# Second derivative test

In geodesy and geophysics, the Bouguer anomaly (named after Pierre Bouguer) is a gravity anomaly, corrected for the height at which it is measured and the attraction of terrain. The height correction alone gives a free-air gravity anomaly.

## Anomaly

The Bouguer anomaly is related to the observed gravity $g_{obs}$ as follows:

$g_{B}=g_{obs}-g_{\lambda }+\delta g_{F}-\delta g_{B}$ $g_{B}=g_{F}-\delta g_{B}$ Here, $g_{B}$ is the Bouguer anomaly, $g_{F}$ is the free-air gravity anomaly, $g_{\lambda }$ is the correction for latitude (because the Earth is not a perfect sphere) and $\delta g_{F}$ is the free-air correction and $\delta g_{B}$ is the correction for terrain called the Bouguer reduction.

A Bouguer reduction is called simple or incomplete if the terrain is approximated by an infinite flat plate called the Bouguer plate. refined or complete Bouguer reduction removes the effects of terrain precisely. The difference between the two, the differential gravitational effect of the unevenness of the terrain, is called the terrain effect. It is always negative.

## Simple reduction

The gravitational acceleration $g$ outside a Bouguer plate is perpendicular to the plate and towards it, with magnitude 2πG times the mass per unit area, where $G$ is the gravitational constant. It is independent of the distance to the plate (as can be proven most simply with Gauss's law for gravity, but can also be proven directly with Newton's law of gravity). The value of $G$ is 6.67 × 10−11 N m² kg−2, so $g$ is 4.191 × 10−10 N m² kg−2 times the mass per unit area. Using 1 Gal = 0.01 m/s² we get 4.191 × 10−5 mGal m² kg−1 times the mass per unit area. For mean rock density (2.67 g/cm³) this gives 0.1119 mGal/m.

The Bouguer reduction for a Bouguer plate of thickness $H$ is

$\delta g_{B}=2\pi \rho GH,$ where $\rho$ is the density of the material and $G$ is the constant of gravitation. On Earth the effect on gravity of elevation is 0.3086 mGal/m decrease when going up, minus the gravity of the Bouguer plate, giving the Bouguer gradient of 0.1967 mGal/m.

More generally, for a mass distribution with the density depending on one Cartesian coordinate z only, gravity for any z is 2πG times the difference in mass per unit area on either side of this z value. A combination of two equal parallel infinite plates does not produce any gravity inside.