Measurement and Reduction of
Gravity Observations
The acceleration due to Gravity typically is
measured in one of two ways:
Absolute: Weight drop, pendulum
Relative: Mass on a spring
Pendulums
The simple pendulum has a period related to its length: T  2
L
g
The problem is that this length is a “virtual length” to the center of mass
of the system, which is not easy to determine accurately.

One way around this is to use a clever device called a
compound pendulum such as that devised by Henry Kater
(1777-1835) in 1818.
It consists of a metal bar with knife edges attached near the
ends and two weights that can slide between the knife edges.
The bar is pivoted from each knife edge in turn and the
positions of the weights are adjusted so that the period of the
pendulum is the same with both pivots. The period is then given
by the formula for the simple pendulum, which enables g to be
calculated to a reasonable precision.
Kater's Pendulum in the Science Museum in London. It was designed to make gravity
surveys in the field so that it is packed in a fitted mahogany case to protect it in its travels.
Pendulums are not used much anymore but were
instrumental (so to speak) in some of the major oil
discoveries of the early 20th century (Saudi Arabia,
for example).
This is how Cecil Green got started.
Weight drop: drop a weight many times in a vacuum and
measure time history using laser interferometry. Very accurate
measure of absolute gravity, but quite expensive.
Relative Measurements using a Mass on a Spring
At first glance this seems simple. We just use Hooke’s law:
F = mg = -kz
g = -(k/m)z
So that a change in g creates a change in z:
dg = -(k/m)dz
But the problem is sensitivity! dg/g = 10-8 = dz/z
So we need to have a really good way to measure the mass
displacement.
We could increase sensitivity by making k/m small, but
this would mean a combination of a big mass and a
weak spring. Not good for field work!
Additionally the system will oscillate with a period
m
T  2
k
which is very long for a sensitive device! You would

have to wait forever to make a reading.
Nevertheless, remember that a very sensitive device
should have a very long period.
A clever way around this problem was
discovered in 1934 by Lucien Lacoste, who
was trying to design a stable long period
seismometer.
He used a combination of geometry and a
device called a “zero-length spring”
Here’s how it works:
In the suspension suspension shown above, there are two torques:
gravitational and spring.
If these two torques balance each other for any angle of the beam, the
system will have infinite period. In this case, the smallest change in
vertical acceleration (or gravity) will cause a large movement.
Let’s see what’s required to meet this condition.
The torque due to gravity is
Tg = Wd sin q
Where W is the weight (=mg) and d is the
distance from the mass to the beam's
hinge. The torque due to the spring is the
product of the pull of the spring and the
springs lever arm, s.
s = a sin b
The length of the spring is r and by the
law of sines:
r = bsinq/sinb
If the spring constant is k and the length of
the spring without force is n, The spring
force is illustrated by this graph.
The torque due to the spring is then:
Ts = -k(r-n)s = kns -k(bsinq/sinb) a sin b = kns - kabsinq
The total torque is then
To = Tg + Ts = Wdsinq + kns - kabsinq = kns + (Wd - kab)sinq
This equation yields zero torque for all
angles q if:
(1) n = 0 and
(2) Wd - kab = 0
The second part is easy and just involves
geometry. For n to equal zero, we must have
a "zero length spring". That is, a spring whose
force-length graph passes through the origin or, at least, points toward
the origin. The turns of a helical spring of zero unstressed length would
bump into each other before the spring actually reached zero length. By
making a helical spring whose turns press against each other when there
is no force on the spring, a "zero length spring" can be made.
There are several ways to make a zero length spring. A simple
zero-length spring is a flat spiral spring. The mechanical properties
of a spiral spring are not as convenient as a helical spring. To make
a zero-length helical spring, the spring wire can be wound onto a
mandrel. As the wire is wound, it can be twisted.
Another method is to hold the wire at an angle and with tension
while winding it on a rotating mandrel.
A problem that still needs to be faced is the sensitivity of the spring
“constant” k to changes in pressure and temperature.
Pressure can be managed by evacuating the box.
Temperature effects are minimized with by keeping the box heated
to a temperature greater than any you are likely to encounter in the
field (50o C).
This last part means that the gravity meter consumes power - often
not a convenient situation.
Nevertheless, the meter is very accurate and the drift rate is quite
small, or in any event is linear and can be corrected for easily.
Views of the
Lacoste-Romberg
Gravity Meter
An second option is the Worden meter, which relies totally on
geometry to increase sensitivity, has an evacuated chamber, and uses
differentially expanding materials to deal with temperature
variations. It works ok, but the drift must be monitored closely
when using it in the field.
Gravity measurements can be made at sea or in the air if you put it on a
gyroscopically controlled inertial table that corrects for the accelerations
of the boat or plane.
Reductions of Gravity Data
Gravity Measurements are not difficult to make, but the flip side is that
once you’ve made your measurements you need to do a fair amount
of work before you can interpret them.
Here is an outline of the corrections you need to make to a raw
measurement:
1. Latitude: Correction for N-S distance
2. Free-Air: Correction for elevation above the data plane
3. Bouguer: Correction for excess mass above the data plane
4. Terrain: Correction for variations in topography
5. Tides:
Attraction of Sun and Moon
6. Eötvös:
Correction for moving vehicle
7. Isostacy: Variations in crustal thickness
Latitude Correction
This is nothing more than accounting for the reference spheroid as
predicted by the International Gravity Formula.
Recall that this formula (or variants on it) looks like:
g = 978.0318(1 + 0.005302sin2fg - 0.0000059sin22fg) cm/sec2
Note that
dg/df = 978.0318(0.0106048sinfgcosfg - 0.000235sin2fgcos2fg)
At 45o degrees latitude, dg/df = 5 gals/radian, or, because 1 radian
= 57o degrees and 1o degree = 111 km:
dg/df = 0.8 mgal/km = 1.3 mgal/mile
Typically, we just calculate the reference gravity and subtract if
from our readings, but this relation can be convenient.
Remember that gravity increases as you go towards the poles, so
we add this correction to our readings as we move to the equator.
Free-Air Correction
Often we make observations at different elevations, and we know that
gravity will decrease as we get farther from the center of mass of the
Earth.
Therefore, we choose a reference elevation (typically sea level) and
adjust our readings to be what they would be at that elevation. To
compute this correction, recall that
g = GM/r2
And so
dg/dr = -2GM/r3 = -2g/r = -0.3085 mgals/m = -0.09406 mgals/ft.
Most land surveys are done above sea level, so the free air correction
will generally increase the gravity reading.
Bouguer Correction
Of course, now that you have taken your reading down to sea level, you
have to account for all the mass between your original elevation and sea
level.
This correction is called the “Bouguer Correction” (after our friend
Pierre) and is calculated by assuming an infinite slab of material of
thickness Dh lies between the station and sea level. We derived the
expression for this before:
Dgslab = 2GDrDh
Assuming a typical density for crustal rocks (r = 2.67 gr/cm3), we arrive
at
Dg = 0.112 mgals/m = 0.0341 mgals/ft
Again, because land surveys are usually above sea level, the Bouguer
correction usually decreases the gravity reading.
Terrain Corrections
Unless you do your survey in Kansas,
you have to correct for variations in
topography that you missed with your
infinite slab. This is called a terrain
correction.
Terrain corrections traditionally are
made by estimating the differences
between the elevation of the station and
that of the topography surrounding the
station.
Remember that terrain always reduces
the gravity reading: mass excess above
the station (mountains) pull up on the
station, while mass deficiencies below Thus, a terrain corrected reading
(valleys) “push” (the correction is for a will always be greater than it
originally was.
negative mass).
Hammer Charts
Terrain corrections
traditionally were made
with something called a
“Hammer” chart -named
after the guy who
introduced them, and
they are truly the low
point of gravity work.
These charts divide the regions around the station into pie shaped
cylindrical sections.
The analyst centers the chart on the station location on a topo map and
estimates the average elevation difference (note that this is in an absolute
value sense!) for all elevations within the section.
You multiply this number by a theoretical value for that section, and then
add them all up.
Fortunately, we don’t have to do that much work
anymore, because Digital Elevation Maps (DEMs)
exist for much of the world, and for the US they are
quite detailed (spacing of 10 m over much of the
country).
Hence, most of this correction can be calculated by
suitable software; you only need to worry about any
terrain very close to the station (for example, if you
happen to be right by a cliff).
Aren’t you lucky?
Tides
The Sun and Moon can contribute 0.2-0.3 mgals of attraction and so we
need to correct for this if we are in the field for more than an hour or so
(depends on desired accuracy, of course).
Some people advocate
reoccupying a base station
several times to determine
this correction, but in fact
we know how to calculate
it better than most of us
can measure!
You should use the
theoretical correction.
Eötvös
If you are making measurements in a moving vehicle, you have to
correct for E-W velocity. As we discussed before, the vertical
acceleration of a Eastward moving vehicle is:
Dac= 2wvEcos(l)
The effect is to decrease the reading taken from an eastward moving
vehicle, and increase the reading from a westward moving one.
Isostatic correction
Gravity studies can be useful in constraining crustal thickness, but often
we are not interested in structures that we can predict theoretically, so in
mountainous terrain we often
make a correction for isostatic
changes in crustal thickness
due to elevation.
There are a number of ways
to do this, from a pure Pratt
to pure Airy type of
correction.
Note that there is a strong
relation between the
Isostatic anomaly and the
Free Air anomaly.
Why should this be?
Name that Anomaly!
When reducing an observed gravity datum gobs, we always make
corrections for latitude (dgL), tide (dgD) and Eotvos (dgE) (if we were in a
vehicle is moving east-west).
If we also correct for elevation (using the free air correction = dgFA) the
resulting anomaly is called the Free Air Anomaly (FAA):
FAA = dgobs + dgL + dgD + dgE + dgFA
If we also correct for the infinite slab (the Bouguer correction dgB) we
have the Simple Bouguer Anomaly (SBA):
SBA = FAA - dgB
If we correct for terrain (dgT), we have the Complete Bouguer Anomaly
(CBA):
CBA = FAA - dgB + dgT
Finally, making an isostatic correction gives the Isostatic Anomaly.
Removing the Regional Field
The Isostatic correction brings up a common problem: what to do when
what you are looking for is hidden in a big signal from some mass
anomaly far from you? Crustal thickness changes are a classic
example.
This “background” field is usually called the Regional Field because it
is a long wavelength background that is typical of an entire “region”.
There are some strategies for dealing with regionals, but be aware that
nothing is foolproof.
Most of these strategies involve removing a smooth background by
filtering or polynomial fitting.
Simple Example
More involved example (two-dimensional)
Example using Polynomials: Which is the Signal?
Upward and Downward Continuation
One interesting characteristic of gravity data is that if you know the
field at one elevation you can in theory calculate or “continue” what it
should be at any other elevation.
For example, you can take your ground observations and predict what
you would measure in an airplane (upward continuation) or vice versa
(downward continuation)
Upward Continuation is
always stable, but downward
continuation is not (Can you
guess why this is?).
In general, downward
continuation should be
avoided.
Determining Densities
Both the Bouguer and terrain corrections depend on the density of the
materials in the vicinity of the station.
Generally it will be a good idea to
collect some samples and measure
these in the lab (how would you
do this, by the way?)
There is also a technique
(Nettleton) which works in certain
circumstances - when the
topography is not closely related
to subsurface structure - that
allows you to do this with your
gravity measurements.
Generally it will be better to use
some kind of ground truth,
however.
Examples of typical density
variations are shown to the
right.
Note that variations in
density are not much more
than a factor of 2 for most
rocks you are likely to
encounter.