Gravitation is the force of attraction one mass has for another. Gravity is the gravitational attraction of the Earth. According to Newton's law of gravitation, the force increases with increasing mass. The force of the attraction also increases as we approach the centre of mass. If one geological body is denser than another, it will have a greater mass per unit volume and a greater gravitational attraction. Measurements of gravity yield little direct geological information, other than to represent the Earth's oblate spheroidal shape, unless corrections are made to account for variations in the Earth's shape and topography.

As the Earth's diameter is approximately 20 km smaller from pole to pole than through the equator, the force of gravity increases the closer we get to the poles. In addition, the Earth's rotation results in a slightly smaller measured gravity at the equator than near the poles. In order to isolate the effect of lateral variations in density within the Earth, the bulk gravity effects of the Earth due to latitude must be removed.

The theoretical gravity is given in milligals (10^-5m·s^-2) by the International Gravity Formula :

g_t = 978032.7(1.0+0.0053024 sin²(θ) + 0.0000058 sin²(2θ))

based on the 1980 Geodetic Reference System, where θ is the latitude in degrees of any point on the Earth. The effect of latitude is removed by subtracting the theoretical value of gravity from the observed values.

To correct for variations in elevation, the vertical gradient of gravity (vertical rate of change of the force of gravity, 0.3086 mGal·m^-1) is multiplied by the elevation of the station and the result is added, producing the free-air anomaly. The free-air gravity anomaly is given by the formula:

FA = g_o - g_t + (δg/δz) h

where:
g_o = observed gravity (mGal)
g_t = theoretical gravity (mGal)
δg/δz = vertical gradient of gravity (0.3086 mGal·m^-1)
h = elevation above mean sea level (m).

To isolate the effects of lateral variations in density on gravity, it is also necessary to correct for the gravitational attraction of the slab of material between the observation point and the mean sea level. This is the Bouguer gravity anomaly, which is given for static land measurements by the formula

BA = g_o - g_t + (δg/δz - 2πGρ_c) h

where:
g_o = observed gravity (mGal)
g_t = theoretical gravity (mGal)
δg/δz = vertical gradient of gravity (0.3086 mGal·m^-1)
G = gravitational constant (6.672 x 10^-11 m³·kg^-1s^-2 or 6.672 x 10^-6 m²·kg^-1·mGal
ρ_c = density of crustal rock (kg·m^-3)
h = elevation above mean sea level (m).

Figure 1: Geometry of isostatic roots in Airy-Heiskanen model larger image [GIF, 12.5 kb, 504 X 271, notice]

The principle of isostasy states that mass excesses, represented by topographic loads at the surface, are compensated by mass deficiencies at depth which are referred to as isostatic roots. The effect of these mass deficiencies are not accounted for in the Bouguer reduction and there exists an inverse correlation between broad Bouguer anomaly lows and positive topography. The isostatic correction removes the gravity effect of the isostatic roots. The depth of the roots can be estimated based on the Airy-Heiskanen model (Simpson et al., 1986).

Land areas
The depth of the root is defined for land areas by the formula

d = d_s + e (ρ_t / δρ)

where:
d = depth to the bottom of the root (m)
d_s = the depth of compensation for sea level compensations (30 000m)
e = elevation (m)
ρ_t = density of the topographic load (2670 kg·m^-3), and
δρ = density contrast between the root and underlying mantle material (600 kg·m^-3, see Figure 1).

Oceanic areas
For oceanic areas, a negative topographic load exists, since lower density water replaces higher density rock. The depth of the root over oceanic areas is defined by the formula:

d = d_s - d_w ((ρ_t - ρ_w) / δρ)

where:
d_w = depth of water
ρ_w = the density of water (1030 kg·m^-3)

Gravitational effect
Heiskanen (1953) defines the gravitational effect of a root at a point, O, by the formula

g_c = G m ((a²/2R) + d cos α) / ( a² + d² - 2 ad sin α/2) ^3/2

Figure 2: Gravitational effect of an isostatic root on a point at sea level larger image [GIF, 11.9 kb, 515 X 371, notice]

where:
g_c = the gravitational effect (mGal)
m = mass deficiency of the root (kg)
R = radius (m) of the Earth
a = distance (m) between the observation point and the point at sea level above the root
d = the distance (m) between the centre of mass of the root and the point at sea level above the root
α = the angle between lines extending from the observation point and the point at sea level above the root through the centre of the Earth (see Figure 2).

In areas of rough terrain, a correction for the effect of nearby masses above (mountains) or mass deficiencies below (valleys) the gravity measurement point can be calculated and applied. The final Bouguer gravity anomaly reflects lateral variations in rock density.

Simpson, R.W., Jachens, R.C., Blakely, R.J., and Saltus, R.W. 1986. A New Isostatic Residual Gravity Map of the Conterminous United States With a Discussion on the Significance of Isostatic Residual Anomalies. Journal of Geophysical Research. V. 91, No 138, p. 8348-8372.

Heiskanen, W. 1953. Isostatic reductions of the gravity anomalies by the aid of high-speed computing machines. Annales Academiae Scientiarum Fennicae, Series A, III. Geologica - Geographica, number 33.