Apparatus for the measurement of gravitational fields

Apparatus for measuring gravitational fields comprising a superconducting string (1) fixed at both ends and forming part of a closed superconducting loop inductively coupled to two driving solenoids (L.sub.d1, L.sub.d2). Displacement of the string in response to a gravitational field is sensed by two magnetic flux transformers each comprising a signal coil and two pick-up coils ((L.sub.p1, L.sub.p2). Pairs of pick-up coils lie in two perpendicular planes providing two independent channels of measurements. The two arms of each flux transformer are balanced to convert only the amplitudes of the string's antisymmetric natural modes into an output voltage. The output voltage of each channel is used to produce a feed-back current distribution (L.sub.y1, L.sub.y2) proximate and parallel to the string. By adjusting the feed-back current, the effective relaxation time and resonant frequency of the first antisymmetric mode of the string can be adjusted, while leaving the symmetric modes unchanged, thus increasing the apparatus' sensitivity to gravity gradients.

This invention relates to the measurement of gravitational fields, 
particularly to gravity gradiometry, and more particularly to a method for 
measuring absolutely off-diagonal components of the gravity gradient 
tensor. 
The gravity gradient tensor is a two-dimensional matrix of the second 
partial derivatives of a gravitational potential, V, with respect to the 
Cartesian co-ordinates, x, y, z, of some arbitrary reference frame. It 
represents how the gravity vector itself in each of these directions 
varies along the axes. 
Accurate absolute measurements of the components of the gravity gradient 
tensor .GAMMA..sub.ij =.differential..sup.2.sub.ij V (ij=x,y,z), taken at 
some local coordinate frame OXYZ are very important to progress in the 
fields of geological prospecting, mapping of the Earth's gravitational 
field, and space, sea and underwater navigation. 
A method of absolute measurement of gravity gradient tensor components was 
invented first by Baron Roland von Eotvos as early as 1890, utilising a 
torsion balance with proof masses hung at different heights from a 
horizontal beam suspended by a fine filament. The gravity gradients give 
rise to differential forces being applied to the masses which result in a 
torque being exerted on the beam, and thus to angular deflection of the 
masses which can be detected with an appropriate sensor. A sensitivity of 
about 1 E (1 E=1 Eotvos=10.sup.-9 s.sup.-2) can be reached but measurement 
requires several hours at a single position due to the necessity to 
recalculate the gravity gradient components from at least 5 independent 
measurements of an angular deflection each with a different azimuth angle. 
Practical devices, which have been built in accordance with this basic 
principle, are large in size and have low environmental noise immunity, 
thus requiring specially prepared conditions for measurements which 
excludes any possibility of using them on a moving carrier. 
A method for absolute measurement of gravity gradient tensor components 
which enhances the above method was invented by Forward in the middle of 
the sixties (see U.S. Pat. No. 3,722,284 (Forward et al) and U.S. Pat. No. 
3,769,840 (Hansen). The method comprises mounting both a dumbbell 
oscillator and a displacement sensor on a platform which is in uniform 
horizontal rotation with some frequency .OMEGA. about the axis of the 
torsional filament. The dumbbell then moves in forced oscillation with 
double the rotational frequency, whilst many of the error sources and 
noise sources are modulated at the rotation frequency or not modulated 
(particularly 1/f noise). The forced oscillation amplitude is at a maximum 
when the rotation frequency satisfies the resonance condition 
2.OMEGA.=.omega..sub.0, where .omega..sub.0 is the angular resonant 
frequency, and the oscillator quality factor Q tends to infinity. Unlike 
the non-rotating method, this method enables one to determine rapidly the 
quantities .GAMMA..sub.yy -.GAMMA..sub.xx and .GAMMA..sub.xy by separating 
the quadrature components of the response using synchronous detection with 
a reference signal of frequency 2.OMEGA.. 
The same principles can be directly used, as proposed by Metzger (see U.S. 
Pat. No. 3,564,921), if one replaces the dumbbell oscillator with two or 
more single accelerometers properly oriented on such a moving platform. 
There are no new features of principle in this solution to compare with 
the previous one except that the outputs of the pairs of accelerometers 
require additional balancing. 
Devices have been built according to this method, but they have met more 
problems than advantages, principally because of the need to maintain 
precisely uniform rotation and the small displacement measurement with 
respect to the rotating frame of reference. The devices have reached a 
maximum working accuracy of about a few tens of Eotvos for a one second 
measuring interval, and they are extremely sensitive to environmental 
vibrational noise due to their relatively low resonant frequencies. The 
technological problems arising in this case are so difficult to overcome 
that the existing developed designs of rotating gravity gradiometers are 
so far only at the stage of prototypes whose measurement accuracy is much 
lower than the limiting theoretical estimates. 
In a paper by A. Nicolaidis and A. Taramopoulos (Il Nuovo Cimento, Vol. 
107B, N.11, pages 1261-1266, November 1992), the theoretical motion of a 
string with fixed ends under the influence of a plane monochromatic 
time-varying gravitational wave is discussed. According to this document, 
a string with fixed ends may be excited to resonance provided certain 
conditions, dependent on the length and orientation of the string and the 
wavelength of the gravitational wave, are met. It is suggested that 
Fourier analysis of the motion of the string could be used to extract the 
direction and energy of the incident wave. The document specifically 
avoids any discussion of the technical implementation of the theory, but 
it does suggest that strings a few meters or a few kilometers long should 
be used for the detection of cosmological radiation or gravitational 
radiation from a black hole or supernova, as the length of the string 
should be comparable to the wavelength of the gravitational waves. For the 
theoretical detector to work requires the gravitational field to oscillate 
in the form of a gravitational wave, which would not be the case for the 
gravitational fields associated with massive bodies such as the Earth. 
Superconducting gravity gradiometers are known (see U.S. Pat. No. 4,841,772 
and Australian patent application 48185/90) utilizing a pair or more of 
sufficiently separated superconducting accelerometers. Even after greatly 
reducing the intrinsic and environmental thermal noise factor, using 
stable persistent super-currents to balance the outputs of the 
accelerometers, and the most sensitive displacement sensors based on 
SQUIDs (Superconducting Quantum Interference Devices), they cannot measure 
the gravity gradient tensor components in their absolute units because 
they are incapable of fixing a position of the accelerometer's proof mass 
which is free of all forces. Therefore, only relative displacements of the 
proof masses can be measured. Rotating designs of such superconducting 
gravity gradiometers are not known. 
Patent Abstracts of Japan vol. 009 No. 117 (P-375) and JP-A-60 050476 
disclose a device for measuring the acceleration due to gravity, wherein a 
weight is suspended from a string. A current passing through the string 
causes the string to vibrate in the magnetic field of a permanent magnet. 
An amplified electrical signal corresponding to this vibration is fed back 
to the string and the string oscillates under self-excitation at a set 
frequency. The acceleration due to gravity is measured from this 
frequency. 
It is an object of the present invention to provide an apparatus for the 
measurement of gravitational fields with improved sensitivity, portability 
and noise immunity over the above known systems. 
It is a further object of the present invention to provide a novel 
apparatus for the absolute measurement of off-diagonal components of the 
gravity gradient tensor, in which the effect of rotation is replaced by 
parametric interaction between the sensitive element and active force 
feed-back connections, whereby enhanced sensitivity and vibrational noise 
immunity are attained. 
It is another object of the present invention to provide a simple 
technological realisation of the above apparatus utilising the advantages 
of the standard superconducting techniques which have shown an ability to 
reach a maximum sensitivity for mechanical displacement measurements and 
to keep intrinsic noise at a minimum level. 
To achieve these objects the present invention provides apparatus for the 
measurement of quasi-static gravitational fields, comprising: a string 
held under tension; and output means for producing an output which is a 
function of the gravitational field, characterised over the disclosure of 
Patent Abstracts of Japan vol. 009 No. 117 (P-375) and JP-A-60 050476 in 
that: the string is fixed at both ends; the apparatus comprises sensing 
means for detecting the transverse displacement of said string from an 
unperturbated position due to a gravitational field acting on said string; 
and the output means are responsive to the detected displacement to 
produce said output which is a function of the gravitational field. 
By "string" no particular limitation as to material or construction is 
intended. Any elongate tension element is included which is capable of 
being transversely deflected by a gravitational field and of providing a 
restoring force. 
An unperturbated stretched flexible string with fixed ends forms an 
absolute straight line in space going through the points where the ends of 
the string are fixed. This line can be identified as one of the axes of 
the local coordinate frame, say Z, and the other two axes, X and Y, are 
chosen to lie in the transverse (to the string) plane. Any string 
deflection from this line is caused by absolute values of the transverse 
components of the force per unit length which is applied to each unit 
element of the string. 
Viewed from another aspect the invention provides a method of measuring 
quasi-static gravitational fields, comprising: providing a string held 
under tension; producing an output, said output being a function of said 
gravitational field, characterised in that: the string has fixed ends; the 
method further comprises detecting the transverse displacement of said 
string from an unperturbated position due to a gravitational field acting 
on said string; and the output which is a function of the gravitational 
field is produced in response to the detected displacement. 
The string's deflection from its unperturbated position can be easily 
detected, by any suitable displacement sensing device. 
Preferably the string is formed of conductive, most preferably 
superconductive material. In this case, if an electric current flows 
through the string, a magnetic field distribution is produced in the 
transverse plane and along the string's direction. If the string is made 
of a superconducting material, a maximum current can be carried, and a 
consequent maximum sensitivity to the deflection can be reached. A d.c. or 
an a.c. current may be produced in the string by incorporating the string 
into a current-carrying circuit directly or by an inductive coupling with 
a pumping circuit(s), provided that the string forms part of a closed 
conducting or superconducting loop. An a.c. current may be induced in the 
string, for example by means of one or more, preferably longitudinally 
symmetrically positioned, coils, which may possibly be superconducting. 
The use of an a.c. current is advantageous in that it allows synchronous 
detection of the output signal. 
When the string carries a current, the transverse magnetic field around the 
string may interact with other conductors, or superconductors, by 
inductive coupling. The amplitude of the current induced in a conductor 
adjacent the string will be directly related to the distance of the string 
from that conductor. Thus, in a preferred embodiment of the invention one 
or more fixed pick-up coils are arranged along the length of the string to 
act as displacement sensing means, the current induced in each coil being 
directly related to the string's displacement from its unperturbated 
position. 
In a preferred embodiment of the invention the sensing means comprises at 
least two sensors, possibly pick-up coils, positioned symmetrically, in 
the longitudinal direction, with respect to the mid point of the string. 
In a particularly advantageous embodiment, displacement sensors, for 
example pick-up coils, are arranged adjacent the string in two 
non-parallel preferably orthogonal, planes, so as to be capable of 
measuring the string's displacement in two transverse directions 
simultaneously. 
It will be understood that the displacement of a string of length l from 
its unpertubated position, for example, in the y-direction of the above 
local coordinate frame as a function of the z-position of a unit element 
and time, y(z,t), can be described by the following differential equation 
##EQU1## 
with boundary conditions corresponding to the fixed ends of the string, 
i.e. y(0,t)=y(1,t)=0. In this equation .eta. denotes the string's mass per 
unit length, h is the friction coefficient per unit length, the parameters 
Y, A and .DELTA.l/1 are the string's Young modulus, the area of its cross 
section and the string's strain respectively. The quantities g.sub.y (0,t) 
and .GAMMA..sub.yz (0,t) are the absolute values of the y-component of the 
total acceleration and the corresponding gravity gradient tensor component 
along the string, both taken at the center of the local coordinate frame 
chosen. The function f.sub.L (z,t) represents the Langevin random force 
per unit length acting on the string due to its interaction with the 
thermostat having the absolute temperature T, with the following 
correlation function 
EQU f.sub.L (z,t)f.sub.L (z',t')=4k.sub.B Th.delta.(z-z').delta.(t-t')(2) 
where k.sub.B =1.4 10.sup.-23 JK.sup.-1 is the Boltzmann constant and 
.delta.(x-x') is the delta-function. 
In this description, the y-direction has been chosen as an arbitrary 
example to simplify the explanation of the invention. However, the 
foregoing and following analysis is equally applicable to any direction 
transverse to the string or any number of directions. 
Applying Fourier analysis to the complex shape of the string caused by its 
interaction with the gravitational field, the function y(z,t), can be 
described, in the range z=0 to z=1, by an infinite sum of sinusoidal 
functions of period 2l, with appropriate coefficients c.sub.y (n,t). Thus 
a solution of Eq.(1), which satisfies the boundary conditions shown above, 
can be represented by the following sum wherein each term in n corresponds 
to one of the string's natural vibrational modes 
##EQU2## 
By substituting Eq.(3) into Eq.(1) and by multiplying its left-hand and 
right-hand sides by sin(.pi.n'z/1), and then by integrating both sides 
over z from 0 to 1, one can obtain the differential equation for c.sub.y 
(n,t) 
##EQU3## 
where the quantities 
##EQU4## 
represent the string's natural frequencies; .tau. and .rho. are the 
relaxation time and the volume mass density of the string respectively. 
When n takes an even value, i.e. for those terms of the infinite sum in Eq. 
3 corresponding to vibrational modes of the string having a node at z=1/2 
(antisymmetric modes), the term involving g.sub.y (0, t) is equal to zero. 
Thus, for n even, c.sub.y is dependent only on .GAMMA..sub.yz (and thermal 
noise). 
In practice this means that the amplitude, c.sub.y, of the antisymmetric 
sinusoidal components of the deflection of the string in the y-direction, 
y(z,t), is dependent only on the magnitude of the gravity gradient tensor 
component .GAMMA..sub.yz. 
The mid point of the string, z=1/2, is the position of a node in all 
antisymmetric vibrational modes of the string. If sensors are positioned 
symmetrically in the longitudinal direction with respect to this point, it 
will be possible to identify displacements of the string corresponding to 
the string's natural antisymmetric vibrational modes while discounting 
displacements corresponding to symmetric modes, the magnitude of which is 
not only affected by the gravity gradient tensor component .GAMMA..sub.yz 
but also the absolute acceleration due to gravity in the y-direction, 
g.sub.y. 
It is particularly advantageous if displacement sensors are positioned at 
z=1/4 and z=31/4, positions corresponding to the antinodes of the first 
antisymmetric vibrational mode of the string, n=2. At these points the 
displacement of the string corresponding to the n=2 mode is at a maximum 
and thus the sensing signal will also be at a maximum, giving optimum 
sensitivity. 
According to a further development of the invention a conductor may be 
provided adjacent the conductive string. The conductor may carry a current 
directly related to the output of the sensing means, by the use of a 
positive feedback loop. The current may be activated continuously or 
periodically, for example in an "off-on" manner. In this case, a small 
deflection of the string due to a gravitational field will be amplified by 
the magnetic interaction of the string and conductor. In other words, the 
conductor will "push" (or "pull") the string into further deflection in 
direct response to a small deflection caused by a gravitational field 
acting on the string. This is clearly advantageous in that the 
displacement of the string is greater by virtue of the magnetic 
interaction with the conductor and is therefore more readily measurable, 
improving the sensitivity of the apparatus. 
In a particularly advantageous embodiment of this development, two or more 
conductors, possibly superconductors, are positioned longitudinally 
symmetrically about the mid-point of the string so that they amplify the 
antisymmetric modes of the string. 
In overview, a preferred embodiment of the invention provides a novel 
apparatus for measuring absolutely and simultaneously a pair of 
off-diagonal components of the gravity gradient tensor by means of a 
flexible superconducting current-carrying string with fixed ends, 
comprising active parametric force feed-back connections. The string is 
the coherent sensitive element whose symmetric natural transverse modes 
are caused by the total acceleration in the transverse plane, whilst the 
antisymmetric modes are caused only by absolute values of the gravity 
gradient components along the string's direction. 
In this embodiment the string forms a low-inductance part of a closed 
superconducting loop which is inductively coupled to a high-inductance 
driving solenoid(s) carrying an a.c. reference current from an external 
pumping source with some frequency .OMEGA.. The string is also inductively 
coupled to two superconducting magnetic flux transformers each comprising 
a signal coil and two pick-up coils wherein the pairs of pick-up coils lie 
in two perpendicular planes the cross-line of which coincides with the 
unperturbated string, thus forming two independent channels of 
measurements. The two arms of each superconducting flux transformer are 
balanced to convert only the string's antisymmetric natural modes into 
signal current in the signal coil to be measured with SQUID's 
(Superconducting Quantum Interference Devices) electronics. The output 
voltage of each channel, deeply modulated with the frequency .OMEGA., is 
then proportional to the amplitudes of the antisymmetric natural modes of 
the string. This voltage is passed through a differentiating and summing 
amplifier, and then used to load the feed-back circuit to produce an 
in-line feed-back current distribution proximate and parallel to the 
string. By adjusting the feed-back current, the effective relaxation time 
and the resonant frequency of the first antisymmetric natural mode of the 
string (whose amplitude depends only upon the gravity gradient along the 
string's direction) can be increased and decreased respectively, while the 
same parameters for the symmetric natural modes (whose amplitudes depend 
upon the total acceleration in the transverse-to-string plane) are not 
changed. In use, the feed-back circuit shifts the Brownian and vibrational 
noise level to far below the sensitivity required for industrial 
applications.

A single channel prototype of a device according to the invention (see FIG. 
2) has a flexible string 1. The string is preferably formed of a 
superconducting material such as Niobium (Nb). Niobium wire is the best 
choice, having optimum elastic properties, which have been proven to be 
usable at 4.2 K. The string forms the low-inductance part L.sub.o of a 
superconducting closed loop which is inductively coupled to 
high-inductance driving solenoid(s) L.sub.d carrying an a.c. reference 
current I.sub.d (t) from an external pumping source with some frequency 
.OMEGA.. The rest of the loop is provided by the casing of the device 
2,2',3,4,5. 
The string has, in this embodiment, a length l=24 cm, is 1 mm in diameter 
and is fixed at its ends by two Nb cups 2,2' of cylindrical shape each 
having a hole of 1 mm diameter at its centre. The cups 2,2' close tightly 
the ends of a Nb cylinder comprising three parts 3,4,5 connected together 
with two Nb cylindrical rings 6,6' carrying a fine thread. The parts 3 and 
5 also carry threads to engage other elements of the construction. The 
string's tension is produced by two Nb nuts 7,7' of 1 mm fine thread. 
The whole construction forms a closed superconducting cylindrical cavity 
with the string axially positioned. There are three spaces 10, 11, 12 
inside this volume electromagnetically insulated as much as possible from 
each other by Nb partitions 9. In two of them 10, 12, driving toroidal 
solenoids L.sub.d2 and L.sub.d2, wound with 0.01 mm Nb wire and connected 
in series, are placed, thus forming a large mutual inductance M.sub.d 
between L.sub.d =L.sub.d1 +L.sub.d2 and the inductance of the cylindrical 
cavity L.sub.o which is of the order 10.sup.-7 H for the sizes chosen. The 
ratio M.sub.d /L.sub.o is about 5.times.10.sup.2, so if the a.c. pumping 
current I.sub.d (t) in the driving solenoids has an amplitude of about 100 
mA then the induced a.c. supercurrent I.sub.o carried by the string is 
about 50 A peak to peak. In this case, the corresponding circular 
component of the magnetic induction B at the string's surface is nearly 
200 Gauss which is approximately four times smaller than the Niobium first 
critical field B.sub.c1. 
The two rectangular-type pick-up coils L.sub.p1 and L.sub.p2 of the 
superconducting flux transformer, and the active force feed-back circuit 
are mounted together on a Titanium tube 8 placed inside the central space 
of the construction shown in FIG. 2. Titanium is chosen because it has a 
thermal expansion coefficient matching that of Niobium. The active force 
feed-back circuit comprises two arms of 0.5 mm insulated copper wire 
stretched parallel to the string and carrying the feed-back current 
I.sub.y =I.sub.y1 +I.sub.y2. 
This design has particular advantages; for instance, the closed 
superconducting configuration gives optimum shielding against external 
varying electromagnetic fields. Also, the cylindrically symmetric 
configuration has a small radial size which, including all integral parts 
of the prototype, is no more than 3.8 cm diameter. Thus, it is possible to 
utilize a standard commercial 100 liter liquid helium vessel having an 
input opening of about 4 cm diameter to cool the construction down with a 
standard probe. Special helium cryostats, which have been used for known 
devices, exclude the possibility of removing the device from the 
cryostat's inner volume, for example if something goes wrong, to readjust 
it under field conditions. Removal of a device from a cryostat requires a 
long period, of for example several hours, to warm the cryostat contents 
to atmospheric temperature so that the contents do not explode under rapid 
thermal expansion. This is one of the major disadvantages of known 
constructions. However, the small input opening of the standard commercial 
100 liter liquid helium vessel prevents such an explosion occurring, which 
means that the apparatus according to the present invention can be removed 
from the vessel and adjusted under field conditions. 
The string is deflected by a non-uniform quasi-static gravitational field 
and interacts with a variable feed-back current distributed close to and 
substantially parallel to the string. The distribution is optimum when the 
feed-back current is injected at or taken from the point in the feed-back 
circuit opposite the mid-point of the string (see FIG. 1). In the local 
coordinate frame chosen this point is z=1/2. Another requirement for 
optimum operation is that the two arms of the feed-back circuit are 
substantially equal and grounded at their ends. In this case, there is no 
electromagnetic coupling between the feed-back current and the closed 
superconducting loop in which the string is incorporated. 
The current I.sub.o (t) flowing through the string and interacting with the 
feed-back current distribution I.sub.y (z,t) gives rise to the following 
transverse component of the force per-unit-length f.sub.y (z,t) acting on 
the string 
##EQU5## 
where .mu..sub.0 =4.pi. 10.sup.-7 Hm.sup.-1 is the magnetic vacuum 
permeability, d is the distance between the center of the unperturbated 
string and the center of the wire carrying the feed-back current and the 
phase of the pumping current source 15 is chosen to be zero. The sign + or 
- is determined by the output buffer of the differentiating and summing 
amplifier 16 shown in FIG. 1. The transverse motion of the unit element of 
the string in the OYZ plane is then described by the following 
differential equation 
##EQU6## 
which will be seen to correspond to Eq.(1) with the addition of the term 
of Eq.(6). Consequently, Eq.(7) also has solutions of the form of Eq.(3). 
Thus, following the same algebraic manipulation as for Eq.(1), a 
differential equation for c.sub.y (n,t) can be obtained for this 
embodiment. This is Eq.(8) which corresponds to Eq.(4) but with the 
addition of a feedback term. 
##EQU7## 
The quantity .epsilon..sub.n relates to the characteristics of the 
transducer system of the feedback loop; the longer the length of the arms 
of the feedback circuit, the larger the quantities .epsilon..sub.n are. If 
the arms of the feed-back circuit are absolutely identical then the 
quantities .epsilon..sub.n are equal to zero for all odd n=1,3,5 . . . . 
Their particular values for the sizes shown in FIG. 2 are determined by 
##EQU8## 
So, for the properly adjusted feed-back circuit, only the antisymmetric 
natural modes of the string interact with the feed-back current. However, 
only the antisymmetric natural modes of the string are sensitive to the 
absolute value of the gravity gradient tensor component to be measured, as 
is seen from Eq.(4) and (7). 
The superconducting pick-up coils L.sub.p1 and L.sub.p2 are placed near the 
string and cause two arms of the superconducting magnetic flux transformer 
to convert, if perfectly balanced, only the antisymmetric natural modes 
into the signal current I.sub.i to be detected with the SQUID's 
electronics 13 (see FIG. 1). One uses SQUID's (Superconducting Quantum 
Interference Devices) 13 as they are the most sensitive variable current 
and magnetic flux sensors currently available. In the prototype shown in 
FIG. 2, the pick-up coils are made in the form of two rectangular-type 
single loops of Nb wire placed symmetrically with respect to the midpoint 
of the string and connected in parallel with the signal coil L.sub.i. If 
the symmetry is perfect and the areas of the loops are absolutely 
identical, then the symmetric natural modes do not produce any signal 
current I.sub.i or feed-back current I.sub.y. The same effect can be 
achieved for slightly non-identical pick-up coils with the accuracy 
required, if one uses the additional inductance(s) L.sub.b connected in 
parallel and/or series with one or both of the pick-up coils. The 
inductance(s) L.sub.b can be tuned to balance the two arms of the 
superconducting flux transformer. The residual "zero-model" current in the 
signal coil L.sub.i corresponding to the unperturbated position of the 
string can be compensated directly inside the SQUID by an additional 
coupling (not shown) to the pumping current source. If the balancing 
conditions are satisfied, then the output voltage of the SQUID's 
electronics 13 is determined by 
##EQU9## 
where K is the total flux to voltage transfer function and L.sub.s is the 
SQUID's inductance. The quantities .beta..sub.n depend on the physical 
design and position of the pick-up coils and are equal to zero if n=1,3,5 
. . . . The function .PHI..sub.N (t) is the equivalent-to-noise random 
magnetic flux inside the SQUID loop, whose spectral density S.sub..PHI. 
(.OMEGA.) determines the intrinsic instrumental limit of the accuracy of 
measurements. The feed-back current I.sub.y (t) is formed from the output 
voltage V.sub.y (t) by passing it through a differentiating and summing 
amplifier 16 which is loaded by resistance R.sub.y. In this case, the 
feed-back current I.sub.y (t) can be represented by 
##EQU10## 
where p, q and .tau.* are constant parameters which depend upon the design 
of the differentiating and summing amplifier 16. 
It must be noted that a mismatch between the two arms of the feed-back 
circuit always exists. The design shown in FIG. 2 uses two identical 
feed-back resistances, R.sub.y1 and R.sub.y2, one for each arm. In this 
case the mismatching can be easily compensated by tuning one of the 
resistances, say R.sub.y2, to obtain the optimum case. 
Equations (7) and (11) represent a closed infinite set ii of differential 
parametric-type equations. Careful analysis has shown that we can ignore 
the terms involving the quantities c.sub.y (n,t) with n&gt;2 in the 
right-hand side of Eq.(11). The reason is that just one mode can be made 
"soft" i.e. the most sensitive to the gravity gradient, namely c.sub.y 
(2,t). If the string's natural frequencies are high enough and separated 
by single octave gaps, only second-order corrections are required which 
can be easily taken into account along with analysis of other instrumental 
errors. Then, as follows from Eq. (7), the self-consistent equation for 
the gravity gradient sensitive mode, n=2, including unavoidable 
fundamental noise sources can in turn be written in the form 
##EQU11## 
and it is assumed that the true sign of the feed-back current has been 
chosen. 
If some easily carried out conditions are satisfied, which are 
##EQU12## 
then one can show that the self-consistent output voltage is 
##EQU13## 
where under "brownian noise" the combination of thermal and back-action 
noises is implied. 
It is of interest to estimate the limiting accuracy of measurements of this 
embodiment of the invention, which can be represented by the value of a 
minimum detectable gravity gradient 
##EQU14## 
where .tau..sub.eff =.tau./(1-.alpha..tau./4) is the effective relaxation 
time, m is the total mass of the string, and E.sub..PHI. (.OMEGA.) is the 
energy resolution of the SQUID. Using the following practical parameters: 
l=0.24 m, m.congruent.1.6 10.sup.-3 kg, .tau..sub.eff .congruent.10.sup.4 
s, .beta..sub.2 .congruent.4.times.10.sup.3 m.sup.-1, L.sub.s 
.congruent.5.times.10.sup.-11 H, I.sub.o .congruent.50 A, 
(.omega..sub.2.sup.2 -.omega..sup.2 /2).sup.1/2/ 2.pi..congruent.2 Hz, 
.omega..sub.2 /2.pi..congruent.40 Hz, 
.OMEGA./2.pi..gtoreq.2.times.10.sup.2 Hz, E.sub..PHI. 
(.OMEGA.).congruent.2.times.10.sup.-31 J/Hz (d.c. biased SQUID), one can 
obtain from Eq.(16) 
##EQU15## 
It can be shown, that a range of the parameters .tau., .omega..sub.2, 
.omega., .alpha. and .OMEGA. exists where the string's response described 
by Eq. 12 is stable. For example, for quasistatic gravity gradients and 
sufficiently high pumping frequency .OMEGA. one can ignore the oscillating 
terms containing Cos(2.OMEGA.t) and Sin(2.OMEGA.t) in the right side of 
Eq. 12. 
There are a number of detecting strategies which can be employed by the 
present invention at this stage, which are dependent on the initial 
mechanical parameters of the string and the application for which the 
apparatus is intended. It is preferable to use a string with a high 
mechanical stiffness and a short relaxation time in order to increase 
immunity to vibrational noise, which is the main noise source in 
industrial applications, particularly in mobile gravity gradiometry. On 
the other hand, the stiffer the string, the stronger the feedback force 
that has to be applied to the string to soften the signal mode, and the 
larger the back-action noise associated with the feedback current. 
Additionally, the shorter the string's relaxation time, the stronger the 
influence of thermal fluctuations of the string on the measuring accuracy 
since the mass per unit length of the string will normally be quite small. 
To overcome both of these problems a best mode of carrying out gravity 
gradient measurements according to another embodiment of the present 
invention uses variable feedbacks in an "off-on" manner. In this case, the 
feedback force is initially not applied to the string for an `off-period` 
during which the string reaches thermodynamic equilibrium. The feedback 
force is then quickly activated for an `on-period` during which the 
effective natural frequency 
##EQU16## 
and the effective relaxation time 
##EQU17## 
become substantially smaller and longer respectively compared to the 
corresponding initial parameters of the string. The feedback is adjusted 
in such a way that the effective relaxation time becomes much longer than 
the on-period. Measurements are carried out during the on-period only, in 
which the string never reaches thermodynamic equilibrium. For example, the 
fluctuation dissipation theorem is no longer applicable to the string 
during the period of measurements and its response to all external noise 
sources is changed (see V. B. Braginsky and A. B. Manukin, Measurement of 
Weak Forces in Physics Experiments, Ed. by D. H. Douglass, University 
Press of Chicago, 1977). 
One can show that in this case the least gravity gradient detectable by 
this embodiment of the invention can be represented by 
##EQU18## 
.tau..sub.m is the measurement time (on-period), m is the total mass of 
the string, E.sub..PHI. (.OMEGA.) is the energy resolution of the SQUID at 
the frequency .OMEGA. and .delta. is a statistical error of the first 
kind. The value of .delta. is the likelihood that the equivalent gravity 
gradient noise will exceed the value represented by the left side of Eq. 
20 for the period of measurement. 
Using the following practical parameters: l=0.24 m, M=1.6.times.10.sup.-3 
kg, .tau.=0.5 s, .tau..sub.m =1 s, .tau..sub.eff .congruent.10.sup.4 s, 
.beta..sub.2 .congruent.4.times.10.sup.3 m.sup.-1, L.sub.S 
.congruent.5.times.10.sup.-11 H, I.sub.o .congruent.50 A, .omega..sub.eff 
/2.pi..congruent.3 Hz, .omega..sub.2 /2.pi..congruent.80 Hz, 
.OMEGA./2.pi..gtoreq.10.sup.4 Hz, E.sub..PHI. 
(.OMEGA.).congruent.5.times.10.sup.-32 J/Hz (500 d.c. biased SQUID), one 
can obtain from Eq.(20) 
EQU .GAMMA..sub.min =0.02 Eotvos 
In both the above embodiments, the desired signal is obtained from the 
output voltage by synchronous detection with a reference signal taken from 
the pumping source 15, and the invention allows calibration of the desired 
signal in gravity gradient absolute units without rotation as has been 
proposed for known rotating gravity gradiometers. As for rotating designs, 
the invention allows the movement of the noise spectrum to a frequency 
range at which 1/f contribution is sufficiently small. Natural vibrations 
of the string, which occur during the time of measurement (on-period), do 
not cause a problem since they can be filtered out from the desired signal 
provided that the on-period is chosen to be much longer than the period 
(2.pi./.omega..sub.eff) of such vibrations. 
Vibrational noise immunity is improved by the factor (.omega..sub.eff 
/.omega..sub.1).sup.2 which can be made as small as 10.sup.-2. 
One must consider inductive cross-coupling between the feedback currents 
and each pair of the pick-up coils and cross-coupling between the pick-up 
coils themselves, both of which act like negative feedback loops. On the 
one hand this leads to unnecessary renormalisation of the amplitudes of 
the output signals until the gain of the SQUID's electronics exceeds some 
critical value. On the other hand in the case of double-channel 
measurements, the output signal of each channel contains a linear 
combination of each gravity gradient component to be measured. It can be 
shown that each of such components can, nevertheless, be measured 
separately and simultaneously, if a proper data acquisition system is 
used. The effect can be easily eliminated by organising additional 
positive feedback to counteract this negative feedback, for example by 
connecting, via a weak inductive coupling, each feedback current with each 
SQUID. 
In practice, the apparatus according to the invention can be used to 
determine in absolute units the off-diagonal components of the gravity 
gradient. By conducting a gravity survey over an area, small differences 
in absolute gravity gradient can be detected. Such small changes may 
indicate variations in local geological features, for example, the 
presence of minerals, gas or oil. 
Repeated readings over time at a single locality could indicate changing 
geological status of an area, such as rising magma. Clearly the invention 
enhances prospecting and other data gathering pursuits where accurate 
gravitational field measurement is required. Use of absolute values 
enhances the information that can be determined from the data measured. A 
gradiometer according to the invention can be used while moving, which 
allows the gradiometer to be used on vehicles whether land, sea or air 
vehicles. For example, the device can be suspended from a helicopter and 
used while the helicopter traverses a selected area.