Lecture 5: Gravity and Geoid

1. Explain the relationship between geodesy and gravity. Why is the study of gravity essential to geodesy?

Answer:
Geodesy is the science of determining the Earth’s size, shape, orientation in space, and gravity field. Gravity is essential because it defines the physical reference surfaces used for measuring elevations and vertical datums. The Earth’s gravity field determines the geoid, which represents mean sea level and serves as the zero-elevation surface for all height systems. Without understanding gravity, accurate measurements of height, sea level, and mass distribution would not be possible. In short, gravity ties the geometric Earth (ellipsoid, coordinates) to its physical reality (mass and potential).

2. Discuss the causes of spatial variations in gravity over the Earth’s surface.

Answer:
Gravity varies spatially due to several factors:

1. Earth’s shape: Because Earth is an oblate spheroid, points at the poles are closer to the center of mass than those at the equator, resulting in stronger gravity at the poles.

2. Rotation: The centrifugal force from Earth’s rotation reduces effective gravity, especially near the equator.

3. Topography and subsurface density: High mountains, deep basins, or local density differences in the crust and mantle cause small but measurable gravity anomalies.

4. Isostatic and tectonic adjustments: Redistribution of mass within the crust and mantle also changes local gravity.
   Together, these create variations ranging from 9.78 to 9.83 m/s² across the globe.

3. Define and differentiate between gravitational attraction, centrifugal acceleration, and Coriolis acceleration.

Answer:

• Gravitational attraction is the fundamental pull of the Earth’s mass on objects, directed toward the Earth’s center.

• Centrifugal acceleration arises because the Earth rotates, creating an apparent outward acceleration perpendicular to the rotation axis, which slightly reduces the net gravity, especially at the equator.

• Coriolis acceleration affects moving bodies (air, water, or projectiles), causing deflection due to rotation.
  In geodesy, total gravity (g) is the vector sum of the gravitational and centrifugal accelerations; the Coriolis term mainly matters for dynamic processes.

4. What is gravity potential, and why is it a more useful concept in geodesy than gravitational force?

Answer:
Gravity potential (W) is the total potential energy per unit mass due to both gravitational and centrifugal effects:

W=V+ΦW

where V is the gravitational potential and Φ the centrifugal potential.
It is preferred in geodesy because it is a scalar quantity, making it easier to model mathematically. The direction and magnitude of gravity are derived from its gradient (g=∇W). Equipotential surfaces (constant W) represent levels of equal potential energy — one of which is the geoid. Thus, potential theory allows a unified description of gravity and Earth’s shape.

5. Define the geoid and describe its significance in geodetic measurements.

Answer:
The geoid is the equipotential surface of Earth’s gravity field that best fits mean sea level globally. It is not a perfect mathematical shape but an irregular, smooth surface reflecting variations in mass distribution. The geoid serves as the true zero-elevation surface, meaning that all orthometric heights (physical elevations) are measured above it.
In practice, it provides the link between geometric coordinates from GNSS (which use an ellipsoid) and physical heights used in mapping, engineering, and hydrology.

6. Differentiate between ellipsoidal height, orthometric height, and geoid undulation.

Answer:

• Ellipsoidal height (h): Distance measured vertically from the reference ellipsoid to a point on Earth’s surface. Obtained directly from GNSS or satellite positioning.

• Orthometric height (H): Height above the geoid, representing “height above mean sea level.” Determined through leveling and gravity observations.

• Geoid undulation (N): The separation between the geoid and ellipsoid.

They are related by:

h=H+N

Thus, GPS gives h, the geoid model gives N, and their difference yields the orthometric height H.

7. Explain why the geoid is an equipotential surface. What does this imply about the direction of gravity?

Answer:
The geoid is an equipotential surface because the gravity potential W has a constant value on it. This means the potential energy of a unit mass is the same everywhere along the surface. As a result, no work is done when moving along the geoid.
Since gravity is the gradient of potential (g=∇W), the direction of gravity is always perpendicular to the geoid at every point. This perpendicularity is why leveling instruments and plumb lines align with the local gravity direction.

8. Describe how satellite and terrestrial gravity data are combined to produce a geoid model.

Answer:
Geoid determination uses both global and local datasets:

1. Satellite data (e.g., from missions like GRACE or GOCE) measure the long-wavelength structure of Earth’s gravity field.

2. Terrestrial gravity measurements (ground-based gravimeters and leveling) refine the short-wavelength details.

3. Mathematical modeling uses spherical harmonic expansions and potential theory to merge these datasets.
   The result is a global model such as EGM96 or a regional model like Philippine Geoid, both of which provide geoid undulations NNN at specific grid intervals. These models convert GPS heights to orthometric heights.

9. Explain isostasy and its importance in understanding gravity anomalies.

Answer:
Isostasy is the concept that the Earth’s crust is in gravitational balance — lighter crustal blocks “float” on the denser, deformable mantle, much like wood floating on water. Regions with thicker crust (mountain ranges) are compensated by deeper “roots,” while thinner crust (oceans) is compensated by shallower depths.
When equilibrium is disturbed (e.g., by erosion, sedimentation, or melting of glaciers), isostatic adjustment occurs. These mass redistributions cause local gravity anomalies, which geodesists can measure to infer subsurface density and tectonic processes.

10. What does the Hudson Bay gravity anomaly reveal about post-glacial rebound and isostatic adjustment?

Answer:
The Hudson Bay region in Canada exhibits a negative gravity anomaly — lower gravity values compared to surrounding areas. This reflects the post-glacial rebound process: during the last ice age, massive ice sheets depressed the crust. When the ice melted, the crust began to rise slowly to reestablish isostatic balance.
Geophysical studies show that this crustal uplift and the viscous mantle response explain about 25–45% of the observed anomaly. The remaining part is attributed to deeper mantle flow. The anomaly is thus evidence of ongoing Earth deformation even after thousands of years.

11. Compare global and regional geoid models. Give examples and describe when each is used.

Answer:
Global geoid models (e.g., EGM96, EGM2008) are derived primarily from satellite data and provide worldwide coverage at moderate resolution (typically 15'×15' grids). They are useful for large-scale or international applications.
Regional geoid models (e.g., AusGeoid, Philippine Geoid, OSU89A) integrate dense local gravity and leveling data, offering higher accuracy for engineering or mapping in a specific country.
Global models establish consistency across continents; regional models improve precision where local gravity data are available.

12. Discuss the effects of Earth’s rotation on measured gravity values.

Answer:
Earth’s rotation introduces a centrifugal force that opposes gravity, reducing its apparent magnitude. The centrifugal acceleration acts outward from the rotation axis and is maximum at the equator, zero at the poles. Consequently, measured gravity is lower at the equator and higher at the poles.
Rotation also causes the Earth to bulge at the equator (oblate spheroid shape), which further increases the difference in gravity. These effects are accounted for in the theoretical model of normal gravity used for comparisons and corrections in geodesy.

13. Why is it necessary to correct GPS-derived heights using a geoid model?

Answer:
GPS provides ellipsoidal heights relative to a mathematical ellipsoid, not mean sea level. Engineers and surveyors, however, require orthometric heights—heights above the geoid (sea level reference). Because the ellipsoid and geoid do not coincide, a correction using the geoid undulation (NNN) is necessary:

H=h−N

Applying this correction ensures that height data correspond to real physical elevations, maintaining consistency with maps, topographic data, and hydrological models.

14. Explain how variations in subsurface density affect gravity readings and what information geodesists can derive from them.

Answer:
Gravity depends directly on mass distribution. Denser subsurface materials (e.g., basalt or ore bodies) cause positive gravity anomalies—higher measured gravity. Less dense materials (e.g., sediments, voids, or magma chambers) produce negative anomalies.
By analyzing these anomalies, geodesists and geophysicists can infer the presence of geological structures such as faults, mineral deposits, or crustal thickness variations. Thus, gravity surveys are not only geodetic tools but also powerful instruments in geological exploration.

15. Discuss the importance of understanding the geoid for modern satellite positioning systems.

Answer:
Modern satellite systems like GPS, GLONASS, and Galileo measure positions relative to a global ellipsoid. However, practical applications — mapping, engineering, navigation, and hydrology — require elevations relative to mean sea level (the geoid).
Understanding the geoid allows conversion between geometric and physical height systems, ensuring compatibility between GNSS data and existing vertical datums. Without geoid knowledge, height data from satellites could be off by tens of meters, leading to serious errors in construction, flood modeling, or sea-level monitoring.

Summary of Key Takeaways