A gravitational eye: a method for extracting maximum information from gravitational potentials

Gravity measurements have uses in a wide range of fields including geological mapping and mine-shaft inspection. The specific application under consideration sets limits on the survey and the amount of information that can be obtained. For example, in a conventional gravity survey at the Earth’s surface a gravimeter is translated on a two-dimensional planar grid taking measurements of the vertical component of gravity. If, however, the survey points cannot be chosen so freely, for example if the gravimeter is constrained to operate in a tunnel where only a one-dimensional line of data could be taken, less information will be obtained. To address this situation, we investigate an alternative approach, in the form of an instrument which rotates around a central point measuring the gravitational potential or its radial derivative on the boundary of a sphere. The ability to record additional components of gravity by rotating the gravimeter will give more information than obtained with a single measurement traditionally taken at each point on a survey, consequently reducing ambiguities in interpretation. We term a device which measures the potential, or its radial derivatives, around the surface of a sphere a gravitational eye. In this article we explore ideas of resolution and propose a thought experiment for comparing the performance of diverse types of gravitational eye. We also discuss radial analytic continuation towards sources of gravity and the resulting resolution enhancement, before finally discussing the possibility of using cold-atom gravimetry and gradiometry to construct a gravitational eye. If realised, the gravitational eye will offer revolutionary capability enabling the maximum information to be obtained about features in all directions around it.


Introduction
There is a long history of using sensitive gravity instruments to detect features below the ground in a diverse range of Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.applications, including mineral exploration [1], geological mapping [2], civil engineering [3], archaeology [4,5], environmental studies [6], petroleum and hydrocarbon mining [7] and unexploded-ordnance management [8].While for most of these the survey is performed above the ground (either airborne or on the surface), some surveys are performed underground or within built structures.This includes applications such as mining [9], mine-shaft inspection [10], coalmine safety [11], tunnel wall inspection [12] and within boreholes for monitoring carbon capture and storage sites [13,14].In this paper we consider a new sensing approach which could provide more information than traditional approaches, particularly in restricted environments such as underground, where the gravitating objects of interest could lie in any direction, including above the instrument.The instrument is then restricted in its ability to locate a gravitating object because it is unable to scan a large region around that object.
Here we examine a concept for an instrument which sequentially measures the gravitational potential or its radial derivative, at points on a sphere surrounding the instrument.We term this sphere the data sphere.The resulting data provide an image of gravitating objects exterior to the data sphere.This allows, in principle, the extraction of maximum information regarding the local gravitational field, through reconstruction of the field and all of its derivatives at any point within the data sphere.This has potential to allow better localisation of features and reduction of ambiguity in circumstances where the instrument is constrained in how it can be moved.We term this device a gravitational eye 1 .In the ideal case, a gravitational eye is an instrument which measures the gravitational potential at all points on the data sphere.
For the ideal instrument, treating the data as specifying a Dirichlet boundary condition we may solve Laplace's equation for the potential inside the data sphere in a standard way in terms of the solid harmonics.The resulting expansion then gives a way of carrying out analytic continuation outside the data sphere, leading to a resolution enhancement of the data.We make this idea more precise in the next section.We base our ideas on related concepts in satellite-based geodesy, the obvious difference being that we consider operation below ground rather than in the microgravity environment of space.
From a practical point of view, it is difficult to measure the gravitational potential directly.However, the concept of the gravitational eye can also be applied to sensors which measure gravitational acceleration (gravimeters) and gravity gradients (gradiometers).There are a number of different gravitysensing technologies which are currently used [16].These include masses on springs [17], magnetically levitated superconducting spheres [18], MEMS [19] and cold atoms [20].In this paper we will consider possible implementations of the gravity eye based on cold atoms, which have some properties which make them suited for use as a gravitational eye due to their ability to measure gravity in any direction [20], ability to suppress common-mode noise between gravimeters [21], lack of mechanical coupling and associated issues [22].
In this article we examine the capabilities of a gravitational eye for the simplest case where the device is static at a single point in space rather than in a moving survey.We first, in section 2, investigate an idealised gravitational eye which measures the potential directly.We consider two different notions of resolution, and we see how resolution may be enhanced by analytic continuation to a larger concentric sphere which is closer to the sources.Then, in order to discuss a more practical device, in section 3 we consider a gravitational eye based on an accelerometer rotating around a central point.This, essentially, measures the radial derivative of the potential on the data sphere.In section 4 we examine gravitational eyes based on gradiometers.Selecting a particular configuration, we carry out a resolution analysis based on realistic performance estimates of cold-atom gradiometers.Once again, resolution enhancement through analytic continuation is considered.In section 5 we look at how a gravitational eye can be realised using cold-atom technology.Section 6 contains a brief description of various correction terms and section 7 contains the discussion and conclusions.We use the spherical harmonics throughout the main text as a way of analysing resolution.Our conventions for spherical coordinates and spherical harmonics are given in appendix A, with the more general case of an ellipsoidal region enclosing the gravitational eye briefly discussed in appendix B.

The idealised gravitational eye
The idealised gravitational eye measures the potential at every point on the data sphere, the interior of which is matter-free apart from the instrument itself.As a first step, assuming that the self-gravity of the instrument can be calibrated out, we wish to solve Laplace's equation within the data sphere.Given that the Laplacian is an elliptic operator this may be accomplished using either Dirichlet or Neumann boundary conditions on the data sphere.The former are appropriate to the idealised gravitational eye.
Let us carry out the Fourier expansion of the potential on the data sphere, which we assume to have radius a, following the nomenclature in appendix A: where the P m l are associated Legendre functions.This gives us the Dirichlet boundary condition.
The potential at any point inside the data sphere, at a radius r, may then be written, in spherical-polar coordinates [23], as Hence, for example, if we know the values of U on the data sphere of radius a (i.e.we know all the coefficients C lm and S lm ) we can easily find the values on a smaller concentric sphere of radius r.The expansion (2) is the expansion of the potential in regular solid harmonics (see appendix A), i.e. eigenfunctions of the Laplacian in sphericalpolar coordinates.
We thus deduce that all the information about surrounding gravitational sources that one can gather from any instrument lying within the sphere of radius a comes from the set of values of the gravitational potential on this sphere.If the gravitating objects were to lie on a larger concentric sphere (once the background field has been removed) then the potential on the sphere of radius a would correspond to a blurred version of the sources.However, in general the situation would be much more ambiguous, as with conventional gravity surveys, because the objects would not all lie at a single radial distance.
Though the ideal device is currently purely theoretical we can still make some comments about its resolving power.We start with two-point resolution.

A thought experiment on two-point resolution
In common with resolution in other areas of physics, we expect the resolution in the data to depend on the ratio of signal to measured noise in the data.Hence the resolution can be improved either by increasing the signal strength or by reducing the noise.Throughout this paper we adopt the philosophy that the resolution estimate for the data must correspond to a physical object, i.e. the test object must be constructible from physical materials; so, although one could consider a point mass, this is not physically realistic.Fortunately, however, a spherically-symmetric mass produces the same gravitational potential as a point mass located at its centre.Therefore, for a solid ball the gravitational potential is simply Gm/q, where G is the gravitational constant, m is the mass of the ball and q is the distance between the centre of the ball and the observation point.As such, the signal strength is dependent on q and the total mass of the ball.
For this thought experiment, we consider two solid lead balls as sources, surrounded by air.We can vary the signal strength by changing the diameter of the lead balls, provided they do not overlap.We also choose the balls to have the same mass and to lie equidistant from the centre of rotation of the gravitational eye.It is clear that when the data is recorded on a spherical surface the maximum value of the potential when only a single lead ball is present will be when q has its minimum value.This will correspond to the point where the surface of the data sphere intersects a line from its centre to the centre of the lead ball.Thus, the gravitational eye reveals the direction in which the lead ball lies.
To investigate resolution with two lead balls, we choose, for simplicity, the two balls to lie in the equatorial plane, since for any positions of their centres we can find a great circle passing through them and a suitable change of coordinates can then bring them both to lie in this plane.
Let us choose the sensor to record data on a sphere of radius 1 m and let the centres of the two lead balls lie on a concentric sphere of radius 2 m.Let us further choose them to have mass 20 kg and the angle between them to be π/2 radians; thus, the centres of the spheres are 283 cm apart and their radii are 7.49 cm (using a density of 11.34 g cm −3 for lead).We make these choices so that when we come to consider the gradiometer version of the gravitational eye the dip between the peaks will have a convenient, if challenging, value of 1 E. The resulting potential is shown in figure 1.It can be seen that there are two peaks, corresponding to the signals from the two lead balls, but they are merged so that there is only a relatively small dip between them.The separation between the minimum of the dip and the peaks is 0.1324 nJ kg −1 .If we assume, for the sake of argument, the measured noise to have a standard deviation at this level then this gives a rough guide to the achievable (two-point) angular resolution for this level of noise and this particular signal strength.Alternatively, one can consider other criteria for when the two peaks are resolved, such as in the standard approaches to two-point resolution for incoherent optics [24].
If one wishes for a more practical view of this thought experiment, then the same magnitude of signal can be achieved with a spherical sinkhole of radius 0.139 m in chalk of density 1.79 g cm −3 positioned at the same distance from the gravitational eye as in the thought experiment.

A resolution limit for the data in terms of the spherical harmonics
An alternative viewpoint for the resolution limit in the data is to consider a spherical-harmonic expansion of the data using (1).The higher-order terms in the expansion represent structures that have smaller angular separations.A truncated expansion, where terms with l greater than some maximum value L are omitted, will represent a 'blurred' version of the data.If, however, the truncation is chosen such that the root mean square of the omitted terms has the same order of magnitude as the noise on the data, one can argue that the blurred version represents all of the useful information about the object and thus determines the resolution.In our case this means that the resolution in the data is roughly determined by the angular distance between neighbouring zero-crossings of the spherical harmonics Y Lm , where L is the band-limit (highest useful degree) of the system.This approach can also be applied to gravity surveys where data is taken in a horizontal plane; in that case, if the object also lies in a horizontal plane, a 2D Fourier expansion can be used and the relationship between object and data is similar to the modulation transfer function in optics.For an early use of the Fourier approach in gravity surveys, see [25].In this subsection we consider how this bandlimit, L, may be estimated from the situation where the object is a point mass.We emphasise that this remains purely theoretical and is included for didactic purposes.Note also that, given a sum of the lowest-degree spherical harmonics up to and including those of degree L, there is no limit in principle as to how close a pair of neighbouring zeros can be, if the zeros further away are more widely spaced.Therefore, the argument only works on average.
As an example, then, let us consider the expansion in spherical harmonics of a point-mass potential V, on a sphere of given radius, r: where m is the mass of the point mass, which is positioned on the same sphere of radius r [26].The primed quantities denote the position of the point mass.
To estimate the band limit, suppose that we measure the potential corresponding to (3) on a smaller concentric data sphere of radius a.In the absence of noise, the theoretical expression for this is obtained by substituting (1) by (3) and then calculating the resulting potential on the smaller sphere using (2).Suppose we limit the number of spherical harmonics in this exact expression for the noiseless data.As explained earlier, an estimate for the band limit can be obtained by adjusting the number of terms until the difference between the exact and approximate expressions falls below the system noise level and the angular resolution can be obtained from consideration of the spacing of the zero crossings of the highest remaining spherical-harmonic function.
As an aside, once we have a band limit the question of data sampling can be addressed.Although not straightforward, this is discussed, for example, in [27].Their theorem 3 for evaluating spherical-harmonic coefficients involves over-sampling but is simple to use.In this paper we just assume sufficiently fine sampling to capture the highest-degree spherical harmonics of interest.

Resolution enhancement through analytic continuation
Solution of the inverse problem in gravimetry typically runs up against the problem of a dimension mismatch.The recorded data lie in a space of lower dimension than the reconstructed object.This makes the inverse problem rather ill-defined.As an alternative to solving the inverse problem one can carry out an analytic continuation of the data in the direction of the gravitating object, leading to a resolution enhancement.We have already seen an example of analytic continuation at the beginning of this section where we saw that the potential on a sphere defined the potential at any point within the sphere, if the sphere contained no gravitating sources.Here we use a similar approach to analyse resolution.
Considering first the case of a two-dimensional survey carried out at the Earth's surface on a horizontal plane (i.e.we have two-dimensional data), if one believes that the object under the surface that one is trying to image is roughly horizontal and two-dimensional then one can argue that the data form a sort of blurred image of the object.Resolution enhancement can then be attempted by analytically continuing the data downwards to a plane closer to the object, after removing the effects of the intervening material.Alternatively, if the data are contaminated by high-frequency noise then analytically continuing in the opposite direction can smooth out this noise, to some extent.The resolution enhancement problem is ill-posed, that is, it is unstable with respect to noise on the data.The smoothing problem is well-posed since it decreases the effects of noise on the data.In this paper we consider the former case of resolution enhancement [28].Now let us look at the case of the data recorded on a sphere and analytically continued to a concentric sphere.This analytic continuation can be accomplished by various means, the simplest being a straightforward Taylor expansion.However, this does have drawbacks which are discussed, for example, in [29].In our case, given that we will typically only record gravity or its first radial derivative, this does not bode well for a Taylor-series approach.An alternative, which is routinely used for problems involving the potential, is to expand the data in terms of spherical harmonics.To us, the appeal of this method is that the spherical-harmonic coefficients are calculated by integrating over the sphere on which the data are collected.We can then use the radial dependence of the solid harmonics in the analytic continuation.
In figure 2 we show the two cases of analytic continuation in spherical polar coordinates.Figure 2(A) shows the exterior potential problem: we measure the data on a sphere enclosing the source.We note that the radial dependence of the solid harmonics involves either a positive or negative power of the radial variable.The solution of the exterior potential problem must be bounded as the radial variable tends to infinity and hence only the irregular solid harmonics can be used in the expansion of the potential.In figure 2(B) we show the interior potential problem.Here the solution must be bounded at zero radius and hence only the regular solid harmonics can be used.
In both cases, analytic continuation is achieved by varying the radial variable in the solid-harmonic expansion.
In this paper we are interested in the interior problem.We assume that all gravitating sources lie outside both spheres and that the data are recorded on the inner sphere.The larger sphere should be chosen to be as large as possible without actually touching any of the surrounding sources.This is our equivalent of the usual Brillouin sphere in geodesy which is chosen to lie as close as possible above the planet's surface without touching it.We transform the potential on one sphere to the potential on the other via analytic continuation using a solidharmonic expansion as in (2).
For our problem, moving from the Brillouin sphere to the data sphere is equivalent to moving further away from the sources, leading to higher angular frequencies in the potential being damped out [23].Conversely, in (2) we see that if one moves from the data sphere to the Brillouin sphere, the higher 'frequencies' are enhanced, leading to an apparent resolution improvement.Since the measured coefficients for these higher-degree terms are relatively smaller and therefore have a lower signal-to-noise ratio, this outward-continuation process also leads to noise amplification.A simple way to mitigate this is to truncate the solid-harmonic expansion at the band-limit, the latter being dependent on the signal-to-noise ratio.
In order to obtain a point-spread function on the data sphere we can model a (non-physical) point potential on the Brillouin sphere using an L 2 version of the δ-function in terms of the spherical harmonics and centred at (θ, ϕ ) [23]: In what follows we use the symbol δ to represent the expression in (4), even though it is not, strictly speaking, a deltafunction and is more closely related to a resolution of the identity.Like the true delta-function it is dimensionless and is designed to be used under an integral sign.An arbitrary potential on the Brillouin sphere can then be modelled as a weighted integral of these point potentials.Let us assume the Brillouin sphere has radius b and the data sphere, radius a.In the absence of instrumental noise, the data, corresponding to this point potential, then form, using a solid-harmonic expansion as in (2), a point-spread function given by In this, we note that higher-degree terms are progressively more strongly damped.In the presence of noise this means that eventually these terms will be lost in the noise.We term the degree L at which this happens the bandwidth.Hence, we have an effective point-spread function on the inner sphere given by a sum truncated at degree ( This then determines the innate resolution of the data. We turn now to the resolution improvement due to analytic continuation of the data back to the Brillouin sphere.From the foregoing we can see that the effect of analytically continuing ( 5) outwards is to change our approximation to the δfunction on the Brillouin sphere to The width of the central peak of this function then gives us a measure of resolution after the analytic-continuation process.Note that this can be simplified using the spherical-harmonic addition theorem: where γ is the angle subtended at the centre of the unit sphere by the two points (θ, ϕ ) and (θ ′ , ϕ ′ ) and and P l is the lth Legendre polynomial, normalised so that P l (1) = 1 [30].We note that the addition theorem only involves the angle γ between the two points (θ, ϕ ) and (θ ′ , ϕ ′ ).The resulting band-limited δ-function then takes the form Hence, we can get the band-limited point-spread function for the analytic continuation by adding up all the Legendre polynomials up to the band-limit.We show in figure 3 the variation of this point-spread function with the number of terms in the expansion.
In what follows we choose a relatively small number of terms.In satellite geodesy a very large number of spherical harmonics are included; in the gravity model WGM2012, data from GOCE satellite measurements contributes to harmonics up to a degree of 250 [31].In our case, the signals are expected to be very much smaller relative to the measurement noise.Throughout this paper we assume that, for the purposes of analytic continuation, we start with a minimum bandwidth of L = 4.We will see in section 4 that for the gradiometer version of the gravitational eye and the setup in our thought experiment (extended to involve four lead balls) a fourfold symmetry can be identified if the sensitivity of the instrument is roughly 1E.It should be noted that a bandwidth of 4 will be much more difficult to achieve for the idealised gravitational eye and also that based on a single gravimeter, but given that we discuss these largely for didactic purposes, for the sake of simplicity we have started with the same minimum bandwidth as for the gradiometer version of the eye.The higher bandwidths in our simulations are included to show how the resolution improves with higher signal-to-noise ratios, The latter may be obtained by increasing the size of the lead balls or reducing their distance from the instrument.
In the remainder of this paper we shall consider gravitational eyes based on currently practical technology: gravimeters and gradiometers.First, however, we will comment briefly on the possibility of measuring the gravitational potential directly.The known way of doing this is to use general relativistic gravitational redshift.The sensitivity of this approach is given by [32], where ∆W is the sensitivity of the potential and ∆f f is the relative frequency shift.Such an approach is used in relativistic geodesy [33] (also known as chronometric levelling).The frequency shift can be measured using a transportable optical clock [34].The best relative frequency shift from a transportable optical clock appears to be around 7 × 10 −17 in [35].In the laboratory, performance of 2.5 × 10 −19 has been achieved [36].We note, however, that a relative frequency shift of 10 −19 corresponds to a ∆W of 10 −3 m 2 s −2 , which falls far short of the performance required in figure 1.We thus conclude that, at the moment, the use of the gravitational redshift is not appropriate to our thought experiment for the idealised gravitational eye based on measurement of the potential.

A gravitational eye based on a single gravimeter
Having discussed the idealised gravitational eye which measures the potential we now look at a more practical instrument.Consider, as a simple gravimeter-based gravitational eye, using a single gravimeter which can be rotated around a central point (shown in figure 4 for the case where the gravimeter is rotated in the equatorial plane of the spherical polar coordinate system).The gravimeter, denoted by 2, records a radial derivative (along the line joining 1 and 2) of the potential, i.e. we measure acceleration rather than potential.
The response of the gravimeter is given by where m is the mass of the lead ball.By simple geometry we have that and

Two-point resolution
To investigate the two-point resolution of this device we follow the approach for the idealised gravitational eye we have already discussed.We assume in figure 4 that ∆r = 1 m and r = 1 m.We show in figure 5 the response to two 20 kg lead balls lying in the equatorial plane with an angle between them of π/2 radians.Their centres are 2 m from the centre of rotation of the instrument.If we were to have a measured noise of standard deviation about 0.08 µgal (the difference between the peak and the dip between the two peaks), then for this size of signal we can argue that this represents the limit of (two-point) resolution.

Resolution enhancement through analytic continuation for a single-gravimeter-based eye
Let us assume that we know the potential on the data sphere of radius a and that we wish to find the radial derivative on a concentric smaller sphere of radius r.The radial derivative of the potential on this smaller sphere is found, at least formally, by differentiating (2): ) l lr l−1 P m l (cos θ) (C lm cos mφ where we note that, compared to the idealised gravitational eye, the l = 0 term has now dropped out.Although we have not justified the term-wise differentiation of an infinite series, the presence of the inevitable band-limit means that the series is effectively truncated at a finite number of terms and therefore this operation is legitimate.We now consider analytic continuation of the data recorded on the data sphere of radius a to the Brillouin sphere of radius b.To assess the resolution enhancement, let us again use a 'δ-function' on the Brillouin sphere.This now represents an impulse in the space of radial derivatives of the potential. First of all, we need to see how analytic continuation works for the first radial derivative of the potential.From (9) we have that, on the data sphere, Defining new coefficients S lm , we can then rewrite (9) as This is the equivalent of (2) for gravimetry.Hence, we see that to continue the first radial derivative from the data sphere to either a smaller or greater concentric sphere of radius r, one just multiplies the terms of degree l by (r/a) l−1 .Returning to a representation of a δ-function on the Brillouin sphere, from (10) we can see that the radial derivatives on the Brillouin sphere of radius b lie in a subspace of L 2 ( S 2 ) specified by the l = 0 term being zero.Analogously to (4) we can define a 'δ-function' on this subspace given by where the sum over l now starts at l = 1.We treat this function as a point radial derivative of the potential.We consider first how this 'point function' is spread out when measured on the data sphere.To do this we simply multiply the term of degree l by (a/b) l−1 as follows from (11).This gives We now assume that the data are band-limited (due to the presence of noise) to a maximum degree of L. This means that, to within the level of the noise, the point-spread function P may be replaced by Analytically continuing to the Brillouin sphere yields an overall point-spread function given by Use of the Addition Theorem then puts this in the form This is shown in figure 6 for 4, 6 and 9 terms.

Gradiometer configurations
There are various arrangements of multiple gravimeters which could also be used to realise a practical gravitational eye for the gravity gradient.Note that in all arrangements we choose to take differences of gravimeter outputs to reject commonmode noise.
In this configuration two gravimeters are rotated about their joint centre of mass, as shown in figure 7(A), to cover all polar angles and azimuths.For each point on the imaginary sphere shown one records the difference of the two gravimeter outputs.Hence this is not really a variant of the gravitational eye.
There is an obvious ambiguity, after the background has been subtracted off, between a positive-mass anomaly in one direction and a negative-mass anomaly in the opposite direction.
In this configuration the centre of rotation is fixed at the centre of one gravimeter and both gravimeters are allowed to vary over all polar angles and azimuths (shown in figure 7(B)).The ambiguity for the first configuration is removed by placing a gravimeter at the centre of rotation.In this paper, for simplicity, we will focus on this configuration.
In this configuration we have a reference gravimeter in the centre of the sphere.The centre of rotation is the central gravimeter.Differences are taken between each of the outer gravimeters and the central one (shown in figure 7(C)).This configuration avoids the technical difficulty associated with the previous one of having to turn the structure upside down.It also has the advantage over the previous one that one can record data twice as quickly because both differences are recorded simultaneously.Clearly one can view the previous configuration as one half of this one.A three-gravimeter device, capable of measuring vertically has already been proposed based on cold atom technology [37].

The measured data
The data measured in the configuration in figure 7(B) involve the difference of two radial derivatives of the potential.As in the previous section, let us assume that we know the potential on a sphere of radius a. From this we wish to find the radial derivative on a sphere of radius r, minus the same at r = 0. We start with the expansion of the radial component of gravity in (9) on the sphere of radius r.Subtracting off the same expansion, but evaluated at the centre of rotation (i.e.r = 0), gives the (noiseless) data on the sphere of radius r ) l lr l−1 P m l (cos θ) (C lm cos mφ where the expansion over l now starts at l = 2. Should we wish to find the data which would be recorded on the data sphere it is now simply (10) but with the l = 1 term omitted.

Basic two-point resolution
We consider first the response to a single lead ball.For our gravitational eye we consider the configuration in figure 7(B), i.e. we have one central gravimeter with a second one rotating around it.The geometry when the gradiometer is rotated through an angle ϕ is that of figure 4. The small circles now both correspond to the gravimeters, with gravimeter 1 lying at the centre of rotation.The larger circle corresponds to the lead ball.We assume that ∆r = 1 m and that, when ϕ = 0 the distance, r, between gravimeter 2 and the centre of the lead ball is also 1 m.For simplicity we assume that gravimeter 2 rotates around gravimeter 1 in the equatorial plane and that the lead ball lies in the equatorial plane.We can always rotate the coordinate system, if required, to arrange this.The radial derivatives (i.e. the derivatives along the line joining the two gravimeters) of the gravitational potential at the two gravimeters are given by where m is the mass of the lead ball and G is the gravitational constant.The angle ψ is given by ( 7) and the distance R by (8).
In the equatorial plane we can then plot out g r (2) − g r (1) as a function of ϕ to show the response to a single point source.We choose a lead ball of radius 7.49 cm, i.e. of mass 19.981 kg, as our source.This gives a convenient maximum response of 1 Eötvös (E).The resulting response is shown in figure 8.
Consider now the response to a pair of lead balls lying in the equatorial plane.We consider an analogue of the Rayleigh resolution criterion in incoherent optics.We assume that we now have two lead balls, each of mass 19.981 kg, with their centres positioned at π and 3π/2 radians.We show the response in figure 9(A).For curve (A) the trough between the peaks is roughly of depth 1 E. Let us assume the noise has a standard deviation of 1 E. Then for a signal-to-noise ratio of unity we can roughly say that the two spheres are resolved when they are spaced π/2 radians apart.Clearly, increasing the size of the spheres means that the angle between their centres can be reduced and hence the resolution can be improved, up to the point where they are just touching.For curve (B) we see that the trough between the peaks is almost 1 E, indicating that, if we can reach this sensitivity, then the minimum bandwidth must be L = 4.

Resolution enhancement through analytic continuation
We now mimic the resolution analysis of the single-gravimeter gravitational eye for the gradiometer in figure 7(B).We use the same representation of an impulse in the space of radial derivatives of the potential on the Brillouin sphere as in (12).Analytically continuing inwards to the data sphere and bandlimiting to a band-limit L yields a recorded point-spread function on the inner sphere, analogous to (13), of where we emphasize that the expansion over l now starts at 2, as in ( 14), since the l = 1 term in the expansion of the radial derivative is cancelled out due to subtracting off the signal in gravimeter 1.
After analytic continuation back to the Brillouin sphere we have, for the overall point-spread function: where, again, the expansion in l starts at 2. By analogy with the earlier procedure we can write this band-limited δ-function as This is plotted in figure 10 for 3, 5 and 8 terms.
In its simplest implementation an atom interferometer can be made by taking a cloud of two-level atoms initially in their ground state, allowing them to fall under gravity, then, as they fall, subjecting them to a sequence of three laser pulses (see figure 11(A)).The first pulse puts the atoms into a superposition of ground and excited states with equal population (a π/2-pulse) [61].After a time T, another laser pulse is applied such that an atom in the ground state has a 100% probability of absorbing a photon, and atoms in the excited state have a 100% probability of emitting a photon by stimulated emission.This pulse therefore causes the ground and excited state populations to swap (a π-pulse).The associated momentum  transfers make the atomic trajectories converge, such that they intersect after a time T. At this point another π/2-pulse overlaps the trajectories, creating interference visible in the populations in the ground and excited state of the interferometer.The change in interferometer phase ∆φ is determined by the gravitational acceleration experienced by the atoms and can be read out by counting the number of excited-state atoms versus the number of ground-state atoms at the end of the interferometry sequence.
The sensitivity of an atom interferometer is fundamentally linked to the space-time area enclosed during the interferometry sequence as well as the number of atoms undergoing the interferometry sequence.The size of the space-time area enclosed by the atom interferometer is proportional to the square of the time between interferometry pulses (T) and is linearly proportional to the amount of momentum applied to the atoms by the laser pulses used to create the superposition state.The atom shot noise (ASN) limit is given by where N is the total number of atoms, C is the contrast (given by C = n f /N, where n f is the number of atoms that contributes to the interference signal) and k eff is the effective wavenumber of the light, which determines the momentum transfer.Increasing the T time improves sensitivity but also increases the distance for which the atoms drop, and therefore their propensity to fall out of the interferometry beam.Other means of improving sensitivity could be used in conjunction with a shorter T time in order to reduce the drop distance and therefore allow operation at higher polar angles.Possible methods are high-data-rate interferometry [62,63] and largemomentum-transfer [64] coupled with high-data-rate interferometry.Note that a cold-atom gravimeter does not measure gravity at a point: during the fall of the atoms they are subject to a varying gravitational field and strictly speaking this needs to be taken into account.Roughly, what is measured is an average value of gravity over the distance of the drop.Reducing the drop distance may yield the additional benefit of making operation closer to the ideal of a point measurement.
A cold-atom gravimeter measures gravity along the direction of the interferometry beam (the laser beam applying the laser pulses), which allows for gravity to be measured over a range of angles, relative to the vertical [21].This is an essential requirement to realise a gravitational eye.When designing a device to operate at either the full range of polar angles or a subset, there are various issues that will need to be addressed to prevent a loss of sensitivity, primary among them being contrast loss due to inhomogeneities in the laser intensity across the atom cloud [65].For the purposes of this paper we will ignore these effects and consider only cloud expansion and ASN.In the typical implementations of an atom interferometer (see figure 11(B) for an example) the system's sensitivity is largely a function of the T time used in the atom interferometer.In this configuration the most significant issue, in realising a useful gravitational eye, is that of achieving the required sensitivity while keeping the cold-atom cloud within the interferometry beam as it travels through space under the effect of gravity.See figure 11(C) for an illustration of this.While the standard implementation (shown in figure 11(B)) of A simple sequence consists of the following steps; once having generated a cloud of atoms they will be allowed to fall under gravity at a time t 0 , then at later time t 1 the first interferometry pulse will be applied followed by the second after a time T at time t 2 , with the final interferometry pulse applied at t 3 after another time T. (C) If this experimental system is tilted and the same experimental sequence performed, as the atom cloud falls under gravity it will fall at an angle to the interferometry beam.If the angle is sufficiently large then the atom cloud will leave the beam before all of the pulses can be applied, rendering the measurement invalid.
an atom interferometer is not ideal for implementation of the gravitational eye, there are several possible variations, which would enable a gravitational eye to be built.Already there exist working cold-atom systems for both θ = 0 • [50,66], θ = 45 • [67] and, in a gradiometer configuration, θ = 90 • [68].The differential sensitivity of the latter device is quoted as 4.2 × 10 −9 g/ √ Hz over 0.7 m which translates to about 60 E/ √ Hz.A simple design to realise an atom interferometer that would work at any angle would be to use as large a diameter interferometry beam as possible and adjust the T time such that the atom cloud will remain in the beam regardless of orientation.The possible drop time available in an atom interferometer with a beam length of 600 mm, a beam radius of 100 mm (1/e 2 ), initial atom cloud radius of 5 mm (1/e 2 ) with a temperature of 3 µK can be seen in figure 12(A).
To provide an idea on the sort of variation in sensitivity that could be expected in such a system we calculate the ASN limited sensitivity for this system, assuming that during the first 18 ms of the drop the atoms are cooled and prepared in a magnetically insensitive state, after which the atoms are subjected to a π/2 − π − π/2 pulse sequence in which 10 6 atoms participate (as shown in figure 11(B)), before finally undergoing detection via fluorescence.Figure 12(C) shows the sensitivity that could be achieved using a two-gravimeter configuration as shown in figure 7(B) separated by a baseline of 1 m.The variation in maximum T time with angle will mean that the system has an angle-dependent ASN sensitivity (as shown in figures 12(B) and (C)), meaning that to achieve the same precision in measurements over all angles longer measurement times will be required further away from the vertical.At its optimal angle a precision of 1 E can be achieved in 25 measurements, while at its least optimal in 255 measurements.While this may seem like an experimentally simple solution, care would need to be taken to reduce contrast loss which can occur in atom interferometers by atoms falling at an angle through a Gaussian beam.
While this simple design would allow for a gravitational eye to be realised, there are several more sophisticated experimental schemes possible using atom interferometry which could be more practical (e.g.requiring less laser power or resulting in a smaller device).The types of scheme can be split broadly into three categories: implementing a sequence such that the atom cloud will remain in the beam regardless of the sensor orientation, delivering each of the interferometry beams separately such that the cloud trajectory will pass through them regardless of angle, and keeping the atoms within the interferometer beam via a confinement technique.
The second of these schemes can be achieved by launching the atoms perpendicular to the beams, such as in a continuous atomic-beam interferometer [69], in which a beam of atoms is used instead of a cloud.This beam of atoms passes through three beams of light, each providing one of the pulses used in atom interferometry.By changing the distance between each of these beams of light as the system is rotated it would be possible to account for the changes in the atom beam's velocity due to gravity.
There are also a number of atom-guiding techniques [70][71][72][73][74] which could be used to confine the atoms to the interferometry beam.This means that, regardless of orientation, the atoms will be confined within the trap and therefore stay within the interferometry beam.Examples of atom-guiding techniques, which have been used in conjunction with atom interferometry include trapping the atoms with a dipole trap inside a hollow-core fibre [75] or techniques to hold the atoms in place [76].
In addition to keeping the atom cloud within the interferometry beam, there are a number of systematic effects which would need to be addressed to prevent a reduction in sensitivity or a bias.Possible sources of bias include the Coriolis effect, contrast loss and phase shifts due to passing at an angle through the interferometry beam, self-gravitation and the Quadratic Zeeman effect [77].There are currently a number of methods used or in deployment to deal with these systematic effects which could be implemented in a gravitational eye [78][79][80][81].

Various corrections and time-lapse imaging
In order to process gravity data to determine the structure of an anomaly one first has to carry out various corrections.These include the terrain corrections (see [82], for a review).For a spherical geometry, as discussed in this paper, the equivalent of the terrain corrections necessary in the planar case includes the deviation from spherical symmetry of the edge of the void outside the Brillouin sphere.To some extent one can estimate the effect of this deviation by using a laser scanner to record the shape of the surrounding void.If one has knowledge of the density of the surrounding rock one can then remove the effects of the deviation from sphericity by modelling.In addition, there is the equivalent of a Bouguer correction for removing the effect of mass between the Brillouin sphere and the objects of interest.
In the ideal case of time-lapse imaging we effectively eliminate terrain corrections, unless they happen to change with time.This means that we can outwardly continue as far as the closest source of interest, provided this is varying with time.
There are also various other corrections in conventional gravity surveys, see, for example in [28,83].Of these the latitude correction can be ignored since our survey is carried out at a single point on the Earth's surface.The centrifugal correction will pick up a dependence on the angle between the local normal to the Earth's rotation axis and the interferometry-beam axis.The Bouguer and the free-air corrections will pick up cosine dependencies on the angle between the local vertical and the interferometry-beam axis.

Discussion and conclusions
The gravitational eye is a method of extracting more information from a local gravitational field than is possible with a standard gravimeter or gradiometer, and will be particularly beneficial in surveys where measurements are restricted along a single axis.However, it is not the only way of extracting multi-axis information; in particular, full-tensor gradiometers (FTGs) have been developed and deployed over many years [84] so it is worth comparing this approach to the gravitational eye.Firstly, one of the major applications of FTGs is for airborne mineral prospecting, which is very different from the application we are considering for the gravitational eye.However, one method of signal processing that has been suggested for FTG data [85][86][87] is worth commenting on here because there is some potential commonality with the measurements in underground environments envisaged in this paper.Consider the case where a 1D line of data is produced, this could be in a borehole gravity survey, for example.If this is made with a FTG, the following processing could be used: take the eigenvalue decomposition of the gradient tensor, if the anomaly is a point source with positive mass, the principal eigenvector points in the direction of this source.If the source is a simple, but extended, anomaly the principal eigenvector points to its centre of mass.Hence, with this method, the gradient-tensor approach gives one direction at each observation point and thus affords the possibility of triangulating to find the centre of mass.While it is possible to use all the invariants of the gradient tensor [88] to yield more sophisticated information about the locations of sources, building up a clear picture of the surroundings is still challenging.We note, that a triangulation approach to locating a simple anomaly could also be used with the gravitational eye.One could do this by finding the direction of the strongest peak in the sensor response for each measurement position.
It is also worth noting that, in practice, the gradient measurements in a FTG are done by using spatially-separated gravity sensors, so the result is only an approximation to a point measurement.For applications where the gravitating source is sufficiently far away this is a reasonable approximation; however, for sources that are close, this approximation may degrade performance if the signal processing assumes that what is measured is the gradient.The gravitational eye, on the other hand, takes into account the fact that the measurement is a difference over a finite interval, so does not suffer from this issue.
In this paper we have looked at the concept of a gravitational eye which can measure either the potential or its radial derivative over the surface of a sphere.In order to study the resolution, we have looked at the innate two-point resolution in the data and also, by analogy with conventional gravity surveys and satellite geodesy, how the one-point resolution is enhanced by analytic continuation in the direction of the sources.
We have considered how such a system, could be constructed using cold-atom accelerometers, which can be tilted relative to the vertical and might operate as a gravitational eye.While all three of the possible implementations discussed here could be used to create a gravitational eye, a full design study is needed to evaluate which architecture would yield the best field instrument.Alternatively, instead of requiring a solution that could work for the full range of possible angles, a reduced range of angles could be explored by the gravitational eye, allowing more information to be gained than a single measurement in the vertical would, but not as much as a full gravitational eye.We term this concept the restricted vision gravitational eye and it will be the subject of further work.
We have seen it is possible to discern information about the surroundings without having to translate the system towards, or away from, a feature of interest.This ability will be particularly important in several application areas including CCS, oil and gas, speleology, geothermal, tunnel inspection and border control.
The normalisation constant A lm is given by With this normalisation convention we have that the functions X lm satisfy for a given order m.We note further that the spherical harmonics are eigenfunctions of the Laplace-Beltrami operator on the 2-sphere: with eigenvalues −l (l + 1).The solid harmonics are solutions to Laplace's equation in spherical polar coordinates and are given by r l Y lm , r −(l+1) Y lm the former being the regular solid harmonics and the latter the irregular ones.

Appendix B. Analytic continuation in ellipsoidal coordinates
In case one views the use of spherical analytic continuation as overly restrictive, in this appendix we sketch the case of data measured on a sphere and analytically continued outwards to a tri-axial ellipsoid.This is inspired by problems in satellite geodesy where the Brillouin surface is sometimes chosen to be an ellipsoid, i.e. we have satellite data recorded on a sphere and then continued downwards to a Brillouin ellipsoid which fits the object of interest (for example the Martian moon Phobos or the asteroid 433 Eros [89]) better than a Brillouin sphere would.
Here we consider the opposite problem where the data are recorded on a sphere which does not enclose any gravitating object.The data are then analytically continued outwards to a tri-axial ellipsoid which better fits the shape of, say, an enclosing underground chamber.Loosely speaking, this helps because the analytic continuation is not totally dominated by the air gap between the surface one is continuing onto and the surrounding rock wall.This should then make it easier to identify sources of interest within the rock.Note that if the ellipsoid is sufficiently elongated then it may be better to perform the survey using a combination of rotations and translations.
The reader should note that, unlike the satellite-geodesy problem, we are unlikely to be able to measure many sphericalharmonic coefficients and this may put a limit on the effectiveness of the analytic continuation in ellipsoidal coordinates.
It is easier to analyse this problem in terms of the potential.This is because our data for the gravimeter-based systems involves radial derivatives, which reflect the coordinate system (spherical polars) that is most appropriate for the data recording.However, if we are doing analytic continuation in ellipsoidal coordinates then it seems inappropriate to try to analytically continue a quantity pertinent to a different coordinate system.Hence, as a first step one should consider integrating with respect to radius the recorded gravity vector, or the radial gravity gradient, to get a potential, albeit one missing the first term (the constant) or the first two terms in the case of the gradient.
We start by considering analytic continuation in ellipsoidal coordinates between ellipsoidal surfaces.This is more complicated than the spherical analogue, mainly because of the difficulty of calculating the ellipsoidal harmonics and also calculating the expansion coefficients.Both internal and external potential problems for ellipsoidal coordinates are discussed in [90].
The equivalent of figure 2 in the main text is shown in figure B1.

B.1. Ellipsoidal coordinates
We denote the ellipsoidal coordinates by (ρ, µ, ν).The first of these refers to the size of a particular ellipsoid and the remaining two are variables describing position on this ellipsoid.There are various definitions for the ellipsoidal coordinates, for example that in [23].We use the convention in [89] and [91].Suppose we have positive real numbers h < k.Then the ellipsoidal coordinates corresponding to Cartesian coordinates (x, y, z) are solutions to the equation Multiplying out by the denominators leads to a cubic equation for λ 2 for which there are three real roots satisfying These roots then satisfy the equations so that surfaces of constant λ 1 correspond to ellipsoids, surfaces of constant λ 2 correspond to one-sheeted hyperboloids and surfaces of constant λ 3 correspond to two-sheeted hyperboloids.These surfaces intersect orthogonally.
Let us fix ρ so that we have an ellipsoid with semi-axes a > b > c.We then have that h and k are the focal lengths:

B.2. Laplace's equation in ellipsoidal coordinates-the Lamé functions
Laplace's equation in ellipsoidal coordinates separates into three copies of the same equation-Lamé's equation-where the three ellipsoidal coordinates correspond to different portions of the same functions, namely the Lamé functions.Lamé's equation can be written in the form (written here for the first ellipsoidal coordinate) Here p takes the role of an eigenvalue.The solutions to this equation are the Lamé functions.The integer n is the degree.
The Lamé functions fall into four classes, denoted K, L, M and N.For each class there are two kinds of function.For functions of the first kind and for a given degree n the corresponding functions are given by (for the first ellipsoidal coordinate) where u K , u L , u M , u N are particular polynomials corresponding to the given class.There are multiple polynomials in each class, with differing degrees.One finds that, counting the different polynomials in all four classes, there are 2 n + 1 for each degree n.
The Lamé functions of the first kind are suitable for the interior problem.The Lamé functions of the second kind are equivalent to the irregular solid harmonics but are not representable in the same way.They are used for the exterior problem and are more difficult to calculate than the functions of the first kind.Reference [89], contains a good description of ellipsoidal harmonics.Section 3.3 of [90] describes the degree and order structure of the Lamé functions.
Since we are concerned with the interior potential problem we can write the potential, within the Brillouin ellipsoid, as where the E m n are Lamé functions of the first kind.For the case of the exterior potential problem, the first of the Lamé functions in the product on the right-hand side is replaced by the equivalent Lamé function of the second kind (see Hu [section 3.3,89]).Here n is the degree of the Lamé function and m is equivalent to the order of the spherical harmonics, different values of m corresponding to different values of the eigenvalue p as well as different classes.
The coefficients c mn are obtained by integrating the potential over the ellipsoid of size ρ.Introducing auxiliary variables and the numbers if we have a function f (µ, ν) given by the expansion then the expansion coefficients are given by where the normalisation coefficient is given by Having determined the values of the expansion coefficients of the potential at a particular value of ρ, analytic continuation is accomplished by varying ρ in (B1).

B.3. Analytic continuation with differently-shaped surfaces
Analytic continuation is usually done between surfaces of the same shape.Suppose we measure the data on a surface of one shape and wish to continue to data on a surface of a different shape.From our perspective, if we consider the interior problem and we assume that we have a sensor which measures data on a sphere, if the cavity in which the instrument sits is more ellipsoidal than spherical, which in general it will be, then analytically continuing to the Brillouin ellipsoid may give better results than continuing to the Brillouin sphere.
We show the exterior and interior potential problems in figure B2.
Referring now to the interior potential problem shown in figure B3, given an estimate of the potential on the inner sphere this may be analytically continued in spherical polar coordinates to give its value on the bounding ellipsoid.Following this the analytic continuation in the first ellipsoidal coordinate to the Brillouin ellipsoid may be carried out.The caveat here is that the two analytic continuations are both in unstable directions.As an alternative strategy, it may be possible to estimate the ellipsoidal harmonic coefficients from the values of the spherical-harmonic coefficients on the data sphere.This is discussed in [92].

Figure 1 .
Figure 1.Potential on the equator of two lead balls spaced π/2 radians apart, with one ball located at π radians and the other at 3 π/2 radians.

Figure 2 .
Figure 2. Cross-section of problem showing data sphere, Brillouin sphere and gravitating object of interest; exterior potential problem (A) and interior potential problem (B).

Figure 3 .
Figure 3. Variation of point-spread function for analytic continuation with number of terms, L + 1, in (6).

Figure 4 .
Figure 4. Example of the operation of the single gravimeter-based gravitational eye.The gravimeter (denoted by 2) is rotated in the equatorial plane around the point 1.This measures the radial derivative of the potential at 2 along the line joining 1 and 2. The angle ϕ represents azimuth.We measure the response to a lead ball inserted in the equatorial plane at an azimutof zero radians.

Figure 5 .
Figure 5. Radial gravity component due to two 20 kg lead balls spaced π/2 radians apart in the equatorial plane, with one sphere located at π radians and the other at 3π/2 radians.

Figure 6 .
Figure 6.Variation of point-spread function of the radial derivative of the potential for analytic continuation with number of terms, L.

Figure 7 .
Figure 7. (A) Simple two-gravimeter configuration, (B) two-gravimeter configuration with centre of rotation at one gravimeter, (C) three-gravimeter configuration with centre of rotation at the central gravimeter.

Figure 8 .
Figure 8. Response to a single lead ball, located at π radians.

Figure 9 .
Figure 9. (A) The response to two lead balls with a dip of 1E between the peaks, with one ball located at π radians and the other at 3π/2 radians.(B) The response to four lead balls positioned at 0, π/2, π and 3 π/2 radians.

Figure 10 .
Figure 10.Variation of point-spread function for analytic continuation of a gradiometer signal with number of terms, L − 1.

Figure 11 .
Figure 11.Cold-atom gravimetry based on light pulse interferometry, with g showing the direction of gravitational acceleration.(A) Operation of an atom interferometer as a gravity sensor.The solid lines show the atomic trajectory in the absence of gravity, while the dotted lines show the atomic trajectory in the presence of gravity.(B) In a typical implementation of an atom interferometer, a cloud of atoms (black dot) is generated and the beam used to generate the interferometry pulses is delivered from above or below and reflected by a mirror.A simple sequence consists of the following steps; once having generated a cloud of atoms they will be allowed to fall under gravity at a time t 0 , then at later time t 1 the first interferometry pulse will be applied followed by the second after a time T at time t 2 , with the final interferometry pulse applied at t 3 after another time T. (C) If this experimental system is tilted and the same experimental sequence performed, as the atom cloud falls under gravity it will fall at an angle to the interferometry beam.If the angle is sufficiently large then the atom cloud will leave the beam before all of the pulses can be applied, rendering the measurement invalid.

Figure 12 .
Figure 12.Implementation of a gravitational eye in which the T time of the system is adjusted such that the atom cloud will remain in the beam regardless of orientation.Insets show the trajectory of the atom cloud (black circle) in the beam at different angles from the vertical.(A) The possible time the atoms can fall for without leaving the interferometry beam (B) The atom shot noise sensitivity which could be achieved at each angle for a gravitational eye based on a single gravimeter.(C) The atom shot noise sensitivity which could be achieved at each angle for a gravitational eye based on two gravimeters, separated by a 1 m baseline.

Figure B1 .
Figure B1.Cross-section of the problem showing data ellipsoid, Brillouin ellipsoid and gravitating object of interest; (A) exterior potential problem, (B) interior potential problem.

Figure B2 .
Figure B2.Data recorded on a sphere with analytic continuation to the Brillouin ellipsoid, showing both Brillouin sphere and Brillouin ellipsoid; (A) exterior potential problem, (B) interior potential problem.

Figure B3 .
Figure B3.The two analytic continuations needed.Analytic continuation from the inner sphere to the bounding ellipsoid followed by analytic continuation from the bounding ellipsoid to the Brillouin ellipsoid.