This is a descriptive article on stable and unstable gravimeters. The article is mainly focused on LaCoste-Romberg and Worden gravimeters. Also, it includes marine gravity survey shortly.

1. Unstable (astatic) Gravimeters
Description:
These instruments have an additional negative restoring force operating against the restoring
spring force, that is, in the same sense as Gravity. Since the 1930s, unstable gravimeters have
been used far more extensively than their stable counterparts. In a stable device, once the
system has been disturbed it will return to its original position, whereas an unstable device will
move further away from its original position. For example, if a pencil lying flat on a table is lifted
at one end and then allowed to drop, the pencil will return to being flat on the table. However,
if the pencil starts by being balanced on its end, once disturbed, it will fall over; i.e. it becomes
unstable, rather than returning to its rest position. The main point of the instability is to
exaggerate any movement, so making it easier to measure, and it is this principle on which the
unstable gravimeter is based. Various models of gravimeter use different devices to achieve the
instability.
The Principleof An Astatic Gravimeter:
An almost horizontal beam hinged at one end supports a mass at the other. The beam is
attached to a main spring which is connected at its upper end to a support above the hinge. The
spring attempts to pull the beam up anticlockwise by its turning moment, which is equal to the
restoring force in the spring multiplied by the perpendicular distance from the hinge (d). This
turning moment is balanced by the gravitational turning moment which attempts to rotate the
beam in a clockwise manner about the hinge and is equal to the weight of the mass (mg) times
the length of the beam (I) multiplied by the cosine of the angle of the beam from the horizontal
(i.e. mgl cos….). If gravity changes, the beam will move in response but will be maintained in its
new position because the main spring is a 'zero-length' spring. One virtue of such a spring is
that it is pretensioned during manufacture so that the tension in the spring is proportional to its
length. This means that if all forces were removed from the spring it would collapse to zero
length, something which is impossible in practice. Another virtue of the zero-length spring is
that it results in an instrument which is linear and very responsive over a wide range of gravity
values. Astatic gravimeters do not measure the movement of the mass in terms of changes in
gravity but require the displaced mass to be restored to a null position by the use of a
micrometer. The micrometer reading is multiplied by an instrumental calibration factor to give
values of gravity, normally to an accuracy within 0.1 g.u. 10.01 mGal) and in some specialist
devices to within 0.01 g.u. (0.001 mGal ).

2. Types:
There are many types of unstable gravimeter used for relative gravity measurement. Such as-
 LaCoste and Romberg Gravimeter
 Thyssen Gravimeter
 Worden Gravimeter
 Sodin Gravimeter
 Vibrating String Gravimeter
Figure 1: Unstable or Astatic Gravimeter

3. LaCoste and Romberg Gravimeter
Description:
LaCoste and Romberg gravity meters were for a long time the state-of-the-art survey tool for
measuring the acceleration due to gravity. They are ‘relative’ meters (rather than ‘absolute’
meters) and are small enough for a single operator to manage.
The mechanism involves a beam, supported by a ‘zero-length’ spring at an angle of 45 degrees.
The meter is read by nulling the mass position, i.e., adding or subtracting a small amount of
force to restore it to a set position. This is achieved by rotating the large metal knob until the
beam is level. The reading is then simply read off the counter. It is a highly precise instrument;
the unit measures in milligals: one milligal is equivalent to 0.00001 meters per second squared.
Geophysicists use gravity surveys to investigate subsurface variations in density. Changes in
density due to different lithologies allow scientists to understand what is happening
underground. Gravity surveys are used for mineral, oil and gas exploration, civil engineering
applications, as well as a variety of environmental purposes.
The Principleof an LaCoste and Romberg Gravimeter:
A zero-length spring is one in which the tension is proportional to the actual length of the
spring, that is, if all external forces were removed the spring would collapse to zero length. The
advantage of the zero-length spring is that if it supports the beam and mass M (figure 2b) in the
horizontal position, it will support them in any position. Zero-length-springs are built with initial
tension so that a thresh-old force is required before spring extension begins(as with a door
spring).
To derive the expression for the sensitivity of this gravimeter, we write k(s-c) for the tension in
the spring when its length is s; thus, c is a small correction for the fact that the spring is not
truly zero length. Taking moments about the pivot in Figure 2b, we get
Mga cos 𝜃= k (s-c) b sin 𝛼
=k(s-c) b (y cos 𝜃)/s
Using the law of sines. Thus
g= (k/M) (b/a) (1-c/s) y
when g increases by δ g, the spring length increases by δ s where
δ g= (k/M) (b/a) (c/s) (y/s) δ s

4. For a given change in gravity δ g, we can make δ s as large as we wish by decreasing one or
more of the factors on the right-hand side; moreover, the closer the spring is to the zero-
length spring, the smaller c is and the larger δ s becomes.
In operation this is a null instrument, a second spring being used, which can be adjusted to
restore the beam to the horizontal position. The sensitivity of gravimeters in use in surface
exploration is generally 0.01 mGal. The instrument requires a constant temperature
environment, usually achieved by keeping it at a constant temperature that is higher than
the surroundings.
Figure 2: LaCoste and Romberg Gravimeter
(a) (b)

5. Worden Gravimeter
Gravimeters are extremely precise instruments that measure the earth’s gravity at a specific
location. Gravimeters are often used by prospectors to locate subterranean deposits of
valuable natural resources (mainly petroleum) as well as by geodesists to study the shape of the
earth and its gravitational field. Differences in topography, latitude, or elevation—as well as
differences in subterranean density—all affect the force of gravity. Commonly, gravimeters are
composed of a weight hanging on a zero-length spring inside a metal housing to negate the
influence of temperature and wind. Gravity is then measured by how much the weight
stretches the spring. Texas Instruments introduced the Pioneer gravimeter in 1960, describing it
as a Worden instrument "ideally suited for gravity programs in areas of limited latitude and
temperature variations." Its design is covered by three patents, all assigned to Texas
Instruments. One (#2,674,887), granted to Sam P. Worden in 1954, described an instrument
"which is of very simple construction and which, at the same time, will permit very delicate
measurements, and is smaller in size and of less weight and more rugged than conventional
types of gravity meters now in use." The patent went on to say that this instrument "is of such
construction and size that the working parts may be more efficiently insulated," and that it
incorporated "a compensating device which dispenses with the necessity of a thermostatic
control." The second patent (#2,738,676), granted to Worden and Boyd Cornelison in 1956,
described a "Large Range Gravity Sensitive Instrument." The third (#2,732,718) was granted to
Cornelison in 1956. Texas Instruments donated this example to the Smithsonian in 1963.
The Principleof Worden Gravimeter:
A simplified schematic is shown in figure 3.2. The moving system is similar to the LaCoste-
Romberg meter. The arm OP’ and beam OM are rigidly connected and pivot about O, changing
the length of the main spring P’C, which is fixed at C. We have the following relations:
∠OCP’= ∠OP’C = 𝜋/2 – ( 𝛼+ 𝜃/2)
RP ⊥ CP ; P’P ⊥ OP
So
RPP’ = 𝜋/2 – 𝛼
s=CP ; δ s= CP’-CP ≈ b𝜃 sin ( 𝜋/2 – 𝛼)
∴ 𝜃 ≈δ s/ (b cos 𝛼)
The correction factor c that appeared in the treatment of the LaCoste-Romberg meter is
negligible for the Worden meter. Taking moments about the pivot for the case where 𝜃 =0, we
get

6. Mga = ksb cos 𝛼
When g increases to (g+δ g), P movesalongthe circle toP’ and
M(g+ 𝛿g) a cos 𝜃= kb (s+ 𝛿s) cos ( 𝛼+ 𝜃/2)
When 𝜃=0, to the firstapproximationthisbecomes
M (g+ 𝛿g) = kb (s+ 𝛿s) {cos𝛼- ( 𝜃/2) sin 𝛼}
= kb (s+ 𝛿s) {cos𝛼- ( 𝛿s/2b) tan 𝛼}
= kb {s cos𝛼- 𝛿s(s/2b) tan 𝛼+ 𝛿s cos𝛼}
Subtracting the first moment equation to eliminate g, we get
Ma 𝛿g= kb {cos𝛼-(s/2b) tan 𝛼} 𝛿s
Using the relation sin 𝛼=(s/2b), we finally get
𝛿g = (k/m) (b/a) (cos2 𝛼/ cos 𝛼) 𝛿s
As in the LaCoste-Romberg meter, the sensitivity can be increased by decreasing the factors
(k/M) and (b/a); in addition, the factor (cos2𝛼/ cos𝛼) approaches zero when 𝛼 approaches 45
degree, thus furnishing another method of obtaining high sensitivity. In practice, the sensitivity
is about 0.01 mGal.

7. Figure 3.1: Worden Gravimeter
(a) (b)
Figure 3.2: Basic Principle of Worden Gravimeter

8. Marine Gravity Survey
Description:
Marine gravity measurements allow scientists to better understand the Earth's crust by
providing quantitative constraints on the structure of the crust and, in combination with other
co-registered data sets and forward models, improve our ability to interpret the geological
processes occurring within oceanic crust.
There are differentmethodsof surveyingmarine gravity.3typesof methodwill be discussedbelow to
overviewthe marine gravitysurveysystem
a) Locating marine stations: Considerable gravity has been done on the surface of water-
covered areas and also on the sea floor. Locating the marine station is usually done by using a
radio-navigation system such as Shoran, Raydist or RPS.
b) Remote control systems: Standard gravimeters have been adapted for operation on the sea
floor to depths of 200 m. this method of measurement is suitable for most unland waters and
coastal areas.
although the high sensitivity of this equipment is an advantage, operation in deep water is slow
because the assembly must be raised to the surface between stations.
c) Shipboard operations: Shipboard gravimeters are used for most gravity measurements at
sea. These meters are mounted on an elaborate gyro-stabilized platform located in the part on
a ship where there is minimum movement to roll and pitch.
Figure 4: Marine Gravity Survey

9. References:
1. Australian society of exploration geophysicists
2. https://www.aseg.org.au/equipment-museum/lacoste-and-romberg-
gravity-model-g-meter
3. "Worden" Gravity Meter Operating Instruction Manual No. 81537-4
(Houston: Texas Instruments, 1961)
4. https://learninglab.si.edu/resources/view/46222#more-info
5. National Museum of American History, Smithsonian Institution.
(2015, October 31). Smithsonian Learning Lab Resource: Worden
Gravimeter. Retrieved July 29, 2019.
6. An Introduction to Applied and Environmental Geophysics book,
copyright by John M. Reynolds in 1997, published by John wiley & sons
in 1997.
7. Applied Geophysics second edition published by the press syndicate of
the University of Cambridge.