1. Gravity of the Earth
Gravitational acceleration a distance r from a sphere of density ρ is
This result is independent of radial density variations

2. Gravity of the Earth
The earth is not a perfect sphere but is approximately an oblate spheroid – an ellipse rotated about the short axis
The ellipticity is given by

3. Gravity of the Earth The radius of an oblate spheroid is
Centrifugal acceleration due to rotation of the earth decreases gravitational acceleration felt by an observer on the surface

4. Gravity of the Earth
These two effects are accounted for in the reference gravity formula adopted by the International Association of Geodesy which gives the gravitational acceleration as a function of latitude

5. Gravity of the Earth The earth is not a perfect oblate spheriod
We define a reference surface called the geoid which is the gravitational equipotential
Over the oceans, this is simply the mean sea level
A plumb bob points perpendicular to the local equipotential surface – in the direction of the potential gradient
The reference spheroid is the theoretical equipotential that is the closest fit to Earth’s field

7. Gravity of the Earth
Shape of the geoid can be determined from gravity measurements
We can measure the geoid very accurately from space
Altimetry satellites can measure the geoid over the oceans
Gravity anomalies are very small compared to the mean surface gravity
g = 981 gal, but we usually work in mgal or μgal

9. Gravity anomaly (mgal)

10. Ice loss from the Greenland and Antarctic Ice Sheets
The large ice sheets are not in equilibrium
More mass is being lost through ice bergs calving into the ocean than is being gained from winter snows inland
Contributing to sea level rise

11. Gravity Anomalies Isostasy
French surveying expedition under Pierre Bouguer found that the Andes mountains deflected their plumb bob line much less than expected
the Great Trigonometric Survey of India, completed under Sir George Everest, found the same result due to the Himalayas
1855 J.H. Pratt and Sir George Airy proposed two separate hypothesis to explain this observation which depend on Archimedes principal – termed isostasy

12. Gravity Anomalies
Calculate a gravity anomaly by subtracting the reference gravity value
Compute the free-air correction (correct for elevation change or Δr

13. Gravity Anomalies Now the free-air gravity anomaly is given by
The Bouguer correction compensates for density of rocks between h and sea level

14. Assignments due next tuesday (10/20)
Reading
Larsen, C.F., Motyka, R.J., Freymueller, J.T., Echelmeyer, K.A., Ivins, E.R., 2005, Rapid viscoelastic uplift in southeast Alaska caused by post-Little Ice Age glacial retreat: Earth and Planetary Science Letters, v. 237,
Homework
Chpt 5. – 14, 19, 21, 22, 24

15. Gravity Anomalies
Terrain correction compensates for devations from a horizontal plane – very small except for mountainous regions and often ignored
The Bouguer anomaly combines all of these corrections
The Bouguer anomaly is then the difference between the observed value and the theoretical value at particular latitude, referenced to sea level

16. Gravity Anomalies and Isostasy
The free air anomaly is primarily used to determine if a region is in isostatic equilibrium
If a region is in equilibrium there should be no excess or lack of mass above the compensation depth and hence no gravity anomaly
The Bouguer anomaly indicates mass variations that are not due to topography

17. Gravity Anomalies
Mid Atlantic
Ridge
UK continental margin

18. Gravity Anomalies
Oahu

19. Gravity Anomalies and Isostasy
If the structure is about 10 times the compensation depth the free air anomaly will be very small away from the edges
Uncompensated structures have zero Bouguer anomaly
Compensated structures have negative Bouguer anomaly – why?

20. Gravity anomalies and Isostasy

21. Density anomalies under the mid atlantic ridge
All four density models (c, e, f, g) describe the observed gravity anomaly. There are actually an infinity number of density models that will produce the same gravity anomaly. This in another example of non-uniqueness.
What are the differences between each of the density models?
Model c is most consistent with what is understood about spreading centers.

22. Geoid height anomalies
Lateral variations in density distribution produce variations in the geoid height. This anomaly is given by:
A trough in the geoid height anomaly is associated with a negative gravity anomaly (low density) and a peak is associated with a positive gravity anomaly (high density)

23. Geoid height anomalies
Overall geoid height anomalies are small indicating that isostatic equilibrium is typical for large features – This implies
The mantle is weak (has finite viscosity) on a geologic time scale
The crust is strong and can support small scale features

25. Geoid height anomalies
The geoid height anomaly for an isostatic density distribution is not zero
Where g is the reference gravity value, Δρ(z) is the anomalous density distribution at depth z beneath point P, and D is the compensation depth

26. Geoid height anomaly
Hawaiian island ridge bends the crust causing flexure
Long wavelength topography caused by mantle upwelling
Increase in topography more than compensates for decreased mantle density and produces long wavelength positive gravity anomaly