GELE Geodesy — The Geoid, Gravity and HeightsExam Answer Templates

Marks

1

Topic

The Geoid

Difficulty

easy

Template Id

T1

Examiner Tip

The word 'equipotential' is the key technical term. Answers without it rarely earn the full mark. Keep the answer to one precise sentence — do not pad.

Model Answer

The geoid is the equipotential surface of the Earth's gravity field that best fits global mean sea level, and it serves as the reference surface for orthometric (levelled) heights.

Question Type

very_short_answer

Answer Structure

• Single sentence: define the geoid as an equipotential gravity surface coinciding with mean sea level [1 mark]

Scoring Breakdown

Common Mark Deductions

• Saying 'mean sea level itself' without mentioning gravity or equipotential — too vague
• Confusing the geoid with the ellipsoid
• Omitting that it is the reference for orthometric heights

Key Phrases To Include

• equipotential surface
• gravity field
• mean sea level
• orthometric heights

Marks

1

Topic

Geoid Undulation

Difficulty

easy

Template Id

T2

Examiner Tip

Examiners specifically look for the sign convention. Many students lose this mark by omitting it. One sentence with the sign rule is sufficient.

Model Answer

Geoid undulation N is the vertical distance between the geoid and the reference ellipsoid at a given point; it is positive when the geoid is above the ellipsoid and negative when the geoid is below the ellipsoid.

Question Type

very_short_answer

Answer Structure

• Single sentence: define N as the vertical separation between geoid and ellipsoid, including sign convention [1 mark]

Scoring Breakdown

Common Mark Deductions

• Reversing the sign convention (positive when ellipsoid is above geoid)
• Defining N as the orthometric height — a fundamental confusion
• No mention of sign convention

Key Phrases To Include

• vertical distance
• geoid above/below ellipsoid
• positive/negative
• geoid height

Marks

1

Topic

The Three Heights

Difficulty

easy

Template Id

T3

Examiner Tip

Write the equation first, then define variables. This is the most important single equation in this chapter — memorize it exactly.

Model Answer

The fundamental height relationship is: h = H + N, where h is the ellipsoidal height above the reference ellipsoid, H is the orthometric height above the geoid, and N is the geoid undulation.

Question Type

very_short_answer

Answer Structure

• Write the equation h = H + N and define all three variables [1 mark]

Scoring Breakdown

Common Mark Deductions

• Writing H = h + N (wrong rearrangement as the primary equation)
• Not defining the variables
• Using incorrect symbols

Key Phrases To Include

• h = H + N
• ellipsoidal height
• orthometric height
• geoid undulation

Marks

2

Topic

The Three Heights

Difficulty

easy

Template Id

T4

Examiner Tip

Structure as two clear paragraphs — one per height type. The examiner awards one mark per definition. Naming the reference surface (geoid vs ellipsoid) and measurement method is essential.

Model Answer

Orthometric height (H) is the height of a point above the geoid (the equipotential gravity surface approximating mean sea level), determined by spirit levelling combined with gravity observations. It represents the practical 'elevation above sea level' used in engineering and mapping. Ellipsoidal height (h) is the height of a point above the reference ellipsoid (e.g., GRS80/WGS84), measured along the ellipsoidal normal. It is obtained directly from GNSS observations but has no physical relationship to sea level or water flow. The two are related by: h = H + N, where N is the geoid undulation.

Question Type

short_answer

Answer Structure

• Line 1–2: Define orthometric height H with reference surface (geoid/MSL) and measurement method (spirit levelling) [1 mark]
• Line 3–4: Define ellipsoidal height h with reference surface (ellipsoid/WGS84) and measurement method (GNSS) [1 mark]

Scoring Breakdown

Common Mark Deductions

• Stating orthometric height is from GNSS — a serious conceptual error
• Not naming the reference surface for each height type
• Omitting the connecting equation h = H + N

Key Phrases To Include

• above the geoid
• mean sea level
• spirit levelling
• above the ellipsoid
• GRS80 or WGS84
• GNSS
• h = H + N

Marks

2

Topic

The Three Heights — Numerical

Difficulty

medium

Template Id

T5

Examiner Tip

The double negative trap is intentional in board exam problems. Always write H = h − (N) fully expanded before computing. Never mentally skip the sign step.

Model Answer

Given: Ellipsoidal height, h = 52.30 m Geoid undulation, N = −30.10 m Using the fundamental height relationship: h = H + N H = h − N Substituting: H = 52.30 − (−30.10) H = 52.30 + 30.10 ∴ H = 82.40 m above mean sea level

Question Type

numerical

Answer Structure

• Line 1–2: State given values with correct units and signs [0.5 mark]
• Line 3: Write the formula H = h − N [0.5 mark]
• Line 4–5: Substitute values and handle double negative correctly [0.5 mark]
• Line 6: State final answer with correct units and reference surface [0.5 mark]

Scoring Breakdown

Common Mark Deductions

• H = 52.30 − 30.10 = 22.20 m (not handling the double negative — most common error)
• Correct formula but no units in final answer
• Calling the result an ellipsoidal height instead of orthometric height

Key Phrases To Include

• H = h − N
• H = 52.30 − (−30.10)
• 82.40 m
• above mean sea level

Marks

2

Topic

The Three Heights — Numerical

Difficulty

easy

Template Id

T6

Examiner Tip

Always state the physical meaning of N — 'geoid is 12.30 m above the ellipsoid here.' This interpretation line often earns the second mark or bonus acknowledgement.

Model Answer

Given: Orthometric height, H = 112.20 m Ellipsoidal height, h = 124.50 m Using the height relationship: h = H + N N = h − H Substituting: N = 124.50 − 112.20 ∴ N = +12.30 m Interpretation: The geoid is 12.30 m above the reference ellipsoid at this benchmark location.

Question Type

numerical

Answer Structure

• Line 1–2: State given values [0.5 mark]
• Line 3: Rearrange formula to N = h − H [0.5 mark]
• Line 4: Substitute and compute [0.5 mark]
• Line 5: State answer with sign and physical interpretation [0.5 mark]

Scoring Breakdown

Common Mark Deductions

• N = H − h = −12.30 m (wrong sign/subtraction order)
• No sign stated for N
• No interpretation of what the sign means physically

Key Phrases To Include

• N = h − H
• N = 124.50 − 112.20
• +12.30 m
• geoid is above the ellipsoid

Marks

3

Topic

The Three Heights — Conceptual

Difficulty

medium

Template Id

T7

Examiner Tip

Three-mark conceptual questions require three distinct, developed points — one per paragraph. Mentioning EGM2008 or a Philippine geoid model by name demonstrates professional knowledge and impresses examiners.

Model Answer

GNSS receivers determine the position of a point by measuring distances to satellites. The heights obtained (h) are ellipsoidal heights — measured above the reference ellipsoid (WGS84/GRS80), which is a smooth mathematical surface with no physical relationship to gravity or water flow. Engineering elevations (orthometric heights, H) are measured above the geoid — the equipotential gravity surface that approximates global mean sea level. Water flows from points of high H to low H; engineering design (road gradients, drainage, flood levels) depends on these physical heights. The two heights are related by: h = H + N, so H = h − N. The geoid undulation N varies from approximately −75 m to +85 m globally (in the Philippines, N is typically in the range of −10 m to +20 m). Without knowing N at each survey point, converting h to H is impossible. A geoid model (e.g., EGM2008, or the Philippine geoid model) provides N values, enabling the conversion. Using h directly as an elevation introduces systematic errors equal to N — potentially tens of metres — rendering the survey useless for engineering purposes.

Question Type

short_answer

Answer Structure

• Paragraph 1: Explain what GNSS provides — ellipsoidal height h above WGS84 ellipsoid [1 mark]
• Paragraph 2: Explain what engineering needs — orthometric height H above geoid/MSL and why [1 mark]
• Paragraph 3: Quantify the error and state the role of the geoid model using h = H + N [1 mark]

Scoring Breakdown

Common Mark Deductions

• Vague statements like 'GNSS is inaccurate' — misses the conceptual point entirely
• Not mentioning the geoid as the engineering reference surface
• Not quantifying or contextualising the error N

Key Phrases To Include

• ellipsoidal height
• WGS84
• orthometric height
• geoid
• mean sea level
• h = H + N
• geoid undulation N
• geoid model (EGM2008)

Marks

3

Topic

Gravity

Difficulty

medium

Template Id

T8

Examiner Tip

Quoting approximate g values (9.78 and 9.83 m/s²) demonstrates quantitative knowledge and distinguishes a good answer from a vague one. Always connect the three paragraphs: gravity varies → geoid is equipotential → so geoid undulates with mass.

Model Answer

Gravity variation with latitude: Gravity increases from the equator to the poles. At the equator, g ≈ 9.78 m/s²; at the poles, g ≈ 9.83 m/s². This occurs because the Earth is flattened at the poles (shorter distance to the centre) and because centrifugal acceleration due to Earth's rotation reduces effective gravity at the equator. Gravity variation with elevation: Gravity decreases with increasing elevation above the surface (free-air effect). For every 1 km increase in elevation, g decreases by approximately 3.086 mGal per metre. The Bouguer correction additionally accounts for the gravitational attraction of the rock mass between the observation point and the geoid. Role in defining the geoid: The geoid is an equipotential surface of Earth's gravity field — meaning gravity acts perpendicular to it at every point. Because mass density varies (e.g., heavy crustal rocks vs. ocean water), the gravity field is irregular, and the equipotential surface undulates accordingly. Gravity observations (from gravimeters, satellite missions like GRACE) map these variations, enabling computation of geoid undulation N through Stokes' integral and related methods. Without gravity data, the geoid — and hence orthometric heights — cannot be determined.

Question Type

short_answer

Answer Structure

• Paragraph 1: Gravity variation with latitude — equator vs. poles, values, and cause [1 mark]
• Paragraph 2: Gravity variation with elevation — free-air effect, Bouguer correction [1 mark]
• Paragraph 3: Gravity defines the geoid as an equipotential surface; mass variations cause undulations; gravity observations are needed for geoid models [1 mark]

Scoring Breakdown

Common Mark Deductions

• Saying gravity is constant everywhere — a factual error worth zero marks
• Mixing up free-air and Bouguer corrections
• Not connecting gravity to the geoid definition

Key Phrases To Include

• 9.78 m/s² at equator
• 9.83 m/s² at poles
• free-air correction
• Bouguer correction
• equipotential surface
• mass density variations
• gravity observations
• geoid undulation

Marks

2

Topic

The Three Heights — Numerical

Difficulty

easy

Template Id

T9

Examiner Tip

Board exams frequently set N as negative. Always write the full substitution H = 215.6 − (−28.4) to show the sign handling explicitly. Never simplify mentally without writing the step.

Model Answer

Given: Ellipsoidal height, h = 215.6 m Geoid undulation, N = −28.4 m Formula: H = h − N Substituting: H = 215.6 − (−28.4) H = 215.6 + 28.4 ∴ H = 244.0 m above mean sea level

Question Type

numerical

Answer Structure

• List given data with signs and units [0.5 mark]
• State formula H = h − N [0.5 mark]
• Show substitution with double-negative expansion [0.5 mark]
• Final answer with unit and reference surface [0.5 mark]

Scoring Breakdown

Common Mark Deductions

• H = 215.6 − 28.4 = 187.2 m — not handling negative N sign
• Omitting units from the answer
• No formula shown — process marks lost

Key Phrases To Include

• H = h − N
• 215.6 − (−28.4)
• 244.0 m
• above mean sea level

Marks

2

Topic

The Three Heights — Numerical

Difficulty

easy

Template Id

T10

Examiner Tip

The physical interpretation is a hallmark of a professional answer. A point near sea level (H ≈ 5 m) but with h ≈ 47 m means the geoid is ≈ 42 m above the ellipsoid there — this makes intuitive sense and earns you interpretation credit.

Model Answer

Given: Orthometric height, H = 5.0 m Ellipsoidal height, h = 47.2 m Formula: h = H + N → N = h − H Substituting: N = 47.2 − 5.0 ∴ N = +42.2 m Interpretation: The geoid lies 42.2 m above the WGS84 ellipsoid at this location, which is physically consistent with a near-sea-level site (H = 5.0 m) having a large positive GNSS height (h = 47.2 m).

Question Type

numerical

Answer Structure

• State given values [0.5 mark]
• Rearrange to N = h − H [0.5 mark]
• Compute N = 42.2 m [0.5 mark]
• Interpret the result physically [0.5 mark]

Scoring Breakdown

Common Mark Deductions

• N = H − h = −42.2 m (reversed subtraction)
• Omitting the positive sign on N
• No interpretation of what a large positive N means

Key Phrases To Include

• N = h − H
• 47.2 − 5.0
• +42.2 m
• geoid above the ellipsoid

Marks

3

Topic

The Three Heights — Conceptual

Difficulty

medium

Template Id

T11

Examiner Tip

This is a classic board exam scenario question. The key insight is that both observations (h = 40 m and H ≈ 0 m) are BOTH correct — they just use different reference surfaces. Stating this clearly at the start earns you the first mark immediately.

Model Answer

There is no discrepancy — the GNSS height and the 'sea level' observation are measuring different things relative to different reference surfaces. The GNSS receiver reports an ellipsoidal height h = 40 m, measured above the WGS84 reference ellipsoid — a smooth mathematical surface that does NOT coincide with the sea surface. The field observation that the point is 'at sea level' means the orthometric height H ≈ 0 m (above the geoid, which approximates mean sea level). Applying the height relationship h = H + N: N = h − H = 40 − 0 = +40 m This implies that at this coastal location, the geoid lies approximately 40 m above the WGS84 ellipsoid (N ≈ +40 m). This is physically consistent: in many parts of Southeast Asia, including the Philippines, the geoid is significantly above the ellipsoid. Conclusion: Without a geoid model providing N, GNSS ellipsoidal heights cannot be interpreted as elevations. The apparent '40 m height' at sea level is fully explained by the geoid undulation of approximately +40 m.

Question Type

short_answer

Answer Structure

• Sentence 1: Identify that two different reference surfaces are involved [1 mark]
• Sentences 2–3: Explain h (ellipsoidal, WGS84) vs. H (orthometric, geoid/MSL) [1 mark]
• Sentences 4–6: Apply h = H + N to compute N ≈ +40 m and explain physically [1 mark]

Scoring Breakdown

Common Mark Deductions

• Saying the GNSS reading is 'wrong' — shows fundamental misunderstanding
• Not applying the formula h = H + N
• Not identifying the reference surfaces of both measurements

Key Phrases To Include

• WGS84 ellipsoid
• geoid
• orthometric height H ≈ 0
• h = H + N
• N = +40 m
• geoid above the ellipsoid
• geoid model required

Marks

3

Topic

The Three Surfaces

Difficulty

medium

Template Id

T12

Examiner Tip

Organise as numbered points — one per surface. Examiners marking a 3-mark question are looking for three distinct, correct elements. Numbering them makes marking unambiguous and shows organisation.

Model Answer

Geodesy recognises three fundamental surfaces: 1. The Reference Ellipsoid (e.g., GRS80/WGS84): A mathematically defined oblate spheroid that approximates the overall shape of the Earth. It has a smooth, regular surface defined by its semi-major axis (a = 6,378,137.0 m for WGS84) and flattening (f = 1/298.257). The height measured above the ellipsoid is the ellipsoidal height (h), obtained directly from GNSS. 2. The Geoid: The equipotential surface of Earth's gravity field that best fits global mean sea level. It is irregular and undulates (±100 m globally) relative to the ellipsoid due to variations in subsurface mass density. The height measured above the geoid is the orthometric height (H), determined by spirit levelling. H is the practical engineering 'elevation above sea level.' 3. The Physical Terrain (Topography): The actual ground surface. No height is measured above the terrain itself; rather, heights of points on the terrain are measured above either the geoid (giving H) or the ellipsoid (giving h). Relationship: h = H + N, where N (geoid undulation) is the vertical separation between the geoid and the ellipsoid. For engineering and land administration (PD 1529, CA 141), H is the legally required height; GNSS surveys must apply a geoid model to obtain H from the measured h.

Question Type

short_answer

Answer Structure

• Point 1: Describe the ellipsoid — shape, mathematical definition, WGS84 parameters, h measured above it [1 mark]
• Point 2: Describe the geoid — equipotential, gravity, MSL reference, undulates ±100 m, H measured above it [1 mark]
• Point 3: Describe terrain and state no height is measured above terrain; summarise h = H + N and PD 1529 relevance [1 mark]

Scoring Breakdown

Common Mark Deductions

• Reversing geoid and ellipsoid — stating the geoid is the mathematical surface
• Not naming the type of height associated with each surface
• No mention of the h = H + N connecting equation

Key Phrases To Include

• ellipsoid
• GRS80/WGS84
• ellipsoidal height h
• geoid
• equipotential surface
• mean sea level
• orthometric height H
• physical terrain
• h = H + N
• geoid undulation N

Marks

5

Topic

Comprehensive — All Topics

Difficulty

hard

Template Id

T13

Examiner Tip

Five-mark long answers are marked using a section-by-section rubric. Use Roman numeral or bold headings to make each section unmistakably clear. A worked example is non-negotiable — it demonstrates application, not just recall. Philippine legal references (PD 1529, RA 8560) in Section V immediately signal a professionally prepared, board-level answer.

Model Answer

I. THE THREE SURFACES IN GEODESY Geodesy defines three reference surfaces: (1) the reference ellipsoid, (2) the geoid, and (3) the physical terrain. The Reference Ellipsoid (GRS80/WGS84) is a smooth oblate spheroid defined by: semi-major axis a = 6,378,137.0 m and flattening f = 1/298.257223563. It is purely mathematical and serves as the geometric reference for GNSS and coordinate systems (e.g., PRS92, PPCS/UTM in the Philippines). The Geoid is the equipotential surface of Earth's gravity field that most closely coincides with global mean sea level. It is irregular and undulates between approximately −105 m and +85 m relative to the ellipsoid, driven by spatial variations in subsurface mass density. The geoid is the physical reference for orthometric heights. II. HEIGHT SYSTEMS Three height types are defined: • Ellipsoidal height (h): height above the reference ellipsoid; provided directly by GNSS. • Orthometric height (H): height above the geoid; the engineering 'elevation above sea level'; determined by spirit levelling. • Geoid undulation (N): vertical separation of geoid above ellipsoid (positive = geoid above ellipsoid). Fundamental relationship: h = H + N → H = h − N → N = h − H III. GRAVITY AND ITS VARIATIONS Gravity g is not constant. It varies: (a) With latitude: g ≈ 9.780 m/s² at equator, g ≈ 9.832 m/s² at poles. The Earth's oblateness and reduction of centrifugal effect toward the poles cause this. (b) With elevation: g decreases with height (free-air effect ≈ 0.3086 mGal/m). The Bouguer correction accounts for the mass of rock between the station and the geoid. Gravity observations define the geoid. Because gravity is perpendicular to an equipotential surface, precise g measurements across a region allow computation of N via Stokes' integral. Global gravity satellite missions (CHAMP, GRACE, GOCE) have produced global geoid models such as EGM2008. IV. WORKED EXAMPLE A GNSS survey at BM-Mayon (near Legazpi City) gives: h = 52.30 m (WGS84 ellipsoidal height) N = −30.10 m (from EGM2008 geoid model) Required: Orthometric height H H = h − N = 52.30 − (−30.10) = 52.30 + 30.10 H = 82.40 m above mean sea level V. PRACTICAL SIGNIFICANCE IN THE PHILIPPINES Under PD 1529 (Property Registration Decree), land surveys must express elevations as orthometric heights H above mean sea level. CA 141 (Public Land Act) similarly requires MSL-referenced elevations for public land boundaries. RA 8560 (Geodetic Engineering Act) mandates that Geodetic Engineers apply proper geoid models when converting GNSS data to engineering elevations. In Philippine GNSS surveys, WGS84/PRS92 coordinates (φ, λ, h) are transformed to PPCS/UTM horizontal coordinates and MSL elevations using the national or global geoid model. Without applying a geoid model, height errors equal to N (ranging from −10 m to +20 m in the Philippines) will propagate into all elevation-dependent engineering design (flood modelling, road gradients, cadastral surveys). Conclusion: The geoid, gravity, and height systems are inseparable. GNSS provides geometric positioning; the geoid model connects geometry to physics. Every practising Geodetic Engineer must understand h = H + N and apply it correctly to avoid systematic errors in elevation surveys.

Question Type

long_answer

Answer Structure

• Section I: Define and compare the three surfaces (ellipsoid, geoid, terrain) [1 mark]
• Section II: State and explain all three height types with h = H + N and its rearrangements [1 mark]
• Section III: Explain gravity variations (latitude, elevation) and how gravity defines the geoid [1 mark]
• Section IV: Present a complete worked numerical example [1 mark]
• Section V: Discuss practical significance in Philippine context — PD 1529, RA 8560, PPCS/UTM, geoid model [1 mark]

Scoring Breakdown

Common Mark Deductions

• No worked numerical example — misses the Section IV mark entirely
• No Philippine context or legal references — misses the Section V mark
• Not defining all three height types separately
• Incorrect gravity values or no mention of free-air/Bouguer corrections
• Vague statements without technical precision throughout

Key Phrases To Include

• GRS80/WGS84
• equipotential surface
• mean sea level
• h = H + N
• geoid undulation N
• orthometric height H
• ellipsoidal height h
• g ≈ 9.780 m/s² (equator)
• free-air correction
• EGM2008
• PD 1529
• RA 8560
• PPCS/UTM
• PRS92

Marks

2

Topic

Gravity and Height Systems

Difficulty

hard

Template Id

T14

Examiner Tip

This is an 'advanced' 2-mark question. The formula C = W₀ − Wₚ earns the first mark. The concept of path independence and connection to dynamic/orthometric heights earns the second. Include both.

Model Answer

A geopotential number C is the difference in geopotential (gravity potential) between a point P and the geoid, expressed in geopotential units (gpu) or m²/s²: C = W₀ − Wₚ where W₀ is the geopotential of the geoid and Wₚ is the geopotential at P. Geopotential numbers are used in precise height determination because they are physically rigorous — they account for the actual distribution of gravity along the levelling path. Unlike raw levelled height differences, geopotential numbers are path-independent; they yield the same value regardless of the route taken between two benchmarks. Dynamic heights and orthometric heights for geodetic levelling networks are both derived from geopotential numbers by dividing by a reference or mean gravity value respectively.

Question Type

short_answer

Answer Structure

• Line 1–2: Define geopotential number C = W₀ − Wₚ with units [1 mark]
• Line 3–4: Explain why geopotential numbers are needed — path independence, gravity-rigorous, basis for dynamic/orthometric heights [1 mark]

Scoring Breakdown

Common Mark Deductions

• Defining geopotential number as simply 'the height measured using gravity' — too vague
• Not mentioning path independence
• Confusing geopotential number with orthometric height directly

Key Phrases To Include

• geopotential number
• C = W₀ − Wₚ
• geopotential units (gpu)
• path independent
• dynamic heights
• orthometric heights
• gravity distribution

Marks

3

Topic

The Three Heights — Multi-part Numerical

Difficulty

medium

Template Id

T15

Examiner Tip

Label each part clearly (i), (ii), (iii). State the relevant rearrangement of h = H + N before substituting — this earns you method marks even if arithmetic is wrong. Part (ii) is deliberately designed with negative N to test sign handling.

Model Answer

(i) Geoid undulation N at Point A: Using N = h − H: N = 180.50 − 210.80 N = −30.30 m (The geoid is 30.30 m below the ellipsoid at Point A.) (ii) Orthometric height H at Point B: Using H = h − N: H = 95.30 − (−18.40) H = 95.30 + 18.40 H = 113.70 m above MSL (iii) Ellipsoidal height h at Point C: Using h = H + N: h = 75.00 + 5.20 h = 80.20 m

Question Type

numerical

Answer Structure

• Part (i): State formula N = h − H, substitute, and give answer with sign [1 mark]
• Part (ii): State formula H = h − N, substitute carefully with negative N, give answer [1 mark]
• Part (iii): State formula h = H + N, substitute, give answer [1 mark]

Scoring Breakdown

Common Mark Deductions

• Part (i): N = 210.80 − 180.50 = +30.30 m (subtraction reversed)
• Part (ii): H = 95.30 − 18.40 = 76.90 m (ignoring negative sign on N)
• Any part: Formula not shown before substitution

Key Phrases To Include

• N = h − H
• H = h − N
• h = H + N
• −30.30 m
• 113.70 m
• 80.20 m
• double negative