1. Instruments used in gravity prospecting
Fundamental design of almost all gravity instruments uses a mass on a spring:
A change in gravity should cause a change
in length given by
Problems:
measurement must be accurate to one part in 108.
restoring force overcompensates, mass overshoots equilibrium point, system oscillates
Solution:
a system with no effective restoring force, an “unstable gravimeter”
such a system has infinite periodicity, and mechanical amplification
2.4. Instruments in gravity prospecting
Almost all instruments designed for measuring the local variation in the gravity field are based on masses on springs. Schematically a simple device for measuring gravity might look like this: Hooke’s law tells us that a change in gravity acceleration, δg, should give us a corresponding change in the length of the spring, δs according to the relationship mδg = kδs, (2.9) where k is the spring constant. Given a measurement of the change in the spring length we can solve for the gravity anomaly δg = (k/m)δg. The problem is that this measurement must be accurate to one part in 108. In the case of an oscillating spring system this is virtually impossible to achieve without more careful design.
The problem was comprehensively solved in the 1930’s, when the LaCoste-Romberg gravimeter was developed. This is an instrument known as an “unstable gravimeter”. A spring system such as the one above tends to oscillate because the spring acts as a restoring force — any movement away from equilibrium is resisted by the spring. In contrast, an unstable gravimeter needs to be brought to equilibrium manually, and any movement away from the equilibrium position is mechanically amplified — the system tends to move away from equilibrium.
The advantage of an unstable gravimeter is twofold: 1. If there is no restoring force then the oscillation period of the system is infinite (in fact, the design of the gravimeter was inspired by the design of very long period seismometers). 2. The instability provides a mechanical amplification of the system that increases its sensitivity.

2. Lacoste-Romberg gravimeter
At equilibrium, zero torque implies:
Sine rule for triangles:
Substituting:
The operation of the Lacoste-Romberg gravimeter is illustrated in Figure 2.4. To operate the gravimeter it is moved into position, levelled, unclamped and thermally stabilized. The beam will no longer be in equilibrium (horizontal) because the gravity force will be changed, and the spring length, s will have changed. The adjusting screw is turned until the beam is perfectly horizontal and in equilibrium, and the distance, y
is measured. The spring force is given by k(s − z), where z is the “zero length” of the spring. If the meter is in equilibrium, we know that the total torque on the beam is zero, i.e., the torques from the spring and from the force of gravity are equal and opposite:
But the angles θ and α are related through the sine rule for triangles:
. (2.11)
Substituting:
, (2.12)
or, solving for the required value, g
sy. (2.13)
If the distances, spring constants and mass of the system are known, the equation will yield the local gravity value.
Solving for g:

3. Lacoste-Romberg gravimeter
At equilibrium, zero torque implies:
Sine rule for triangles:
Substituting:
IF z=0:
The operation of the Lacoste-Romberg gravimeter is illustrated in Figure 2.4. To operate the gravimeter it is moved into position, levelled, unclamped and thermally stabilized. The beam will no longer be in equilibrium (horizontal) because the gravity force will be changed, and the spring length, s will have changed. The adjusting screw is turned until the beam is perfectly horizontal and in equilibrium, and the distance, y
is measured. The spring force is given by k(s − z), where z is the “zero length” of the spring. If the meter is in equilibrium, we know that the total torque on the beam is zero, i.e., the torques from the spring and from the force of gravity are equal and opposite:
But the angles θ and α are related through the sine rule for triangles:
. (2.11)
Substituting:
, (2.12)
or, solving for the required value, g
sy. (2.13)
If the distances, spring constants and mass of the system are known, the equation will yield the local gravity value.
If the beam is stable in the horizontal position, it is stable in any position!

4. Zero length spring
Sketch a graph of spring force k(s − z) vs spring length, s for a real “zero length spring”.
Force, f
Graph starts at z>0, ramps up steeply then flattens to a straight line in the “operating range”
Spring length, s

5. Zero length spring
Sketch a graph of spring force k(s − z) vs spring length, s for a real “zero length spring”.
Graph starts at z>0, ramps up steeply then flattens to a straight line in the “operating range”

6. Lacoste-Romberg gravimeter
The spring is
therefore pretensioned during manufacture (Figure 2.5) such that the tensioned spring behaves as if
the “zero length” were effectively zero.

7. Lacoste-Romberg gravimeter
Equation for g:
Sensitivity: Minimize the amount that g has to change, before the spring length, s will change a given amount:
Minimize
Sensitivity: We want to maximize the change in spring length, δs for a given change in gravity, δg. This
means we want to maximize δs/δg, or minimize
s2 . (2.14)
One way to do this is to make z, the unstretched spring length as small as possible.
Thus, the instrument sensitivity is maximized if z, the unstretched spring length is as small as possible. This re-enforces the importance of the “zero length spring”

8. Worden gravimeter
Description: (description by Bob Neese, current owner of the Worden Gravity Meter Company) The Worden Gravity Meter measures gravity differences of the earth and can detect 1 part in 100,000,000 of the normal gravity of the earth. Under ideal conditions, this meter can measure gravity differences of 0.01 milligal or 1 inch in elevation change. This particular type of Worden meter continues to be in use today for exploration. It is a very portable and accurate gravity meter. It is one of over 1500 Worden Gravity Meters that have been manufactured. This is not quite as old as Gravity Meter SN 131 (Item N ) and was manufactured after Texas Instruments acquired the business from Sam Worden (Houston Technical Laboratories) in Worden Gravity Meters - A General Overview The Worden Gravity Meter is based on an elastic system constructed of quartz. It is a three spring device employing a pretension of zero-length mainspring to produce the necessary sensitivity. The basic mass is only five milligrams, and the moment of inertia is very low. The low mass, together with the almost perfect elastic qualities of quartz, makes the Worden Gravity Meter a rugged instrument. Sam Worden developed the Worden Gravity Meter in the late 1940's. In 1953, Worden Gravity Meter manufacturing (Houston Technical Laboratories) was sold, by Worden, to Texas Instruments, and was Texas Instruments' first entry into the geophysical equipment market. This made the Worden Gravity Meter a very special item in the eyes of Texas Instrument's executives, for many years. In the mid 1980's, Texas Instruments sold its geophysical equipment production facilities to Halliburton Geophysical Services. Halliburton previewed all the products it acquired and decided to seek buyers for the rights to manufacture and sell the instruments that were not considered mass production or assembly-line products. The Worden Gravity Meter Division was transfered in 1990 to Bob Neese who established the Worden Gravity Meter Company in Richmond Texas in conjunction with other gravity operations that had been founded by his father, Urban Neese. The manufacture and service of the Worden Gravity Meter continues today as it has for over 50 years. More than 1500 Worden Meters have been manufactured -- more than any other type of gravity meter.

9. Worden gravimeter
Description: (description by Bob Neese, current owner of the Worden Gravity Meter Company) The Worden Gravity Meter measures gravity differences of the earth and can detect 1 part in 100,000,000 of the normal gravity of the earth. Under ideal conditions, this meter can measure gravity differences of 0.01 milligal or 1 inch in elevation change. This particular type of Worden meter continues to be in use today for exploration. It is a very portable and accurate gravity meter. It is one of over 1500 Worden Gravity Meters that have been manufactured. This is not quite as old as Gravity Meter SN 131 (Item N ) and was manufactured after Texas Instruments acquired the business from Sam Worden (Houston Technical Laboratories) in Worden Gravity Meters - A General Overview The Worden Gravity Meter is based on an elastic system constructed of quartz. It is a three spring device employing a pretension of zero-length mainspring to produce the necessary sensitivity. The basic mass is only five milligrams, and the moment of inertia is very low. The low mass, together with the almost perfect elastic qualities of quartz, makes the Worden Gravity Meter a rugged instrument. Sam Worden developed the Worden Gravity Meter in the late 1940's. In 1953, Worden Gravity Meter manufacturing (Houston Technical Laboratories) was sold, by Worden, to Texas Instruments, and was Texas Instruments' first entry into the geophysical equipment market. This made the Worden Gravity Meter a very special item in the eyes of Texas Instrument's executives, for many years. In the mid 1980's, Texas Instruments sold its geophysical equipment production facilities to Halliburton Geophysical Services. Halliburton previewed all the products it acquired and decided to seek buyers for the rights to manufacture and sell the instruments that were not considered mass production or assembly-line products. The Worden Gravity Meter Division was transfered in 1990 to Bob Neese who established the Worden Gravity Meter Company in Richmond Texas in conjunction with other gravity operations that had been founded by his father, Urban Neese. The manufacture and service of the Worden Gravity Meter continues today as it has for over 50 years. More than 1500 Worden Meters have been manufactured -- more than any other type of gravity meter.

10. Field operations for gravity surveys
We may divide gravity surveys into:
land operations
marine operations
sea bed measurements
ship-borne surveys
airborne gravity
satellite altimetry
(This is arranged in order of decreasing accuracy and precision, and increasing costs, and spatial dimensions of the target)
Field operations for gravity surveys may be divided into the following categories:
1. Land operations
2. Marine operations, further divided into
a) sea bed measurements, and
b) ship-borne surveys
3. Airborne gravity
4. Satellite altimetry
Generally speaking these are in order of decreasing accuracy and precision: land and sea-bed operations typically have a precision of the order of 0.01 mgal, ship borne surveys typically claim a precision of the order of 1 mgal, while airborne surveys have an accuracy of the order of 5 mgal. By the same token, seaborne and airborne surveys generally have targets of much larger sizes (both spatially and in terms of magnitude).

11. Land operations Desired gravity precision ±0.01mgal.
Elevation precision ±3 cm
Latitude precisions ±10 m
Typical station spacing:
Very large scale regional survey 20 km
Oil and gas exploration 1 km
Mineral exploration m
Geotechnical 1-5 m
Archaeological < 1 m.
Drift - small change in the instrument response with time, caused by spring inelasticity and thermal dependencies in the system.
Earth tides - changes in Earth gravity due to the relative movement of the Sun and Moon. These cause changes of the order of ±0.3 mgal over a 24 hour period.
In order to achieve precisions of the order of 0.01 mgal not only must the gravity meter be a very precise instrument, but very precise measurements are also required of the location of the survey — both latitude and elevation have a strong effect on the gravity readings that must be removed before interpretation. The required precisions for location (to achieve 0.01 mgal precision in gravity) are:
Elevation precision ±3 cm Latitude precisions ±10 m
Because each station must be “surveyed in”, these survey costs must be factored into the overall cost of a gravity survey - the terrain surveying portion of the gravity survey may end up being the most expensive component.
In order to keep costs down, the operations designer may decide to use a coarser station spacing (i.e., to sample the gravity field less frequently). Generally the station spacing is target-dependent, and care must be taken not to lose the signature of the target by undersampling. Generally, the deeper the target the more gentle is the anomaly and the wider is the requir ed station spacing. Typical station spacings might be …
Additional measurement difficulties are caused by:
Drift - essentially a small change in the instrument response with time, caused by a variety of effects, mainly spring in-elasticity and thermal dependencies in the system. Earth tides - changes in Earth gravity due to the relative movement of the Sun and Moon. These cause changes of the order of ±0.3 mgal over a 24 hour period. Usually the two effects (drift and Earth tides) are taken into account simultaneously, by re-occupying a base station periodically during the survey. From the base station recordings, the combined drift/Earth tide effect on the survey vs time is estimated, and then removed. The process is illustrated in Figure 2.6.

12. Land gravity survey plan
Drift - essentially a small change in the instrument response with time, caused by a variety of effects, mainly spring in-elasticity and thermal dependencies in the system. Earth tides - changes in Earth gravity due to the relative movement of the Sun and Moon. These cause changes of the order of ±0.3 mgal over a 24 hour period. Usually the two effects (drift and Earth tides) are taken into account simultaneously, by re-occupying a base station periodically during the survey. From the base station recordings, the combined drift/Earth tide effect on the survey vs time is estimated, and then removed. The process is illustrated in Figure 2.6.

13. Marine operations An underwater gravimeter system
i) Sea bed measurements
Procedures for taking measurements with gravimeters from sea bottom recorders are equivalent to, but more complicated than land operations (and more costly). Generally these operations are only carried out in shallow water, they are not widely used. Sea bottom recorders are lowered to the sea bottom and levelled by remote control. Accurate bathymetry information is crucial.
An underwater gravimeter system

14. Marine operations
ii) Shipborne measurements
Ship-board gravimeters are widely used in oil and gas exploration as a \piggyback" to seismic operations. Precision measurements are a problem due to the eect of wave motion, which can generate accelerations of several thousand mgals. Suitable gravimeters require gyro-stabilized gimbaled platforms to reduce the effects of wave action, and measurements must be averaged over long periods of time to average out the wave effects. The resultant precision of ship-board gravimeter systems is of the order of 1mgal.
An air/sea gravity meter, showing the gyro-stabilized platform

15. Airborne gravity only widely available for the last decade
depends on dramatic improvements in GPS estimation of acceleration
requires:
Gyro-stabilized platforms, see Figure 2.8
GPS-derived acceleration corrections
Use of gravity “gradiometry” to minimize effects of sensor movement
In order to achieve precisions of the order of 0.01 mgal not only must the gravity meter be a very precise instrument, but very precise measurements are also required of the location of the survey — both latitude and elevation have a strong effect on the gravity readings that must be removed before interpretation. The required precisions for location (to achieve 0.01 mgal precision in gravity) are:
Elevation precision ±3 cm Latitude precisions ±10 m
Because each station must be “surveyed in”, these survey costs must be factored into the overall cost of a gravity survey - the terrain surveying portion of the gravity survey may end up being the most expensive component.
In order to keep costs down, the operations designer may decide to use a coarser station spacing (i.e., to sample the gravity field less frequently). Generally the station spacing is target-dependent, and care must be taken not to lose the signature of the target by undersampling. Generally, the deeper the target the more gentle is the anomaly and the wider is the requir ed station spacing. Typical station spacings might be …
Additional measurement difficulties are caused by:
Drift - essentially a small change in the instrument response with time, caused by a variety of effects, mainly spring in-elasticity and thermal dependencies in the system. Earth tides - changes in Earth gravity due to the relative movement of the Sun and Moon. These cause changes of the order of ±0.3 mgal over a 24 hour period. Usually the two effects (drift and Earth tides) are taken into account simultaneously, by re-occupying a base station periodically during the survey. From the base station recordings, the combined drift/Earth tide effect on the survey vs time is estimated, and then removed. The process is illustrated in Figure 2.6.

16. Airborne gravity
Within the last decade contractors have begun to oer airborne gravity surveys. The ability to do this largely derives from the dramatic improvements in the measurement of aircraft accelerations from GPS data. Commercial airborne surveys usually feature
Gyro-stabilized platforms such the one shown in Figure 2.7.
GPS-derived acceleration corrections (see Figure 2.8)
Use of gravity gradiometry" to minimize the effects of sensor movement.

17. Eötvös correction (required for seaborne, airborne measurements)
Apparent gravity is affected by motion of moving platforms
Eastward ship (aircraft) travel: adds to the earth's rotation, increases centrifugal forces and decreases the gravity readings.
Westward travel: increases the gravity reading.
North-south travel: is independent of rotation, and decreases the gravity reading in either case.
Correction required:

18. Next lecture: Gravity Data Reduction
In decreasing order, gravity is affected by
Latitude (due to shape and rotation
Elevation
Local topography
Lunar and solar tidal forces
Local density variations
The first four must be corrected for before we can “see” the local
density variations in the gravity data
All but the first of these effects must be corrected for before we can \see" the local density variations in the
gravity data.