Diffraction tomography system and methods

• US 4,562,540 A
• Filed: 11/12/1982
• Issued: 12/31/1985
• Est. Priority Date: 11/12/1982
• Status: Expired due to Term

• Current Assignee: Schlumberger Technology Corporation (SLB)
• Original Assignee: Schlumberger Technology Corporation (SLB)
• Inventors: Devaney, Anthony J.
• Primary Examiner(s): Smith, Jerry
• Assistant Examiner(s): Jablon, Clark A.
• Application Number:
  US06/441,323
• Time in Patent Office: 1,145 Days
• Field of Search: 250/256, 250/253, 250/269, 378/901, 378/4, 73/602, 364/420, 364/414, 364/422, 364/400, 364/421, 128/660, 367/14
• US Class Current: 700/90
• CPC Class Codes: G01S 15/8977 using special techniques for image reconstruction, e.g. FFT, geometrical transformations, spatial deconvolution, time deconvolution
• CPC Class Codes: G06T 11/006 Inverse problem, transformation from projection-space into object-space, e.g. transform methods, back-projection, algebraic methods
• CPC Class Codes: G06T 2211/421 Filtered back projection [FBP]
• CPC Class Codes: Y10S 128/916 Ultrasound 3-D imaging
• CPC Class Codes: Y10S 378/901 Computer tomography program or processor

Abstract

Diffraction tomography systems and methods are disclosed for the reconstruction of physical properties of two and three-dimensional objects from information collected in one or more diffraction tomographic procedures. A diffraction tomographic 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 object, a processing and filtering system for processing and filtering the measured signals according to a filtered backpropagation technique thereby generating arrays which are used to reconstruct 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 transmission tomography. The methods and systems apply filtered backpropagation techniques which are generalizations of the filtered backprojection technique of X-ray computed tomography to cases where diffraction of the insonifying beam is taken into consideration.

Claims

1. A method for the reconstruction of an object which has diffracted at least one wave of energy propagating through it, said wave of energy being selected from a group consisting of sonic and electromagnetic waves, comprising the steps of:

(a) obtaining signals which are a function of at least one of the phase and amplitude of the diffracted propagating wave after said wave has interacted with said object; and

(b) converting said signals by means of a filtered backpropagation technique into an array representing a partial reconstruction of the object.

2. A method for the reconstruction of an object according to claim 1 wherein said at least one wave of energy propagating through said object propagates at at least one view angle on a plane further comprising:

(c) separating said obtained signals into sets of signals, each set of signals representing the signals obtained from the propagation of an energy wave at a separate view angle, wherein by said converting step each set of signals is converted into a separate array representing the partial reconstruction of the object at the separate view angle; and

(d) interpolating said partial reconstruction arrays onto a master array, wherein the master array represents the two-dimensional reconstruction of the object.

3. A method for the reconstruction of an object according to claim 2 further comprising:

preprocessing signals after said obtaining step and before said converting step, wherein said signals of said converting step are obtained signals which have been both preprocessed and separated according to said separating and preprocessing steps.

6. A method according to claim 2 further comprising:

displaying said two-dimensional reconstruction of said object.

4. A method for the reconstruction of a three-dimensional object which has diffracted waves of energy propagating through it at at least one view angle on a plane and for at least one planar orientation, comprising the steps of:

(a) obtaining signals which are a function of at least one of the phase and amplitude of each propagating wave after said wave has interacted with said object;

(b) preprocessing and separating said obtained signals to obtain sets of preprocessed signals, each set representing the signals obtained from the propagation of an energy wave at a separate view angle;

(c) converting said sets of preprocessed signals by means of a filtered backpropagation technique into arrays representing the partial reconstruction of the object; and

(d) interpolating each partial reconstruction onto a master array to obtain the two-dimensional reconstruction of a planar projection of the object.

5. A method for the reconstruction of a three-dimensional object according to claim 4 wherein said preprocessing and separating step further comprises:

(e) additionally separating said obtained signals into second sets of signals, each second set of signals representing the signals obtained from the propagation of an energy wave at a separate planar orientation, wherein by said converting step, the two-dimensional planar projected reconstruction obtained is for each planar orientation, and a three-dimensional reconstruction of the object can be obtained from a plurality of two-dimensional planar projection reconstructions.

7. A method according to claim 4 further comprising:

displaying said two-dimensional planar projection reconstruction of said object.

8. A method according to claim 5 further comprising:

displaying a reconstruction of a planar section of said object which was reconstructed from said two-dimensional planar projections.

9. A system for the reconstruction of an object which has diffracted at least one wave of energy propagating through it, said wave of energy being selected from a group consisting of sonic and electromagnetic waves, comprising:

(a) means for obtaining signals which are a function of at least one of the phase and amplitude of the diffracted propagating wave after said wave has interacted with said object; and

(b) filter means for converting said signals by means of a filtered backpropagation technique into an array representing a partial reconstruction of the object.

10. A system for the reconstruction of an object according to claim 9 wherein said at least one wave of energy propagating through said object propagates at at least one view angle on a plane, further comprising:

(c) means for separating said obtained signals into sets of signals, each set of signals representing the signals obtained from the propagation of an energy wave at a separate view angle, wherein said filter means filters and converts each set of signals into a separate array representing the partial reconstruction of the object at the separate view angle; and

(d) interpolation means for interpolating said partial reconstruction arrays onto a master array, wherein the master array represents the two-dimensional reconstruction of the object.

11. A system for the reconstruction of an object according to claim 10 further comprising:

preprocessing means for preprocessing signals which have been obtained by said obtaining means but have not yet been converted by said filter means wherein said signals converted by said filter means are obtained signals which have been both preprocessed and separated by said means for separating and means for preprocessing.

14. A system according to claim 10 further comprising:

display means for displaying said two-dimensional reconstruction of said object.

12. A system for the reconstruction of a three-dimensional object which has diffracted waves of energy propagating through it at at least one view angle on a plane for at at least one planar orientation, comprising:

(a) means for obtaining signals which are a function of at least one of the phase and amplitude of each propagating wave after said wave has interacted with said object;

(b) preprocessing and separating means for separating and preprocessing said obtained signals to obtain sets of preprocessed signals, each set representing the signals obtained from the propagation of an energy wave at a separate view angle;

(c) filter means for converting said sets of preprocessed signals into arrays representing the partial reconstruction of the object by means of a filtered backpropagation technique; and

(d) interpolation means for interpolating each partial reconstruction array onto a master array to obtain the two-dimensional reconstruction of a planar projection of the object.

13. A system for the reconstruction of a three-dimensional object according to claim 12 wherein said preprocessing and separating means further comprise:

(e) means for separating said sets of preprocessed signals into second sets of signals, each second set of signals representing the signals obtained from the propagation of an energy wave at a separate planar orientation, wherein the two-dimensional planar projected reconstructions obtained are for each planar orientation, and by said filtering a three-dimensional reconstruction of the object can be obtained from a plurality of two-dimensional planar projected reconstructions.

15. A system according to claim 12 further comprising:

displaying means for displaying said two-dimensional planar projected reconstruction of said object.

16. A system according to claim 13 further comprising:

display means for displaying a reconstruction of a planar section of said object which was reconstructed from said two-dimensional planar projections.

17. A method for the reconstruction of a subsurface geological formation which has diffracted at least one wave of energy propagating through it, said wave of energy being selected from a group consisting of sonic and electromagnetic waves, comprising the steps of:

(a) obtaining signals which are a function of at least one of the phase and amplitude of the diffracted propagating wave after said wave has interacted with said formation; and

(b) converting said signals by means of a filtered backpropagation technique into an array representing a partial reconstruction profile of the formation.

18. A method according to claim 17 wherein said at least one wave of energy approximates a cylindrical wave at at least one phase tilt, further comprising:

(c) separating said obtained signals into sets of signals, each set of signals representing the signals obtained from the propagation of an energy wave at a separate phase tilt, wherein by said converting step each set of signals is converted into a separate array representing the partial reconstruction of the formation at the separate phase tilt; and

(d) interpolating said partial reconstruction array onto a master array, wherein the master array represents the two-dimensional reconstruction of said formation.

19. A method according to claim 18 further comprising:

preprocessing signals after said obtaining step and before said converting step, wherein said signals of said converting step are obtained signals which have been both preprocessed and separated according to said separating and preprocessing steps.

20. A method according to claim 18 further comprising:

displaying said two-dimensional reconstruction of said subsurface formation.

21. A system for the reconstruction of a subsurface geological formation which has diffracted at least one wave of energy propagating through it, said wave of energy being selected from a group consisting of sonic or electromagnetic waves, comprising:

(a) means for obtaining signals which are a function of at least one of the phase and amplitude of the diffracted propagating wave after said wave has interacted with said formation; and

(b) filter means for converting said signals by means of a filtered backpropagation technique into an array representing a partial reconstruction of the formation.

22. A system according to claim 21 wherein said at least one wave of energy approximates at least one cylindrical wave at at least one phase tilt, further comprising:

(c) means for separating said obtained signals into sets of signals, each set of signals representing the signals obtained from the propagation of an energy wave at a separate phase tilt, wherein said filter means filter and converts each set of signals into a separate array representing the partial reconstruction of the formation at the separate phase tilt; and

(d) interpolation means for interpolating said partial reconstruction array onto a master array, wherein the master array represents the two-dimensional reconstruction of said formation.

23. A system according to claim 22 further comprising:

preprocessing means for preprocessing signals which have been obtained by said obtaining means but have not yet been converted by said filter means wherein said signals converted by said filter means are obtained signals which have been both preprocessed and separated by said means for separating and means for preprocessing.

24. A system according to claim 22 further comprising:

display means for displaying said two-dimensional reconstruction of said formation.

25. A method for tomographic reconstruction of a two-dimensional object comprising:

(a) directing at a view angle one or more waves of energy toward an object being examained;

(b) detecting as a function of time the one or more fields produced by said one or more waves;

(c) producing one or more signals which are functions of said one or more detected fields; and

(d) filtering said produced signals according to a filtered backpropagation technique to generate a two-dimensional array representing the partial reconstruction of the object.

26. A method for tomographic reconstruction of a two-dimensional object according to claim 25 further comprising:

(e) interpolating said partial reconstruction onto a master array;

(f) changing said view angle; and

(g) repeating steps a-f at different view angles to obtain a two-dimensional reconstruction of the object.

27. A method according to claim 26 further comprising:

(h) displaying said two-dimensional reconstruction of the object.

28. A method for tomographic reconstruction of a two-dimensional object according to claim 26 further comprising:

preprocessing said produced signals before said filtering step, wherein said produced signals filtered by said backpropagation filter are preprocessed signals.

29. A method according to claim 28 wherein said preprocessing step comprises:

converting said one or more produced signals from analog to digital form;

transforming said digital 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.

30. A method according to claim 26 wherein said filtering step comprises:

filtering said preprocessed signals with a standard X-ray tomographic filter; and

backpropagating the filtered preprocessed signals according to a backpropagation filter.

31. A method according to claim 30 wherein:

said preprocessed signals are filtered and backpropagated according to the filtered backpropagation technique

space="preserve" listing-type="equation">H.sub.q,m =h.sub.m e.sup.i(γ

.sbsp.m.sup.-k) (z.sbsp.q.sup.-l.sbsp.o.sup.)

wherein H_q,m is a filtered backpropagation technique for object depth z_q, and m indexes sample values of said preprocessed signals;

h_m is a standard X-ray tomography filter;

γ

_m =k√

1-(m/M)², where M is the number of detectors detecting said fields;

k is the wavenumber associated with the frequency of said directed waves;

z_q is the distance of the investigated object depth q from the axis of rotation of said object; and

l_o is the distance between the axis of rotation of said object and the detector array.

32. A method according to claim 28 wherein said detected wave of energy is a very short pulse and said preprocessing step comprises:

converting said produced signal from analog to digital form;

timing the delay between the directing and detecting of said pulse;

measuring the peak amplitude of said detected fields; and

providing said timed delay and measured peak amplitude for unwrapped complex phase converting said converted signals to produce a one-dimensional array for input into said filtering step.

33. A method for tomographic reconstruction of a three-dimensional object comprising:

(a) directing at a view angle on a plane one or more waves of energy toward an object being examined;

(b) detecting as a function of time the one or more fields produced by said one or more waves;

(c) producing one or more signals which are functions of said one or more detected fields;

(d) preprocessing said produced signals; and

(e) filtering said preprocessed signals according to a filtered backpropagation technique to generate a two-dimensional array representing the partial reconstruction of the object.

34. A method for tomographic reconstruction of a three-dimensional object according to claim 33 further comprising:

(f) interpolating said partial reconstruction array onto a master array;

(g) changing said view angle on said plane; and

(h) repeating steps a-g at different view angles on said plane to obtain a two-dimensional reconstruction of a planar projection of the object.

35. A method for tomographic reconstruction of a three-dimensional object according to claim 34 further comprising:

(i) changing said plane on which said waves of energy are directed at a view angle; and

(j) repeating steps a-i at a sufficient number of different planes to obtain a plurality of planar projections of the three-dimensional object profile from which a full three-dimensional reconstruction of the object profile can be obtained.

36. A method according to claim 34 further comprising:

displaying said planar projection of the three-dimensional reconstruction of the object.

37. A method according to claim 35 further comprising:

displaying a reconstruction of a planar section of said object profile which is obtained from said planar projections.

38. A method according to claim 35 wherein said preprocessing step comprises:

projecting a two-dimensional array of said produced signals onto the line formed by the intersection of the detecting plane with the plane of rotation of the directing-detecting system to produce a one-dimensional array for input into said filtering step.

39. A method according to claim 38 wherein said preprocessing step further comprises:

converting said produced signal from analog to digital form;

transforming said digital 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.

40. A method according to claim 35 wherein said filtering step comprises:

filtering said preprocessed signals with a standard X-ray tomographic filter; and

backpropagating the filtered preprocessed signals according to a backpropagation filter.

41. A method according to claim 40 wherein:

said preprocessed signals are filtered and backkpropagated according to a filtered backpropagation technique

space="preserve" listing-type="equation">H.sub.q,m =h.sub.m e.sup.i(γ

.sbsp.m.sup.-k) (z.sbsp.q.sup.-l.sbsp.o.sup.)

wherein H_q,m is a filtered backpropagation technique for object depth z_q, and m indexes samples values of said preprocessed signals;

h_m is a standard X-ray tomography filter;

γ

_m =k√

1-(m/M)², where M is the number of detectors detecting said fields;

k is the wavenumber associated with the frequency of said directed waves;

z_q is the distance of the investigated object depth q from the axis of rotation of said object; and

l_o is the distance between the axis of rotation of said object and the detector array.

42. A method according to claim 35 wherein said directed wave of energy is a very short pulse and said preprocessing step comprise:

converting said produced signal from analog to digital form;

timing the delay between the directing and detecting of said pulse;

measuring the peak amplitude of said detected fields; and

unwrapped complex phase converting said delay time and peak amplitude measured signals to produce a one-dimensional array for input into said filtering step.

43. A system for tomographic reconstruction of a two-dimensional object comprising:

(a) one or more sources of waves of energy directed at a view angle at an object;

(b) one or more detecting means for detecting as a function of time the one or more fields produced by the waves interacting with said object;

(c) signal producing means for producing signals which are functions of said fields; and

(d) filter means for filtering said produced signals according to a backpropagation technique to generate a two-dimensional array representing the partial reconstruction of the object.

44. A system for tomographic reconstruction of a two-dimensional object according to claim 43 further comprising:

(e) interpolation means for interpolating said partial reconstruction array onto a master array; and

(f) view angle changing means for changing said view angle.

45. A system according to claim 44 further comprising:

(g) display means for displaying said master array so that a two-dimensional reconstruction of the object may be viewed.

46. A system for tomographic reconstruction of a two-dimensional object according to claim 44 further comprising:

preprocessing means for preprocessing said signals produced by said signal producing means.

47. A system according to claim 46 wherein said preprocessing means comprises:

an analog to digital converter for converting said detected signals;

a Fourier transformer for transforming said converted signals;

a normalizer for normalizing said transformed signals;

complex phase converting means for complex phase converting said normalized signals; and

phase unwrapping means for phase unwrapping said complex phase converted signals.

48. A system according to claim 46 wherein said detected 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 time 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.

49. A system according to claim 44 wherein said filter comprises:

standard X-ray tomographic filtering means for implementing the standard X-ray tomographic filter; and

backpropagation filter means for backpropagating the filtered preprocessed signals.

50. The system of claim 49 wherein:

said preprocessed signals are filtered and backpropagated according to the backpropagation technique

space="preserve" listing-type="equation">H.sub.q,m =h.sub.m e.sup.i(γ

.sbsp.m.sup.-k) (z.sbsp.q.sup.-l.sbsp.o.sup.)

wherein H_q,m is a filtered backpropagation technique for object depth z_q, and m indexes sample values of said preprocessed signals;

h_m is a standard X-ray tomography filter;

γ

_m =k√

1-(m/M)², where M is the number of detectors detecting said fields;

k is the wavenumber associated with the frequency of said directed waves;

z_q is the distance of the investigated object depth q from the axis of rotation of said object; and

l_o is the distance between the axis of rotation of said object and the detector array.

51. A system for tomographic reconstruction of a three-dimensional object comprising:

(a) one or more sources of waves of energy directed at a view angle on a plane at an object;

(b) one or more detecting means for detecting as a function of time the one or more fields produced by the waves interacting with said object;

(c) signal producing means for producing signals which are functions of said one or more fields;

(d) preprocessing means for preprocessing said produced signals; and

(e) filter means for filtering said preprocessed signals according to a filtered backpropagation technique to generate a two-dimensional array representing the partial reconstruction of the object.

52. A system for tomographic reconstruction of a three-dimensional object according to claim 51 further comprising:

(f) interpolation means for interpolating said partial reconstruction array onto a master array; and

(g) view angle changing means for changing said view angle, wherein a two-dimensional planar projection reconstruction may be obtained.

53. A system for tomographic reconstruction of a three-dimensional object according to claim 52 further comprising:

(h) plane changing means for changing said plane on which said energy waves are directed at said view angles, wherein a three-dimensional reconstruction may be obtained.

54. A system according to claim 53 further comprising:

display means for displaying said master array so that two-dimensional planar projection reconstruction or a three-dimensional reconstruction may be viewed.

55. A system according to claim 53 further comprising:

display means for displaying a reconstruction of a planar section of said object which is obtained from said planar projection reconstructions.

56. A system according to claim 53 wherein said preprocessing means comprises:

projection calculating means for projecting the two-dimensional array produced by said signal producing means onto a line formed by the intersection of the plane of said detecting means and the plane of rotation of said object of said source and detecting means.

57. A system according to claim 53 wherein said preprocessing means further comprises:

an analog to digital converter for converting said detected signals;

a Fourier transformer for transforming said converted signals;

a normalizer for normalizing said transformed signals;

complex phase converting means for complex phase converting said normalized signals; and

phase unwrapping means for phase unwrapping said complex phase converted signals.

58. A system according to claim 53 wherein said filter comprises:

standard X-ray tomographic filtering means for implementing the standard X-ray tomographic filter; and

backpropagation filter means for backpropagating the filtered preprocessed signals.

59. A method according to claim 58 wherein:

said preprocessed signals are filtered and backpropagated according to the filtered backpropagation technique

space="preserve" listing-type="equation">H.sub.q,m =h.sub.m e.sup.i(γ

.sbsp.m.sup.-k)(z.sbsp.q.sup.-l.sbsp.o.sup.)

wherein H_q, m is a filtered backpropagation technique for object depth z_q, and m indexes sample values of said preprocessed signals;

h_m is a standard X-ray tomography filter;

γ

_m =k√

1-(m/M)², where M is the number of detectors detecting said fields;

k is the wavenumber associated with the frequency of said directed waves;

z_q is the distance of the investigated object depth q from the axis of rotation of said object; and

l_o is the distance between the axis of rotation of said object and the detector array.

60. A system according to claim 53 wherein 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.

61. A method for tomographic reconstruction of a subsurface formation comprising:

(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;

(b) detecting as a function of time the one or more fields produced by said one or more waves;

(c) producing one or more signals which are functions of said one or more detected fields; and

(d) filtering said produced signals according to a filtered backpropagation technique to generate a two-dimensional array representing the partial reconstruction of the subsurface formation.

62. A method for tomographic reconstruction of a subsurface formation according to claim 61 further comprising:

(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.

63. A method according to claim 62 further comprising:

displaying said partial reconstruction of said subsurface formation.

64. A method according to claim 62 wherein:

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.

65. A method according to claim 62 wherein:

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.

66. A method according to claim 62 wherein:

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.

67. A method according to claim 62 wherein:

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.

68. A method according to claim 62 further comprising:

preprocessing said produced signals wherein said produced signals of said filtering step are preprocessed signals.

69. A method according to claim 62 wherein:

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.

70. A method according to claim 68 wherein:

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 detected signals.

71. A method according to claim 68 wherein 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.

72. A method according to claim 68 wherein 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.

73. A method according to claim 68 wherein 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.

74. A method according to claim 62 wherein said filtering step comprises:

filtering said preprocessed signals with a convolutional phase-tilt dependent deblurring filter; and

backpropagating the filtered preprocessed signals according to a backpropagation filter.

75. A method according to claim 74 wherein:

said preprocessed signals are filtered and backpropagated according to a filtered backpropagation technique

space="preserve" listing-type="equation">H.sub.q,m =h.sub.m e.sup.iγ

.sbsp.m.sup.(x.sbsp.q.sup.-l.sbsp.o.sup.) e.sup.ik(s.sbsp.0.sup.·

xl.sbsp.o.sup.-s.sbsp.o.sup.·

r

wherein H_q,m is a filtered backpropagation technique for object depth x_q, and m indexes sample values of preprocessed signals;

h_m is a convolutional, phase-tilt dependent, deblurring filter;

γ

_m =k√

1-(m/M)², where M is the number of detectors detecting said fields;

k is the wavenumber 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 is the x axis unit vector;

l_o is the distance between the axis of rotation of said object and the detector array; and

s₀ ·

r is a scalar product between the unit propagation vector s₀ and the position vector r.

76. A system for tomographic reconstruction of a subsurface earth formation comprising:

(a) one or more sources of waves of energy 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;

(b) one or more detecting means for detecting as a function of time the one or more fields produced by the waves interacting with said earth formation;

(c) signal producing means for producing signals which are a function of said one or more fields; and

(d) filter means for filtering said produced signals according to a filtered backpropagation technique to generate a two-dimensional array representing the partial reconstruction of the subsurface formation.

77. A system for tomographic reconstruction of a subsurface earth formation according to claim 76 further comprising:

(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 format.

78. A system for tomographic reconstruction of a subsurface earth formation comprising claim 77 further comprising:

(g) display means for displaying said two-dimensional reconstruction of the earth formation.

79. A system according to claim 77 wherein:

said one or more sources of waves are located on the surface of said earth formation; and

said one or more detecting means are located in at least one borehole in said earth formation.

80. A system according to claim 77 wherein:

said one or more sources of waves are located in at least one borehole of said earth formation; and

said one or more detecting means are located on the surface of said formation.

81. A system according to claim 77 wherein:

said one or more source of waves are located in at least one borehole in said earth formation formation; and

said one or more detecting means are located in at least one other borehole in said earth formation.

82. A system according to claim 77 further comprising:

preprocessing means for preprocessing said produced signals, wherein said produced signals of said filter means are preprocessed signals.

83. A system according to claim 77 wherein:

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.

84. A system according to claim 82 wherein:

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 for stacking said detected signal.

85. A system according to claim 82 wherein said preprocessing means comprises:

converting means for converting said produced signals from analog to digital form;

Fourier transforming means for transforming said analog signals over time with a Fourier transformer;

normalizing means for normalizing said Fourier transformed signals;

complex phase converting means for complex phase converting said normalized signals; and

phase unwrapping means for phase unwrapping said complex phase converted signals to produce a one-dimensional array for input into said filtering step.

86. A system according to claim 82 wherein 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.

87. A system according to claim 82 wherein said preprocessing means comprises:

an analog 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.

88. The system of claim 76 wherein said filtering means comprises:

convolutional, phase-tilt dependent, deblurring filtering means; and

backpropagation filter means for backpropagating the filtered preprocessed signals.

89. A method according to claim 88 wherein:

said preprocessed signals are filtered and backpropagated according to a filtered backpropagation technique

space="preserve" listing-type="equation">H.sub.q,m =h.sub.m e.sup.iγ

.sbsp.m.sup.(x.sbsp.q.sup.-l.sbsp.o.sup.) e.sup.ik(s.sbsp.0.sup.·

xl.sbsp.o.sup.-s.sbsp.0.sup.·

r)

wherein H_q,m is a filtered backpropagation technique for object depth x_q, and m indexes sample values of preprocessed signals;

h_m is a convolutional, phase-tilt dependent, deblurring filter;

γ

_m =k√

1-(m/M)², where M is the number of detectors detecting said fields;

k is the wavenumber 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 is the x axis unit vector;

l_o is the distance between the axis of rotation of said object and the detector array; and

s₀ ·

r is a scalar product between the unit propagation vector s_o and the position vector r.

90. A method for the reconstruction of an object which has diffracted at least one wave of energy propagating through it, said wave of energy being selected from a group consisting of sonic or electromagnetic waves, comprising the steps of:

(a) obtaining signals which are a function of at least one of the phase and amplitude of the diffracted propagating wave after said wave has interacted with said object;

(b) converting said signals by means of a backpropagation filter into an array representing a partial reconstruction of the object; and

(c) generating a tangible record of said array representing said partial reconstruction of the object.

91. A system for the reconstruction of an object which has diffracted at least one wave of energy propagating through it, said wave of energy being selected from a group consisting of sonic or electromagnetic waves, comprising:

(a) means for obtaining signals which are a function of at least one of the phase and amplitude of the diffracted propagating wave after said wave has interacted with said object;

(b) filter means for converting said signals by means of a backpropagation filter into an array representing a partial reconstruction of the object; and

(c) means for generating a tangible record of said array representing said partial reconstruction of said object.