Gravity anomaly due to a simple-shape buried body The general equation for gravity anomaly is: gZ = V 1 cosdV , 2 r where: is the gravitational constant is the density contrast

r is the distance to the observation point is the angle from vertical V is the volume Example: a sphere 4a 3 1 z gZ = . 2 2 2 2

3 (x + z ) (x + z ) Gravity anomaly due to a simple-shape buried body A horizontal wire of infinite length Starting with : dm cos sin . 2 r Substituting : dm = dl, dgZ =

cos = R /r, sin = Z / R is mass per length R is the distance to the wire r is the distance to an element dl and : r 2 = R 2 + l 2, we get : + g Z = Z

dl 2 2 3/2 (R + l ) 2 Z Z = 2 2 . R2 x + Z2 = Z

l 2 2 + 2 1/ 2 R (R + l ) |= Gravity anomaly due to a simple-shape buried body An infinitely long horizontal cylinder

To obtain an expression for a horizontal cylinder of a radius a and density , we replace with a2 to get: Z gZ = 2a 2 . 2 x +Z 2 It is interesting to compare the solution for cylinder with that of a sphere. cylinder

sphere Gravity anomaly due to a simple-shape buried body A horizontal thin sheet of finite width Starting from the expression for an infinite wire, we write : Z dgZ = 2 2 dx, r where is mass per area. Replacing sin with Z/r : sin dgZ = 2 dx r 2

x2 sin gZ = 2 dx =2 d = 2 . r x1 1 Remarkably, the gravitational effect of a thin sheet is independent of its depth. Gravity anomaly due to a simple-shape buried body A thick horizontal sheet of finite width Starting from the expression for a thin sheet, we write : dgZ = 2dh , with being in units of mass per volume.

Integration with respect to depth : h2 gZ = 2 dh = 2h . h1 station surface Actually, you have seen this expression before z Station of two

Dimensional structure Gravity anomaly due to a simple-shape buried body A thick horizontal sheet of infinite width To compute the gravitational effect of an infinite plate we need to replace with : gZ = 2h . Geoid anomaly Geoid is the observed equipotential surface that defines the sea level. Reference geoid is a mathematical formula

describing a theoretical equipotential surface of a rotating (i.e., centrifugal effect is accounted for) symmetric spheroidal earth model having realistic radial density distribution. Geoid anomaly The international gravity formula gives the gravitational acceleration, g, on the reference geoid: g( ) = gE (1+ sin 2 + sin 4 ) where : gE is the g at the equator

is the latitude = 5.278895 103 = 2.3462 105 Geoid anomaly The geoid height anomaly is the difference in elevation between the measured geoid and the reference geoid. Note that the geoid height anomaly is measured in meters. Geoid anomaly Map of geoid height anomaly: Figure from: www.colorado.edu/geography

Note that the differences between observed geoid and reference geoid are as large as 100 meters. Question: what gives rise to geoid anomaly? Geoid anomaly Differences between geoid and reference geoid are due to: Topography Density anomalies at depth Figure from Fowler Geoid anomaly What is the effect of mantle convection on the geoid anomaly? downwelling

Two competing effects: 1. Upwelling brings hotter and less dense material, the effect of which is to reduce gravity. 2. Upwelling causes topographic bulge, the effect of which is to increase gravity. Figure from McKenzie et al., 1980 upwelling Flow Temp. Geoid anomaly

SEASAT provides water topography Note that the largest features are associated with the trenches. This is because 10km deep and filled with water rather than rock. Geoid anomaly and corrections Geoid anomaly contains information regarding the 3-D mass distribution. But first, a few corrections should be applied: Free-air Bouguer Terrain Geoid anomaly and corrections Free-air correction, gFA:

This correction accounts for the fact that the point of measurement is at elevation H, rather than at the sea level on the reference spheroid. Geoid anomaly and corrections Since: 2 R( ) 2h g( ,h) = g(,0) g(,0) 1 , R( ) + h

R( ) where: is the latitude h isthe topographic height g() is gravity at sea level R() is the radius of the reference spheroid at The free-air correction is thus: 2h . R( ) This correction amounts to 3.1x10-6 ms-2 per meter elevation. gFA = g(,0) g( ,h) = g( ,0)

should this correction be added or subtracted? Question: Geoid anomaly and corrections The free-air anomaly is the geoid anomaly, with the free-air correction applied: gFA = reference gravity - measured gravity + gFA . Geoid anomaly and corrections Bouguer correction, gB: This correction accounts for the gravitational attraction of the rocks between the point of measurement and the sea level.

Geoid anomaly and corrections The Bouguer correction is: gB = 2h , where: is the universal gravitational constant is the rock density height h is the topographic For rock density of 2.7x103 kgm-3, this correction amounts to 1.1x10-6 ms-2 per meter elevation. Question: should this correction be added or subtracted? Geoid anomaly and corrections

The Bouguer anomaly is the geoid anomaly, with the free-air and Bouguer corrections applied: gB = reference gravity - measured gravity + gFA gB . Geoid anomaly and corrections Terrain correction, gT: This correction accounts for the deviation of the surface from an infinite horizontal plane. The terrain correction is small, and except for area of mountainous terrain, can often be ignored. Geoid anomaly and corrections

The Bouguer anomaly including terrain correction is: gB = reference gravity - measured gravity + gFA gB + gT . Bouguer anomaly for offshore gravity survey: Replace water with rock Apply terrain correction for seabed topography After correcting for these effects, the ''corrected'' signal contains information regarding the 3-D distribution of mass in the earth interior.