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.^[1] The height correction alone gives a free-air gravity anomaly.

The Bouguer anomaly ${\displaystyle g_{B}}$ defined as:

$${\displaystyle g_{B}=g_{F}-\delta g_{B}+\delta g_{T}}$$

Here,

• ${\displaystyle g_{F}}$ is the free-air gravity anomaly.
• ${\displaystyle \delta g_{B}}$ is the Bouguer correction which allows for the gravitational attraction of rocks between the measurement point and sea level;
• ${\displaystyle \delta g_{T}}$ is a terrain correction which allows for deviations of the surface from an infinite horizontal plane

The free-air anomaly ${\displaystyle g_{F}}$, in its turn, is related to the observed gravity ${\displaystyle g_{obs}}$ as follows:

$${\displaystyle g_{F}=g_{obs}-g_{\lambda }+\delta g_{F}}$$

where:

• ${\displaystyle g_{\lambda }}$ is the correction for latitude (because the Earth is not a perfect sphere; see normal gravity);
• ${\displaystyle \delta g_{F}}$ is the free-air correction.

A Bouguer reduction is called simple (or incomplete) if the terrain is approximated by an infinite flat plate called the Bouguer plate. A refined (or complete) Bouguer reduction removes the effects of terrain more precisely. The difference between the two is called the (residual) terrain effect (or (residual) terrain correction) and is due to the differential gravitational effect of the unevenness of the terrain; it is always negative.^[2]

The gravitational acceleration ${\displaystyle g}$ outside a Bouguer plate is perpendicular to the plate and towards it, with magnitude 2πG times the mass per unit area, where ${\displaystyle G}$ 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 ${\displaystyle G}$ is 6.67×10⁻¹¹ N m² kg⁻², so ${\displaystyle g}$ is 4.191×10⁻¹⁰ N m² kg⁻² times the mass per unit area. Using 1 Gal = 0.01 m s⁻² (1 cm s⁻²) we get 4.191×10⁻⁵ mGal m² kg⁻¹ times the mass per unit area. For mean rock density (2.67 g cm⁻³) this gives 0.1119 mGal m⁻¹

The Bouguer reduction for a Bouguer plate of thickness ${\displaystyle H}$ is $${\displaystyle \delta g_{B}=2\pi \rho GH}$$ where ${\displaystyle \rho }$ is the density of the material and ${\displaystyle G}$ is the constant of gravitation.^[2] On Earth the effect on gravity of elevation is 0.3086 mGal m⁻¹ decrease when going up, minus the gravity of the Bouguer plate, giving the Bouguer gradient of 0.1967 mGal m⁻¹.

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 parallel infinite if equal mass per unit area plates does not produce any gravity between them.

• Lowrie, William (2004). Fundamentals of Geophysics. Cambridge University Press. ISBN 0-521-46164-2.
• Hofmann-Wellenhof, Bernard; Moritz, Helmut (2006). Physical Geodesy (2nd. ed.). Springer. ISBN 978-3-211-33544-4.

• Bouguer anomalies of Belgium. The blue regions are related to deficit masses in the subsurface
• Bouguer gravity anomaly grid for the conterminous US by the [United States Geological Survey].
• Bouguer anomaly map of Grahamland F.J. Davey (et al.), British Antarctic Survey, BAS Bulletins 1963-1988
• Bouguer anomaly map depicting south-eastern Uruguay's Merín Lagoon anomaly (amplitude greater than +100 mGal), and detail of site.
• List of Magnetic and Gravity Maps by State by the [United States Geological Survey].