WO2008140498A1

WO2008140498A1 - Gravitational tomography technique for determining a mass distribution

Info

Publication number: WO2008140498A1
Application number: PCT/US2007/024453
Authority: WO
Inventors: Colin P. Williams; Igor K. Kulikov
Original assignee: Williams Colin P; Kulikov Igor K
Priority date: 2006-11-28
Filing date: 2007-11-28
Publication date: 2008-11-20

Abstract

Embodiments of a system (such as a computer system), a method, a computer program product (i.e., software) for use with the system, and a circuit are described. These techniques may be used to determine a mass distribution in a region or a volume. In some embodiments, the system measures one or more components of the gravitational field at locations on one or more surfaces exterior to a volume. Next, the system optionally computes the one or more components of the gravitational field at additional locations on the one or more surfaces based on an initial mass distribution within the volume. Then, the system determines the mass distribution based on the one or more measured components of the gravitational field at the locations and the one or more computed components of the gravitational field at the additional locations.

Description

GRAVITATIONAL TOMOGRAPHY TECHNIQUE

FOR DETERMINING A MASS DISTRIBUTION

Inventors: Colin P. Williams and Igor K. Kulikov

FIELD OF THE INVENTION

[0001] The present invention relates generally to an apparatus, and related methods, for processing data, and more specifically, for determining a mass distribution in a region.

CROSS REFERENCE TO RELATED APPLICATIONS

[0002] This application claims priority under 35 U.S.C. Section 119(e) to U.S.

Provisional Application Serial Number 60/861,552, "Gravitational Tomography for Detecting Nuclear Weapons in Shipping Containers," filed on November 28, 2006, the contents of which are herein incorporated by reference.

BACKGROUND

[0003] Nuclear weapons concealed in shipping containers remain a significant threat.

Each year, some 7 million shipping containers are imported, unloaded near major population centers, and distributed nationwide via road and rail networks. A nuclear weapon in just one of these shipping containers could kill millions of people and destroy an entire city. Consequently, there is a strong need to monitor shipping containers. [0004] Unfortunately, there are counter-measures and/or limitations associated with many existing monitoring techniques. For example, radiological scanning of shipping containers can be defeated by lead cladding. Moreover, imaging of shipping container using electromagnetic radiation (such as X-rays or gamma-rays) can damage cargo (thereby impeding legitimate trade) and risks exposing personnel to large doses of radiation. Moreover, the use of electromagnetic radiation to examine a shipping container that did contain a concealed nuclear device could initiate a nuclear detonation. [0005] Hence, what is needed is a method and an apparatus that facilitates examination of shipping containers and, more generally, regions or volumes without the problems listed above.

SUMMARY [0006] Embodiments of a system (such as a computer system), a method, a computer program product (i.e., software) for use with the system, and a circuit are described. These techniques may be used to determine a mass distribution in a region or a volume. In some embodiments, the system measures one or more components of the gravitational field at locations on one or more surfaces exterior to a volume. Next, the system optionally computes the one or more components of the gravitational field at additional locations on the one or more surfaces based on an initial mass distribution (such as a spatial mass distribution) within the volume. Then, the system determines the mass distribution based on the one or more measured components of the gravitational field at the locations and the one or more optional computed components of the gravitational field at the additional locations. [0007] In some embodiments, the measurements are performed using gravimeters, which may be arranged in an array. These gravimeters may include: cold atom gravimeters, superconducting quantum interference device gravimeters, Bose Einstein condensate-based gravimeters, and/or electromagnetically induced transparency-based gravimeters. [0008] In some embodiments, a given surface in the one or more surfaces includes a plane. Moreover, the one or more components may include three components of the gravitational field. Note that the three components may be substantially perpendicular to each other.

[0009] In some embodiments, the one or more surfaces include a pair of surfaces, where a first surface in the pair of surfaces is on an opposite side of the volume than a second surface in the pair of surfaces. Moreover, the surfaces in the pair of surfaces may be approximately symmetrically positioned with respect to a center of the volume. [0010] In some embodiments, the one or more surfaces include three pairs of surfaces. Note that a first surface in a given pair of surfaces may be on an opposite side of the volume than a second surface in the given pair of surfaces. Moreover, the given pair of surfaces may be approximately perpendicular to a given dimension (such as an axis) and the given dimension may be approximately orthogonal to dimensions associated with the other pairs of surfaces. [0011] In some embodiments, the mass distribution is determined by inverting an equation corresponding to: the one or more measured components of the gravitational field at the locations, one or more optional computed components of the gravitational field at the additional locations, and the initial mass distribution. For example, the inverting may include a singular value decomposition of a matrix associated with the equation. Moreover, the number of measurements of the one or more measured components of the gravitational field at the locations may exceed the number of volume elements in the initial mass distribution that are used to compute the one or more optional computed components of the gravitational field at the additional locations so that the equation is overdetermined. [0012] In some embodiments, the mass distribution is determined by multiplying the one or more components of the gravitational field at the locations by an inversion matrix. [0013] In some embodiments, the mass distribution is determined by minimizing the difference between the one or more measured components of the gravitational field at the locations and the one or more optional computed components of the gravitational field at the additional locations. For example, the minimizing may include Maximum Likelihood Estimation and/or least squares minimization.

[0014] In some embodiments, the mass distribution includes a 2-dimensional distribution and/or a 3-dimensional distribution within the volume. [0015] In some embodiments, the volume includes a container, a vehicle, and/or a building.

[0016] In some embodiments, the mass distribution is determined without measuring electromagnetic radiation or particles having mass that pass through or are emitted within the volume.

[0017] However, in other embodiments the mass distribution is determined in conjunction with measurements of electromagnetic radiation or particles having mass that pass through or are emitted within the volume. Moreover, the mass distribution may be determined in conjunction with measurements of one or more components of the gradient of the gravitational field. Consequently, data associated with one or more additional measurements (such as the gravitational gradient, electromagnetic radiation and/or particles having mass that pass through or are emitted within the volume) may used in the equation and, thus, during the inverting operation described above.

[0018] In some embodiments, the initial mass distribution is homogeneous.

[0019] In some embodiments, the measurements are performed simultaneously. [0020] In some embodiments, the locations on a given surface are regularly spaced.

Moreover, the locations may be the same as or different than the additional locations.

[0021] In some embodiments, the volume moves relative to sensors that perform the measurements during and/or between the measurements. [0022] In some embodiments, the system compares the mass distribution to an expected mass distribution to identify differences.

[0023] In some embodiments, the determining is performed in two or more processors or processor cores that execute instructions in parallel.

[0024] In some embodiments, the mass distribution facilitates identification of an amount of an element or a compound within the volume.

[0025] In some embodiments, information corresponding to the one or more components of the gravitational field, such as one or more components of the gravitational acceleration or a scalar potential associated with the gravitational field, are used instead of the one or more components of the gravitational field. [0026] In some embodiments, the initial mass distribution defines volume elements within the volume, and a mass of a given volume element is a variable that is determined when the mass distribution is determined. Consequently, the initial mass distribution may be a hypothesized mass distribution and/or a function.

[0027] Another embodiment relates to a computer program product for use in conjunction with the system. This computer program product may include instructions corresponding to at least some of the above-described operations.

[0028] Another embodiment provides a computer system. This computer system may execute instructions corresponding to at least some of the above-described operations.

Moreover, these instructions may include high-level code in a program module and/or low- level code that is executed by a processor in the computer system.

[0029] Another embodiment provides a method for determining the mass distribution.

This method may perform at least some of the above-described operations.

[0030] Another embodiment provides a circuit that is configured to perform at least some of the above-described operations. [0031] Another embodiment provides an integrated circuit that includes the circuit.

[0032] Another embodiment provides another system that includes an array of sensors and a computational mechanism. This array of sensors may be configured to measure the one or more components of the gravitational field at the locations on the one or more surfaces exterior to the volume. Moreover, the computational mechanism may be configured to optionally compute the one or more components of the gravitational field at the additional locations on the one or more surfaces based on the initial mass distribution within the volume, and may be configured to determine the mass distribution based on the one or more measured components of the gravitational field at the locations and the one or more optional computed components of the gravitational field at the additional locations. [0033] The disclosed embodiments reduce or eliminate the problems described above and provide an analysis technique to: determine a mass distribution in a region or a volume; store the determined mass distribution in a computer-readable medium; and/or display the determined mass distribution on a display.

BRIEF DESCRIPTION OF THE FIGURES

[0034] FIG. IA is a block diagram illustrating an embodiment of a system.

[0035] FIG. IB is a block diagram illustrating an embodiment of a system.

[0036] FIG. 1C is a block diagram illustrating an embodiment of a system. [0037] FIG. 2 is a block diagram illustrating an embodiment of elements in a volume.

[0038] FIG. 3 is a flow diagram illustrating an embodiment of a process for determining a mass distribution.

[0039] FIG. 4A is a graph illustrating a mass distribution.

[0040] FIG. 4B is a graph illustrating measured acceleration as a function of sensor location.

[0041] FIG. 5 A is a graph illustrating computed mass as a function of element number in an object.

[0042] FIG. 5B is a graph illustrating a determined mass distribution.

[0043] FIG. 6 is a block diagram illustrating an embodiment of a circuit. [0044] FIG. 7 is a block diagram illustrating an embodiment of a computer system.

[0045] FIG. 8 is a block diagram illustrating an embodiment of a data structure.

[0046] Table 1 provides some chemical elements and their densities.

[0047] Table 2 provides limiting sensitivities for the gravitational tomography technique. [0048] Note that like reference numerals refer to corresponding parts throughout the drawings. DETAILED DESCRIPTION

[0049] The following description is presented to enable any person skilled in the art to make and use the invention, and is provided in the context of a particular application and its requirements. Various modifications to the disclosed embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be applied to other embodiments and applications without departing from the spirit and scope of the present invention. Thus, the present invention is not intended to be limited to the embodiments shown, but is to be accorded the widest scope consistent with the principles and features disclosed herein. [0050] Embodiments of a system (such as a computer system), a method, and a computer program product (i.e., software) for use with the system, a circuit, and an integrated circuit that includes the circuit are described. These systems, software, and processes may be used to determine a mass distribution in a region or a volume. In particular, the system may measure one or more components of the gravitational field at locations on one or more surfaces exterior to the volume (for example, using gravitometers). Alternatively, the system may obtain measurement results of the one or more components of the gravitational field at the locations, for example, by accessing stored data in a computer-readable memory. Note that in some embodiments at least some of the measurements are performed simultaneously. [0051] Next, the system may optionally compute the one or more components of the gravitational field at the same and/or additional locations on the one or more surfaces based on an initial mass distribution within the volume (for example, using an initial spatial mass distribution). For example, the initial mass distribution within the volume may be homogeneous (such as a density of zero or a finite density). In some embodiments, the initial mass distribution may define volume elements within the volume and the masses associated with these volume elements may be variables that are to be determined in the analysis technique described below. Consequently, the initial mass distribution may be a hypothesized mass distribution and/or a function.

[0052] Then, the system may determine the mass distribution based on the one or more measured components of the gravitational field at the locations and the one or more optional computed components of the gravitational field at the same and/or the additional locations. In particular, the mass distribution may be determined by inverting an equation corresponding to: the one or more measured components of the gravitational field at the locations, one or more optional computed components of the gravitational field at the same and/or the additional locations, and the initial mass distribution. For example, the inverting may include a singular value decomposition of a matrix associated with the equation. Moreover, the number of measurements of the one or more measured components of the gravitational field at the locations may exceed the number of volume elements in the initial mass distribution that are used to compute the one or more optional computed components of the gravitational field at the same and/or the additional locations so that the equation is overdetermined. Note that, in general, the masses associated with at least some of the volume elements in the determined mass distribution are different from each other. [0053] Additionally, the mass distribution may be determined by minimizing the difference between the one or more measured components of the gravitational field at the locations and the one or more optional computed components of the gravitational field at the same and/or the additional locations. For example, the minimizing may include Maximum Likelihood Estimation and/or least squares minimization. [0054] This technique may be implemented separately from or in conjunction with other techniques. For example, additional measurements may be based on: one or more components of the gradient of the gravitational field and/or ionizing radiation (such as electromagnetic waves and/or particles having mass) that passes through and/or is emitted within the volume. However, in some embodiments this technique may be implemented passively, e.g., without providing or measuring ionizing radiation that passes through and/or is emitted within the volume.

[0055] Note that the determined mass distribution may be used to identify the presence of an amount of an element or a compound within the volume. For example, the element or compound may be or may be associated with a radioactive material, such as uranium or plutonium. Moreover, the volume may include: a container, a vehicle, and/or a building.

[0056] In some embodiments, this technique is implemented in software and/or in hardware. Moreover, the software may include high-level code in a program module and/or low-level code that is executed by one or more processors in a computer system. [0057] We now describe embodiments of a system to determine a mass distribution. This system implements a technique that infers the density distribution inside an object (and, therefore, the mass distribution for a given volume associated with the object) using 'gravitational tomography' (which is described further below with reference to FIG. 2). As shown in Table 1 , nuclear materials have higher densities than more common materials, such as metals. Consequently, these materials may possess distinct gravitational signatures within the object (such as a shipping container). Moreover, as gravitational fields cannot be shielded, it may be more difficult to develop a counter-measure for this system.

Table 1

[0058] In some embodiments, this technique is passive, and does not require the use of ionizing radiation, such as electromagnetic waves (for example, X-rays), muons, and/or neutrons. Note that in the discussion that follows ionizing radiation includes electromagnetic radiation or particles having mass that pass through or are emitted within a volume, such as that associated with the object. Moreover, by performing at least some of the measurements and/or the calculations in this technique in parallel, threat/non-threat classification times on the order of fractions of a second with acceptable rates for false-positive and no false- negatives can be obtained. [0059] This technique leverages the sensitivity of modern gravimeters and gravity gradiometers (which include two or more gravimeters that are coupled to each other), which are able to measure gravitational field changes as small as those associated with 10^~13 g. Such sensors can detect a one kg object at a distance often meters. [0060] FIG. IA presents a block diagram illustrating a system 100. In this system, one or more groups of sensors 112 are positioned along one or more surfaces (such as one or more planes, circumferences, or rectangular rings) that are exterior and/or adjacent to a volume 110-1. For example, one group of sensors 112-1 may be positioned a distance 114-1 from a center of the volume 110-1, and another group of sensors 112-2 may be positioned a distance 114-2 from the center of the volume 110-1.

[0061] Note that the sensors 112 may be arranged in arrays with uniform and/or non- uniform spacing. Moreover, these arrays may be 1 -dimensional, 2-dimensional and/or 3- dimensional (for example, inter-digited sensors 112) along the one or more surfaces. Additionally, note that the volume 110-1 may correspond to: an object, a container (such as a shipping container), a vehicle, a building, an animal, and/or a human. Furthermore, the volume 110-1 may be rectangular or may have an arbitrary shape. [0062] As illustrated in FIG. IA, note that the one or more surfaces may include a pair of surfaces, where a first surface in the pair of surfaces is on an opposite side of the volume than a second surface in the pair of surfaces. Moreover, the surfaces in the pair of surfaces may be approximately symmetrically positioned with respect to a center of the volume 110-1 (e.g., the distances 114 may be approximately equal). [0063] Each of the sensors in the groups of sensors 112 may be configured to measure the one or more components of the gravitational field (or the related gravitation acceleration) at a given location on a given surface (relative to the volume 110-1) at a given time or within a time interval. In some embodiments, the sensors 112 include gravimeters and/or gradiometers. Note that the gravimeters and/or the gradiometers may include: cold atom gravimeters, superconducting quantum interference device gravimeters, Bose Einstein condensate-based gravimeters, falling corner cubes, rotating accelerometers, and/or electromagnetically induced transparency-based gravimeters. Moreover, in embodiments where the gravimeters are cold atom gravimeters, a single common laser beam and/or multiple common laser beams may be used. [0064] At least some of the sensors 112 may perform measurements of the one or more components of the gravitational field simultaneously. Moreover, the one or more components may include three components of the gravitational field. Note that the three components may be substantially perpendicular to each other, such as the x-, y-, and z- components of the gravitational field.

[0065] Collectively, the sensors 112 may measure a set of data which includes measurements of the one or more components of the gravitational field at multiple locations on the one or more surfaces. As discussed further below with reference to FIG. 2, a computational mechanism may use this set of data to determine a mass distribution within the volume 110-1. In particular, the computational mechanism (such as circuit 600 in FIG. 6 and/or computer system 700 in FIG. 7) may optionally compute the one or more components of the gravitational field at the multiple locations and/or at other locations on the one or more surfaces based on an initial mass distribution within the volume 110-1 (i.e., may optionally compute another set of data). Note that this initial mass distribution may be pre-determined or may be calculated from the set of data. Then, the computational mechanism may determine a 2-dimensional or 3-dimensional mass distribution within the volume 110-1 using the measured set of data and the other optional computed set of data. [0066] This determined mass distribution may facilitate identification of the presence of an amount of an element or a compound within the volume 110-1. In some embodiments, the mass distribution is compared to an expected or pre-determined mass distribution of the volume 110-1 (or the associated object) to identify differences, such as those associated with the element or compound. Note that the element or compound (and, more generally, a material) may include: an element in the periodic table, an alloy, a chemical compound, a liquid, a mineral, an ore, an organic material, an inorganic material, a radioactive element, a radioactive alloy, uranium, plutonium, gold, and/or platinum.

[0067] In some embodiments, the mass distribution is determined without measuring ionizing radiation, such as electromagnetic radiation or particles having mass that pass through or are emitted within the volume 110-1. However, in other embodiments the mass distribution is determined in conjunction with additional measurements performed by one or more additional sensors 116. Consequently, data associated with one or more additional measurements may used when determining of the mass distribution. These additional measurements may include: measurements of ionizing radiation and/or measurements of one or more components of the gradient of the gravitational field at one or more locations (such as locations on one or more of the surfaces). For example, the one or more additional sensors 116 may measure: muon scattering, the presence of radioactivity, and/or gravitational gradients. [0068] In general, sensors 112 may enclose a given volume on one or more sides.

This is shown in FIG. IB, which presents a block diagram illustrating a system 140 in which two pairs of groups of sensors 112 are positioned on four sides of volume 110-2. [0069] Note that in some embodiments, sensors 112 measure the one or more components of the gravitational field on all sides of the volume 110-2. Thus, in some embodiments the one or more surfaces include three pairs of surfaces. Note that a first surface in a given pair of surfaces may be on an opposite side of the volume than a second surface in the given pair of surfaces. Moreover, the given pair of surfaces may be approximately perpendicular to a given dimension (such as an axis) and the given dimension may be approximately orthogonal to dimensions associated with the other pairs of surfaces. [0070] In some embodiments, the sensors 112 are at fixed positions relative to the given volume. However, in some embodiments the sensors 112 can be displaced either during and/or between measurements. This is shown in FIG. 1C, which presents a block diagram illustrating a system 170. In this system, group of sensors 112-5 may be moved, with a velocity 180, along one or more directions. For example, the group of sensors 112-5 may be attached to a gantry. Other groups of sensors (not shown) may be attached to additional gantries that may be moved separately and/or in conjunction with the group of sensors 112-5. [0071] Note that the one or more directions may be principal axes of volume 110-3. Moreover, the motion may be relative, i.e., either the group of sensors 112-5 and/or the volume 110-3 may move with respect to each other. In some embodiments, the motion is intermittent.

[0072] In some embodiments, the gantry or gantries allow fine positioning of the sensors 112 at particular locations on the one or more surfaces. Moreover, the ability to move the sensors 112 may facilitate an order of magnitude decrease in the amount of time needed to measure the one or more components of the gravitational field at the multiple locations on the one or more surfaces.

[0073] Note that systems 100 (FIG. IA), 140 (FIG. IB) and/or 170 may include fewer components or additional components. Moreover, two or more components can be combined into a single component and/or the position of one or more components can be changed. [0074] As noted previously, the calculation of the mass distribution includes computing the one or more components of the gravitational field at locations on the one or more surfaces. In some embodiments, during this calculation, it is assumed that the given volume (such as the volume 110-3) includes multiple elements, thereby approximating the shapes of the actual objects inside the given volume. This is shown in FIG. 2, which presents a block diagram illustrating an embodiment of elements 210 in a volume 200. Moreover, each of the elements 210 (such as element 210-1) may be assigned an initial mass based on the initial mass distribution.

[0075] In some embodiments, the initial mass distribution is homogeneous. For example, the elements 210 may each have the same mass. However, in other embodiments the set of data is inverted (as described further below) to determine the initial mass distribution. In general, after the mass distribution is determined, the masses associated with the elements 210 will vary from element to element.

[0076] Note that a wide variety of shapes may be used for the elements 210. For example, the shapes may include: rectangular, cubic, spherical, ellipsoidal, and/or a platonic solid (such as a tetrahedra, hexahedra, octahedra, dodecahedra, and/or icosahedra). Moreover, the elements 210 may have the same size and shape, or at least some of the elements 210 may have different sizes and/or shapes. Additionally, in some embodiments a combination of shapes and sizes are used to fill the volume 200 with or without voids. [0077] In some embodiments, the shapes of the elements 210 are chosen such that there is a closed-form expression for the associated components of the gravitational field. This allows the gravitational field contributions from the elements 210 to be summed to obtain a closed-form expression for the gravitational field associated with the volume 200 at given points surrounding the volume 200. As described further below, this expression for the other optional computed set of data may be inverted to allow the mass at each of the elements 210 to be determined based on the measured set of data. [0078] Note that when the elements 210 are small enough, and the set of data is measured over a sufficiently fine grid (e.g., closely spaced locations on the one or more surfaces), the mass distribution with the volume 200 may be determined with sufficient resolution that the presence of anomalously high densities can be identified, thereby allowing the presence of nuclear or other high-density materials to be detected. Moreover, as described further below, the useful size of the elements 210 is related to the accuracy of the measurements of the one or more components of the gravitational field. Further advances in sensor sensitivity will allow the mass distribution to be determined with even finer resolution. [0079] We now describe the mathematical underpinnings of the gravitational tomography technique. We start with the development of a tomographic reconstruction process, which is a mathematical procedure that allows a gravitational tomogram to be obtained. Note that the gravitational field of an object (such as a shipping container) can be obtained from the equation for gravitational acceleration:

where G is the Newtonian gravitational constant, p is the density of the object, and V is the volume of the object. This equation can be expressed as a sum of n integrals over rectangular volumes of elements (such as the elements 210) that each have constant densities. Each of these elements produces a field of gravitational acceleration g = (g_x, g_y, g_z). Moreover, the total gravitational field around the object is the geometrical sum of the gravitational acceleration of the elements. Assuming that each element has sides having lengths (2X, 27, 2Z), Eqn. (1) can be rewritten in component form as:

g,(X) = -G∑F_m(x)p_n. (2)

Note that the functions F_ιn(x) may be written as:

+ H₁(X₁ -X, x₂ -Y, x₃ -Z) -H \(x_x + X, X₂-Y, X₃ -Z) -H₁(X_x -X, X₂+ Y, X₃ -Z)-H₁(X_x -X, x₂ -Y, x₃ +Z)

*^•»(*) = (3) + H₁(X_x +X,x₂ +Y,x₃ -Z) + H₁(X_x +X,x₂ -Y, X₃ +Z) + H₁(X_x -X, x₂ + Y, x₃ +Z)-H₁(X_x +X,x₂ +Y,x₃ +Z)

In Eqn. (3), x, (where i - 1, 2, or 3) are the coordinates of the location, functions H are

and R is given by the equation R(X) = (Xf + X) +

. Note that Eqn. (2) may be expressed as:

g_x (X) = -G∑ F_xn(x)_Pn

g_y(x) = -G∑F_yn(^χ:)p_π (5) n=]

[0080] As described previously in FIGs. 1 A-IC, the set of data (i.e., one or more components of the gravitational field) are measured at locations around the object. For example, in FIG. IA the locations are located in 2-dimensional planes above and below the object. Note that in some embodiments the locations are measured in the six planes around the object.

[0081] Eqn. (5) may be solved. In particular, the system of equations shown in Eqn.

(5) is linear and can be written in a symbolic form as:

where/is a vector constructed from n components of measured gravitational accelerations {gj} and p is the vector consisting of A: element densities. Because this system of equations is linear, the solution to the inversion problem can be expressed symbolically as:

As follows from this equation, the density for each element may be found by applying the inverse operator F^"1 (such as an inversion matrix) to the data array/ Note that the matrix F is defined only by the geometry of the object, the locations of elements and their sizes.

Moreover, if the size of the vector/is smaller than the size of the vector p then the system of the equations in Eqn. (6) will be underdetermined and its solution will not be unique. Consequently, in some embodiments, the length(/) is greater than or equal to the length(p), thereby resulting in a one-to-one or an overdetermined system of equations. Therefore, the data in Eqn. (6) can be fit to the set of data using Maximum Likelihood Estimation (MLE), which is based on minimization of the functional S:

with respect with unknown densities p. The resulting equation is p = (F^TF)-\Ff). (9)

Note that in some embodiments other minimization techniques, such as least squares minimization, is used. Moreover, in some embodiments, the mass distribution is determined using an iterative minimization technique. Additionally, in some embodiments the pseudo- inverse of matrix F is used. [0082] In some embodiments the measured set of data and the calculated set of data include a scalar potential associated with the gravitational field. Moreover, in some embodiments, where the additional locations are different than the locations, an interpolation technique is used to generate measured and/or calculated sets of data at common locations for use in the gravitational tomography technique.

[0083] Note that the measured set of data, the optional computed set of data, the initial mass distribution and/or the determined mass distribution may be represented: in the spatial domain, in the frequency domain (for example, using spatial frequencies), and/or in functional form (for example, using wavelets).

[0084] In some embodiments, a singular value decomposition technique is used to invert the mapping between the densities of the elements 210 (FIG. 2) and the gravitational field values, such that the densities of the elements 210 (FIG. 2) can be expressed as a function of the gravitational field values. Then, the densities of the elements 210 (FIG. 2) can be determined such that the difference between the calculated and measured gravitational field values are minimized. In particular, by using the singular value decomposition of matrix F, the solution of the system of the equations in Eqn. (6) can be written in the form:

p = (VWU^T)f . (10)

Note that Eqn. (10) gives us the mass distribution inside the object and a tomographic image of the spatial variation in mass density.

[0085] We now describe embodiments of a process for determining a mass distribution. FIG. 3 presents a flow diagram illustrating a process 300 for determining a mass distribution, which may be performed by a system. This system may measure one or more components of the gravitational field at locations on one or more surfaces exterior to a volume (310). Next, the system may optionally compute the one or more components of the gravitational field at additional locations on the one or more surfaces based on an initial mass distribution within the volume (312). [0086] Then, the system may determine the mass distribution based on the one or more measured components of the gravitational field at the locations and the one or more optional computed components of the gravitational field at the additional locations (314). In some embodiments, the system optionally compares the mass distribution to an expected mass distribution to identify differences (316). [0087] Note that in some embodiments of the process 300 there may be additional or fewer operations. Moreover, the order of the operations may be changed and/or two or more operations may be combined into a single operation. [0088] We now describe exemplary embodiments of the technique for determining a mass distribution. Note that there are limitations on the size of the elements 210 (FIG. 2). In particular, while the accuracy of the reconstruction of the mass distribution in the interior of the object can be increased as the size is decreased, the size cannot be arbitrarily small because there will be insufficient information in the set of data to determine the mass distribution at such a fine granularity. Moreover, as the size of the elements 210 (FIG. 2) decreases, the number of elements 210 (FIG. 2) increases, which (as discussed further below) can result in a non-unique solution to the inversion problem that is being solved. Consequently, there is a trade-off between the accuracy of the reconstruction of the mass distribution (as well as the uniqueness of a given solution), the number of measurements in the set of data and the total time required to complete the measurements. (As noted previously, the number of unknown masses or elements 210 in FIG. 2 should not be larger than number of measurements in the set of data.) [0089] Additionally, the accuracy of the measurement of gravitational field at each of the locations depends on the measurement time, e.g., how long it takes to measure the one or more components of the gravitational field at each location. In general, the accuracy of the measurement of gravitational field decreases as the measurement time decreases. Moreover, as the number of elements 210 (FIG. 2) increases, the measurement time at each location may decrease. [0090] Table 2 illustrates an exemplary embodiment of the capabilities of the gravitational tomography technique for appropriate trade offs of these factors (which are described further below). In particular, for gravitational sensors, having a sensitivity (in m/s²), placed near a corner of an object, Table 2 indicates the smallest mass (in kg) that can be detected at either the center of the object or the furthest corner of the object away from the gravitational sensors. Moreover, Table 2 also indicates the minimum size (in cm) of one side or edge of the elements 210 (FIG. 2) that can be detected when the object is: water, uranium or plutonium.

Table 2

[0091] As noted previously, to obtain an accurate reconstruction of the mass distribution in the object, a large number of elements 210 (FIG. 2) may be used. For example, for an object having a volume of 30 m³, there are 1920 elements having an edge size of 0.25 m or a volume of 0.015625 m³. With this size element, an object having a volume of 16 m³ has 1024 elements. Moreover, an object having a volume of 1 m³ has 1000 elements having an edge size of 0.1 m or a volume of 0.001 m³. If each measurement takes one second, then the measurement times for these objects are 1920, 1024 and 1000 seconds, respectively. To speed up the data collection, arrays of gravity sensors may be used to perform measurements concurrently. For example, if there are N measurements and each measurement takes a time T, then the total measurement time is NT. If an array often gravity sensors performs measurements in parallel, the total measurement time can be reduced by a factor often.

[0092] For an object having a volume of 30 m and 1920 elements of volume

0.015625 m , the total measurement time may be 192 seconds. Larger arrays may allow the data to be collected even faster. Thus, an array of twenty gravity sensors can be used to collect the data in 96 seconds. Similarly, for the objects having volumes of 16 m³ and 1 m³, respectively, the total measurement time can be 50 seconds.

[0093] At first inspection it may seem difficult to determine a unique gravitational inverse for a given set of data in the vicinity of an object. However, ambiguity in the inversion is only a true problem in the continuum limit. If a finite size is used for the elements 210 (FIG. 2), then there is a definite unique mass distribution that best fits the set of data. [0094] More precisely, the discrete problem considered here approximates the underlying infinite dimensional problem when the number of elements 210 (FIG. 2) becomes infinitely large. Stated differently, discrete problems may become ill-conditioned as they become more accurate. However, problems with ill-posedness (or ill conditioning in the discrete case) can be overcome using regularization techniques (such as Tichonov regularization) and/or by imposing exogenous information regarding the known (predetermined) properties of the actual mass distribution within the object. For example, additional information or constraints on the solution may include: maximum and/or minimum mass or densities; a pre-defined density profile (such as one based on a shipping manifest); and/or a total mass.

[0095] In an exemplary embodiment, a shipping container having a volume of 30 m³ and weight 7000 kg, is filled with a homogeneous material, having the density of light wood, except for three masses in a plane inside of the shipping container. This is shown in FIG. 4 A, which presents a graph 400 illustrating a mass distribution as a function of dimensions 410 of the shipping container. Note that masses 412 include two aluminum slabs and a cross made of dense glass (flint).

[0096] The gravitational field associated with this mass distribution was computed over a fine grid on the surface of the shipping container. This is shown in FIG. 4B, which presents a graph 450 illustrating measured acceleration 460 (in nm/s²) as a function of sensor location 462 along a face of the shipping container.

[0097] This computer- generated data set was then used in conjunction with the gravitational tomography technique (which is also referred to as an inversion technique) to determine the mass distribution that best agreed with the computer-generated data set. During this inversion calculation, there were 1920 elements each having a volume of 0.015625 m³ in the shipping container.

[0098] The determined mass distribution is shown in FIG. 5A, which presents a graph

500 illustrating computed mass 510 (kg) as a function of element number 512 in the object. Note that the gravitational tomography technique determined nine mass concentrations of about 50 kg each, and four mass concentrations of about 40 kg each. Moreover, the spatial coordinates of these mass concentrations are shown in FIG. 5B, which presents a graph 550 illustrating a determined mass distribution as a function of the dimensions 410 of the shipping container. Note the estimated masses 512, which correspond to the masses 412 (FIG. 4A). [0099] We now describe embodiments of a circuit that may determine the mass distribution. FIG. 6 presents a block diagram illustrating an embodiment 600 of a circuit 610. This circuit may implement the analysis (i.e., the gravitational tomography technique) of the measured set of data. In particular, interface 612 may receive the measured set of data (for example, from the system 100 in FIG. IA). Computational manager 614 may distribute portions of the analysis to one or more processors cores or processors 616, and these processors may execute instructions associated with at least a portion of the analysis in parallel (e.g., concurrently). Results of this analysis, such as the determined mass distribution, may be stored in memory 618. [00100] Components and/or functionality illustrated in the embodiments shown in FIG.

6 may be implemented using analog circuits and/or digital circuits. Moreover, components and/or functionality in these embodiments may be implemented using hardware and/or software. [00101] Note that circuit 600 may include fewer components or additional components. Moreover, two or more components can be combined into a single component and/or the position of one or more components can be changed.

[00102] We now further describe systems that may utilize one or more embodiments of the technique for determining a mass distribution. FIG. 7 presents a block diagram illustrating an embodiment of a computer system 700. Computer system 700 includes: one or more processors or processor cores 710 (which are means for processing information), a communication interface 712, a user interface 714, and one or more signal lines 722 coupling these components together. Note that the one or more processors 710 may support parallel processing and/or multi -threaded operation, the communication interface 712 may have a persistent communication connection, and the one or more signal lines 722 may constitute a communication bus. Moreover, the user interface 714 may include: a display 716, a keyboard 718, and/or a pointer 720, such as a mouse.

[00103] Memory 724 in the computer system 700 may include volatile memory and/or non-volatile memory. More specifically, memory 724 may include: ROM, RAM, EPROM, EEPROM, flash, one or more smart cards, one or more magnetic disc storage devices and/or one or more optical storage devices. Memory 724 may store an operating system 726 that includes procedures (or a set of instructions) for handling various basic system services for performing hardware dependent tasks. Memory 724 may also store communication procedures (or a set of instructions) in a communication module 728. These communication procedures may be used for communicating with one or more computers and/or servers, including computers and/or servers that are remotely located with respect to the computer system 700.

[00104] Memory 724 may include multiple program modules (or a set of instructions), including: measurement module 730 (or a set of instructions), computation module 732 (or a set of instructions), and/or comparison module 734 (or a set of instructions). Measurement module 730 may instruct a device or system to perform multiple measurements of one or more components of the gravitational field. These measurements may be performed at multiple locations on one or more surfaces exterior to a region or a volume, which are associated with an object (such as a shipping container). Note that at least some of these measurements may be performed simultaneously. After the measurements are performed, measurement module 730 may store measurements 746, including field components 748 that are measured at the multiple locations. [00105] Based on one or more volumes 736 and one or more initial mass distributions 738, computation module 732 may optionally calculate values of the one or more gravitational field components at the multiple locations and/or at additional locations. Moreover, computation module 732 may store calculations 750, including field components 752 that are optionally computed at the multiple locations and/or at the additional locations. Additionally, computation module 732 may store one or more determined mass distributions 744.

[00106] Note that computation module 732 may instruct one or more of the processors

710 to perform these computations, and that at least some of these computations may be performed in parallel with each other. In some embodiments, computation module 732 determines the one or more mass distributions 744, at least in part, using additional measurements 742 (such as measurements of ionizing radiation and/or one or more components of the gravitational gradient at the multiple locations and /or at other locations). [00107] Comparison module 736 may compare one or more expected mass distributions 740 with the one or more determined mass distributions 744. This comparison may be used to identify the presence of an amount of an element or compound in the one or more volumes 736. Results of the comparison may be displayed on display 716. For example, a displayed image may include: intensity variations to represent materials of different mass density; color coding to represent materials of different mass density; only a subset of the elements in one or more volumes 736 to highlight those of greatest interest; spatial variations of the determined mass distribution(s) in a three dimensional form (such as a hologram); spatial variations of the determined mass distribution(s) 744 in a set of 2- dimensional forms; and/or statistical distributions of the determined mass distribution(s) 744, with or without information about the spatial locations of the associated elements in the one or more volumes 736.

[00108] Instructions in the various modules in the memory 724 may be implemented in a high-level procedural language, an object-oriented programming language, and/or in an assembly or machine language. The programming language may be compiled or interpreted, i.e., configurable or configured to be executed by the one or more processors 710. [00109] Although the computer system 700 is illustrated as having a number of discrete components, FIG. 7 is intended to provide a functional description of the various features that may be present in the computer system 700 rather than as a structural schematic of the embodiments described herein. In practice, and as recognized by those of ordinary skill in the art, the functions of the computer system 700 may be distributed over a large number of servers or computers, with various groups of the servers or computers performing particular subsets of the functions, hi some embodiments, some or all of the functionality of the computer system 700 may be implemented in: one or more ASICs, one or more FPGAs, and/or one or more digital signal processors DSPs. Moreover, in the computer system 700 may include a supercomputer and/or a cluster computer. [00110] Computer system 700 may include: fewer components or additional components. Moreover, two or more components may be combined into a single component and/or a position of one or more components may be changed. In some embodiments the functionality of the computer system 700 may be implemented more in hardware and less in software, or less in hardware and more in software, as is known in the art. [00111] We now describe embodiments of data structures that may be used in the computer system 700. FIG. 8 presents a block diagram illustrating an embodiment of a data structure 800. This data structure may include measured and/or optional computed gravitational field information for one or more locations 810 on a one or more surfaces exterior to a region or a volume, which are associated with an object (such as a shipping container). For a given location, such location 810-1, data structure 800 may include values of one or more gravitational field components 812. Note that that in some embodiments, data structure 800 includes fewer or additional components. Moreover, two or more components may be combined into a single component, and/or a position of one or more components is changed.

[00112] While the preceding discussion has used the gravitational field as an illustrative example, in other embodiments this technique may be applied to: spatial and/or temporal derivatives of the gravitational field; to other fields; and/or to other types of data (e.g., to inversion problems associated with ionizing radiation). Moreover, in some embodiments this technique may be used to detect and track the motion of objects having mass inside of electromagnetically opaque structures.

[00113] The foregoing descriptions of embodiments of the present invention have been presented for purposes of illustration and description only. They are not intended to be exhaustive or to limit the present invention to the forms disclosed. Accordingly, many modifications and variations will be apparent to practitioners skilled in the art. Additionally, the above disclosure is not intended to limit the present invention. The scope of the present invention is defined by the appended claims.

Claims

WHAT IS CLAIMED:

1. A method for determining a mass distribution, comprising: measuring one or more components of the gravitational field at locations on one or more surfaces exterior to a volume; computing the one or more components of the gravitational field at additional locations on the one or more surfaces based on an initial mass distribution within the volume; and determining the mass distribution based on the one or more measured components of the gravitational field at the locations and the one or more computed components of the gravitational field at the additional locations.

2. The method of claim 1, wherein the measurements are performed using an array of gravimeters.

3. The method of claim 1, wherein the measurements are performed using gravimeters.

4. The method of claim 3, wherein the gravimeters include cold atom gravimeters, superconducting quantum interference device gravimeters, Bose Einstein condensate-based gravimeters, electromagnetically induced transparency-based gravimeters.

5. The method of claim 1, wherein a given surface in the one or more surfaces includes a plane.

6. The method of claim 1, wherein the one or more components include three components of the gravitational field.

7. The method of claim 6, wherein the three components are substantially perpendicular to each other.

8. The method of claim 1, wherein the one or more surfaces include a pair of surfaces; and wherein a first surface in the pair of surfaces is on an opposite side of the volume than a second surface in the pair of surfaces.

9. The method of claim 8, wherein the surfaces in the pair of surfaces are approximately symmetrically positioned with respect to a center of the volume.

10. The method of claim 1, wherein the one or more surfaces include three pairs of surfaces; wherein a first surface in a given pair of surfaces is on an opposite side of the volume than a second surface in the given pair of surfaces; and wherein the given pair of surfaces is approximately perpendicular to a given dimension and the given dimension is approximately orthogonal to dimensions associated with the other pairs of surfaces.

11. The method of claim 1, wherein the mass distribution is determined by inverting an equation corresponding to the one or more measured components of the gravitational field at the locations, one or more computed components of the gravitational field at the additional locations, and the initial mass distribution.

12. The method of claim 11, wherein the inverting includes a singular value decomposition of a matrix associated with the equation.

13. The method of claim 11, wherein the number of measurements of the one or more measured components of the gravitational field at the locations exceeds the number of volume elements in the initial mass distribution that are used to compute the one or more computed components of the gravitational field at the additional locations so that the equation is overdetermined.

14. The method of claim 1, wherein the mass distribution is determined by minimizing the difference between the one or more measured components of the gravitational field at the locations and the one or more computed components of the gravitational field at the additional locations.

15. The method of claim 14, wherein the minimizing includes Maximum Likelihood Estimation or least squares minimization.

16. The method of claim 1, wherein the mass distribution includes a 2-dimensional distribution within the volume.

17. The method of claim 1, wherein the mass distribution includes a 3-dimensional distribution within the volume.

18. The method of claim 1, wherein the volume includes a container, a vehicle, or a building.

19. The method of claim 1, wherein the mass distribution is determined without measuring electromagnetic radiation or particles having mass that pass through or are emitted within the volume.

20. The method of claim 1, wherein the mass distribution is determined in conjunction with measurements of electromagnetic radiation or particles having mass that pass through or are emitted within the volume.

21. The method of claim 1, wherein the mass distribution is determined in conjunction with measurements of one or more components of the gradient of the gravitational field.

22. The method of claim 1, wherein the initial mass distribution is homogeneous.

23. The method of claim 1, wherein the measurements are performed simultaneously.

24. The method of claim 1, wherein the locations on a given surface are regularly spaced.

25. The method of claim 1, wherein the volume is moving relative to sensors that perform the measurements during the measurements.

26. The method of claim 1, wherein the locations are different than the additional locations.

27. The method of claim 1, further comprising comparing the mass distribution to an expected mass distribution to identify differences.

28. The method of claim 1, wherein the determining is performed in two or more processors or processor cores that execute instructions in parallel.

29. The method of claim 1, wherein the mass distribution facilitates identification of an amount of an element or a compound within the volume.

30. The method of claim 1, wherein the initial mass distribution defines volume elements within the volume; and wherein a mass of a given volume element is a variable that is determined when the mass distribution is determined.

31. A computer program product for use in conjunction with a computer system, the computer program product comprising a computer-readable storage medium and a computer- program mechanism embedded therein for determining a mass distribution, the computer- program mechanism including: instructions for measuring one or more components of the gravitational field at locations on one or more surfaces exterior to a volume; instructions for computing the one or more components of the gravitational field at additional locations on the one or more surfaces based on an initial mass distribution within the volume; and instructions for determining the mass distribution based on the one or more measured components of the gravitational field at the locations and the one or more computed components of the gravitational field at the additional locations.

32. A computer system to determine a mass distribution, comprising: a processor; memory; a program module, wherein the program module is stored in the memory and configurable to be executed by the processor, the program module including: instructions for measuring one or more components of the gravitational field at locations on one or more surfaces exterior to a volume; instructions for computing the one or more components of the gravitational field at additional locations on the one or more surfaces based on an initial mass distribution within the volume; and instructions for determining the mass distribution based on the one or more measured components of the gravitational field at the locations and the one or more computed components of the gravitational field at the additional locations.

33. A computer system to determine a mass distribution, comprising: means for processing; memory; a program module, wherein the program module is stored in the memory and configurable to be executed by the means, the program module including: instructions for measuring one or more components of the gravitational field at locations on one or more surfaces exterior to a volume; instructions for computing the one or more components of the gravitational field at additional locations on the one or more surfaces based on an initial mass distribution within the volume; and instructions for determining the mass distribution based on the one or more measured components of the gravitational field at the locations and the one or more computed components of the gravitational field at the additional locations.

34. A system, comprising: an array of sensors configured to measure one or more components of the gravitational field at locations on one or more surfaces exterior to a volume; and a computational mechanism configured to compute the one or more components of the gravitational field at additional locations on the one or more surfaces based on an initial mass distribution within the volume, and configured to determine a mass distribution based on the one or more measured components of the gravitational field at the locations and the one or more computed components of the gravitational field at the additional locations.

35. An integrated circuit, comprising: an interface configured to obtain one or more measured components of the gravitational field at locations on one or more surfaces exterior to a volume; and a computational mechanism configured to compute the one or more components of the gravitational field at additional locations on the one or more surfaces based on an initial mass distribution within the volume, and configured to determine a mass distribution based on the one or more measured components of the gravitational field at the locations and the one or more computed components of the gravitational field at the additional locations.