Gravity measurements are a reflection of the earth's gravitational attraction, its
centripetal force,
tidal accelerations due to the sun, moon, and planets, and other applied forces. Gravity gradiometers measure the spatial derivatives of the gravity vector. The most frequently used and intuitive component is the vertical gravity gradient,
Gzz, which represents the rate of change of vertical gravity (
gz) with height (
z). It can be deduced by differencing the value of gravity at two points separated by a small vertical distance, l, and dividing by this distance. : G_{zz} = {\partial g_z\over \partial z} \approx {g_z \bigl(z + \tfrac \ell 2 \bigr) - g_z \bigl(z - \tfrac \ell 2 \bigr )\over \ell} The two gravity measurements are provided by accelerometers which are matched and aligned to a high level of accuracy.
Unit The unit of gravity gradient is the
eotvos (symbol E), which is (). A person at a distance of 2 metres would provide a gravity gradient signal approximately one E. Mountains can give signals of several hundred eotvos.
Gravity gradient tensor Full tensor gradiometers measure the rate of change of the gravity vector in all three perpendicular directions giving rise to a gravity gradient tensor (Fig 1). Let V be the gravitational field potential (defined up to an additive constant). The gravitational field vector field is -\nabla V (more properly, and the gravity gradient
tensor field is the
second derivative . In general, a second-order tensor in \R^3 has 9 free variables, but because -\nabla^2 V is symmetric, it has only 6 free variables. Furthermore, by the
Poisson equation, {{tmath|1= \operatorname{Tr} \Gamma = 4\pi G \rho }}, so in free space, {{tmath|1= \operatorname{Tr} \Gamma = 0 }}, leaving only 5 free variables. In particular, this means that when the equipment performing the gradiometry is in air or vacuum, which is almost always the case, the full gravity gradient tensor \Gamma needs to measure only 5 numbers. == Comparison to gravity ==