1. Gravity II: Gravity anomaly due to a simple-shape buried body
The general equation for gravity anomaly is:
where:
 is the gravitational constant
 is the density contrast
r is the distance to the observation point
 is the angle from vertical
V is the volume
Example: a sphere

2. Gravity anomaly due to a simple-shape buried body
A horizontal wire of infinite length b
is mass per length
R is the distance to the wire
r is the distance to an element dl

3. Gravity anomaly due to a simple-shape buried body
An infinitely long horizontal cylinder
To obtain an expression for a horizontal cylinder of a radius a and density , we replace  with a2 to get:
It is interesting to compare the solution for cylinder with that of a sphere.
cylinder
sphere

4. Gravity anomaly due to a simple-shape buried body
A horizontal thin sheet of finite width
Remarkably, the gravitational effect of a thin sheet is independent of its depth.

5. Gravity anomaly due to a simple-shape buried body
A thick horizontal sheet of finite width
surface
station
Station of two
Dimensional structure z 
Actually, you have seen this expression before

6. Gravity anomaly due to a simple-shape buried body
A thick horizontal sheet of infinite width
To compute the gravitational effect of an infinite plate we need to replace  with :

7. Geoid anomaly
Geoid is the observed equipotential surface that defines the sea level.
Reference geoid is a mathematical formula describing a theoretical equipotential surface of a rotating (i.e., centrifugal effect is accounted for) symmetric spheroidal earth model having realistic radial density distribution.

8. Geoid anomaly
The international gravity formula gives the gravitational acceleration, g, on the reference geoid:

9. Geoid anomaly
The geoid height anomaly is the difference in elevation between the measured geoid and the reference geoid.
Note that the geoid height anomaly is measured in meters.

10. Map of geoid height anomaly:
Geoid anomaly
Map of geoid height anomaly:
Figure from:
Note that the differences between observed geoid and reference geoid are as large as 100 meters.
Question: what gives rise to geoid anomaly?

11. Differences between geoid and reference geoid are due to: Topography
Geoid anomaly
Differences between geoid and reference geoid are due to:
Topography
Density anomalies at depth
Figure from Fowler

12. What is the effect of mantle convection on the geoid anomaly?
Flow
Temp.
upwelling
downwelling
Two competing effects:
Upwelling brings hotter and less dense material, the effect of which is to reduce gravity.
Upwelling causes topographic bulge, the effect of which is to increase gravity.
Figure from McKenzie et al., 1980

13. Geoid anomaly
SEASAT provides water topography
Note that the largest features are associated with the trenches. This is because 10km deep and filled with water rather than rock.

14. Geoid anomaly and corrections
Geoid anomaly contains information regarding the 3-D mass distribution. But first, a few corrections should be applied:
Free-air
Bouguer
Terrain

15. Geoid anomaly and corrections
Free-air correction, gFA:
This correction accounts for the fact that the point of measurement is at elevation H, rather than at the sea level on the reference spheroid.

16. Geoid anomaly and corrections
Since:
where:
 is the latitude
h is the topographic height
g() is gravity at sea level
R() is the radius of the reference spheroid at 
The free-air correction is thus:
This correction amounts to 3.1x10-6 ms-2 per meter elevation.
Question: should this correction be added or subtracted?

17. Geoid anomaly and corrections
The free-air anomaly is the geoid anomaly, with the free-air correction applied:

18. Geoid anomaly and corrections
Bouguer correction, gB:
This correction accounts for the gravitational attraction of the rocks between the point of measurement and the sea level.

19. Geoid anomaly and corrections
The Bouguer correction is:
where:
 is the universal gravitational constant
 is the rock density
h is the topographic height
For rock density of 2.7x103 kgm-3, this correction amounts to 1.1x10-6 ms-2 per meter elevation.
Question: should this correction be added or subtracted?

20. Geoid anomaly and corrections
The Bouguer anomaly is the geoid anomaly, with the free-air and Bouguer corrections applied:

21. Geoid anomaly and corrections
Terrain correction, gT:
This correction accounts for the deviation of the surface from an infinite horizontal plane. The terrain correction is small, and except for area of mountainous terrain, can often be ignored.

22. Geoid anomaly and corrections
The Bouguer anomaly including terrain correction is:
Bouguer anomaly for offshore gravity survey:
Replace water with rock
Apply terrain correction for seabed topography
After correcting for these effects, the ''corrected'' signal contains information regarding the 3-D distribution of mass in the earth interior.