WO2011001154A2

WO2011001154A2 - Gravitational gradiometer

Info

Publication number: WO2011001154A2
Application number: PCT/GB2010/001282
Authority: WO
Inventors: Alexey Veryaskin
Original assignee: Gravitec Instruments Limited; Tomlinson, Kerry, John
Priority date: 2009-07-03
Filing date: 2010-07-02
Publication date: 2011-01-06
Also published as: GB0911650D0; GB2471662B; GB2471662A; WO2011001154A3

Abstract

Apparatus for the measurement of quasi-static gravity gradients comprising: a flexible ribbon (3) held under tension at both longitudinal ends (5,7); sensing means (20) arranged to detect the transverse displacement of the ribbon from an undisturbed position due to the gravitational field acting on said ribbon and to generate a signal representing the displacement; and output means coupled to said sensing means and responsive to said displacement signal to generate an output signal which is a function of the gravitational gradient tensor of the gravitational field. To modulate the response of the apparatus to gravitational gradients, the apparatus further comprises ribbon stiffening means (10) operable to change the ribbon from a normal state to a stiffened state in which the displacement of the ribbon due to the gravitational field is reduced; and control means (13) to periodically operate the stiffening means. In use, the control means periodically operates the stiffening means at regular intervals so as to modulate the stiffness of the ribbon and the displacement signal, and the output means demodulates the displacement signal by synchronous detection using a signal representing the modulation of the ribbon stiffness.

Description

Gravitational Gradiometer

The invention relates to an apparatus for directly measuring components of the gravitational gradient tensor, particularly the off-diagonal components of the tensor, and to a method of measuring said tensor components.

Gravitational gradiometry is the measurement of the gravitational gradient field of differential accelerations between two infinitesimally close spatial points. The gravitational gradient field is described by a second rank tensor, T_tJ >:

wherein i, j = (x, y, z) and the scalar V is the gravitational potential of a local reference frame of orthogonal Cartesian coordinates (x, y, z). Taking the z axis as pointing vertically into the ground, the components of the tensor at some point in the local reference frame (x, y, z), calculated by determining the spatial rate of change along directions x, y and z of the spatial rate of change of the gravitational potential in directions x,y and z, represent the rate of change of acceleration due to gravity along that direction. For example, the component T_y2 represents the rate of change along the y direction of acceleration due to gravity along the direction z towards the ground, and is typically measured in units of Eδtvόs Units (1 Eotvδs = 1 EU =10^'9 s^~2). The tensor consists of nine components, only five of which are totally independent due to their geometrical symmetry (i.e. Ty - 7},, where i≠j) and due to the validity of the Laplace equation (i.e. T_xx + T_yy + T_2Z = 0) for gravitational potential fields outside of the extent of gravitational field sources.

Providing apparatus that enables accurate and absolute measurements of the various components of the gravity gradient tensor T_tJ is very important in the fields of oil, gas and mining of various other natural resources. Gravitational gradiometry particularly enables the mapping of variations in the density of subsurface rocks and deposits to assist in the targeting of prospecting, and in increasing the effectiveness of drilling for oil and gas and mining. Gravitational gradiometry finds further application in defence and space industries for navigation and reconnaissance (e.g. void detection), geological prospecting, sub-sea/underwater navigation and exploration, terrestrial and marine archaeology, medicine and space exploration (for example, obtaining density maps of asteroids and other solar system orbital bodies).

For many gravity gradiometry applications, it is the T₂₂ component (i.e. the second order derivative of gravitational potential in the vertical direction) that many gradiometers aim to measure, whether by direct measurement, or by measuring at least some of the other tensor components and recalculating T₂₂ from their dependent relationship, or both. However, in their paper 'On the combined gravity gradient modeling for applied geophysics', Journal of Geophysics and Engineering, 2008, VoI 5, pp 348-356, Veryaskin and McRae show that by measuring and using the two off- diagonal gravity gradient tensor components T_x2 and T_y2, it is possible to obtain more information about anomalous subsurface density contrasts than by measuring and using the vertical gravity gradient component T₂₂. To retrieve this subsurface density information, a gradiometer arrangement is required that is capable of simultaneously producing real-time data sets of direct measurements of both the T_∞ and Ty₂ tensor components.

A method of absolute measurement of gravity gradient tensor components was invented first by Baron Lorand von Eδtvδs 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 EU 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 five independent measurements of an angular deflection each with a different azimuth angle.

Practical devices, which have been built in accordance with this basic method of Eotvδs, are large in size, bulky and have low environmental noise immunity, thus requiring specially prepared conditions for measurements. This excludes any possibility of using them on a moving carrier or for many practical applications where there are weight or space constraints, such as in the confined environment of a borehole, and in airborne drones, space launcher payloads, satellites, and

extraterrestrial rovers.

Another method for absolute measurement of gravity gradient tensor components which enhances the above method was invented by Forward in the 1960s (see US patents 3,722, 284 (Forward et al) and 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 Ω 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 2Ω = oo, where ωo 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 T_n, - T_xx and T_x^ by separating - A - the quadrature components of the response using synchronous detection with a reference signal of frequency 2Ω.

The same principles can be directly used, as proposed by Metzger (see US patent 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 Eδtvδs 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 show a measurement accuracy which is much lower than the limiting theoretical estimates.

In WO-A-96/10759 a method and apparatus for the measurement of two off- diagonal components of the gravity gradient tensor is described. According to this document, the transverse deflection of a stationary flexible string with fixed ends in its second fundamental mode of oscillation (the 'S' mode) is coupled to an off diagonal gravity gradient, whilst its deflection in its first fundamental mode of oscillation (the 'C mode) is coupled to an effective (i.e averaged with a weight function along the string's length) transverse gravitational acceleration. In other words, a string with fixed ends is bent into its 'S' mode by a gravity gradient only, provided that it does not experience any angular movements. Therefore, by measuring absolutely the mechanical displacement of such a string which corresponds to the ¹S' mode it is possible to measure absolutely an off-diagonal component (i.e. T_x2 or T_y2, for a string aligned along the z axis) of the gravity gradient tensor. While this document teaches the use of a one-dimensional 'string', any generic element having a width and depth much smaller than its length, for example, a flat ribbon, is suitable.

In this design for a gradiometer having a current-carrying string, or ribbon, of length / aligned along the z axis and having a uniform mass distribution per unit length along its extent, the displacement, y{z,i), of the string from its undisturbed position (i.e. the straight line joining its fixed points at both ends), for example in the y-direction of the local coordinate frame as a function of the z-position of a unit element, and of time, t, can be described by the following force balancing equation for a vibrating string. (N.B. A similar equation and following analysis is applicable to the orthogonal direction transverse to the string and to any number of other directions). η

+ I(t)B_x(0,t)- WK(0,t)z

+ thermal noise (3)

The components on the right hand side of the equation represent the forces acting on the string (including gravitational and magnetic forces) in the y direction, and the components on the left hand side of the equation represent the restoring string forces in the y direction.

The equation has the boundary conditions corresponding to the fixed ends of the string, i.e. y(0,ή =y(l,t) = 0. In this equation η denotes the string's mass per unit length, h is the friction coefficient per unit length, the parameters Y, A and AUl are the string's Young modulus, the area of its cross section and the string's strain respectively. The quantity g_y(0,t) is the absolute value of the y-component of the gravitational acceleration and 7^(0,t) the corresponding gravity gradient tensor component along the string, both taken at the centre of the local coordinate frame chosen (i.e. z = 0). The quantity I(t) is the current flowing through the string. It is well known that a conductor carrying a current I(t) in a non-uniform magnetic vector field of flux density B(x, y, z) is subject to force F =1 (t) {n x B(x, y, z)}, where n is the unit vector in the direction of current flow, in this case the z direction. The quantities B_x(O, t) and B_x2(O, t) therefore represent the absolute values of the x- component of the magnetic field and the corresponding magnetic gradient tensor component along the string, respectively, both taken at the centre of the local coordinate frame chosen.

Since the string is subject to Brownian fluctuations, the corresponding thermal noise driving source is included on the right side of equation (3).

Of the gravitational force components of the equation (3), - ηg_y(O,t) represents the force in the y direction on the unit element of the string due to the acceleration due to gravity, and - ηT_yz(§,t)z represents the force in the y direction on the unit element of the string due to the change along the z direction in the

acceleration due to gravity.

Applying Fourier analysis to the complex shape of the string caused by its interaction with the gravitational and magnetic field, the function y(z,t), can be described, in the range z = 0 to z = /, by an infinite sum of sinusoidal functions of period 21, with appropriate coefficients c_y(n,t). Thus a solution of force balance vibration equation (3), which satisfies the boundary conditions shown above, can be represented by the following sum (4) wherein each term in n corresponds to one of the string's natural vibrational modes.

By substituting equation (4) into equation (3) and by multiplying its left-hand and right-hand sides by sin(π«'z//), and then by integrating both sides over z from 0 to /, one can obtain the following differential equation (4) for c_y(n,t).

+ -

η WB _β(0,θl J

+ thermal noise

(4) where the quantities

represent the string's natural frequencies; and τ and p are the relaxation time and the volume mass density of the string respectively.

When n takes an even value (i.e. for those terms c_y(n,t) of the infinite sum in equation (3) corresponding to anti-symmetric vibrational modes of the string having a node at z=//2, the midpoint of the string), the force component of equation (4) involving g_y(0,f) and B_x(O, t) is equal to zero and the force component being a function of the gravitational gradient tensor component Ty₂ and magnetic field gradient tensor component B₁₂(O, t) remains. Thus, for anti-symmetric modes of the string (i.e. n = even), c_y is dependent only on T_y2 and B_x2(O, t) (and thermal noise).

In practice this means that the amplitude, c_y, of the anti-symmetric 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 T_y and the magnetic field gradient tensor component Bx₂(O, t).

The string has an effective mechanical bandwidth of oscillation limiting its displacement response to oscillations below a few kHz (even for extremely stiff strings). The force on the string due to the magnetic field gradient is dependent on the current carried in the string. Therefore, by not pumping the string with any current at all or by pumping the string with an alternating current well outside the mechanical bandwidth of the string, the string will effectively not be sensitive to magnetic field gradients because oscillations at such frequencies are damped. In this way a string sensitive only to the gravity gradient tensor component T_y is provided.

The mid point of the string, z=//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 anti-symmetric vibrational modes while discounting displacements corresponding to the string's symmetric vibrational modes.

It is particularly advantageous if displacement sensors are positioned at z=l/4 and z=3//4, positions corresponding to the antinodes of the first anti-symmetric vibrational mode of the string, n=2 (the 'S' mode). At these points the displacement of the string corresponding to the 'S' mode is at a maximum and thus the gradiometer sensing signal will also be at a maximum, giving optimum sensitivity.

In WO 96/10759, two rectangular type pick-up coils in the form of a

Superconducting Quantum Interference Device (SQUID) are arranged to detect the transverse displacement in a superconducting Niobium string held under tension at its ends inside a superconducting casing, the whole apparatus being cooled to 4.2K or less in a cryogenic liquid helium vessel. Solenoids arranged symmetrically at either end of the string are driven by an alternating signal having frequency Ω to induce an AC super-current in the string also having frequency Ω. The superconducting casing excludes the external magnetic field from the casing such that no magnetic field forces act on the string and the displacement of the string from its straight line configuration is in response to the gravitational field only. The two coils of the SQUID device are positioned proximate to the string and are located at symmetrical longitudinal positions one on either side of the mid-point of the string and are arranged in a circuit as two arms of a superconducting magnetic flux transformer. The AC super-current carried by the string induces a current in each coil of the SQUID device proportional to the displacement of the string at that point from its undisturbed position. If the positions and responses of the two coils are arranged such that the two arms of the magnetic flux transformer are perfectly balanced either side of the mid-point of the string, the response is in 'anti-phase' such that the symmetrical modes of the string (i.e. n = odd, including the dominant ¹C mode) do not produce any signal current in the flux transformer. For the anti-symmetric modes, the displacement response of the string is dominated by the n = 2 'S' mode and all higher modes can be ignored (or factored in to error sources); then it follows that the output voltage of the SQUID is an AC signal having frequency Ω and an amplitude that is proportional to the displacement of the string in the first anti-symmetric 'S' mode only, and hence, to the off-diagonal gravitational gradient component (in the example given above, T_yz(^Q,ή). The amplitude of this SQUID output signal is obtained by synchronous detection of the signal using the alternating signal driving the solenoids as a reference. A force feedback circuit is also provided which takes as an input the voltage output of the SQUID and induces in the string a feedback current formed from this voltage output to increase the sensitivity of the device to the gravitational gradient component. For a gradiometer of this design having typical practical parameters, the - theoretical minimum gravity gradient detectable is calculated as being 0.02 EU. The string-based gravitational gradiometer device is less sensitive to vibrational noise than the earlier rotating gradiometer designs and lends itself to deployment on a mobile platform where measurements can be taken to retrieve high resolution data of local differences in gravity gradient. However, deployment is problematic in that the linear and angular accelerations of the mobile platform affect the deformation of the string and the output of the device.

In WO 03/27715 the string based gradiometer design is developed further by providing a gravity gradiometer in which the string is in the form of a uniform metal strip or ribbon and is constrained to its rest position at its mid-point, with, for example, a rigid knife-edge mounted to the casing and touching the ribbon but not exerting any force thereon. This knife-edge restricts any movement of the ribbon at that point and adds another boundary condition with the effect that deformation of the string into all symmetric modes (i.e. n = odd) is limited while deformation into all anti-symmetric modes (i.e. n = even), including the dominant 'S' mode, is permitted. Notably, deformation of the ribbon into the otherwise dominant first order symmetric 1C mode is significantly limited. This use of a ribbon arrangement in place of a string is such that the ribbon is more constrained in its movement making the output of the device less dependent on linear accelerations exerted on the device and more manageable. This makes the device more suitable for operation on mobile platforms. The device operates in a liquid nitrogen cryogenic bath at 77K which reduces the effects of thermal noise and increases mechanical stability. In place of a SQUID device, two pick-up coils are provided positioned symmetrically about a mid-point of - l i the ribbon and arranged as two arms of a resonant bridge circuit tuned to the

frequency of an alternating carrier signal supplied to the ribbon as an alternating current. The frequency of the AC carrier current pumped to the ribbon is above the mechanical bandwidth of the tensioned ribbon such that the ribbon's displacement response due to interaction forces with the ambient magnetic field is damped and the detected signal is dependent on the gravitational field only. The two coils are located at positions directly adjacent the antinodes of the first anti-symmetric mode of the ribbon (i.e. at 2= L/4 and z= 3L/4) which correspond to the maximum displacement and increases the sensitivity of the response. A voltage signal is induced in the bridge circuit having the same frequency as the carrier signal, and having an amplitude that is a measure of the average deflection of the ribbon over a region situated around the L/4 and 3 L/4 positions. By synchronously detecting the voltage amplitude of the induced signal with reference to the carrier signal, the amplitude of the local off diagonal gravity gradient component can be retrieved. The response of the ribbon is modulated with a square wave by indirectly changing its ability to deform away from its rest position due to gravity gradients between a sensitive state and an insensitive state low value. This is achieved by using a square wave signal to switch a negative feedback circuit arranged to periodically produce in the ribbon a current signal proportional to the output of the bridge circuit but in anti-phase or quadrature therewith such that the ribbon is forced to its rest position! In this negative feedback or 'insensitive' state, the response of the detector to the gravity gradients is low, and in its relaxed or 'sensitive' state the response of the detector to the gravity gradients is high. This modulated output is retrieved using a lock-in amplifier. Three sets of four single axis gradiometer modules are provided in an 'umbrella' arrangement to remove . the effect of angular accelerations on the output of the combined device, which is capable of providing absolute and direct measurement of all gravitational gradient tensor components.

In the ^"conventional, non-string-based, gravitational gradiometers known in the art, the signal-to-noise ratio of the gradiometers response is typically very low making them impractical for quick and useful gravity gradient measurements of a sensitivity required for the desired practical applications. To improve the signal-to-noise ratio, it is known to modulate the response of the gradiometer by rotating the sensing element and demodulating the signal output from the gradiometer by synchronous detection of the signal output together with the modulation signal. However, the rotation of these devices creates many engineering problems that makes these devices impractical and can introduce further noise sources. Further, the increased signal-to-noise ratio of these devices still does not provide a sensitivity required for the desired practical applications.

A similar modulation technique is known in the art to be applied in string- based gravitational gradiometers by periodically applying a negative feedback force to the string equal to displacement signal by means of a feedback circuit and a conductor provided adjacent the string (see WO-A-96/10759). A feedback signal is applied to the conductor such that an electromagnetic force is induced in the current-carrying string that forces the string to return to its undisturbed or 'rest' position. This electronic modulation of the string's response between a sensitive state and an insensitive state and subsequent synchronous detection and demodulation of the output signal has the effect of increasing the signal-to-noise ratio.

This modulation method achieves an increase in signal-to-noise ratio by moving the output signal away from DC which, by synchronous detection, reduces the effect of frequency dependent thermal noise (commonly referred to as 1/f noise due to the spectrum of the noise), an inherent noise source in electronic and mechanical systems, on the output signal.

However, the electronic modulation method has been found by the inventors to potentially produce electromagnetic interference in the system, which can create noise in the pick-up coils and disrupt the detected gravity gradient signal.

Specifically, the limitations for using the electronic feedback to modulate the sensitivity of the sensor arise mainly from the high voltage required to apply the necessary force to the ribbon. In order to overcome the force of gravity an

appreciable feedback force is needed to act on the ribbon, and this requires a high feedback voltage at a reasonably high frequency (of the order of 10 kHz). This high voltage signal interferes with the local electronics in the sensor. The high voltage also needs to be very stable and controlled, otherwise the noise on the control voltage would affect the ribbon and generate noise difficult to distinguish from gravity gradient signals. Such a controlled and stable high frequency and high voltage signal is very difficult to produce. High voltage electronics also dissipate more heat, making cooling in the sensor more problematic.

Further, in cases where the string does not carry a current (for example, where a non-electromagnetic string displacement detection system is used and no string current is therefore needed), the negative feedback circuit would not be effective as no electromagnetic force can then be exerted on the string to push it back into its rest position.

It is an object of the present invention to provide a string-based apparatus for the measurement of gravitational gradiometers in having a high signal-to-noise ratio.

Viewed from one aspect, the present invention provides apparatus for the measurement of quasi-static gravity gradients comprising: a flexible ribbon held under tension at both longitudinal ends; sensing means arranged to detect the transverse displacement of the ribbon from an undisturbed position due to the gravitational field acting on said ribbon and to generate a signal representing the displacement; output means coupled to said sensing means and responsive to said displacement signal to generate an output signal which is a function of the gravitational gradient tensor of the gravitational field; ribbon stiffening means operable to change the ribbon from a normal state to a stiffened state in which the displacement of the ribbon due to the gravitational field is reduced; and control means to periodically operate the stiffening means; wherein, in use, the control means periodically operates the stiffening means at regular intervals so as to modulate the stiffness of the ribbon and the displacement signal; and the output means demodulates the displacement signal by synchronous detection using a signal representing the modulation of the ribbon stiffness.

In accordance with this aspect of the invention the sensitivity of the ribbon to gravitational gradient forces is modulated from a high value (in the normal state) to a low value (in the stiffened state) as the force required to displace the ribbon from its undisturbed position is varied. This causes the signal output from the sensing means to be modulated at the stiffness modulation frequency, which can then be

demodulated by synchronous detection. By modulating the sensitivity of the gradiometer in this way moves the signal away from the DC such that the amplitude of the thermal or 1/f noise from the mechanical and electrical elements of the gradiometer in the detected signal is significantly reduced and the signal-to-noise ratio is increased, resulting in a practical gravitational gradiometer having a high sensitivity. Further, by modulating the response of the ribbon by modulating its stiffness, the effect of thermal noise can be reduced without needing complicated negative feedback loop electronics to force the sensing element back towards its undisturbed position without changing its stiffness, as in the prior art.

The stiffening means may be arranged to operate to stiffen the ribbon by increasing the tension holding the ribbon. The stiffening means may comprise:

floating spring flexures arranged to hold both longitudinal ends of the ribbon under tension and to be operable to move between a normal configuration, in which the element is held under normal tension, and a stiffened configuration, in which the element is held under increased tension; and actuators arranged to operate the floating spring flexures. The actuators may be piezoelectric actuators or magnetostriction devices. The actuators and floating spring flexures may combine to, in use, apply a tensioning force to the ribbon that is aligned with the plane of the ribbon. The apparatus may further comprise spring mounted rollers arranged to engage the ribbon at its boundary condition ribbon-end locations and retain the ribbon at its undisturbed position there.

In accordance with this preferred arrangement, the means for stiffening the device is provided by mechanical actuation to periodically apply tension through the length of the string. This arrangement is particularly suitable for implementation in the ribbon-based gradiometer of the invention and provides effective modulation of the ribbon's displacement response without otherwise negatively affecting the operation of the device.

The ribbon may be arranged to be stiffenable in response to an electric or magnetic or electromagnetic stimulus; and the stiffening means may be operable to supply said stimulus to the ribbon. The ribbon may comprise a material that increases or decreases in stiffness in response to the stimulus. In accordance with this alternative preferred arrangement, the stiffness of the ribbon is modulated without requiring any moving parts or actuators.

The control means may be configured to periodically stiffen the ribbon at a frequency equal to a mechanical resonant frequency of the ribbon so as to increase the deflection of the ribbon in its normal (non-stiffened) state. The control means may be configured to periodically stiffen the ribbon at a frequency equal to twice that of the mechanical resonant frequency of the ribbon's ¹S' mode of oscillation. In this preferred arrangement the gradiometer response experiences parametric resonance.

The apparatus may further comprise rotating means arranged to, in use, rotate the ribbon about its length axis. The rotating means may be configured to rotate the ribbon at a rotational frequency equal to the mechanical resonant frequency of the ribbon's 'S' mode of oscillation.

In this preferred arrangement the gradiometer response experiences

(degenerated) parametric amplification and the signal-to-noise ratio is improved.

The output means may be arranged to generate the output signal as being representative of an absolute measure of the gravity gradient tensor. To achieve this, the output means may be arranged to base the output signal on the difference between the displacement signal generated during the stiffened state of the ribbon and the displacement signal generated during the normal state of the ribbon.

In accordance with this preferred arrangement, the calibration of the displacement signal in the normal state of the ribbon against the displacement signal in the stiffened state provides an output signal that can be taken as an absolute measure of the gravitational gradient detected by the gradiometer. This absolute output is practically useful in the intended applications of the gradiometer device. Viewed from another aspect, the present invention provides a method of measuring quasi-static gravity gradients comprising: holding a flexible ribbon under tension at both ends; operating a ribbon stiffening means arranged to change, periodically, the ribbon from a normal state to a stiffened state in which the displacement of the ribbon due to the gravitational field is reduced; arranging sensing means to detect the transverse displacement of the ribbon from an undisturbed position due to the gravitational field acting on said ribbon and to generate a signal representing the displacement; and generating, by demodulating the displacement signal by synchronous detection using a signal representing the modulation of the ribbon stiffness, an output signal which is a function of the gravitational gradient tensor of the gravitational field.

According to the method and apparatus of the invention, the signal-to-noise ratio of the gradiometer device may be significantly greater compared to a

gradiometer in which the response of the sensitive element is not modulated. The mechanical stiffness modulation of the ribbon's response provides a useful and in many cases advantageous alternative modulation mechanism to the electromagnetic negative force-feedback response modulation mechanism known in the string-based gradiometers of the prior art. The increased signal-to-noise ratio of the gradiometer of the invention makes it suitable for practical deployment and more useful in the abovementioned practical applications.

Further, the tensioning modulation scheme according to the invention uses high voltage but low current at an extremely low frequency which makes the modulation control low power and thus less likely to interfere with the local electronics. Further advantages of the modulated tensioned ribbon arrangement of the invention are that monitoring and control of the ribbon sensing element is provided. In the invention, the ribbon is 'plucked' by the mechanical modulation and the oscillation of the plucked ribbon gives information about the ribbon S and W mode frequencies. This allows the ribbon tensions to be controlled by monitoring the ribbon frequencies. In a changing local environment (especially considering local temperature changes), monitoring and control of the ribbon sensing element is a distinct advantage.

Certain preferred embodiments of the invention will now be described by way of example only, and with reference to the accompanying drawings, in which:

Figure 1 is a schematic of a gravitational gradiometer according to first embodiment of the present invention; and

Figure 2 is a schematic of a gravitational gradiometer according to a second embodiment of the present invention.

The sensing element of the first embodiment of a gravitational gradiometer device 1 according to the invention shown in Figure 1 is provided by a long ribbon 3 having a width and a depth much smaller than its length. The ribbon 3 has a length, L, of the order of tens of centimetres and has a width, W, that is greater than its depth, D, such that the ribbon 3 resembles a length of tape (i.e. L » W > D). In the embodiment shown in Figure 1, the length, L, of the ribbon 3, between its two fixed endpoint boundary conditions 5, 7, is 300mm, the width, W, of the ribbon is 5.0mm, and the depth, D, of the ribbon is 0.3mm.

This ribbon shape means that the displacement of the ribbon 3 is constrained to the direction orthogonal to the plane of the ribbon's major extent (in its length and width directions) and the gradiometer is thus only sensitive to forces causing the ribbon 3 to be displaced in this direction (the direction being the depth direction of the ribbon).

The ribbon 3 is held under tension between two fixed points ^"5, 7 at its longitudinal ends. The two fixed points 5, 7, towards the ends of the ribbon are held stationary in the depth direction of the ribbon at their undisturbed or 'rest' positions by spring mounted rollers 9. This provides the boundary conditions at the end-points of the ribbon. The ribbon 3 is, however, permitted to roll past the rollers 9 in the length direction due to variations in applied tension.

Between these two fixed points 5, 7, movement preventing means 4 is provided as a 'knife-edge' device mounted at the mid-point M of the ribbon 3 at its rest position to touch the ribbon 3 but not exert any force thereon. The movement preventing means 4 provides an additional boundary condition at the mid-point of the ribbon 3.

The ribbon 3 is otherwise free to move such that it can be displaced away from the straight line joining the two points under the influence of any external force acting on the ribbon 3, such as a gravitational force acting on the ribbon 3 (causing it to deform into its ¹W mode of oscillation, which is the linear sum of all remaining symmetric mode deflections) and a differential gravitational gradient across the ribbon 3 (causing it to deform into its 'S' mode, as shown by the dashed line in Figure 1 , and its higher anti-symmetric modes).

At the ends of the ribbon 3 there is provided a ribbon stiffening means 10 operable to change the ribbon between a normal state to a stiffened state. The ends of the ribbon are clamped upon floating spring flexures 11 which are arranged to apply tension to the ribbon 3. The spring flexures 1 1 are operable to move between a normal configuration, in which the ribbon 3 is held under normal tension, and a stiffened configuration, in which the ribbon 3 is held under increased tension.

Changing the tension applied to the ribbon 3 in its length direction L, By elastically deforming the ribbon 3 by changing its length, the stiffness of the ribbon in its depth direction increases. In this way the force, for example the gravitational force, required to overcome tension and displace the ribbon 3 away from its rest position can be altered. Thus the sensitivity of the ribbon 3 to applied gravitational forces can be modulated, which is achieved in the embodiment in the following way.

Connected to the spring flexures 11 are actuators 12, provided as piezoelectric actuators, which are arranged to operate the floating spring flexures 11 between their normal and stiffened configurations in response to an input signal provided by the mechanical modulation control means 13 in the form of a square wave generated therein. The ribbon stiffening means 10 are thus arranged so as to apply two different tensioning forces in the plane of the ribbon 3 in an on-off manner in response to the square wave input from the mechanical modulation control means 13. This is achieved by the control means input signal being such that, at the square wave peaks, a control voltage is applied to the piezoelectric actuators 12 to push the spring flexures 11 towards each other to provide the 'normal' tensioning state of the ribbon, and at the square wave troughs, the control voltage reduced to zero is applied to the piezoelectric actuators 12 such that the spring flexures 11 pull the ribbon 3 back to its 'stiffened' tensioning state. The actuators 12 may alternatively be provided as magnetostriction devices.

The amplitude of the push-pull movement of the spring flexures 11 between the two configurations, and thus the amplitude of the stretching and relaxing motion of the ribbon 3 is extremely small, typically on the order of tens of microns, and about 15 microns in the embodiment shown in Figure 1. This push and pull action of the ribbon stiffening means 10 is aligned with the plane of the ribbon 3 such that the movement between the 'normal' and 'stiffened' configurations does not introduce any displacements to the ribbon other than that parallel to the ribbon's surface. In view of this motion, the ribbon 3 itself is also formed to be a straight flat plane to the degree of accuracy required for the apparatus (typically micron accuracy is required), which is possible using precision engineering techniques. However, if the desired accuracy can not be obtained then monitoring and compensation can be used to improve results.

In the stiffened configuration in response to a square wave signal input peak, the ribbon stiffening means 10 applies a high tension to the ribbon such that the ribbon 3 has a natural frequency (i.e. in its 'S' mode of oscillation) typically of around 50-80 Hz. When the tension is released over a short period of time by the ribbon stiffening means 10 going into their normal configuration in response to a square wave input signal trough, the natural frequency (again, in its 'S' mode of oscillation) of the ribbon 3 drops to around 10-20 Hz.

The mechanical displacement of the ribbon 3 due to gravitational forces (or any other forces) is inversely proportional to the ribbon's stiffness. Therefore the magnitude of displacement of the ribbon 3 in its anti-symmetric modes due to the gravitational gradient across the ribbon 3 is greater when the ribbon stiffening means 10 is in its 'normal' configuration than when it is in its 'stiffened' configuration for the same gravitational gradient. The magnitude of displacement of the ribbon due to other external forces is also likewise affected.

This means that the effective gravitational 'gain' of the sensitivity of the gradiometer 1 to gravitational gradients on the ribbon 3 goes periodically very quickly from a high value (when the ribbon stiffening means 10 are in their normal configuration) to a low value (when the ribbon stiffening means 10 are in their stiffened configuration). Specifically, the gravitational gain falls off as the inverse square of the natural frequency of the ribbon. Thus, for a typical soft S-mode frequency of 20 Hz there is roughly a 16-25 times reduction in gravity gradient sensitivity between soft and firm mode.

The modulation rate of the stiffness of the string (i.e. the frequency of the square wave) is typically around 5-1 OHz (an order of magnitude greater than the rotational modulation of the non-string-based gradiometers known in the art).

However, the mechanical modulation control means 13 may provide a stiffness modulation of significantly higher frequency, up to the order of 10² Hz. Since the gradiometer signal is modulated by the mechanical modulation at a frequency away from DC, the 1/f signal noise at the modulation frequency is significantly reduced compared to the non-modulated arrangement. With less electrical noise, the signal to noise ratio of the device improves.

Sensing means 20 is provided to detect the transverse displacement of the ribbon 3 from an undisturbed position S due to the gravitational gradient acting on the ribbon and to generate a signal representing this displacement. The sensing means 20 is provided generally in the form known in the art described above. Two pick-up coils 25 are positioned symmetrically about the mid-point M of the ribbon 3 and are electrically connected to form the two arms of a resonant bridge circuit (not shown) and to a control and detection system 27. Control and detection system 27 is also electrically connected to the ribbon 3 via the spring mounted rollers 9 to pump the ribbon 3 with an AC carrier signal generated therein, and having a frequency above the mechanical bandwidth of the ribbon (such that magnetic forces acting on the ribbon are damped). The frequency of the resonant bridge circuit is tuned to that of the AC carrier signal such that a corresponding signal is generated in each pick-up coil 13, 15 having a strength that increases as the distance between the ribbon 3 and the pick-up coil 13, 15 decreases. The pick-up coils 13, 15 and bridge circuit are balanced in an anti-phase arrangement such that the signal output from the bridge circuit is sensitive to displacement of the ribbon 3 in its anti-symmetric modes of oscillation (and primarily the ¹S' mode of oscillation) but is not sensitive to

displacement of the ribbon 3 in its symmetric modes of oscillation. In this way a signal is generated in the resonant bridge circuit that is representative of the displacement of the ribbon 3 due to the gravitational gradient causing the ribbon 3 to oscillate in its sensing direction.

As the displacement response of ribbon 3 to gravitational gradient is square wave-modulated by a the periodic mechanical stiffening of the ribbon 3, the displacement signal output from the bridge circuit is therefore modulated by the square wave signal generated by the mechanical modulation control means 13. The displacement signal is also modulated by the AC carrier signal generated in the control and detection system 27.

The displacement signal output from the bridge circuit is retrieved in control and detection system 27 by synchronous detection and demodulation together with the AC carrier signal and the square wave signal provided to the control and detection system 27 by the mechanical modulation control means 13. The demodulation can be achieved using hardware or software based signal processing.

By amplifying and processing the displacement signal, the control and detection system 27 generates an output signal which is a function of the gravitational gradient tensor of the gravitational field. The output signal resulting from the mechanical stiffness modulation of the gradiometer at a frequency away from the DC reduces the 1/f noise in the output signal and thus increases the signal-to-noise ratio of the gradiometer device.

Mechanically modulating the stiffness of the ribbon 3 allows an absolute measure of the gravitational gradient to be determined from the gradiometer in the following way. Any external forces, such as those from gravity gradient fields and uniform gravity fields, have much weaker influence on the ribbon 3 in its stiffened state than in its normal state. Thus the displacement measurement of the ribbon 3 taken in the stiffened state of the ribbon 3 can be thought of as an effective zero-point reading taken for the ribbon 3 in the absence of gravity forces (i.e. the ribbon 3 has a severely reduced sensitivity to gravity forces). In the normal state of the ribbon 3 when it has lower stiffness the ribbon is allowed to relax and deform under external forces and the displacement measurement taken then is the offset measurement. The gravity gradient reading is therefore the difference between the tensioned (zero reading) and relaxed (gravity gradient reading) states of the ribbon. Using this method the gravity gradiometer 1 becomes an absolute meter, since it measures the difference in deflection between gradient sensitive/gradient insensitive states. The difference in readings between these two states constitutes the absolute measurement of the local gravity gradient.

By modulating the stiffness of the ribbon at a frequency at or approaching the mechanical resonance of the ribbon (i.e. typically above 30 Hz), the resonance of the ribbon 3 can be excited which has the effect of amplifying the displacement produced in the ribbon 3 due to the same gravity gradient. This parametric amplification of the displacement signal has the effect of amplifying the gradiometer signal output by the device 1 and further increasing the signal-to-noise ratio of the device by as much as one to two orders of magnitude and allowing the gradiometer 1 to achieve a sensitivity down to as little as 0.1-1 EU.

A gravitational gradiometer device 100 according to another embodiment of the invention shown in Figure 2. The gravitational gradiometer 100 of the second embodiment is identical in its arrangement to the gradiometer 1 of the first embodiment, with the addition that the gradiometer has a rotatable housing 101 which is operable to rotate about the central length axis of the ribbon 3. Rotation means 103 are provided to rotate the housing 101 and the ribbon 3 at a rotational frequency determined by a signal generated by rotation control means 105. When the ribbon 3 is being rotated, the signal generated in the resonant bridge circuit is further modulated at the rotational frequency of the ribbon. The displacement signal is then retrieved in the control and detection system 27 by synchronous detection and demodulation together with the AC carrier signal, the square wave signal provided to the control and detection system 27 by the mechanical modulation control means 13, and the rotational frequency signal provided to the control and detection system 27 by the rotation control means 105.

By setting the rotation control means 105 to rotate the ribbon 3 at a rotational frequency equal to the resonant frequency of the 'S' mode of the ribbon 3 and setting the mechanical modulation control means 13 to modulate the stiffness of the ribbon at a frequency twice that of the resonant frequency of the 'S' mode of the ribbon 3, the conditions are such that the gravitational gradiometer is provided as a degenerated parametric transducer. This arrangement provides a further enhanced signal-to-noise ratio.

Modulation of the ribbon stiffness can be achieved by means other than the mechanical action of two peizo-devices attached to the ends of the ribbon to apply a variable tension to the ribbon. These include, but are not limited to, materials that stiffen/soften in the presence of stimuli such as light, current or electric or magnetic fields. All that is required is that the ribbon changes its sensitivity to gravity forces between zero (or low) sensitivity and full sensitivity at a reasonable rate (5-10 Hz typically, faster for some advanced applications).

Claims

Claims:

1. Apparatus for the measurement of quasi-static gravity gradients comprising: a flexible ribbon held under tension at both longitudinal ends;

sensing means arranged to detect the transverse displacement of the ribbon from an undisturbed position due to the gravitational field acting on said ribbon and to generate a signal representing the displacement;

output means coupled to said sensing means and responsive to said

displacement signal to generate an output signal which is a function of the

gravitational gradient tensor of the gravitational field;

ribbon stiffening means operable to change the ribbon from a normal state to a stiffened state in which the displacement of the ribbon due to the gravitational field is reduced; and

control means to periodically operate the stiffening means;

wherein, in use, the control means periodically operates the stiffening means at regular intervals so as to modulate the stiffness of the ribbon and the displacement signal; and

the output means demodulates the displacement signal by synchronous detection using a signal representing the modulation of the ribbon stiffness.

2. Apparatus as claimed in claim 1 , wherein the stiffening means is arranged to operate to stiffen the ribbon by increasing the tension holding the ribbon.

3. Apparatus as claimed in claim 2, wherein the stiffening means comprises: floating spring flexures arranged to hold both longitudinal ends of the ribbon under tension and to be operable to move between a normal configuration, in which the ribbon is held under normal tension, and a stiffened configuration, in which the ribbon is held under increased tension; and

actuators arranged to operate the floating spring flexures.

4. Apparatus as claimed in claim 3, wherein the actuators are piezoelectric actuators or magnetostriction devices.

5. Apparatus as claimed in claim 3 or 4, wherein the actuators and floating spring flexures combine to, in use, apply a tensioning force to the ribbon that is aligned with the plane of the ribbon.

6. Apparatus as claimed in claim 3, 4 or 5, further comprising spring mounted rollers arranged to engage the ribbon at its boundary condition ribbon-end locations and retain the ribbon at its undisturbed position there.

7. Apparatus as claimed in claim 1, wherein the ribbon is arranged to be stiffenable in response to an electric or magnetic or electromagnetic stimulus; and the stiffening means is operable to supply said stimulus to the ribbon.

8. Apparatus as claimed in claim 7, wherein the ribbon comprises a material that increases or decreases in stiffness in response to the stimulus.

9. Apparatus as claimed in any preceding claim, wherein the control means is configured to periodically stiffen the ribbon at a frequency equal to a mechanical resonant frequency of the ribbon so as to increase the deflection of the ribbon in its normal state.

10. Apparatus as claimed in claim 9, wherein the control means is configured to periodically stiffen the ribbon at a frequency equal to twice that of the mechanical resonant frequency of the ribbon's 'S' mode of oscillation.

11. Apparatus as claimed in any preceding claim, further comprising rotating means arranged to, in use, rotate the ribbon about its length axis.

12. Apparatus as claimed in claim 11, wherein the rotating means is configured to rotate the ribbon at a rotational frequency equal to the mechanical resonant frequency of the ribbon's 'S' mode of oscillation.

13. Apparatus as claimed in any preceding claim, wherein the output means is arranged to generate the output signal as being representative of an absolute measure of the gravity gradient tensor.

14. Apparatus as claimed in claim 14, wherein the output means is arranged to base the output signal on the difference between the displacement signal generated during the stiffened state of the ribbon and the displacement signal generated during the normal state of the ribbon.

15. A method of measuring quasi-static gravity gradients comprising:

holding a flexible ribbon under tension at both ends;

operating a ribbon stiffening means arranged to change, periodically, the ribbon from a normal state to a stiffened state in which the displacement of the ribbon due to the gravitational field is reduced;

arranging sensing means to detect the transverse displacement of the ribbon from an undisturbed position due to the gravitational field acting on said ribbon and to generate a signal representing the displacement; and

generating, by demodulating the displacement signal by synchronous detection using a signal representing the modulation of the ribbon stiffness, an output signal which is a function of the gravitational gradient tensor of the gravitational field.