for the Philippines but appears in board problems set in other regions.
Scenario
EXERCISES FROM CHAPTER: (1) h = 215.6 m, N = −28.4 m — find H. (2) H = 5.0 m, h = 47.2 m — find N.
Solution
(1) H = h − N = 215.6 − (−28.4) = 215.6 + 28.4 = 244.0 m (2) N = h − H = 47.2 − 5.0 = 42.2 m
Applications
- GPS/GNSS-controlled surveys for cadastral and engineering projects in the Philippines.
- Vertical datum transformation when converting between old MSL benchmarks and new GNSS-based coordinates.
- Setting up sea-level-referenced control for flood mapping, coastal zone management, and reclamation projects.
- Checking consistency between GNSS and spirit-leveling results on geodetic control networks.
- Computing undulation grids for national geoid modeling by NAMRIA.
Misconceptions
- MISCONCEPTION: 'H = h always, or H ≈ h for practical purposes.' CORRECTION: N can be 10–100 m; ignoring it causes errors of the same magnitude — catastrophic for engineering design.
- MISCONCEPTION: 'If N is negative, you subtract its absolute value from h to get H.' CORRECTION: H = h − N. If N = −30, then H = h − (−30) = h + 30. Always use the algebraic sign.
- MISCONCEPTION: 'The geoid model gives exact H.' CORRECTION: Geoid models have residual errors (cm to dm level); for precise work, combine geoid modeling with spirit leveling.
Related Concepts
- Spirit leveling (differential leveling)
- GNSS positioning
- Geoid modeling (EGM2008, PHL geoid models)
- Vertical datum
- Geopotential numbers and dynamic heights
Common Exam Questions
Example
Given h = 85.00 m and N = +18.50 m, find H. Answer: H = 85.00 − 18.50 = 66.50 m.
Approach
Given two of h, H, N — solve for the third. Always write h = H + N first, then rearrange.
Question Type
Direct computation (most common)
Example
If N = −20 m, is H greater or less than h? Answer: H = h − (−20) = h + 20, so H > h.
Approach
Determine whether H > h or H < h given the sign of N.
Question Type
Sign interpretation
Example
GNSS measures h (ellipsoidal); engineering needs H (orthometric). Without N from a geoid model, H cannot be determined.
Approach
Explain in words why a geoid model is needed to convert GNSS heights to elevations.
Question Type
Conceptual explanation
Key Points To Remember
- MASTER EQUATION: h = H + N (memorize this — it appears on nearly every board exam).
- Rearrangements: H = h − N and N = h − H.
- h is from GNSS; H is from spirit leveling; N is from a geoid model.
- Sign of N matters critically — check whether geoid is above (+) or below (−) the ellipsoid.
- When N is negative (geoid below ellipsoid), H will be LARGER than h — do not be confused.
- For the Philippines, NAMRIA provides the official geoid undulation model used in practice.
- Board exams often give two of the three values and ask for the third — always set up h = H + N first.
Gravity: Variation, Measurement, and Role in Height Systems
Gravity is the combined effect of gravitational attraction and centrifugal force due to Earth's rotation. It defines the geoid and, consequently, the orthometric height system. MAGNITUDE OF GRAVITY: • At the equator: g ≈ 9.780 m/s² (centrifugal effect reduces g) • At the poles: g ≈ 9.832 m/s² (closer to Earth's center, no centrifugal reduction) • Standard gravity: g₀ = 9.80665 m/s² (international standard) • Units: m/s² or Gal (1 Gal = 0.01 m/s²; 1 mGal = 10⁻⁵ m/s²) FACTORS AFFECTING GRAVITY: 1. LATITUDE EFFECT: Gravity increases from equator to poles due to (a) Earth's oblate shape (poles are closer to the center) and (b) reduced centrifugal effect at higher latitudes. The International Gravity Formula (or Somigliana formula for GRS80) models this. 2. FREE-AIR EFFECT: Gravity decreases with elevation because the point is farther from Earth's center. Free-air gradient ≈ −0.3086 mGal/m (gravity decreases ~0.3086 mGal for each meter of elevation gain). 3. BOUGUER EFFECT: The mass of rock between the observation point and the geoid (Bouguer slab) attracts the plumb bob upward, increasing g. Bouguer correction removes this effect to isolate the free-air anomaly. 4. TERRAIN EFFECT: Irregular topography around a station causes additional perturbations corrected by terrain (or topographic) reduction. GRAVITY ANOMALIES: • Free-air anomaly = observed g − theoretical g (latitude) + free-air correction • Bouguer anomaly = free-air anomaly − Bouguer plate correction ± terrain correction • Gravity anomalies indicate subsurface mass irregularities and determine geoid undulation N via Stokes' integral. ROLE IN HEIGHT SYSTEMS: Because gravity varies from point to point along a leveling route, simply adding or subtracting rod readings (geometric leveling) does not give truly consistent 'heights above MSL.' Two points at the same water surface level have the same geopotential number (C) but may have slightly different orthometric heights because g differs. This motivates the distinction between: • Orthometric height H: geometric distance from geoid to point along the plumb line (C/ḡ where ḡ is mean gravity along the plumb line) • Dynamic height: C/γ₄₅ (uses standard gravity at 45° latitude — removes loop misclosures in large networks) • Normal height: used in some European systems; reference surface is the quasi-geoid
Examples
The free-air correction is positive when reducing to a lower elevation (sea level) because gravity increases as you go down toward the Earth's center. This corrected value can then be compared to theoretical gravity to compute the free-air anomaly.
Scenario
The observed gravity at a hilltop station in the Cordillera is g = 9.7850 m/s². The elevation above MSL is H = 1500 m. Compute the free-air corrected gravity (reduce to sea level using the free-air gradient of 0.3086 mGal/m).
Solution
Free-air correction = +0.3086 mGal/m × 1500 m = +462.9 mGal = +0.004629 m/s² g_FA = 9.7850 + 0.004629 = 9.78963 m/s²
This is the fundamental reason why precise leveling networks use geopotential numbers or dynamic heights rather than raw geometric height differences — to maintain physical consistency over large areas.
Scenario
Why do two water-surface points in Laguna Lake have the same geopotential number but potentially different orthometric heights?
Solution
Both points are on the same equipotential surface (geopotential number C is identical — water is level). However, H = C/ḡ, and ḡ (mean gravity along the plumb line) varies with latitude and local mass. Hence H may differ by a few millimeters even though both points are physically at the same water level.
Applications
- Gravity surveys for geoid modeling and national vertical datum establishment by NAMRIA.
- Geophysical exploration for mineral resources (gravity prospecting) relevant to MGB/DENR work.
- Establishing the Philippine Vertical Datum referenced to mean sea level at Intramuros tide gauge.
- Correcting leveling loops spanning large latitude ranges in national geodetic networks.
- Free-air and Bouguer anomaly mapping for crustal studies in the Philippine archipelago.
Misconceptions
- MISCONCEPTION: 'Gravity is constant everywhere on Earth.' CORRECTION: Gravity varies by about 0.5% from equator to poles and decreases with elevation.
- MISCONCEPTION: 'Bouguer anomaly = free-air anomaly.' CORRECTION: Bouguer anomaly = free-air anomaly minus the Bouguer plate correction (for the rock mass between station and geoid).
- MISCONCEPTION: 'Orthometric height = geometric height difference from spirit leveling.' CORRECTION: For precise work, observed height differences must be weighted by gravity to give orthometric heights.
Related Concepts
- Geopotential number C
- Normal gravity (Somigliana formula)
- Free-air correction
- Bouguer correction
- Stokes' integral for geoid computation
- Plumb line and deflection of the vertical
Common Exam Questions
Example
Station elevation = 800 m, observed g = 9.7920 m/s². Free-air corrected g = 9.7920 + (0.3086 × 800)/100000 = 9.7945 m/s².
Approach
Apply free-air correction ΔgFA = +0.3086 × H mGal to reduce observed gravity to sea level, or subtract to reduce to elevation.
Question Type
Numerical gravity reduction
Example
At which location is gravity larger — Zamboanga City (7°N) or Baguio City (16°N)? Answer: Baguio City, because higher latitude → larger g.
Approach
State that g increases from equator to poles and give approximate values.
Question Type
Conceptual: gravity variation with latitude
Example
Express g = 9.80 m/s² in Gal. Answer: 9.80 m/s² ÷ 0.01 m/s²/Gal = 980 Gal.
Approach
Convert between m/s², Gal, and mGal.
Question Type
Units conversion
Key Points To Remember
- g ≈ 9.780 m/s² at equator; g ≈ 9.832 m/s² at poles; standard g₀ = 9.80665 m/s².
- Gravity unit: Gal (1 Gal = 0.01 m/s²); survey precision in mGal (10⁻⁵ m/s²).
- Free-air gradient: gravity decreases ~0.3086 mGal per meter of elevation gain.
- Bouguer correction accounts for the gravitational attraction of the rock mass between station and geoid.
- Gravity anomalies (free-air, Bouguer) are used to compute geoid undulations via Stokes' integral.
- Orthometric height H = geopotential number C ÷ mean gravity ḡ along the plumb line.
- Dynamic heights eliminate loop misclosures in precise leveling networks spanning large areas.
- For most Philippine engineering surveys (not geodetic precision), geometric leveling is sufficient.
Height Systems: Orthometric, Ellipsoidal, and Dynamic Heights
Different applications require different height systems. The PRC board exam tests the ability to distinguish these systems, know their references, and understand when each is used. ORTHOMETRIC HEIGHT H: Definition: Distance from the geoid (MSL) to the point, measured along the curved plumb line. Reference surface: The geoid. How obtained: Spirit (differential) leveling + gravity observations. Use: All Philippine engineering, legal, and cadastral work. Required for slope computations, flood mapping, drainage design. Formula: H = C / ḡ (C = geopotential number; ḡ = mean gravity along plumb line) Practical approximation: H ≈ Σ(Δn) for short lines where g variation is negligible. ELLIPSOIDAL HEIGHT h: Definition: Distance from the reference ellipsoid to the point, measured along the ellipsoidal normal. Reference surface: The reference ellipsoid (WGS84 for GPS; PRS92 for Philippine national surveys). How obtained: GNSS/GPS positioning. Use: 3D positioning, satellite geodesy, GNSS-based control surveys. Note: h has no physical meaning (does not relate to gravity or sea level directly). GEOID UNDULATION N: Definition: Height of the geoid above (+) or below (−) the ellipsoid, along the ellipsoidal normal. How obtained: N = h − H (from combined GNSS + leveling at a point); or from global geoid models such as EGM2008. Role: The bridge between h and H via h = H + N. DYNAMIC HEIGHT H_D: Definition: H_D = C / γ₄₅ (geopotential number divided by normal gravity at latitude 45°, γ₄₅ = 9.80629 m/s²) Reference: Same as orthometric height (the geoid), but uses a constant divisor. Use: Precise national leveling networks where loop misclosures must be eliminated. Points on the same water surface have identical dynamic heights. Note: Numerically close to H but not identical; not commonly tested in Philippine board exams at basic level. NORMAL HEIGHT H*: Definition: C / γ̄ (geopotential number divided by mean normal gravity along the normal plumb line to the telluroid). Reference surface: The quasi-geoid (co-geoid). Use: European vertical datums (e.g., EVRS); not currently the standard in the Philippines. SUMMARY TABLE FOR BOARD EXAM: h (ellipsoidal): GNSS, ellipsoid, no physical meaning. H (orthometric): Spirit leveling, geoid/MSL, engineering use. N (undulation): Geoid model, geoid-ellipsoid separation, bridge between h and H. H_D (dynamic): Leveling + gravity, no loop misclosure.
Examples
This scenario is realistic for reclaimed areas in Metro Manila or Pampanga floodplains. Using the raw GNSS height of +12.50 m would give a completely wrong impression — the road is actually below sea level and at serious flood risk.
Scenario
A surveyor reports the elevation of a road centerline point as h = 12.50 m using a GNSS receiver. The geoid undulation from NAMRIA's model is N = +16.80 m. Is the road above or below MSL, and what is its correct elevation?
Solution
H = h − N = 12.50 − 16.80 = −4.30 m The road is 4.30 m BELOW mean sea level.
The key is that spirit leveling is a physical measurement in the gravity field — it naturally produces results closest to orthometric heights, unlike GNSS which gives ellipsoidal heights.
Scenario
Distinguish: What height does a precise spirit-leveling survey give — orthometric, ellipsoidal, or dynamic?
Solution
Spirit leveling gives geometric height differences. When multiplied by mean gravity, these become geopotential numbers, from which orthometric heights H are computed. For basic surveys (short lines), the distinction between geometric and orthometric difference is negligible. For national geodetic networks, gravity corrections are applied to get true H.
Applications
- Flood inundation mapping: orthometric heights H determine which areas are below flood stage.
- Vertical datum modernization in the Philippines: shifting from old tide-gauge MSL to GNSS-compatible geoid-based heights.
- Engineering design for LRT/MRT alignments, airport approaches, and port facilities requiring MSL-referenced elevations.
- Tsunami inundation modeling: requires accurate MSL-referenced coastal elevations.
- Interoperability between GNSS survey results and legacy leveling-based control networks.
Misconceptions
- MISCONCEPTION: 'GNSS-derived heights can be used directly as elevations on engineering plans.' CORRECTION: Philippine survey regulations (NAMRIA standards, PD 1529 implementing rules) require MSL-referenced orthometric heights H on official plans.
- MISCONCEPTION: 'Dynamic height = orthometric height.' CORRECTION: Dynamic heights use a constant gravity divisor (γ₄₅) while orthometric heights use the actual mean gravity along the plumb line; they differ by small amounts.
- MISCONCEPTION: 'WGS84 and PRS92 are the same ellipsoid.' CORRECTION: Both use GRS80 parameters (virtually identical geometry) but have different datum realizations — WGS84 is global (GNSS); PRS92 is the Philippine national realization.
Related Concepts
- Spirit (differential) leveling procedures
- PRS92 and WGS84 reference ellipsoids
- NAMRIA and Philippine vertical datum
- Geopotential number C
- Normal height and quasi-geoid
- EGM2008 global geoid model
Common Exam Questions
Example
A GPS survey gives a height of 35.60 m at a BM. This is: (a) orthometric height (b) ellipsoidal height (c) dynamic height (d) geopotential number. Answer: (b) ellipsoidal height.
Approach
Given a scenario (GNSS survey vs. spirit leveling vs. geoid model), identify which height type is being measured.
Question Type
Classification/identification
Example
Ellipsoidal heights h do not account for gravity and do not align with mean sea level — two points at the same h value may have water flowing between them.
Approach
Explain why ellipsoidal heights cannot directly replace leveled elevations in engineering plans.
Question Type
Practical implication
Key Points To Remember
- Orthometric height H: from geoid, obtained by spirit leveling, used in all Philippine engineering.
- Ellipsoidal height h: from ellipsoid, obtained by GNSS, no physical gravity meaning.
- Geoid undulation N: separation between geoid and ellipsoid; N = h − H.
- Dynamic height H_D: same geopotential reference as H but uses constant gravity (γ₄₅) — eliminates loop misclosure.
- For most board exam problems, focus on h, H, N and the equation h = H + N.
- PRS92 is the Philippine ellipsoidal datum; GNSS heights in the Philippines are typically referenced to WGS84 then transformed to PRS92.
- All PD 1529 and CA 141 survey plans must show orthometric heights H, not ellipsoidal heights h.
Practice Problems
The 0.48 m discrepancy may be due to (1) error in the old leveling data, (2) error in the geoid model, (3) land subsidence since the original leveling, or (4) datum inconsistency. In Philippine practice, such discrepancies are investigated before accepting either value.
Problem
PROBLEM 1 (Direct Computation): A GNSS survey at a cadastral monument in Iloilo City gives an ellipsoidal height h = 18.74 m (WGS84). The NAMRIA geoid model gives N = +14.22 m. (a) Compute the orthometric height H. (b) If the monument is supposed to be at H = 5.00 m per the old leveling data, what is the discrepancy?
Solution
(a) H = h − N = 18.74 − 14.22 = 4.52 m (b) Discrepancy = 5.00 − 4.52 = +0.48 m (old leveling gives 0.48 m higher than GNSS-derived H)
Negative N means the geoid is below the ellipsoid, so subtracting N (a negative value) ADDS to h. The result H = 82.40 m is significantly larger than h = 52.30 m. Always substitute the algebraic sign — do not drop the negative sign.
Problem
PROBLEM 2 (Negative N — common board trap): A GNSS rover unit reads h = 52.30 m at a BM in a highland area of Bukidnon. The geoid undulation at this location is N = −30.10 m (geoid is BELOW the ellipsoid). Find the orthometric height H.
Solution
H = h − N = 52.30 − (−30.10) = 52.30 + 30.10 = 82.40 m
This is the standard field method for calibrating a geoid model: occupy a benchmark with known H using GNSS to get h, then compute N. Multiple such points allow building or verifying a local geoid model — a key NAMRIA activity for the Philippine national geodetic infrastructure.
Problem
PROBLEM 3 (Finding N from GNSS + Leveling): A survey team occupies a first-order benchmark with known orthometric height H = 248.60 m. The GNSS receiver simultaneously observes an ellipsoidal height h = 263.15 m at the same point. Compute the geoid undulation N and state whether the geoid is above or below the ellipsoid.
Solution
N = h − H = 263.15 − 248.60 = +14.55 m Since N > 0, the geoid is ABOVE the ellipsoid by 14.55 m at this location.
This is a practical staking-out scenario using GNSS for grade control. The engineer programs the target h into the data collector, which alerts when h = 168.75 m is reached — equivalent to the required H = 150.00 m elevation.
Problem
PROBLEM 4 (Forward check): A design engineer needs to set a GNSS stake at a required orthometric height H = 150.00 m in an area where N = +18.75 m. What ellipsoidal height h should the GNSS receiver display when the stake is at the correct elevation?
Solution
h = H + N = 150.00 + 18.75 = 168.75 m The GNSS receiver should show h = 168.75 m when the stake is at the design elevation of 150.00 m MSL.
Free-air correction is positive when reducing upward to sea level because gravity increases as you descend toward the Earth's center. The correction is proportional to elevation. This type of unit conversion (m/s² ↔ Gal ↔ mGal) appears in board exams testing basic gravity reductions.
Problem
PROBLEM 5 (Gravity unit conversion and free-air correction): A gravity station in Benguet has an elevation H = 1200 m and observed gravity g = 9.7841 m/s². (a) Convert g to Gal. (b) Apply the free-air correction to reduce g to MSL (use gradient = 0.3086 mGal/m). (c) Express the free-air correction in m/s².
Solution
(a) g = 9.7841 m/s² ÷ 0.01 m/s²/Gal = 978.41 Gal (b) Free-air correction = +0.3086 mGal/m × 1200 m = +370.32 mGal = +0.37032 Gal g_MSL = 978.41 + 0.37032 = 978.78 Gal (approx) (c) 370.32 mGal × (10⁻⁵ m/s² per mGal) = 0.0037032 m/s² g_MSL = 9.7841 + 0.0037032 = 9.7878 m/s²
This problem integrates all three concepts: computing N, interpreting its sign, and using it for a second independent height determination. The 2 cm difference is typical of GNSS precision and would be acceptable for most engineering surveys.
Problem
PROBLEM 6 (Three-surface conceptual): At a coastal point in Cebu, a surveyor measures: GNSS height h = 22.80 m; spirit-leveled height from a nearby BM gives H = 8.15 m. (a) Compute N. (b) Is the geoid above or below the ellipsoid? (c) If a second GNSS measurement at the same point gives h = 22.78 m, what is the implied H using the same N?
Solution
(a) N = h − H = 22.80 − 8.15 = +14.65 m (b) N > 0: geoid is ABOVE the ellipsoid by 14.65 m (c) H = h − N = 22.78 − 14.65 = 8.13 m Difference = 8.15 − 8.13 = 0.02 m = 2 cm discrepancy (within typical GNSS precision)
Exam Preparation Tips
- MASTER h = H + N FIRST: This equation appears in some form in nearly every PRC board exam covering geodesy. Know all three rearrangements: H = h − N, N = h − H, h = H + N. Practice substituting with both positive and negative N.
- WATCH THE SIGN OF N: The most common error is mishandling negative N. When N is negative, H = h − N gives H > h (subtraction of a negative = addition). Always write the full algebraic substitution — never drop signs.
- KNOW THE THREE SURFACES AND THEIR HEIGHTS: Ellipsoid → h (GNSS); Geoid → H (spirit leveling, engineering elevation); Terrain → the actual ground. Be able to sketch and label these three surfaces and their relationships.
- MEMORIZE GRAVITY REFERENCE VALUES: g ≈ 9.780 m/s² (equator), g ≈ 9.832 m/s² (poles), g₀ = 9.80665 m/s² (standard). Free-air gradient = 0.3086 mGal/m. Know unit conversions: 1 Gal = 0.01 m/s²; 1 mGal = 10⁻⁵ m/s².
- UNDERSTAND WHY GNSS HEIGHT ≠ ELEVATION: This conceptual question is asked frequently. Answer: GNSS gives ellipsoidal h; engineering needs orthometric H; the difference is N from a geoid model. Without N, GNSS heights have no engineering value.
- LEARN PHILIPPINE CONTEXT: In the Philippines, N is generally positive (+10 to +22 m over most of the archipelago), meaning GNSS-derived h values are numerically smaller than the corresponding elevations H. NAMRIA provides the official geoid model. PRS92 is the national ellipsoidal datum.
- PRACTICE UNIT CONVERSIONS FOR GRAVITY: Board exams test Gal-to-m/s² conversions and free-air correction calculations. These are straightforward if you memorize: 1 Gal = 0.01 m/s² and free-air gradient = 0.3086 mGal/m.
- REVIEW PHILIPPINE LAWS: PD 1529 (Property Registration Decree) and CA 141 (Public Land Act) require surveys with proper elevations. RA 8560 modernized the practice of geodetic engineering in the Philippines. RA 4374 established NAMRIA. Survey plans must show orthometric heights.
- AVOID CONFUSION BETWEEN h AND H ON EXAM PROBLEMS: Read every problem carefully — some problems give h and ask for H, while others give H and ask for h. Identify which variable is known and which is unknown before computing.
- FOR MULTIPLE-CHOICE EXAMS: If you forget the formula, use dimensional reasoning: h and H are heights (m), N is a separation (m), so the only sensible relationship is additive: h = H + N. Then solve for the unknown.
- DRAW A SKETCH: For any height problem, quickly sketch the three surfaces (ellipsoid at bottom if N > 0, geoid above it, point on terrain above the geoid). Label h, H, and N on the sketch. This visual check prevents sign errors.
- DYNAMIC VS ORTHOMETRIC HEIGHTS: For the board exam, know that dynamic heights eliminate loop misclosures in leveling networks and that H_D = C/γ₄₅. You need not compute dynamic heights for basic-level problems, but you should be able to distinguish them from orthometric heights conceptually.
In summary
The geoid, gravity, and heights chapter is foundational to modern geodetic engineering in the Philippines. Every GNSS-based survey, every engineering elevation, and every cadastral plan depends on understanding three surfaces (ellipsoid, geoid, terrain), three heights (h, H, N), and their fundamental relationship h = H + N. For the PRC Geodetic Engineer Licensure Examination, the priority is: (1) master the equation H = h − N and handle both positive and negative N correctly; (2) distinguish which height type each measurement instrument produces (GNSS → h; spirit leveling → H; geoid model → N); (3) understand gravity's role in defining the geoid and orthometric heights; and (4) know the Philippine legal and institutional context — PRS92 as the national datum, NAMRIA as the geoid-model authority, and PD 1529/CA 141/RA 8560 as the governing laws. In Philippine practice, N is generally positive (geoid above ellipsoid, +10 to +22 m), meaning GNSS heights h are numerically smaller than orthometric heights H over most of the archipelago. Misapplying this relationship — or ignoring the geoid entirely — leads to elevation errors of 10–20 m or more, which can have catastrophic consequences for flood modeling, drainage design, and infrastructure siting. With consistent practice of the board-type problems in this chapter, correct application of the sign convention, and clear understanding of why each height system exists, examinees will be well-equipped to answer any geodesy height question confidently and accurately.
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