DIFFRACTION TOMOGRAPHY SYSTEMS AND METHODS

Abstract: Diffraction tomography systems and methods are dis­ closed for the reconstruction of physical properties of two and three-dimensional objects from information collected in one or more diffraction tomographic procedures. A diffraction tomo­ graphic system is comprised of a continuous or pulsed source of wave energy, a detecting system for the measurement of the amplitude and/or phase of the scattered field resulting from the wave energy interacting with a two or three-dimensional ob­ ject, a processing and filtering system for processing and filter­ ing the measured signals according to a filtered backpropaga­ tion technique thereby generating arrays which are used to re­ construct the object properties, and a graphics display system for displaying obtained reconstructions. Specific embodiments include ultrasound transmission computed tomography; well-­ to-well electromagnetic and sonic tomography; subsurface electromagnetic or seismic exploration; and X-ray transmis­ sion tomography. The methods and systems apply filtered backpropagation techniques which are generalizations of the filtered backprojection technique of X-ray computed tomo­ graphy to cases where diffraction of the insonifying beam is taken into consideration.

Description: BACKGROUND OF THE INVENTION This invention relates to systems and methods for reconstructing acoustic and/or electromagnetic properties of two or three-dimensional objects using diffraction tomographic procedures. The systems and methods are useful in the geophysical prospecting arts, such as well-to-well tomography and subsurface seismic prospecting. In conventional parallel beam transmission computed tomography, an object profile O(x,y) representing, for example, an X-ray attenuation coefficient, is reconstructed from the object's projections where (ξ,η) denote the coordinates in a cartesian coordinate system rotated by the angle ϑ relative to the (x,y) system as shown in Figure 1a. The reconstruction of O(x,y) from P ϑ (ξ) (which is called the "reconstruction of the object" or the "reconstruction of the object profile" by those skilled in the art) is made possible by the projection-slice theorem which states that the one-dimensional Fourier transform of P ϑ (ξ) is a slice at angle ϑ through the two-dimensional Fourier transform ∼ / O(K x ,K y ) of O(x,y) as shown in Figure 1b. Thus, the projections P ϑ (ξ) yield, upon Fourier transformation, an ensemble of slices of the two-dimensional Fourier transform ∼ / O(K x ,K y ) of O(x,y). O(x,y) can then readily be reconstructed from this ensemble of slices via its Fourier integral representation expressed in circular polar coordinates. In particular, one finds that where is the one-dimensional Fourier transform of P ϑ (ξ). In Equation (2), the K integration is limited to the interval -W to W since, in any application, ∼ / P ϑ (K) will be known only over a finite bandwidth which is taken to be -W to W. Because of this restriction, the resulting reconstruction is a low pass filtered version of O(x,y). In practice, in place of the Fourier integral representation given in Equation (2), a technique known in the art as the filtered backprojection algorithm is used to reconstruct O(x,y). Defining the filter function which, when convolved with a low pass filter, is known in the art as the standard X-ray tomography deblurring filter, it can be shown (see, for example, A.C. Kak, "Computerized Tomography with X-ray Emission and Ultrasound Sources"; Proc. IEEE 67, pp. 1245-1272 (l974) that Equation (2) can be written in the form where Q ϑ (t), called "filtered projections", are given by The process of reconstructing O(x,y) according to the prior art thus consists of first filtering the projections with the filter function h(t) and then backprojecting the filtered projections. The process of backprojection used in the prior art consists of assigning any given pixel value (x,y) within the image array the value of the filtered projection Q ϑ (t) at the point Backprojection at any given view angle ϑ thus results in a partial image consisting of parallel straight lines as defined by Equation (7) with the grey level of each line being assigned the value Q ϑ (t). Following the process of backprojection at a fixed view angle, the entire process (filtering followed by backprojection) is repeated at different angles and the resulting partial image arrays are added pixel by pixel onto the previously existing partial image (hence, the integral over d ϑ in Equation 5). In this manner, the profile of the object is reconstructed. In practice, of course, the filtered projection is only known for a finite number of angles so that a discrete approximation (e.g. a summation) to the backprojection integral Equation (5) is employed. The success of transmission computed tomography clearly depends on the availability of the projections P ϑ (ξ). In X-ray tomography, these projections are available in the form of measurements of the log amplitude [1/2 log(intensity)] of the electromagnetic field produced when a plane, monochromatic X-ray beam propagating along the η axis is incident to the object as is illustrated in Figure 2a. At X-ray wavelengths (≈1A °) the incident plane wave suffers very little scattering (refraction or diffraction) in passing through the object. Absorbtion does occur, however, so that the negative of the log amplitude of the field along the line η=1₀ is simply the integrated value of the X-ray absorbtion coefficient µ(x,y) along the travel path (i.e. along the straight line ξ=constant). The object profile O(x,y) in this case is then µ(x,y) and the projections P ϑ (ξ) are simply the negative of the log amplitude of the electromagnetic field as measured along the line η=1₀. In ultrasonic tomography, the wavelength is considerably greater (≈1mm.) than that of X-rays so that an acoustic wave experiences scattering, in the form of refraction and diffraction, in the process of propagating through an object. The log amplitude of the acoustic field along the line η=1₀ resulting from an insonifying plane wave propagating along the η axis is then not simply the projection of the acoustic absorbtion coefficient along the line ξ=constant as is the case in X-ray tomography. This breakdown in the relationship between measured log amplitude and projections of the absorbtion coefficient is one of the major reasons why conventional computed tomography is of limited use in ultrasonic applications. Although it is not possible, in general, to obtain an exact expression for the acoustic field generated by the interaction of an insonifying plane wave with an acoustic (diffracting) object O(x,y), expressions within the so-called first Born and Rytov approximations are available. The Born and Rytov approximations are essentially "weak scattering" approximations and, as such, will be valid for cases where the deviations of the object profile O(x,y) from zero are small. Which approximation, Born or Rytov, is employed depends upon the specific application, although the Rytov approximation is generally thought to be superior in medical ultrasound tomography (See, for example, M. Kaveh, M. Soumekh and R. K. Mueller, "A Comparison of Born and Rytov Approximations in Acoustic Tomography", Acoustical Imaging , Vol. II Plenum Press New York, (1981).) A more complete treatment of the Born and Rytov approximations is found in A. J. Devaney, "Inverse Scattering Theory Within the Rytov Approximation," Optics Letters 6 , pp. 374-376 (1981). Within these approximations, it has been shown by the inventor in "A New Approach to Emission and Transmission CT" 1980 IEEE Ultrasonics Symposium Proceedings , pp. 979-983 (1980); "Inverse Scattering Theory Within the Rytov Approximation" Optics Letters 6 , pp. 374-376 (1981); and by R.K. Mueller, et al., "Reconstructive Tomography and Application to Ultrasonics", Proc. IEEE 67 , pp. 567-587 (1979), that for an incident monochromatic plane wave propagating along the η axis, the profile O(x,y) is related to the processed signals (hereinafter referred to as "data") taken along the line η=1₀ via the equation where the variable κ can assume all values in the range[-k,k]. Here ϑ₀ = ϑ + π/2 is the angle that the wave vector of the insonifying plane wave makes with the x axis, ω is the angular frequency, and k=2π/λ is the wavenumber of the insonifying acoustic field. The data are shown in the case of the Born approximation, to be simply related to the complex amplitude of the scattered field portion of the acoustic field evaluated along the line η=1₀ and, in the case of the Rytov approximation, to the difference in the complex phase of the total field and incident field evaluated along this same line. For discussion regarding diffraction tomography, it is convenient to introduce two unit vectors; where ξ⋄and and η⋄and are unit vectors along the ξ, η axes. The unit vector s ₀ is simply the unit propagation vector of the insonifying plane wave. For values of |κ|<k, Equation (8) can be expressed in terms of s and s ₀ in the form From Equation (10) it may be concluded that the one-dimensional Fourier transform of is equal to the two-dimensional Fourier transform ∼ / O(K x ,K y ) of the profile O(x,y) evaluated over the locus of points defined by where the unit vector s assumes all values for which s · s ₀>0. Equation 10 may thus be seen as the diffraction tomography equivalent of the projection slice theorem. The K values satisfying Equation (11) for fixed s ₀ with s · s ,>0, are seen to lie on a semicircle, centered at -k s ₀ and having a radius equal to k. By changing the direction of propagation of the insonifying plane wave, it is thus possible to sample ∼ / O(K x ,K y ) over an ensemble of semicircular arcs such as illustrated in Figure 2b. The ultrasonic tomographic reconstruction problem then consists of estimating O(x,y) from specification of ∼ / O(K x ,K y ) over this ensemble of arcs. The problem of reconstructing an object profile from specification of its Fourier transform over an ensemble of circular arcs such as shown in Figure 2b is an old one, with proposed solutions having first occurred in X-ray crystallography, H. Lipson and W. Cochran, The Determination of Crystal Structures (Cornell University Press, Ithica, New York, 1966), and later in inverse scattering applications using both X-rays , B.K. Vainshtein, Diffraction of X-rays by Chain Molecules (John Wiley, New York, 1977), and visible light, E. Wolf, "Three-dimensional Structure Determination of Semi-Transported Objects from Holographic Data", Optics Comm. , pp. 153-156 (1969). More recently, these inverse scattering solutions have arisen in ultrasonics, (see e.g. S.J. Norton and M. Linzer, "Ultrasonic Reflectivity Imaging in Three Dimensions: Exact Inverse Scattering Solutions for Phase, Cylindrical and Spherical Apertures," IEEE Trans Biomed. Eng. BME-28, pp. 202-220 (1981)) and also in ultrasonic tomography which has become known as diffraction tomography to distinguish it from conventional tomography where diffraction effects are not taken into account. In all of these applications, the reconstruction of the profile O(x,y) is made possible by the fact that by varying s o over the unit circle, one can determine ∼ / O(K x ,K y ) at any point lying within a circle, centered at the origin and having a radius equal to √ 2k . A low pass filtered version O LP (x,y) of O(x,y) can then be obtained by means of the Fourier integral representation where vector notation is used such that r =(x,y), K =(K x ,K y ) and d²K stands for the differential surface element in K . The article of M.F. Adams et al published in Acoustical Imaging vol. 10, 1982 p. 365-380 "Tomography from Multiview Ultrasonic Diffraction Data : comparison with image reconstruction from projections" describes the reconstruction of an image from ultrasonic diffraction data based on synthetic aperture imaging. An image is reconstructed at any depth of the object by focusing the data at said depth and performing an Inverse Fourrier transform. The focusing at depth z r is achieved by multiplication with a quadratic phase factor dependent on z r . In any application, the profile O( r ) will vanish beyond some finite range so that a Fourier series expansion for O LP ( r ) can be used in place of the Fourier integral (12). According to the solutions proposed in the art, in the Fourier series expansion, the transform ∼ / O(K) must then be known for K values lying on a regular, square sampling grid in K space. To obtain the requisite sample values therefore requires that either one interpolate ∼ / O(K x ,K y ) from the sample values given on arcs over which this quantity is known (interpolation method) or alternatively, that one select propagation directions s ₀ such that at least one arc falls on every equired sample point in the array (direct sampling method). Discussion and examples of the interpolation method can be found in W.H. Carter, "Computerized Reconstruction of Scattering Objects from Holograms", J. Opt. Soc Am 60. , pp. 306-314 (1976), while the direct sampling method procedure is followed in e.g. A. F. Fercher et al., "Image Formation by Inversion of Scattered Field Data: Experiments and Computational Simulation," Applied Optics 18 , pp. 2427-2431 (1979). Both the interpolation and direct sampling methods for obtaining sample values of O and( K ) over a regular square sampling grid have serious drawbacks. Interpolation of ∼ / O( K ) from sample values given on semicircular arcs suffers from the same limitations as does interpolation from slices in K space. Reconstructions based on two-dimensional Fourier Transforms from interpolated sample values in conventional tomography are well known to be inferior in accuracy to those obtained via filtered backprojection. Direct sampling methods, on the other hand, are undesirable because they generally require an extremely large number of s ₀ values (i.e., many experiments). In addition, both procedures place severe if not insurmountable demands on the experimentalist in that they require extreme accuracy in registering data collected from repeated tomographic procedures. The present invention is a great improvement over the prior art in that a method and system are provided which allow the low pass filtered object profile O LP (r) to be reconstructed directly from processed signals without the necessity of first determining ∼ / O( K ) over a regular square sampling grid in K space as was required in the prior art. The provided methods and systems employ the so-called filtered backpropagation algorithm (so named by the inventor) which is closely analogous to the filtered backprojection algorithm of conventional transmission tomography. The two techniques differ in one important respect: in the backprojection process the filtered projections Q ϑ (t) are continued back through the object space along parallel straight line paths; whereas in the process of backpropagation, the filtered phases are continued back in the manner in which they are diffracted. In other words, in the backprojection process, the points along a wave of energy may each be considered to follow straight parallel paths through the object. In the backpropagation process, the points are each considered to undergo scattering throughout the object. For example, the Rytov and Born approximations permit an analysis of the problem by suggesting that only first scatterings be taken into account, and that secondary scatterings may be ignored as insignificant. Thus, in backpropagation, quantities ∼ / Q(t), related to the data through the filtering operation of equation (6), are mapped onto the object space via an integral transform that is the inverse of the transform that governs phase propagation within the Rytov approximation. It is natural then to call this process backpropagation since it corresponds mathematically to the inverse of the usual forward propagation process. The relationship which is employed and embodied in the filtered backpropagation technique has been conveniently denoted by the inventor as the filtered backpropagation algorithm to emphasize its formal similarity to the filtered backprojection algorithm of conventional tomography. Thus, for purposes of this application and the claims herein, the term "backpropagation" shall be defined to mean that operation that is the inverse or approximate inverse of a forward propagation process. The term "filtered backpropagation technique" shall be defined to describe any diffraction tomographic technique for the partial or complete reconstruction of an object where a filtered real or complex amplitude and/or filtered real or complex phase of a wave is backpropagated into the object space; i.e., is propagated back into object space according to the inverse or approximate inverse of the way in which the wave was originally diffracted. The filtered backpropagation technique is usually implemented in the form of a convolution of filters. For purposes of brevity, such an implementation will identically be called the filtered backpropagation technique. A "backpropagation filter" shall be defined to describe that filter of the filtered backpropagation technique which accounts for diffraction in the backpropagation of the phase; e.g. in ultrasound diffraction tomography, the filter of the filtered backpropagation technique which is not the standard x-ray tomographic filter (or a variation thereon). A "filtered backpropagation operation" shall be defined as any procedure which employs the filtered backpropagation technique. SUMMARY In accordance with the invention, a method and a system for tomographic reconstruction of a subsurface earth formation are provided according to claims 1 and 13. A set of embodiments of the invention is provided by electromagnetic or sonic well-to-well tomography. In these geophysical embodiments, one or more sources of electromagnetic or sonic waveforms are placed in a borehole. One or more detectors of electromagnetic or sonic waveforms are placed in a second borehole. The waveforms received in the second borehole are then used to reconstruct the velocity and/or attenuation profile of the formation between the two boreholes. In the borehole to borehole (well to well) situation, only a two-dimensional reconstruction is possible due to the impossibility of rotating either the object (formation) or the sources and detectors. Another set of embodiments of the present invention involves subsurface electromagnetic or seismic exploration where electromagnetic or sonic sources are placed on the earth's surface in the immediate vicinity of a borehole and one or more detectors are placed in the borehole. (A second configuration has the detectors on the surface and the sources in the borehole.) The waveforms received by the detectors are used to reconstruct the velocity and/or attenuation profile of the formation existing between the sources and the detectors. Again, due to rotation limitations, only a two-dimensional image may be reconstructed according to the procedures detailed above. When spherical waves, which are generated by point sources, are used in well-to-well tomography and subsurface seismic exploration, the analog electrical signals produced by the detecting system are converted to digital signals and processed in such a way as to obtain the signal that would have been detected if the transmitted wave were a continuous, cylindrical waveform having a linear phase shift along the transmitter axis. This later signal is then processed so as to obtain its unwrapped complex phase. The unwrapped complex phase is input to the subsystem implementing the filtered backpropagation technique. This subsystem is similar to that used for ultrasound transmission computed tomography using plane waves, but employs a variation of the ultrasound backpropagation filter. Additionally, the standard X-ray deblurring filter is altered substantially. Following the filtered backpropagation operation, the partial image array is transferred to a master image array which may be displayed by suitable means and the entire sequence of steps is repeated for a different choice of linear phase shift. The final two-dimensional image obtained is a rendition of the electromagnetic or acoustic velocity and attenuation profile of the plane section of the formation lying between the two boreholes, or the borehole and the surface, whichever the case may be. It is thus an object of the invention to provide diffraction tomography methods and systems for sonic and/or electromagnetic reconstruction of subsurface earth formations. BRIEF DESCRIPTION OF THE DRAWINGS The present invention may be better understood and its numerous objects and advantages will become apparent to those skilled in the art by reference to the detailed description of the preferred embodiments and by reference to the representative drawings wherein: Figure 1a illustrates the projection obtained in a cartesian coordinate system when conventional parallel beam transmission computed tomography is applied to an object. Figure 1b illustrates a slice through the two-dimensional Fourier transform of an object obtained via the projection slice theorem in conventional parallel beam transmission computed tomography. Figure 2a illustrates the projection of an object in the prior art X-ray tomography. Figure 2b illustrates the diffraction tomography equivalent of the conventional projection slice theorem where the two-dimensional Fourier transform of an object must be evaluated on an ensemble of circular arcs. Figure 3 is a partially schematic and partially block diagram of an ultrasound transmission computed tomographic system useful to understand the present invention. Figure 4 is a block diagram of the preprocessing performed on detected signals in the two-dimensional object system embodiment. Figure 5 is a block diagram of the preprocessing performed on detected signals in an alternative embodiment. Figure 6 is a block diagram of the preprocessing performed on detected signals in the three-dimensional object system embodiment. Figure 7 is a block diagram of the filtered backpropagation technique. Figure 8a is an illustration of an array obtained by the ultrasound preprocessing of received signals from an ultrasound transmission computed tomographic system. Figure 8b is an illustration of the results obtained after the arrays of Figure 8a are filtered by the filtered backpropagation technique of Figure 7. Figure 9 is an illustration of the rotation-interpolation operations used to provide a more complete reconstruction. Figure 10 illustrates the rotation-interpolation operations in block diagram form. Figure 11 is a block diagram illustrating the general method for tomographic reconstruction of an object using ultrasound transmission computed tomography. Figure 12A is a partially schematic, partially block diagram of a geophysical tomographic system of the invention depicting a source-detector and signal preprocessing arrangement. Figure l2b is an illustration of an alternate preferred geophysical tomographic system embodiment. Figure l2c illustrates the borehole to borehole geophysical tomographic system embodiment. Figure 13 illustrates the formation of a cylindrical wave of a determined phase from a series of spherical waves. Figure 14 is a block diagram of the preprocessing and filtering performed on the signals in a geophysical tomographic system embodiment. Figure 15 illustrates the geometry and the associated coordinate system for the geophysical tomography embodiments. Figure 16 illustrates the regions in K space over which the object's Fourier transform is known for geophysical and ultrasonic tomographic procedures. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS ULTRASOUND DIFFRACTION TOMOGRAPHY As seen in Figure 3, an object 10 interposed in a fluid bath 12 is investigated by an ultrasound transmission computed tomographic system useful to explain the present invention. An ultrasound source 14 and a detector plane 16 are located on opposite sides of the object. The source 14 is such that the acoustic wave entering the object 10 is approximately a plane wave. This can be accomplished by appropriately positioning a small disk-shaped single transducer element, or by means of a suitably phased planar array of transducer elements, or by any other suitable means. The ultrasound signal leaving the source can be of either a continuous wave or pulsed type. The center frequency of the source is optimally such that the wave length of the pressure field induced in the fluid batch at this frequency is significantly (i.e., on the order of one magnitude or more) smaller than the dimensions of the object being investigated. The detector plane 16 is preferably aligned parallel to the plane wavefront of the insonifying plane wave and contains one or more detector elements 18. In a typical system the transducer elements used for the source 14 and detectors 18 will be piezo-electric crystals. The system is configured so as to allow rotation between the object, and the detector and source elements of the system, about an axis 0 passing through the object and parallel to the detector plane. For purposes of identification, the plane which is perpendicular to the axis of rotation and which bisects the source, is denoted as the plane of rotation of the system. Thus, in Figure 3, the plane of rotation is the plane of the paper. Likewise, the angle 20 formed between the direction of propagation of the insonifying plane wave 22, and an arbitrary reference line 24 fixed relative to the object and passing through the axis of rotation in the plane of rotation, is denoted as the view angle ϑ₀. After the insonifying wave propagates through the object, the resulting diffracted field is detected by one or more detector elements 18 on the detector plane 16. The detectors 18 are preferably transducers which output electrical signals e m (t) (the voltage at the m'th detector over time). These signals are input into computer and graphics systems 25 which implement a filtered backpropagation technique, generate a partial reconstruction of the object, and display the results. One configuration of the systems disclosed herein is represented in Figure 4, where the acoustic properties of an object do not vary appreciably along the axis of rotation of the system. In these two-dimensional objects, the axis of symmetry of the object is the axis of rotation of both the object and the source-detector system. The detecting system in this case consists of a linear array of M elements evenly spaced over the measurement line -- this line being formed by the intersection of the detector plane 16 with the plane of rotation of the system. The detector elements 18a are separated by a distance approximately equal to the fluid bath wavelength λ₀ corresponding to the center frequency f o of the insonifying plane wave. As shown in Figure 4, the output signals e m (t) from transducers 18a are preprocessed by inputting them, if required or desired, into analog electronic circuitry and a buffer storage system 28a for prefiltering, calibration and temporarily storing the analog electrical signals in preparation for analog to digital conversion. A/D converter 30a converts the stored analog signals into digital format. The digitized electrical signals are then Fourier transformed in time and normalized by suitable means 32a according to the equation where U o is the amplitude of the isonifying wave at its center frequency f o , and k is the wavenumber in the fluid bath at this frequency. The frequency (transform variable) has been set equal to the center frequency f o of the insonifying waveform as those skilled in the art will recognize, because the insonifying wave output by source 14 may be a pulse which is comprised of many sine waves at different amplitudes, phases and frequencies. By choosing the center frequency, a constant frequency is employed which will provide the amplitude and phase of the signal at an optimal signal to noise ratio. After transformation and normalization, electrical output signals E m are further preprocessed by converting them to a complex phase  m by suitable means 34a according to the following operations: where the principle value of the arc tangent is computed. For each value of m, the quantity  m represents the principle value of the complex phase of the pressure field at frequency f o at the spatial position of the m'th detector element. The usefulness of complex phase converter 34a may be understood by realizing that when an energy wave goes through a diffracting body, the body has two effects: It attenuates the waves and thus changes the amplitude; and it changes the phase of the wave. Thus, the complex phase converter 34a manipulates the incoming data so that the amplitude information is contained in Real  m and the phase shift is found in Imag  m . After the electrical signals are converted into complex phases  m , these complex phases are phase unwrapped at 36a and the unwrapped phase array p m is used to perform a partial reconstruction of the object. Phase unwrapping is desirable because complex phase converter 34a treats input values such as 0, 2π and 4π alike, and therefore the output from converter 34a does not necessarily reflect the proper signal. Phase unwrapper 36a, however, as seen in figure 4 by its additional inputs, utilizes the knowledge that the shift between adjacent points cannot be too large (i.e. >2π), and therefore corrects the phase array. Any standard phase unwrapper may be employed for these purposes. Another manner of preprocessing signals e m (t) concerns the cases where the pulse duration of the insonifying plane wave is very short. In this situation the electrical signals e m (t) from the detector array are processed and stored at 28a, A/D converted at 30a and then input into a preprocessing system which may utilize software to estimate the total delay time ξ m between the pulse received at the detector and the transmitted pulse. The peak amplitude of the received pulse A m is also estimated. A block diagram of the preprocessing steps in this embodiment is shown in Figure 5. After ξ m and A m are calculated at 29 and 31 respectively, the unwrapped complex phase p m is approximately determined at 35 via the relationships where ω o = 2πf o is the center frequency of the insonifying pulse. The above two relationships are only approximate but, nevertheless, provide good approximations in cases where the total pulse duration is very short. In addition, the approximation obtained for Imag p m can be used, if desired, to aid in the phase unwrapping operation 36a required in the embodiment discussed above. Yet another manner of preprocessing signals e m (t) concerns cases where the Born approximation for the interaction between the insonifying wave and object is employed. For such cases the normalized, Fourier transformed transducer output signals E m are processed according to the equations where k is the wavenumber of the insonifying acoustic field and ℓ o is the perpendicular distance from the center of rotation of the object to the detector array, to yield the array of complex numbers of p m . A second configuration of the systems disclosed herein applies to objects whose acoustic properties vary appreciably along the axis of rotation of the system (3-D objects). In the preferred embodiment of this configuration, as seen in Figure 6, the detecting system consists of a rectangular NXM array of transducer elements 18 evenly spaced over the detector plane 16. The centers of the detector elements are separated by a distance on the order of the wavelength λ ϑ corresponding to the center frequency f o of the insonifying wave. The electrical signals generated by the detector elements are input into the analog electronic circuitry and a buffer storage system 28b which prefilters, calibrates and temporarily stores the analog electrical signals in preparation for the A/D conversion. After A/D conversion at 30b, the digitized electrical signals from the transducer elements are used to calculate the two-dimensional unwrapped phase array P nm of the pressure field at the detector locations. Again, as in the case of two-dimensional objects, the preprocessing of the digitized signals can be performed in three different preferred manners. In one of the preferred manners, the signals are digitized, Fourier transformed in time and normalized at 32b and the principle valued complex phase  nm corresponding to each detector location on the detector plane is calculated at 34b in the same manner as  m was calculated in the two-dimensional configuration. After the signals are converted into complex phases  nm , the two-dimensional array of complex phases are phase unwrapped at 36b to yield the unwrapped phase array P nm . A second preferred embodiment of the three-dimensional configuration applies to cases where the pulse duration of the insonifying pulse is very short. In this embodiment, the digitized electrical signals e nm (t) from the detector array are preprocessed so as to yield the delay time ξ nm and peak amplitude A nm of the received pulses. These quantities are then processed according to the equations to yield approximate values of the unwrapped complex phase array P nm . A third preprocessing method applies when the Born approximation is employed. For this case, the complex array P nm is calculated according to the equations and the P nm are then further processed as described below. Following the calculation of the two-dimensional unwrapped complex phase array P nm , this array is projected onto the line forced by the intersection of the detector plane with the plane of rotation of the system. The one-dimensional array representing the projected unwrapped complex phases in the three-dimensional system is denoted by p m and is calculated by summing the unwrapped phase array column fashion according to as seen at 38. This one-dimensional array p m is used to perform a partial reconstruction of the object. The projection of the two-dimensional array to arrive at the one-dimensional array p m is required in performing either a reconstruction of a three-dimensional object profile or a reconstruction of a two-dimensional planar slice of an object. Additionally, it should be noted that the one-dimensional array used in partially reconstructing the object is identically denoted p m in both the two-dimensional and three-dimensional configurations. While the notation is identical, as a one-dimensional array is used in the filtered backpropagation technique, the arrays are arrived at differently and contain different information. Those skilled in the art will appreciate that if the three-dimensional embodiment is used on a two-dimensional object to generate array p m , the result after suitable normalization would be identical to the p m array generated by the preferred two-dimensional system. After the above-discussed signal preprocessing, the array of complex numbers p m (m = 1, 2 ..., M) is input into the backpropagation subsystem 50 as shown in Figure 7. Subsystem 50 implements the filtered backpropagation technique. In this embodiment, the quantities p m (m = 1, ..., M) are Fourier transformed at 52 according to the relationship (where j is a dummy variable) to yield array ∼ / p m (m = 1, ..., M). The M complex numbers ∼ / p m are then input to the filter bank 54 which performs the filtered backpropagation operation (i.e. implements the filtered backpropagation technique) and generates the two-dimensional array ∼ / H qm ∼ / p m (q = 1, 2,..., L, m = 1, 2,...,M) where L is the total number of depth samples desired for the object reconstruction. Finally, L one-dimensional Fourier transforms are performed to yield the two-dimensional filtered and backpropagated complex phase X qm according to the relationship where, again, j is used as a dummy variable. In the filter bank, the two-dimensional array ∼ / H qm (q = 1, 2, ... L) (m= 1, 2, ...M) is produced according to the filtered backpropagation technique The quantities ∼ / h m are the standard X-ray tomography smoothing filter, i.e. the Fourier coefficients of a low-pass smoothing filter convolved with an X-ray tomography deblurring filter as discussed above in the Background. The remainder of the filter (e i(γ m -k)(z q - l o ) ) is the preferred backpropagation filter which performs the backpropagation operation. γ m are defined as with k being the wavenumber associated with the center frequency of the insonifying plane wave. The quantities z q (q = 1, 2,...L) are the depth coordinates of lines over which the filtered and backpropagated phase is being evaluated, and l o is the distance between the axis O of the object and the detector plane. Thus, it is seen that the preferred backpropagation filter which is one of the filters of the filtered backpropagation technique, is a non-stationary filter, as it is dependent on object depth z q . The backpropagation filter accounts for the diffraction of waves in the object and permits the object profile to be reconstructed directly from the gathered information without having to use interpolation or direct sampling over a square sampling grid. For two-dimensional objects, the X qm array represents the partial reconstruction of the object profile as determined at the particular view angle employed in obtaining the data. For three-dimensional objects, this array represents the partial reconstruction of the planar projection of the object profile onto the plane of rotation of the system. The q index indicates the depth coordinate while the m index indicates the coordinate position along the line of intersection of the plane of rotation of the system with the detector plane. Figure 8(a) illustrates the p m array at the M sample points along the line of intersection for the case of a 45° view angle. For the purposes of the illustration, p m is plotted as a single real array of sample values. In general, p m is a complex array so that two plots (one for the real part of the array and one for the imaginary part) are required. In Figure 8(b), a filtered and backpropagated phase array X qm is shown for selected object depth values. Note, that again, X qm is plotted as being a single two-dimensional real array whereas, in practice a two-dimensional array would be required. Referring to Figure 8(b), it is seen that the orientation of the q and m axes will always be the same, independent of the view angle. However, the partial reconstruction of the object represented by the X qm array will depend on the view angle. Thus, as seen in Figure 9 it is necessary to rotate the X qm array about the axis of rotation of the system by the angle -ϑ₀ and then interpolate the rotated array onto a master image array which is fixed relative to the object in the plane of rotation of the system. For two-dimensional objects, this master image array represents the ultrasound velocity and attenuation profile of the object. For three-dimensional objects, this array represents the projection of the object's three-dimensional velocity and attenuation profile onto the plane of rotation of the system. A block diagram of the rotation-interpolation operation is given in Figure 10 wherein X qm is rotated by angle -ϑ₀ at 111, and interpolated onto the master array at 113. In practice, any interpolation scheme may be employed. The nearest neighbor interpolation technique is well-suited for these purposes. Finally, by repeating the entire sequence of operations for a new view angle and adding the rotated and interpolated X nm array pixel by pixel onto the master array, a new and improved partial reconstruction of the object (or its planar projection) will be obtained. By repeating this for a number of view angles, a complete rendition of the object (or its planar projection) will be generated. For three-dimensional objects, the plane of rotation of the system can then be changed and the entire sequence repeated to obtain a projection of the object onto a new plane. In this way, any number of planar projections can be obtained from which a complete three-dimensional rendition of a three-dimensional object can be generated using known 3-D tomographic reconstruction techniques. Of course, the master array and if desired, individual p m arrays may be displayed in both two-dimensional and three-dimensional situations on a screen or as a hard copy as desired. For two-dimensional objects, the master image array obtained from the filtered backpropagation technique is a complex valued two-dimensional array. Its real part is a rendition of the real part of the object profile while its imaginary part is a rendition of the imaginary part of the object profile. The real part of the object profile carries information primarily about the acoustic velocity profile of the object and may be displayed on a hard copy or video display system by simply displaying the real part of the master image array. The imaginary part of the object profile carries information primarily about the acoustic attenuation profile of the object and may be displayed on a hard copy or video display system by simply displaying the imaginary part of the master image array. Alternatively, the real (velocity) and imaginary (attenuation) profiles may be displayed jointly using a color graphics display system with different colors being used for velocity and attenuation. It should be noted that both the acoustic velocity and acoustic attenuation profiles of tumors are presently important in medical diagnostics of cancer (see, for example, J. F. Greenleaf and R. C. Bah, "Clinical Imaging With Transmissive Ultrasound Computerized Tomography," IEEE Trans. Biomed. Eng. , BME··28 , 177-185 (1981).) As in the case of two-dimensional objects, the master image array obtained for any given plane of rotation for a three-dimensional object and system is a complex valued two-dimensional array. Its real part is a rendition of the projection of the real part of the three-dimensional object profile onto the plane of the rotation of the system while the imaginary part is the projection of the imaginary part of the three-dimensional object profile onto the plane of rotation of the system. The real and imaginary parts of the two-dimensional master image array may be displayed separately or jointly on hard copy or video graphics display systems as can be done for two-dimensional objects. The real part of the array carries information primarily about the acoustic velocity profile of the three-dimensional object while the imaginary part carries information primarily about the acoustic attenuation profile of the object. It is sometimes desirable to display sections (planar slices) of objects such as biological media rather than planar projections of such media. Sections can be reconstructed from a sufficient number of planar projections by means of the three-dimensional projection slice theorem. In the present invention, the planar projections of the three-dimensional object profiles as generated by the filtered backpropagation technique can be used in conjunction with the three-dimensional projection-slice theorem to yield planar sections of the object profile of the three-dimensional body being examined. The planar sections so obtained can then be displayed on a graphics display system in a way entirely analogous to that employed for two-dimensional object profiles. The planar projections of the object profiles resulting from the filtered backpropagation technique can also be used to obtain an approximate reconstruction of the full three-dimensional object profile. The real and imaginary parts of the three-dimensional profile can then be displayed using known three-dimensional display methods either separately or jointly. Again, as was the case for two-dimensional objects, the real part of the three-dimensional profile carries information primarily about the three-dimensional acoustic velocity profile while the imaginary part carries information primarily about the three-dimensional acoustic attenuation profile. Figure 11 depicts in block diagram form the general method for tomographic reconstruction of an object using ultrasound transmission computed tomography. The methods are integrally connected with the systems described above as will be recognized by those skilled in the art. At step 114, a wave of energy is emitted from a suitable ultrasound source toward the object 115 being examined. The diffracted pressure field produced by the interaction of the wave with the object is detected at step 118 by one or more detectors. As the detectors are typically transducers, the detectors produce an analog signals which are functions of the detected field over time. The produced signals are preprocessed in steps 121, 128, 130, 131, 132, 134, 136, and 138 where they are respectively calibrated, stored, converted into a digital signal, normalized and Fourier transformed, complex phase calculated, phase unwrapped and for three-dimensional objects, projected onto the line formed by the intersection of the detector plane with the plane of rotation of the system. As described above, where the pulse duration of the insonifying wave is very short, steps 132, 134 and 136 are omitted, and in their place delay times and peak amplitude are measured and processed to determine the complex phases at 145. Also, for cases involving the Born approximation, these complex phases are readily calculated directly from the output of 132. In all three methods, the preprocessed signals are input into a system which implements the filtered backpropagation technique in steps 152, 154, and 156. Step 152 Fourier transforms the preprocessed signal, which is a one-dimensional array, and step 154 performs the filtered backpropagation technique. Step 156 Fourier transforms the two-dimensional filtered and backpropagated complex phase resulting from step l54 and in doing so generates partial reconstruction of the object as determined at the particular view angle employed in obtaining the data. In steps 160, 162, and 164, the two-dimensional array which is the partial reconstruction of the object is rotated, interpolated, and stored on the master array. By repeating the method described at a number of view angles and adding the resulting partial reconstructions obtianed at each view angle to the master image array, a complete rendition of a two-dimensional object profile or the planar projection of a three-dimensional object profile onto the plane of rotation of the system is obtained. A complete three-dimensional object profile or a planar slice of the object profile can be reconstructed from a sufficient number of planar projections so obtained, if desired. The obtained arrays, slices, projections of profiles, or profiles may be displayed at 166 using suitable display means. Those skilled in the art will recognize that many permutations and variations may be made in the methods and systems disclosed for the ultrasound computed tomography. For example, the detector system may comprise one detector which is scanned over a line for two-dimensional objects or over a plane for three-dimensional objects. Alternatively, the detector system may include a one-dimensional array of detectors for a two-dimensional object or a one-dimensional array that is scanned in a direction perpendicular to the array for a three-dimensional object. A two-dimensional array of detectors may be used for three-dimensional objects. Another possibility would be to use a single large detector which could recognize variations in the diffracted field. Those skilled in the art will recognize other possible detector schemes. While it is preferable that the detectors be aligned parallel to the plane wavefront of the insonifying plane wave, those skilled in the art will recognize that if desired, the detectors can be placed at an angle and the signals detected can be additionally corrected for the deviation. For example, arrangements such as the hereinafter-discussed geophysical tomography embodiments where the sources and detectors are non-parallel and stationary but where the sources are arranged to produce waves of variable phase delay, may be utilized with proper preprocessing in conjunction with the ultrasound diffraction tomography embodiments disclosed herein. It will also be appreciated that the analog to digital conversion of the data is not necessary to carry out the method. If analog signals are used, the filtered backpropagation technique is slightly modified as the Fourier transform and the filter array must be changed to account for the difference in data form. Likewise, other aspects of the preprocessing may be changed, or deleted as desired by those skilled in the art. Additionally, it will be apparent that the filtered backpropagation technique may be altered according to various circumstances. Thus, the filter bank may be split so that array ∼ / p is first filtered by an x-ray tomography deblurring filter and then separately filtered according to a backpropagation filter. Moreover, ∼ / p n can be convolved into a bank of backpropagation filters on a point by point basis rather than as an entire array. The deblurring filtering and/or backpropagation filtering may also be performed in the space domain instead of the frequency domain, as those skilled in the art will appreciate. Further the backpropagation filter may be mathematically altered as required or desired to provide equivalents which may account for changes in the system. GEOPHYSICAL TOMOGRAPHY The geophysical tomography system and method embodiments of the invention are in many ways closely related to the ultrasound tomography embodiments useful in the medical and other arts. The best modes of the geophysical tomography embodiments which may be used to explore aspects of the subsurface terrain between boreholes or between a borehole and the surface utilize sources, detector-transducers, preprocessing systems, backpropagation filters, and master image arrays. The important differences between the geophysical tomography and ultrasound tomography embodiments stem from the physical situations. Clearly, in geophysical tomography, the formation (body) cannot be rotated relative to the sources and detectors. With the detectors and sources on the surface and in a borehole(s), only two-dimensional approximate reconstructions can be accomplished. Another difference between the systems concerns the sources. In geophysical tomography, the sources are sources of sonic or electromagnetic energy and not ultrasonics due to the depth of investigation required, i.e. the distances from borehole to borehole, or borehole to the surface are much greater than the ultrasonics application and hence larger wavelengths are required to obtain large signal to noise ratios at the receivers. Moreover, in geophysical applications, the sources are sources of spherical waves as the formation (body) is larger in relation to the sources. Due to these and other differences which will become apparent, the preprocessing of the detected signals is different from that employed in ultrasonic tomography. Likewise, the filtered backpropagation technique is altered as the standard X-ray filtering function is radically changed due to the different geometries. The backpropagation filter, however, remains essentially the same. More specifically, Figures 12a, l2b and 12c show three of the preferred geophysical tomography source detector arrangements of the present invention. Figures 12a and l2b show two subsurface electromagnetic or seismic exploration arrangements, while Figure l2c depicts an electromagnetic or sonic well-to-well tomography arrangement. As seen in Figure 12a, the transmitter sources 214a may be arranged on the surface of the formation 210a. The sources 214a are typically electromagnetic or sonic energy sources and may be considered to be sources of spherical waves as the sources are in contact with the formation (body) 210a. The sources are preferably pulsed in either of two manners: sequentially; or pulsed together with a variable phase delay between adjacent transmitters 214a. In both cases, the electromagnetic or sonic energy pulse propagates through the formation 210a and is detected by detector elements 218a which are located in borehole 215a. The signals detected at the j'th detector (j = 1, 2, ..., N j ) for the spherical wave transmitted from the ν'th transmitter (ν = 1, 2, ..., Nν) are denoted for the sequentially-pulsed case as P j (t;ν); and for the phase delayed case as P j (t;ϑ n ) wherein ϑ n is the phase tilt of the transmitted cylindrical wave as shown in Figure 13. Where the sources are pulsed together with a variable phase delay, the detected signals are processed, stored, and A-D converted at break 229a in the same manner as discussed in connection with the ultrasonic embodiments. In the embodiment of the present invention where the sources are pulsed with phase delay; the digitized detected signals P j (t; ϑ n ) are Fourier transformed in time and normalized to yield P j (ω o ,ϑ n ) according to the equation where U o is the amplitude of the insonifying cylindrical pulse at its center frequency f o and ω o = 2πf o is the center frequency in radians/second. In this embodiment, the principle value complex phase  j (ϑ n ) of ∼ / P j (ω o ; ϑ n ) is calculated at 234 of Figure 14 using the relations The principle valued complex phase array  j is then phase unwrapped to yield the unwrapped complex phase array P j (ϑ n ) as indicated at 236. In the situation where the pulse duration of the insonifying pulse is very short, the unwrapped complex phase array P j (ϑ n ) is approximately determined by means of the following relations (which are completely analogous to the ultrasonics short pulse situation): In these relationships, A j (ϑ n ) and ξ j (ϑ n ) are, respectively, the amplitude and phase delay of the pulse received at the j'th detector. Yet another embodiment applies when the Born approximation is employed. In this case, the array P j (ϑ n ) is calculated using the equations where χ is the phase of the insonifying cylindrical wave along the measurement line. The unwrapped complex phase array P j (ϑ n ) obtained by either of the above three embodiments is then input to a backpropagation system which will be described below. When the sources are sequentially pulsed, the detected signals are likewise processed, stored and A-D converted. The digital signals are then Fourier transformed in time and normalized at 231a to generate ∼ / P j (ω o ;ν) and the transformed signal is slant stacked at stacker 233a as is well known in the geophysical arts. The signals ∼ / P j (ω o ;ν) are preferably stacked so as to simulate the signal that would be received at the j'th detector if a cylindrical wave had been transmitted (i.e., to simulate ∼ / P j (ω o ;ϑ n )). Thus, the response of a cylindrical wave is artificially generated by stacking the received signal data. The stacker 233a uses a well-known stacking filter F (ϑ n ,ν) where ϑ n is the phase tilt of the transmitted signal along the transmitter axis as shown in Figure 13. Stacker 233a thus generates ∼ / P j (ω o ;ϑ n ) which is the same signal generated by the phase delayed array. The calculated unwrapped complex phase array P j (ϑ n ) j = 1, 2, ..., N j corresponds to sample values along the borehole axis. The total number of sample values N j is determined by the number of detecting elements in the receiving array. In order to obtain a good reconstruction of the geological formation lying between the transmitter and receiver axis, it is preferable that a great many sample values of the unwrapped complex phase be determined. This may be accomplished in different ways. If an array of detectors is used, the array must be translated along the measurement line to new locations in order to obtain many sample values unless the array is large enough to span the length of the formation of interest. If the array is moved, the entire set of operations delineated above must be repeated for each location. For the case of a single receiver (N j =1), the unwrapped complex phase is determined only at a single sample point for any given location of the detector. If a total of M sample values are desired, it is then necessary to perform M procedures with the transducer being successively moved from sample point to sample point between procedures. By means of repeated procedures, the unwrapped complex phase is determined at a total of M sample points along the measurement axis. The M sample values p m (ϑ n ), (m = 1, 2, ..., M) play the same role in geophysical tomography as does the unwrapped phase array p m , m = 1, 2, ... M obtained in ultrasound tomography. In particular, the array p m (ϑ n ) is input into a backpropagation system 250 to generate a reconstruction of the physical properties of the section of geological formation lying between the transmitter and receiver axes. For the case of sonic energy sources and receivers, the reconstruction yields an acoustic profile analogous to that obtained in ultrasound tomography. For the case of electromagnetic tomography, the reconstruction is of the index of refraction profile of the geological formation. As discussed above, the principle differences between subsurface exploration diffraction tomography and ultrasound diffraction tomography stem from the different geometries involved. The difference in geometries requires a change in the filtered backpropagation technique. In particular, for geophysical applications, the mathematical theory of the prior art discussed in the Background with regard to ultrasonics may be conveniently extended to sonic and elctromagnetic applications. Thus, where it is desirable to reconstruct the profile of a region of geological formation, the Born and Rytov approximations may be employed. Within the approximate models, an equation similar to Equation (10) holds true. In particular, processed signals (hereafter referred to as data) taken along the receiver axis (the line x = 1 o ) as shown in Figure 15,are related to the object profile of the formation O (x,y) via the equation In this equation, s ₀ is the unit vector normal to a cylindrical wave radiating out from the transmitter axis (see Figure 13) and ϑ₀ is the angle that s ₀ makes with the positive x axis. The unit vector s is given by while s ₀ is then where again κ is restricted to lie in the interval -k<κ<k. From Equation (28) it follows that the one-dimensional Fourier transform of the data yields the two-dimensional Fourier transform ∼ / O(K x ,K y ) of the object profile over semicircular arcs. However, these semicircular arcs differ from the ones occurring in ultrasound tomography. In particular, the locus of points defined by Equation (11) with s and s ₀ given by Equations (29a) and (29b) are semicircles centered at -k s ₀ but for which s x ≡ x and · s > 0 rather than s · s ₀ > 0 as was the case in ultrasound tomography. It is interesting to note that the angle ϑ 0 can only range from 0 to π/2 radians in the geophysical case. Consequently, the data will not allow ∼ / O (K x ,K y ) to be sampled throughout a region of K space as large as was obtained in the ultrasound tomography case. This region is shown as area 275 in Figure 16, together with the associated region 285 (| K | <√ 2 k) obtained in the ultrasound tomography case. In the backpropagation system 250 of the subsurface embodiment, after Fourier transformation, the unwrapped phase array P m (ϑ n ) is filtered using a filtered backpropagation technique where and where γ m is defined in Equation (23), and ∼ / x is the unit vector along the x axis In this case the convolutional (stationary) deblurring filter ∼ / h m is no longer the standard x-ray deblurring filter, but depends explicitly on the tilt angle ϑ m of the insonifying cylindrical wave. Also, the backpropagation filter, ∼ / r qm , is slightly altered from the ultrasonics case in that it depends on where x q = qδx and y m = mδy are the (x, y) coordinates associated with the m, q sample point in the image array. However, for the special case where ϑ n = 0, corresponding to the measurement line being perpendicular to the unit wavevector s ₀, the filter functions ∼ / h m and ∼ / r qm reduce to those of the ultrasound diffraction tomography. After the signals have undergone the filtering and backpropagation in a filter bank in backpropagation system 250, the signals are inverse Fourier transformed and interpolated onto a master image array. In the geophysical cases, the arrays need not be rotated as part of the interpolation process as the sources and detectors are not rotated relative to the formation. The master array may be displayed as desired By pulsing the sources 214a at a determined tilt and detecting the produced fields with detector 218a along the borehole axis 215a for the entire area of interest, a partial reconstruction profile of the subsurface formation may be created, as discussed above. In order to reconstruct a more complete profile of the subsurface formation, the phase tilt of sources 214a should cover an entire range of 0 to π/2, wherein the signals for each tilt when separately processed provide a partial reconstruction of the subsurface formation. As those skilled in the art will recognize, a phase tilt of 0 causes the cylindrical wave formed by the spherical waves to propagate straight down into the formation. A phase tilt of π/2 causes the wave to propagate toward the borehole in a manner normal to the surface. The range of phase tilt between 0 and π/2 causes waves entering the formation to propagate at different angles. For each different angle, another partial reconstruction is provided, and the interpolation of those reconstructions onto a master array provides a more complete reconstruction of the subsurface formation. As seen in Figure 12b, the location of the sources 214b and the detectors 218b may be reversed from the previously discussed embodiment shown in Figure 12a. The preprocessing at break 229b and thereafter, and the filtering remain identical. The source 214b, however, is moved in the borehole according to well-known techniques. The detectors 218b may be placed on the surface of the formation or they may be buried slightly under the formation surface as desired. Of course, the phase tilt also covers a range of 0 to π/2. Figure 12c shows the borehole to borehole embodiment of the invention. Sources 214c are located in borehole or well 215c and detectors 218c are located in borehole or well 215d. As discussed with regard to the two other geophysical embodiments, the sources may be pulsed with a phase tilt or in a sequential fashion. The preprocessing at 229c and thereafter and the filtered backpropagation technique are identical to the other geophysical cases. Of prime importance in the borehole to borehole (well to well) embodiment is that the phase tilts of the propagating waves range from 0 to π, thereby providing a more complete reconstruction of the formation profile. In fact, as those skilled in the art will appreciate, if both sources and detectors are placed in both boreholes 215c and 215d, a full two-dimensional reconstruction of the formation profile may be obtained. The best mode embodiments of the methods for reconstructing profiles of subsurface formations are integrally connected with the systems and therefore are analogous to the methods for tomographic reconstruction of an object using ultrasound transmission computed tomography. Thus, at least one wave of electromagnetic or sonic energy is directed at a phase tilt into the formation being investigated. The one or more fields produced by the waves are detected as a function of time and signals representative thereof are produced. The produced signals are preprocessed by calibrating, storing, A/D converting, Fourier transforming, normalizing, and, if necessary, stacking. The preprocessed signals are then Fourier transformed and filtered by a filtered backpropagation technique which includes a deblurring filter and a backpropagation filter which depend on phase tilt angle. After another Fourier transformation, the resulting array is a partial reconstruction of the subsurface formation profile. By repeating the steps at different phase tilts and interpolating the resulting arrays onto a master array, a more complete reconstruction may be obtained. Those skilled in the art will recognize that many permutations and variations may be made in both the methods and systems disclosed for the geophysical (surface and borehole to borehole) embodiments For example, not only may detectors and sources be placed in one or more boreholes in the borehole to borehole embodiment, but sources and/or detectors may also be placed on the formation surface. It will be appreciated that if sources are placed on the surface and only detectors are placed in the two boreholes, a 0 to π range of phase tilts may be covered, and the reconstruction profile of the formation might be quite adequate. In this manner, the problems associated with locating sources in the borehole can be avoided.

Claims: A method for tomographic reconstruction of a subsurface earth formation comprising the steps of: a) directing one or more waves of energy into said formation, the totality of said waves approximating one or more cylindrical waves of a given phase tilt (ϑ n ); b) detecting the one or more fields produced by said one or more waves to produce signals as a function of time of said fields; c) preprocessing said time signals to produce one or more complex signals P j (ϑ n ) which are functions of said one or more detected fields, said complex signals having a real part containing the amplitude information and an imaginary part containing the phase information; and d) filtering said preprocessed signals to generate a two-dimensional array representing the partial reconstruction of the earth formation according to a filtered backpropagation technique: ∼ / H q,m = ∼ / h m . ∼ / r q,m for object depth X q , and where m indexes sample values of the preprocessed signals; where M is the number of detectors detecting said fields; k is the wave number associated with the frequency of said directed waves; x q is the distance of the investigated object depth q from the axis of rotation of said object; x and is the x axis unit vector; l o is the distance between the axis of rotation of said object and the detector array; and s o . r is a scalar product between the unit propagation vector s o and the position vector r . A method for tomographic reconstruction of a subsurface earth formation according to claim 1 characterized by: e) interpolating said partial reconstruction onto a master array: f) changing the phase tilt of said directed waves; and g) repeating steps a-f at different phase tilts to obtain a partial two-dimensional reconstruction of the subsurface formation. A method according to claim 2 characterized by: displaying said partial reconstruction of said subsurface formation. A method according to claim 2 characterized in that: said one or more directed waves are directed from the surface of said formation towards at least one borehole in said formation, said detecting occurring in said at least one borehole. A method according to claim 2 characterized in that: said one or more directed waves are directed from at least one borehole in said formation towards the surface of said earth formation, said detecting occurring on said surface. A method according to claim 2 characterized in that: said one or more directed waves are directed from at least one borehole in said formation toward at least one other borehole in said formation, said detecting occurring in said at least one other borehole A method according to claim 2 characterized in that: said directed waves are directed from at least one borehole in said formation toward at least one other borehole in said formation and toward the surface of said formation, said detecting occurring in said at least one other borehole in said formation and on or above said surface. A method according to claim 2 characterized in that: said directed waves are produced by spherical wave point source emitters which approximate a cylindrical wave by pulsing with a variable phase delay between each emitter. A method according to claim 2 characterized in that: said directed waves are produced by spherical wave point source emitters which in conjunction with said preprocessing approximate a cylindrical wave by sequentially pulsing, wherein said preprocessing comprises stacking the detached signals. A method according to claim 2 characterized in that said preprocessing step comprises: converting said produced signals from analog to digital form: transforming said analog signals over time with a Fourier transform; normalizing said transformed signals; complex phase converting said normalized signals; and phase unwrapping said complex phase converted signals.to produce a one-dimensional array for input into said filtering step. A method according to claim 2 characterized in that said directed wave of energy comprises a very short pulse and said preprocessing step comprises: an analog to digital converter; delay timing means for timing the delay time between the directing and detecting of said one or more energy waves; amplitude measuring means for measuring the peak amplitude of said one or more detected fields; and providing said timed delay and measured peak amplitude for unwrapped complex phase converting said timed and measured signals to produce a two-dimensional array for input into said filter means. A method according to claim 2 characterized in that said preprocessing step comprises: converting said produced signals from analog to digital form; transforming said analog signals over time with a Fourier transformer; normalizing said to transform signals; and processing said normalized signals according to the Born approximation. A system for tomographic reconstruction of a subsurface earth formation wherein a tomographic reconstructing system is used comprising one or more sources of waves of energy directed at an object to be reconstructed, one or more detecting means for detecting the one or more fields produced by the waves interacting with said object to produce detected signals, and processing means for filtering and backprojecting said signals to generate a two-dimensional array representing the partial reconstruction of the object, characterized in that: a) said one or more sources (214) of waves of energy are directed at said earth formation, wherein the totality of the waves of said one or more sources approximate one or more cylindrical waves of a given phase tilt (ϑ n ); b) said one or more detecting means (218) produce detected signals as a function of time of said fields produced by the waves interacting with said earth formation; and said processing means comprise: c) preprocessing means (229,231,232,233,234) for converting said time signals to produce one or more complex signals P j (ϑ n ) which are functions of said one or more detected fields, said complex signals having a real part containing the amplitude information and an imaginary part containing the phase information; and d) filter means (250) for filtering said preprocessed signals to generate the partial reconstruction of the earth formation according to a filtered backpropagation technique: ∼ / H q,m = ∼ / h m . ∼ / r q,m for object depth X q , and where m indexes sample values of the preprocessed signals; where M is the number of detectors detecting said field; k is the wave number associated with the frequency of said directed waves; x q is the distance of the investigated object depth q from the axis of rotation of said object; x and is the x axis unit vector; l o is the distance between the axis of rotation of said object and the detector array; and s o . r is a scalar product between the unit propagation vector s o and the position vector r . A system for tomographic reconstruction of a subsurface earth formation according to claim 13 characterized by: e) interpolation means for interpolating said partial reconstruction array onto a master array; and f) phase tilt changing means for changing the phase tilt of said one or more waves of energy directed into said earth formation. A system for tomographic reconstruction of a subsurface earth formation according toclaim 14 characterized by: g) display means for displaying said two-dimensional reconstruction of the earth formation. A system according to claim 14 characterized by: said one or more sources of waves (214a) are located on the surface of said earth formation; and said one more detecting means (218a) are located in at least one borehole in said earth formation. A system according to claim 14 characterized in that: said one or more sources of waves (214b) are located in at least one borehole of said earth formation; and said one or more detecting means (218b) are located on the surface of said formation. A system according to claim 14 characterized in that: said one or more source of waves (2l4c) are located in at least one borehole in said earth formation formation; and said one or more detecting means (218c) are located in at least one other borehole in said earth formation. A system according to claim 14 characterized by: said one or more sources of waves are a plurality of spherical wave point source emitters which approximate a cylindrical wave by emitting with a variable phase delay between each source. A system according to claim 14 characterized by: said one or more sources of waves comprise a plurality of spherical wave point source emitters which in conjunction with said preprocessing means approximate a cylindrical wave by sequentially pulsing, wherein said preprocessing means comprises a stacker (233) for stacking said detected signal. A system accoring to claim 14 characterized in that said preprocessing means comprises; converting means (229) for converting said produced signals from analog to digital form; Fourier transforming means (232) for transforming said analog signals over time with a Fourier transformer; normalizing means (232) for normalizing said Fourier transformed signals; complex phase converting means (234) for complex phase converting said normalizing signals; and chase unwrapping means (236) for phase unwrapping said complex phase converted signals to produce a one-dimensional array for input into said filtering step. A system according to claim 14 characterized in that said directed wave of energy comprises a very short pulse and said preprocessing means comprises; an analog to digital converter; delay timing means for timing the delay between the directing and detecting of said pulse; amplitude measuring means for measuring the peak amplitude of said detected fields; and means for receiving the outputs of said delay timing means and amplitude measuring means and for converting said timed and measured signals into a two-dimensional array for input into said filter means. A system according to claim 14 characterized in that said preprocessing means comprises: an anolog to digital converter for converting said produced signals for analog to digital form; a Fourier transformer for transforming said signals over time; a normalizer for normalizing said transformed signals, and a processor for processing said normalized signals according to the Born approximation.

Inventors: DEVANEY, ANTHONY
Applicants: SCHLUMBERGER LIMITED, SOCIETE DE PROSPECTION ELECTRIQUE  SCHLUMBERGER
Application year: 1983
Main classification: G06F15
IPC codes: G06T11, G01S15