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```SIO 226: Introduction to
Marine Geophysics
Gravity
LeRoy Dorman
Scripps Institution of Oceanography
Winter, 2013
Goals:
To leave you with some tools to help see the
connection between observed gravity
(1)density variations in the earth and
(2)The earth's shape(including topography), which is
the basis for estimation of bathymetry from satellite
altimetry.
Much of the analysis of gravity is based on a
phenomenon called “upward continuation”.
anomalies, and what they show us.
Gravity Reductions
The first task is to remove the earth's main field, which is much
larger than the anomalies we study.
2 The formula of 1967
4 is
g ϕ= 978031.846(1+ 0.005278895sin ϕ+ 0.000023462sin ϕ) mGal
Where phi is latitude. This is based on a rotating ellipsoidal earth.
To correct for altitude difference from the ellipsoid, we use the
derivative, -0.3086 mGal/meter. The result of these two
corrections is the Free Air anomaly.
To correct for the material between the point and observation, we
make the Bouguer plate correction of 2 Gh, which is 0.1119h
mGal/m, for a density of 2670kg/m^3. The result of making
this additional correction is the Bouguer anomaly.  can be
adjusted to optimize topographic suppression, thus estimating ρ.
Eliminate Things We Know

Elevation: Free Air
0.3088 mGal/m

Topography:
Bouguer and
terrain reductions
subtract 0.1967
mGal/m

Net: +0.1121
mGal/m

Isostasy: (later)
The horizontal scale of the previous figure was ~1 km.
The one below is ~1000 km and provides a very different picture.

Things which worked on a local scale leave us
with large
anomalies when applied
to a larger
Continent-scale
Data
data set.
Mantle Bouguer Anomaly

At sea, the plate correction (evaluated for field at the sea
surface for topographic height directly beneath the ship) is a
poor approximation to the field from the actual topography, so
real bathymetric data are used.

To reveal features of the mantle, Kuo and Forsyth (1988)
suggested removal of the gravity field from the crust, using
whatever data are available- from a seismic survey or assuming
a simple constant-thickness crust. The usefulness of this, of
course depends on the accuracy of the estimate of the crustal
field, which, in turn, depends on the survey data used and
other assumptions.
Basics from Physics I
Gravity is a vector force per unit mass caused by the presence
of other masses. For two masses, this force is, from Newton's
law,
m1 m2
F= G
.
l2
The theory which describes gravity is called potential theory, since
this force is the gradient (derivative) of a scalar potential,
Gm
U=
l
and scalars are simpler to deal with. Since taking a derivative of
both sides of an equation leaves us with a valid equation, many
transformations can be used on the potential apply as well to
components of gravity.
Basics from Physics II
F⋅ ds= − 4 πG M
Gauss's Theorem∫ is∫ that
F perp ( x)=
− 2 πG σ( x)
which implies
that
This means that we can choose a σ( x) which
produces any value of F perp ( x) we desire.
Thus we have no hope of extracting the
earth's density structure uniquely from gravity
data alone.
Laplace's Equation
∇ 2U = 0
In mass-free space, Laplace's equation
applies, and this provides a strong constraint
on the potential field U. Writing this out, we
get
∂ 2x U + ∂ 2x U + ∂ 2x U = 0
1
2
3
and after Fourier transformation,
− k 21 U (k )− k 22 U (k )− k 23 U (k )= 0
or,
k 12+ k 22+ k 23= 0
Fourier Transform and
derivatives
In one dimension, the ∞ Fourier Transform of a
function is F (k )= ∫ f ( x)e− 2 πi k x dx.
−∞
∞
The FT of a deriv is F x [ f ' ( x)](k )= ∫ f ' ( x)e− 2 πi k x dx
−∞
Integration by parts, ∫ v du= [uv ]− ∫ u dv
with du= f ' ( x)dx and v= e− 2 πi k x
so u= f ( x) ,and dv= 2 πi k e− 2 πi k x dx
∞
F x [ f ' ( x)](k )= [ f ( x)e− 2 πk x ]− ∫ f ( x)(− 2 πi k e− 2 πi kx dx)
−∞
so multiplication by 2 πi k corresponds to
differentiation, since we can omit the 2nd term.

Fourier Transformation in 2D
The Fourier Transformation is
where the real part of
i k⋅ x the
F (k )=is∫plotted
∫ f ( x)e
1 dxfor
2
kernel
at the dx
right
one value of k. The direction
of k is normal to the wave
“crests” and the magnitude of
k is 2π̸λ, where  is the
wavelength.
Upward Continuation
In geophysical surveys, we normally gather data
in two dimensions, xand
x, 1which
1
k2
transform into
and
. The equation
k1
2
2
2
tells us that if we know any two, we
khowever,
+
k
+
k
1
2
3 = 0,
can calculate the third!
Thus if we have the gravity field at some level in
the earth (such as from a lamina), we can
calculate the field at the surface.
Upward decay of anomaly amplitude
If we take a 2D Fourier transform of a spatial
grid of data, we end up with a 2D array of
amplitudes evaluated at two wavenumbers
k = − (k + k ) where k i = 2 π/ λ i
When we reconstruct the gravity field, thek i
enter the calculation as the exponential kernel
of the Fourier transform. A real value of k
produces a sinusoidal variation but an
imaginary k produces an exponential decay. So
long wavelengths decay slowly and short
wavelengths decay rapidly.
2
3
2
1
2
2
Simple modeling
We can calculate the gravity field of a geological
model in 4 steps.
(1) Approximate the structure using one or more
lamina using the equivalent stratum.
(2) Make a Fourier expansion of the field from the
model.
(3) “Upward continue” the Fourier components.
(4) Synthesize the anomaly from the Fourier
components.
Seamount Example
10km
Water depth:
4km
Ridge Example
Water Depth:
4km
Is this a realistic
depth?
A Simple Isostatic Model

Consider a simple model in
which the topography is
mirrored by a structure of like
density, but opposite sign, at
some (initially one) depth.

This model is consistent with
the the data, both local and
continental, which we have
just seen.
Implications of this model
We represent topography as a lamina with density σ(x)= h(x)
and the Compensation is a lamina with density -(x).
Take one harmonic component at a time, h ( x)= h ei k ⋅ x a
sinusoid with wavelength k =2π/λ, whose gravity field is

1
ik x 1
gThe
field
from hthe
( x)=
2 πρG
e 1compensating
B
lamina is
and the BA is
ik x − k z
e
 gDividing
by h e produces
Q (k )= − 2 πρG
a log scale as a line whose slope is
1
g BA (k )= − 2 πρG h e
g BA
gB
1
1
1
which plots on
c
ik 1 x 1 − k 1 z c
e
Q(k 1 )= − e
− zc
− k 1 zc
Isostatic Response Data

Surveyor data is oceanic
from off Oregon, McNutt,
1978.

USA is from Lewis and
Dorman, 1970.

Australia is Mcnutt, 1978.
Plotted on a log scale

Shows straight lines,
more or less.

Slope of oceanic is
shallower, since
compensation depth is
smaller.

Circles are oceanic,
squares and triangles are
US and Australia.
0
-1
Idealized Models

In terms of out simple model, the classical Pratt and Airy
models look like:
Mass Anomalies and Geodesy

A mass anomaly not
only changes the
gravity, it changes the
shape of the earth's
geoid as well.

Fortunately, the
changes in the geoid
are for  > 400 km.
In The Space Domain

Convolving the gravity
anomaly field with the Stokes
functions produces the geoid
height.

Since these functions are not
localized around the
computation point, gravity at
distant parts of the earth is
important.
Principle of SIO's Bell
Gravimeter

which is a forcebalance
accelerometer.

Its accuracy depends
on a very accurate
digital voltmeter.

Its principle of
operation is shown
at right.
Bell Gravimeter (reality)

In operational reality,
its appearance is a
little different.
Summary:
 Gravity measurements allow us to:



measure the density of the surficial material,
measure the shape of the earth
Show that mountains “float”
```