Free-air gravity anomaly

Survey methods

The free-air correction adjusts measurements of gravity to what would have been measured at mean sea level, that is, on the geoid. The gravitational attraction of Earth below the measurement point and above mean sea level is ignored and it is imagined that the observed gravity is measured in air, hence the name. The theoretical gravity value at a location is computed by representing the Earth as an ellipsoid that approximates the more complex shape of the geoid. Gravity is computed on the ellipsoid surface using the International Gravity Formula.

For studies of subsurface structure, the free-air anomaly is further adjusted by a correction for the mass below the measurement point and above the reference of mean sea level or a local datum elevation. This defines the Bouguer anomaly.

Calculation

The free-air gravity anomaly ${\displaystyle g_F}$ is given by the equation:

${\displaystyle g_F=(g_{obs}+\delta g_F)-g_{\lambda }}$

Here, ${\displaystyle g_{obs}}$ is observed gravity, ${\displaystyle \delta g_F}$ is the free-air correction, and ${\displaystyle g_{\lambda }}$ is theoretical gravity.

It can be helpful to think of the free-air anomaly as comparing observed gravity to theoretical gravity adjusted up to the measurement point instead of observed gravity adjusted down to the geoid. This avoids any confusion of assuming that the measurement is made in free air. Either way, however, the Earth mass between the observation point and the geoid is neglected. The equation for this approach is simply rearranging terms in the first equation of this section so that reference gravity is adjusted and not the observed gravity:

${\displaystyle g_F=g_{obs}-(g_{\lambda }-\delta g_F)}$

Correction

Gravitational acceleration decreases as an inverse square law with the distance at which the measurement is made from the mass. The free air correction is calculated from Newton's Law, as a rate of change of gravity with distance:

${\displaystyle \begin{array}{cc}g& =\frac{GM}{R^2}\\ \frac{dg}{dR}& =-\frac{2GM}{R^3}=-\frac{2g}{R}\end{array}}$

At 45° latitude, ${\displaystyle 2g/R=0.3086}$ mGal/m.

The free-air correction is the amount that must be added to a measurement at height ${\displaystyle h}$ to correct it to the reference level:

${\displaystyle \delta g_F=\frac{2g}{R}\times h}$

Here we have assumed that measurements are made relatively close to the surface so that R does not vary significantly. The value of the free-air correction is positive when measured above the geoid, and negative when measured below. There is the assumption that no mass exists between the observation point and the reference level. The Bouguer and terrain corrections are used to account for this.

Significance

Over the ocean where gravity is measured from ships near sea level, there is no or little free-air correction. In marine gravity surveys, it was observed that the free-air anomaly is positive but very small over the Mid-Ocean Ridges in spite of the fact that these features rise several kilometers above the surrounding seafloor. The small anomaly is explained by the lower density crust and mantle below the ridges resulting from seafloor spreading. This lower density is an apparent offset to the extra height of the ridge indicating that Mid-Ocean Ridges are in isostatic equilibrium.