The general equation for gravity anomaly is:

where: is the gravitational constant is the density contrastr is the distance to the observation point is the angle from verticalV is the volume

Gravity anomaly due to a simple-shape buried body

Example: a sphere

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gZ = γΔρ1

r2cosαdV ,

V

∫

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gZ =4πγa3Δρ

3

1

x 2 + z2( )

z

x 2 + z2( )

.

Gravity anomaly due to a simple-shape buried body

A horizontal wire of infinite length

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Starting with :

dgZ =γdm

r2cosβ sinφ .

Substituting :

dm = λdl,

cosβ = R /r,

sinφ = Z /R

and :

r2 = R2 + l2,

we get :

ΔgZ = γλZdl

R2 + l2( )3 / 2 =

−∞

+∞

∫ γλZl

R2 R2 + l2( )1/ 2

−∞

+∞

| =

2γλZ

R2= 2γλ

Z

x 2 + Z 2.

is mass per lengthR is the distance to the wirer is the distance to an element dl

Gravity anomaly due to a simple-shape buried body

An infinitely long horizontal cylinder

cylinder sphere

To obtain an expression for a horizontal cylinder of a radius a and density , we replace with a2 to get:

It is interesting to compare the solution for cylinder with that of a sphere.

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gZ = 2γπa2ρZ

x 2 + Z 2.

Gravity anomaly due to a simple-shape buried body

A horizontal thin sheet of finite width

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Starting from the expression for an infinite wire, we write :

dgZ = 2γZ

r2 σdx,

where σ is mass per area.

Replacing sinφ with Z/r :

dgZ = 2γσsinφ

rdx ⇒

ΔgZ = 2γσsinφ

rdx =

x1

x2

∫ 2γσ dφφ1

φ 2

∫ = 2γσ φ.

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

surface

stationStation of two

Dimensional structure

z

Actually, you have seen this expression before€

Starting from the expression for a thin sheet, we write :

dgZ = 2γφρdh ,

with ρ being in units of mass per volume.

Integration with respect to depth :

ΔgZ = 2γφρ dhh1

h2

∫ = 2γφρh .

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 :

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gZ = 2πγρh .

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.

The international gravity formula gives the gravitational acceleration, g, on the reference geoid:

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g(λ ) = gE 1+α sin2 λ + β sin4 λ( )

where :

gE is the g at the equator

λ is the latitude

α = 5.278895 ×10−3

β = 2.3462 ×10−5

Geoid anomaly

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:

Note that the differences between observed geoid and reference geoid are as large as 100 meters.

Figure from: www.colorado.edu/geography

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

Figure from McKenzie et al., 1980

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.

What is the effect of mantle convection on the geoid anomaly?

Flow

Temp.

upwellingdownwelling

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:

where:• is the latitude • h is the topographic height• g() is gravity at sea level• R() is the radius of the reference spheroid at

The free-air correction is thus:

This correction amounts to 3.1x10-6 ms-2 per meter elevation.

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g(λ ,h) = g(λ ,0)R(λ )

R(λ ) + h

⎛

⎝ ⎜

⎞

⎠ ⎟2

≈ g(λ ,0) 1−2h

R(λ )

⎛

⎝ ⎜

⎞

⎠ ⎟ ,

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gFA = g(λ ,0) − g(λ ,h) = g(λ ,0)2h

R(λ ) .

Question: should this correction be added or subtracted?

Geoid anomaly and corrections

The free-air anomaly is the geoid anomaly, with the free-air correction applied:

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gFA = reference gravity - measured gravity + δgFA .

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

Geoid anomaly and corrections

The Bouguer correction is:

where: is the universal gravitational constant is the rock densityh is the topographic height

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?

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gB = 2πγρh ,

Geoid anomaly and corrections

The Bouguer anomaly is the geoid anomaly, with the free-air and Bouguer corrections applied:

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gB = reference gravity - measured gravity + δgFA −δgB .

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:

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gB = reference gravity - measured gravity + δgFA −δgB + δgT .

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.