Apparatus and method for imaging with wavefields using inverse scattering techniques

A method for producing an image of an object in a region from wavefield energy that has been transmitted into and scattered by the object, said image comprising a map of selected physical characteristics at selected points within the region, said image being stored in a computer memory, and said method comprising the steps of:

• (a) transducing an electric signal at each of one or more frequencies into wavefield energy propagated from one or more of transmitter transducer positions, each said transmitter transducer position propagating wavefield energy at at least one orientation defined by Euler angles with respect to a selected fixed coordinate system;
• (b) for one or more receiver positions each having at least one orientation defined by Euler angles with respect to said selected fixed coordinate system, detecting at each of said one or more receiver positions and respective orientations thereof said wavefield energy;
• (c) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into one or more reception stored signals stored in said computer memory and corresponding to a scattered wavefield energy detected;
• (d) setting a region characteristics estimate of selected physical characteristics at selected points within the region and storing each said region characteristics estimate in said computer memory;
• (e) performing a convergence step comprising the following steps:
• (1) preparing, for each said one or more frequencies at each said transmitter transducer positions and respective orientations thereof, an estimate of a total wavefield energy at said selected points derived from a selected incident wavefield energy for said selected points stored in the computer memory and said region characteristics estimate for said selected points by the steps of:
• (i) designating a primary set of surfaces of a plurality of selected surfaces and a different secondary set of surfaces of said selected surfaces, each said selected surface intersecting said region;
• (ii) setting the estimate of the total wavefield energy equal to an initial total incident wavefield energy estimate for the primary set of surfaces;
• (iii) computing the estimate of the total wavefield energy on the secondary set of surfaces using the region characteristics estimate on the union of the primary and secondary sets of surfaces and the total wavefield energy on the primary set of surfaces;
• (iv) re-designating the primary set of surfaces to include a sub- set of the secondary set of surfaces and re-designating the secondary set of surfaces to include another set of the selected surfaces;
• and
• (v) repeating steps ((iii) through ((iv)) until the estimate of the total wavefield energy is computed for each of the selected surfaces;
• (2) deriving, for each of said one or more frequencies at each said transmitter transducer position and orientations thereof, a calculated scattered wavefield energy for one or more of said receiver positions and respective orientations thereof from at least one of said region characteristics estimate at said selected points and said estimate of said total wavefield energy for a corresponding transmitter transducer position and orientations thereof at said selected points by designating an external surface situated outside said object, and then approximating an integral on the external surface by the sum of:
• (i) a first quantity times the estimate of the total wavefield energy on said external surface times a derivative of a Green's function constructed to map to one or more of said receiver positions and respective orientations thereof;
• and
• (ii) a second quantity times a derivative of the estimate of the total wavefield energy on said external surface times a Green's function constructed to map to one or more of said receiver positions and respective orientations thereof;
• (3) for each said transmitter transducer position and orientations thereof and for each said receiver position and orientation thereof, comparing said scattered wavefield energy detected to said calculated scattered wavefield energy to derive therefrom a comparator;
• and
• (4) when said comparator is greater than a selected tolerance, determining and storing in said computer memory said region characteristics estimate by computing one or more derivatives of the comparator or approximations thereof with respect to one or more of said selected physical characteristics at one or more of said selected points, and then using said one or more derivatives of the comparator or approximations thereof to compute a region characteristics correction, and then adding said region characteristics correction to each of said region characteristics estimate for each of said one or more of said selected points, wherein said one or more derivatives of the comparator or approximations thereof is computed from one or more of:
• (i) at each said one or more frequencies, said estimate of said total wavefield energy for said selected points for each of said one or more of said transmitter transducer positions and respective orientations thereof;
• (ii) at each of said one or more frequencies, said calculated scattered wavefield energy for said one or more receiver positions and respective orientations thereof, and for each of said one or more of said transmitter transducer positions and respective orientations thereof;
• (iii) at each of said one or more frequencies, said scattered wavefield energy detected for said one or more receiver positions and respective orientations thereof, and for each of said one or more of said transmitter transducer positions and respective orientations thereof;
• and
• (iv) said region characteristics estimate for said selected points;
• (f) repeating said convergence step until said comparator is less than or equal to said selected tolerance, and thereafter storing said region characteristics estimate as said image in the computer memory.

A method for producing an image of an object in a region from wavefield energy that has been transmitted into and scattered by the object, said image comprising a map of selected physical characteristics at selected points within the region, said image being stored in a computer memory, and said method comprising the steps of:

• (a) transducing an electric signal at each of one or more frequencies into wavefield energy propagated from one or more of transmitter transducer positions, each said transmitter transducer position propagating wavefield energy at at least one orientation defined by Euler angles with respect to a selected fixed coordinate system;
• (b) for one or more receiver positions each having at least one orientation defined by Euler angles with respect to said selected fixed coordinate system, detecting at each of said one or more receiver positions and respective orientations thereof said wavefield energy;
• (c) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into one or more reception stored signals stored in said computer memory and corresponding to a scattered wavefield energy detected;
• (d) setting a region characteristics estimate of selected physical characteristics at selected points within the region and storing each said region characteristics estimate in said computer memory;
• (e) performing a convergence step comprising the following steps:
• (1) preparing, for each said one or more frequencies at each said transmitter transducer positions and respective orientations thereof, an estimate of a total wavefield energy at said selected points derived from a selected incident wavefield energy for said selected points stored in the computer memory and said region characteristics estimate for said selected points by the steps of:
• (i) designating a primary set of surfaces of a plurality of selected surfaces and a different secondary set of surfaces of said selected surfaces, each said selected surface intersecting said region;
• (ii) setting the estimate of the total wavefield energy equal to an initial total incident wavefield energy estimate for the primary set of surfaces;
• (iii) computing the estimate of the total wavefield energy on the secondary set of surfaces using the region characteristics estimate on the union of the primary and secondary sets of surfaces and the total wavefield energy on the primary set of surfaces;
• (iv) re-designating the primary set of surfaces to include a sub- set of the secondary set of surfaces and re-designating the secondary set of surfaces to include another set of the selected surfaces;
• and
• (v) repeating steps ((iii) through ((iv)) until the estimate of the total wavefield energy is computed for each of the selected surfaces;
• (2) deriving, for each of said one or more frequencies at each said transmitter transducer position and orientations thereof, a calculated scattered wavefield energy for one or more of said receiver positions and respective orientations thereof from at least one of said region characteristics estimate at said selected points and said estimate of said total wavefield energy for a corresponding transmitter transducer position and orientations thereof at said selected points by approximating an integral on at least a portion of the region of the product of:
• (i) the estimate of the total wavefield energy at said selected points;
• (ii) the region characteristics estimate at said selected points;
• and
• (iii) a Green's function constructed to map to one or more of said receiver positions and respective orientations thereof;
• (3) for each said transmitter transducer position and orientations thereof and for each said receiver position and orientation thereof, comparing said scattered wavefield energy detected to said calculated scattered wavefield energy to derive therefrom a comparator;
• and
• (4) when said comparator is greater than a selected tolerance, determining and storing in said computer memory said region characteristics estimate by computing one or more derivatives of the comparator or approximations thereof with respect to one or more of said selected physical characteristics at one or more of said selected points, and then using said one or more derivatives of the comparator or approximations thereof to compute a region characteristics correction, and then adding said region characteristics correction to each of said region characteristics estimate for each of said one or more of said selected points, wherein said one or more derivatives of the comparator or approximations thereof is computed from one or more of:
• (i) at each said one or more frequencies, said estimate of said total wavefield energy for said selected points for each of said one or more of said transmitter transducer positions and respective orientations thereof;
• (ii) at each of said one or more frequencies, said calculated scattered wavefield energy for said one or more receiver positions and respective orientations thereof, and for each of said one or more of said transmitter transducer positions and respective orientations thereof;
• (iii) at each of said one or more frequencies, said scattered wavefield energy detected for said one or more receiver positions and respective orientations thereof, and for each of said one or more of said transmitter transducer positions and respective orientations thereof;
• and
• (iv) said region characteristics estimate for said selected points;
• (f) repeating said convergence step until said comparator is less than or equal to said selected tolerance, and thereafter storing said region characteristics estimate as said image in the computer memory.

A method for producing an image of an object in a region from wavefield energy that has been transmitted into and scattered by the object, said image comprising a map of selected physical characteristics at selected points within the region, said image being stored in a computer memory, and said method comprising the steps of:

• (a) transducing an electric signal at each of one or more frequencies into wavefield energy propagated from one or more of transmitter transducer positions, each said transmitter transducer position propagating wavefield energy at at least one orientation defined by Euler angles with respect to a selected fixed coordinate system;
• (b) for one or more receiver positions each having at least one orientation defined by Euler angles with respect to said selected fixed coordinate system, detecting at each of said one or more receiver positions and respective orientations thereof said wavefield energy;
• (c) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into one or more reception stored signals stored in said computer memory and corresponding to a scattered wavefield energy detected;
• (d) setting a region characteristics estimate of selected physical characteristics at selected points within the region and storing each said region characteristics estimate in said computer memory;
• (e) performing a convergence step comprising the following steps:
• (1) preparing, for each said one or more frequencies at each said transmitter transducer positions and respective orientations thereof, an estimate of a total wavefield energy at said selected points derived from a selected incident wavefield energy for said selected points stored in the computer memory and said region characteristics estimate for said selected points by the steps of:
• (i) designating a primary set of surfaces of a plurality of selected surfaces and a different secondary set of surfaces of said selected surfaces, each said selected surface intersecting said region;
• (ii) setting the estimate of the total wavefield energy equal to an initial total incident wavefield energy estimate for the primary set of surfaces;
• (iii) computing the estimate of the total wavefield energy on the secondary set of surfaces using the region characteristics estimate on the union of the primary and secondary sets of surfaces and the total wavefield energy on the primary set of surfaces;
• (iv) re-designating the primary set of surfaces to include a sub- set of the secondary set of surfaces and re-designating the secondary set of surfaces to include another set of the selected surfaces;
• and
• (v) repeating steps ((iii) through ((iv)) until the estimate of the total wavefield energy is computed for each of the selected surfaces;
• (2) deriving, for each of said one or more frequencies at each said transmitter transducer position and orientations thereof, a calculated scattered wavefield energy for one or more of said receiver positions and respective orientations thereof from at least one of said region characteristics estimate at said selected points and said estimate of said total wavefield energy for a corresponding transmitter transducer position and orientations thereof at said selected points by performing the steps of:
• (i) for each selected point of a portion of the selected points, said potion of the selected points corresponding to one of said one or more receiver positions and respective orientations thereof, setting said calculated scattered wavefield energy equal to the estimate of the total wavefield energy less said selected incident wavefield energy;
• and
• (ii) computing a sum over said portion of said selected points equal to the sum of the calculated scattered wavefield energy for said portion of said selected points times a function constructed to correspond to one or more of said one or more receiver positions and respective orientations thereof;
• (3) for each said transmitter transducer position and orientations thereof and for each said receiver position and orientation thereof, comparing said scattered wavefield energy detected to said calculated scattered wavefield energy to derive therefrom a comparator;
• and
• (4) when said comparator is greater than a selected tolerance, determining and storing in said computer memory said region characteristics estimate by computing one or more derivatives of the comparator or approximations thereof with respect to one or more of said selected physical characteristics at one or more of said selected points, and then using said one or more derivatives of the comparator or approximations thereof to compute a region characteristics correction, and then adding said region characteristics correction to each of said region characteristics estimate for each of said one or more of said selected points, wherein said one or more derivatives of the comparator or approximations thereof is computed from one or more of:
• (i) at each said one or more frequencies, said estimate of said total wavefield energy for said selected points for each of said one or more of said transmitter transducer positions and respective orientations thereof;
• (ii) at each of said one or more frequencies, said calculated scattered wavefield energy for said one or more receiver positions and respective orientations thereof, and for each of said one or more of said transmitter transducer positions and respective orientations thereof;
• (iii) at each of said one or more frequencies, said scattered wavefield energy detected for said one or more receiver positions and respective orientations thereof, and for each of said one or more of said transmitter transducer positions and respective orientations thereof;
• and
• (iv) said region characteristics estimate for said selected points;
• (f) repeating said convergence step until said comparator is less than or equal to said selected tolerance, and thereafter storing said region characteristics estimate as said image in the computer memory.

A method for calibrating an imaging device and an imaging process used therewith, said imaging device and imaging process using model calibration parameters, said imaging device producing with said imaging process an image of matter in a region, said imaging device using interrogating wavefield energy and employing:

• (i) one or more transmitter transducers at one or more transmitter positions;
• each said one of more transmitter positions having one or more transmitter orientations, each said one or more transmitter orientations being defined by Euler angles;
• and
• (ii) one or more receiver transducers at one or more receiver positions, each said one of more receiver positions having one or more receiver orientations, each said one or more receiver orientations being defined by Euler angles, said model calibration parameters comprising at least one of:
• (a) one or more transmitter equivalent source vectors, said one or more transmitter equivalent source vectors being input to one or more transmitter simulators, said one or more transmitter simulators outputting one or more simulated transmitter wavefields of said interrogating wavefield energy;
• (b) one or more receiver equivalent source vectors, said one or more receiver equivalent source vectors being input to one or more receiver simulators, said one or more receiver simulators outputting one or more simulated receiver sensitivity functions, said one or more receiver sensitivity functions being electrical responses of said one or more receiver simulators to a point source of interrogating wavefield energy placed at substantially any location in said region;
• (c) one or more transmitter simulation positions, each said one of more transmitter simulation positions having one or more transmitter simulation orientations, each said one or more transmitter simulation orientations being defined by Euler angles;
• (d) one or more receiver simulation positions, each said one of more receiver simulation positions having one or more receiver simulation orientations, each said one or more receiver simulation orientations being defined by Euler angles;
• and
• (e) one or more electronic system parameters of said imaging device, said electronic system parameters substantially characterizing the analog and digital electronic functions of said imaging device;
• said method comprising the steps of:
• (I) collecting a received wavefield energy by a process comprising the steps of:
• (A) choosing a substantially known matter distribution in said region from a selected set of one or more substantially known matter distributions;
• (B) choosing one of said one of more transmitter transducers at one of said one or more transmitter positions and at one of said one or more transmitter orientations;
• (C) choosing one of said one of more receiver transducers at one of said one or more receiver positions and at one of said one or more receiver orientations;
• (D) transmitting interrogating wavefield energy from said one of said one of more transmitter transducers at one of said one or more transmitter positions and at one of said one or more transmitter orientations into said region;
• (E) receiving interrogating wavefield energy at said one of said one or more transducer receivers at one of said one or more receiver positions and at one of said one or more receiver orientations;
• (F) repeating steps I(C) through I(E) until a selected portion of said one or more receiver transducers at said one or more receiver positions and orientations thereof have been chosen;
• (G) repeating steps I(B) through I(F) until a selected portion of said one or more transmitter transducers at said one or more transmitter positions and orientations thereof have been chosen;
• (H) repeating steps I(A) through I(G) until all of said selected set of one or more substantially known matter distributions have been chosen;
• (II) constructing an imaging system simulation model of said imaging device and said imaging process used therewith with an input equal to said model calibration parameters and an output equal to a predicted received wavefield energy, said predicted received wavefield energy being an approximation of said received wavefield energy, said imaging system simulation model being derived from at least one of the following quadra-parameters:
• (A) a mathematical theory of said interrogating wavefield energy;
• (B) a simulation model of the analog and digital electronic functions of said imaging device;
• (C) said one or more transmitter simulators;
• and
• (D) said one or more receiver simulators;
• (III) setting a portion of said model calibration parameters equal to selected values;
• (IV) solving for substantially all other portions of said model calibration parameters using the steps of:
• (A) defining a comparator, said comparator being a positive scaler value that is a measure of the difference between said received wavefield energy and said predicted received wavefield energy;
• and
• (B) using an optimization algorithm to substantially minimize said comparator with respect to said substantially all other portions of said model calibration parameters until said comparator is less than a selected tolerance;
• (V) storing said model calibration parameters in a computer memory.

A method for increasing the spatial resolution and the contrast resolution of an image of matter in a region, said image produced by an imaging device, said spatial resolution being quantitatively defined as the reciprocal of the width at half maximum of the point spread function of said imaging device and said matter in said region, said contrast resolution being quantitatively defined as the reciprocal of the smallest contrast of an object imbedded in a background for which said object can be distinguished in said image, said contrast of an object being defined as the ratio of the difference of the intensity of the object and the intensity of the background over the sum of the intensity of the object and intensity of the background, said image consisting of a set of image values, said set of image values corresponding to selected points in said region, said imaging device generating electrical signals that are passed through transmitter beam formers, said transmitter beam formers producing focused electrical signals by applying a set of transmitter time delays and a set of transmitter amplitude scale factors to said generated electrical signals, said focused electrical signals then being transduced into transmitted wavefield energy by one or more transducer arrays, said transducer arrays including one or more transducer elements, said transmitted wavefield energy being transmitted into said region and being scattered from matter in said region, said transducer arrays transducing into receiver electrical signals reflected wavefield energy that was scattered by said matter, said receiver electrical signals being passed through receiver beam formers, said receiver beam formers producing focused receiver electrical signals by applying a set of receiver time delays and a set of receiver amplitude scale factors to said receiver electrical signals, said focused receiver electrical signals then being used to construct said set of image values, said method comprising the steps of:

• (a) defining an image resolution measure function as a scaler value computed from one or more portions of said set of image values, said image resolution measure function defined such that said spatial and contrast resolutions are increased for an increase in said image resolution measure function up to a maximum;
• (b) defining a set of control parameters and defining a control function such that a primary machine focusing set is equal to the sum of the output of said control function having as an input said set of control parameters plus a selected initial machine focusing set, wherein said primary machine focusing set is the union of said set of transmitter time delays, said set of transmitter amplitude scale factors, said set of receiver time delays, and said set of receiver amplitude scale factors, and wherein said selected initial machine focusing set is the union of said selected initial set of transmitter time delays, said selected initial set of transmitter amplitude scale factors, said selected initial set of receiver time delays, and said selected initial set of receiver amplitude scale factors;
• (c) setting said set of control parameters equal to a selected set of starting values, computing said primary machine focusing set by inputting said control parameters into said control function and adding said initial machine focusing set to the output of said control function, and re-computing said image resolution measure function from said one or more portions of said set of image values, setting an initial resolution measure function equal to said image resolution measure function;
• (d) setting a set of auxiliary vectors equal to a set of initial vector values;
• (e) performing an optimization step comprising the following steps of:
• (i) computing a gradient vector of said image resolution measure function with respect to said set of control parameters;
• (ii) computing a search direction vector equal to the sum of:
• (A) said gradient vector;
• plus
• (B) an auxiliary direction vector derived from said set of auxiliary vectors and said gradient vector, when said optimization step is the first step, setting a initial gradient vector equal to said gradient vector and computing a magnitude of said initial gradient vector;
• (iii) update said set of auxiliary vectors using said gradient vector;
• (iv) computing a search length and then adding said search direction vector multiplied by said search length to said set of control parameters, said search length computed such that said image resolution measure function is increased;
• (v) computing said primary machine focusing set by inputting said set of control parameters into said control function and adding said initial machine focusing set to the output of said control function, and re-computing said image resolution measure function from said one or more portions of said set of image values, computing one of:
• (A) an absolute value of the percent change in said image resolution measure function using said image resolution measure function and said initial image resolution function;
• and
• (B) a magnitude of said gradient vector and a percent change in said magnitude of said gradient vector using said magnitude of said gradient vector and said magnitude of said initial gradient vector;
• (vi) repeating steps e(i)1 through e(v)5 when one of:
• (A) said absolute value of the percent change in said image resolution measure function is greater than a selected tolerance;
• and
• (B) said percent change in said magnitude of said gradient vector is greater than a selected tolerance;
• (f) storing said set of image values in a computer memory when one of:
• (A) said absolute value of the percent change in said image resolution measure function is less than or equal to a selected tolerance;
• and
• (B) said percent change in said magnitude of said gradient vector is less than or equal to a selected tolerance.

A method for producing an image of a region from wavefield energy that has been transmitted into and scattered by matter in the region, said region being defined by boundaries, said image comprising a map of selected physical characteristics at selected points within the region, said image being stored in a computer memory, said method comprising the steps of:

• (a) transducing an electric signal at each of one or more frequencies into wavefield energy propagated from one or more of transmitter transducer positions, each said transmitter transducer position propagating wavefield energy at at least one orientation defined by Euler angles with respect to a selected fixed coordinate system;
• (b) for one or more receiver positions each having at least one orientation defined by Euler angles with respect to said selected fixed coordinate system, detecting at each of said one or more receiver positions and respective orientations thereof said wavefield energy;
• (c) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into one or more reception stored signals stored in said computer memory and corresponding to a scattered wavefield energy detected;
• (d) setting a region characteristics estimate of selected physical characteristics at selected points within the region and storing each said region characteristics estimate in said computer memory;
• (e) performing a convergence step comprising the following steps:
• (1) preparing, for each said one or more frequencies at each said transmitter transducer position and respective orientations thereof, each said orientation of each said transmitter transducer position being determined by a fixed orientation vector, an estimate of a total wavefield energy at said selected points, by performing the steps:
• (i) designating a backscattering surface on a portion of the boundaries of said region, said backscattering surface being substantially perpendicular to said fixed orientation vector, and composed of a portion of said selected points, said backscattering surface having a measured total wavefield energy derived from said scattered wavefield energy detected;
• (ii) designating two substantially parallel surfaces that are substantially parallel to said fixed orientation vector, each said substantially parallel surfaces having a measured total wavefield energy derived from said scattered wavefield energy detected;
• (iii) designating a forwardscattering surface on a portion of said boundaries that is substantially parallel to and substantially different from said backscattering surface;
• (iv) setting the normal derivative of the estimate of the total wavefield energy on said backscattering surface equal to a measured total wavefield energy normal derivative;
• (v) setting the estimate of the total wavefield energy on said backscattering surface equal to said measured total wavefield energy for said backscattering surface;
• (vi) setting the estimate of the total wavefield energy on each said two substantially parallel surfaces equal to said measured total wavefield energy on said two substantially parallel surfaces;
• (vii) designating a first primary set of surfaces of a plurality of selected surfaces and a different first secondary set of surfaces of said plurality of selected surfaces, said plurality of selected surfaces being substantially composed of said selected points;
• (viii) determining an estimate of the total wavefield energy on said first secondary set of surfaces using the region characteristics estimate on the union of the first primary and first secondary sets of surfaces and the estimate of the total wavefield energy on the first primary set of surfaces;
• (ix) re-designating the first primary set of surfaces to include a subset of the first secondary set of surfaces and re-designating the first secondary set of surfaces to include another set of the plurality of the selected surfaces;
• (x) performing a low-pass filter on the estimate of the total wavefield energy on the first secondary set of surfaces to obtain a low- pass filtered estimate;
• (xi) setting the estimate of total wavefield energy on the first secondary set of surfaces equal to the low-pass filtered estimate;
• (xii) repeating steps (vii) through (xi) until the estimate of the total wavefield energy is computed for each selected surface of said plurality of selected surfaces;
• (2) subtracting the measured total wavefield energy on said forwardscattering surface from the estimate of the total wavefield energy on said forwardscattering surface to form a difference wavefield energy;
• (3) backpropagating said difference wavefield energy on said forwardscattering surface to the selected points, to obtain an estimate of a backpropagated wavefield energy, by solving an under-determined system of equations derived from derivative information, said derivative information being derivatives of the estimate total energy with respect to said region characteristic estimates;
• and
• (4) backpropagating said difference wavefield energy on said forwardscattering surface to the selected points, to obtain an estimate of a backpropagated wavefield energy by adding said estimate of the backpropagated wavefield energy to said region characteristics estimate on said selected points to obtain an updated region characteristics estimate on said selected points;
• (f) calculating a comparator from said region characteristics estimate and said updated region characteristics estimate on selected points;
• (g) performing said convergence step when comparator is greater than a selected tolerance;
• and
• (h) storing said region characteristics estimate in computer memory.

A method of reconstructing an image of matter in a region using a processing unit programmed to process data derived from an incident wavefield energy that has been transmitted at one or more frequencies, and one or more transmitter positions, each said transmitter position propagating wavefield energy at at least one orientation defined by Euler angles with respect to a selected fixed coordinate system, scattered by matter within said region, and detected by one or more receivers at one or more positions and orientations thereof, said orientations defined by Euler angles with respect to a selected fixed coordinate system, said method comprising the steps of:

• (a) choosing a forward scattering model, which generates a total wavefield energy given one or more physical characteristics of said matter defined at selected points within said region, and said incident wavefield energy;
• (b) propagating said incident wavefield energy, each said incident wavefield energy having one or more frequencies, toward said region from one or more transmitter positions and orientations thereof;
• (c) for one or more receiver positions each having at least one orientation defined by Euler angles with respect to said selected fixed coordinate system, detecting at each of said one or more receiver positions and respective orientations thereof, a detected wavefield energy;
• (d) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into one or more reception stored signals stored in said computer memory and corresponding to said detected wavefield energy
• (e) setting a region characteristics estimate of selected physical characteristics at said selected points within the region and storing said region characteristics estimate in said computer memory;
• (f) said processing unit calculating a fixed target characteristics estimate from the detected wavefield energy, said fixed target characteristics estimate being defined such that when operated on by a selected fixed approximation of the forward scattering model, the detected wavefield energy results;
• (g) said processing unit performing a convergence step comprising the following steps:
• (1) preparing, for each said one or more frequencies at each said transmitter positions and respective orientations thereof, an estimate of said total wavefield energy at said selected points derived from said incident wavefield energy for said selected points stored in the computer memory and said region characteristics estimate for said selected points;
• (2) deriving, for each of said one or more frequencies at each said transmitter position and orientations thereof, a calculated scattered wavefield energy for one or more of said receiver positions and respective orientations thereof from at least one of
• (i) said region characteristics estimate at said selected points, and
• (ii) said estimate of said total wavefield energy at said selected points
• (3) deriving from said calculated scattered field a variable region characteristic by solving a system whose right hand side is said calculated scattered wavefield energy, and whose left hand side is the result of applying said fixed approximation of the scattering model to said variable region characteristics;
• (4) comparing said variable regions characteristic to said fixed target characteristics estimate to derive a comparator;
• (5) when said comparator is greater than a selected tolerance, said processing unit computing and storing in said memory an updated region characteristics estimate from:
• (i) said estimate of said internal field,
• (ii) said fixed target characteristics estimate;
• (iii) said variable region characteristics;
• and
• (iv) the square Jacobian of the variable region characteristics with respect to the region characteristics estimate, said Jacobian utilization being implemented via a method of the family of gradient methods specifically designed for square systems;
• and then setting said region characteristics estimate equal to said updated region characteristics estimate;
• (h) repeating said processing unit convergence step until said comparator is less than or equal to said selected tolerance, said processing unit thereafter storing said updated region characteristics estimate in said memory.

A method for producing an image of matter in a region using a processing unit programmed to process data derived from incident wavefield energy that have been transmitted at one or more frequencies from one or more transmitter positions, each said transmitter position propagating wavefield energy at at least one transmitter orientation defined by Euler angles with respect to a selected fixed coordinate system, wherein said propagated wavefield energy is scattered by matter within said region and is detected by one or more receivers at one or more receiver positions and receiver orientations thereof, said receiver orientations defined by Euler angles with respect to said selected fixed coordinate system, said method comprising the steps of:

• (a) choosing a forward scattering model which generates a total wavefield energy given said incident wavefield energy and one or more physical characteristics of said matter defined at selected points within said region;
• (b) propagating said incident wavefield energy toward said region from said one or more transmitter positions and transmitter orientations thereof, said incident wavefield energy having one or more frequencies;
• (c) detecting detected wavefield energy at each of said one or more receiver positions and respective receiver orientations thereof;
• (d) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into detected scattered wavefield energy that is stored in said computer memory and that corresponds to said detected wavefield energy;
• (e) selecting a region characteristics estimate of selected physical characteristics at said selected points within the region and storing said region characteristics estimate in said computer memory;
• (f) said processing unit calculating a fixed target characteristics estimate from the detected wavefield energy, said fixed target characteristics estimate being defined such that when operated on by a selected fixed approximation of the forward scattering model, the detected wavefield energy results;
• (g) said processing unit performing a convergence step comprising the following steps:
• (1) preparing, for each of said one or more frequencies at each said transmitter position and respective transmitter orientations thereof, an estimate of said total wavefield energy at said selected points derived from said incident wavefield energy for said selected points stored in the computer memory and said region characteristics estimate for said selected points;
• (2) deriving, for each of said one or more frequencies at each said transmitter position and respective transmitter orientations thereof, a calculated scattered wavefield energy for one or more of said receiver positions and respective receiver orientations thereof from at least one of:
• (i) said region characteristics estimate at said selected points, and
• (ii) said estimate of said total wavefield energy at said selected points;
• (3) deriving from said calculated scattered field one or more variable region characteristics by solving a system whose right hand side is said calculated scattered wavefield energy, and whose left hand side is the result of applying said fixed approximation of the scattering model to said one or more variable region characteristics;
• (4) comparing said one or more variable region characteristics to said fixed target characteristics estimate to derive a comparator;
• (5) when said comparator is greater than a selected tolerance, said processing unit computing and storing in said computer memory an updated region characteristics estimate from:
• (i) said estimate of said internal field;
• (ii) said fixed target characteristics estimate;
• (iii) said one or more variable region characteristics;
• and
• (iv) the square Jacobian of the one or more variable region characteristics with respect to the region characteristics estimate, said square Jacobian utilization being implemented via a method of the family of gradient methods specifically designed for square systems, and then setting said region characteristics estimate equal to said updated region characteristics estimate;
• (h) repeating said processing unit convergence step until said comparator is less than or equal to said selected tolerance, said processing unit thereafter storing said updated region characteristics estimate in said computer memory.

A method for producing an image of matter in a region using a processing unit programmed to process data derived from incident wavefield energy that have been transmitted at one or more frequencies from one or more transmitter positions, each said transmitter position propagating wavefield energy at at least one transmitter orientation defined by Euler angles with respect to a selected fixed coordinate system, wherein said propagated wavefield energy is scattered by matter within said region and is detected by one or more receivers at one or more receiver positions and receiver orientations thereof, said receiver orientations defined by Euler angles with respect to said selected fixed coordinate system, said method comprising the steps of:

• (a) choosing a forward scattering model which generates a total wavefield energy given said incident wavefield energy and one or more physical characteristics of said matter defined at selected points within said region;
• (b) propagating said incident wavefield energy toward said region from said one or more transmitter positions and transmitter orientations thereof, said incident wavefield energy having one or more frequencies;
• (c) for said one or more receiver positions, detecting at each said one or more receiver positions and respective receiver orientations thereof, a detected wavefield energy;
• (d) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into detected scattered wavefield energy stored in said computer memory and corresponding to said detected wavefield energy;
• (e) selecting a variable scattered wavefield;
• (f) said processing unit performing a convergence step comprising the following steps:
• (1) deriving from said variable scattered wavefield a set of variable region characteristics by solving a system whose right hand side is said variable scattered wavefield, and whose left hand side is the result of applying a fixed approximation of the forward scattering model to said set of variable region characteristics,
• (2) preparing, for each of said one or more frequencies at each said transmitter position and respective orientations thereof, an estimate of said total wavefield energy at said selected points derived from said incident wavefield energy for said selected points stored in the computer memory and said set of variable region characteristics for said selected points;
• (3) deriving, for each of said one or more frequencies at each said transmitter position and orientations thereof, a calculated scattered wavefield energy for said one or more of said receiver positions and respective receiver orientations thereof from at least one of:
• (i) said set of variable region characteristics at said selected points, and (ii) said estimate of said total wavefield energy at said selected points;
• (4) comparing said calculated scattered wavefield energy determined by said processing unit to said detected scattered wavefield energy to derive a comparator;
• (5) when said comparator is greater than a selected tolerance, said processing unit computing an updated variable scattered wavefieldom:
• (i) said estimate of said total wave field;
• (ii) said detected scattered wavefield energy;
• (iii) said variable region characteristics;
• and
• (iv) the square Jacobian of the calculated scattered wavefield with respect to the varible scattered wavefield, said square Jacobian utilization being implemented via a method of the family of gradient methods specifically designed for square systems, and then setting said region characteristics estimate equal to said updated region characteristics estimate;
• (h) repeating said processing unit convergence step until said comparator is less than or equal to said selected tolerance;
• and
• (i) deriving from said calculated scattered wavefield a region characteristics estimate by solving a system whose right hand side is said calculated scattered wavefield, and whose left hand side is the result of applying said fixed approximation of the forward scattering model to said region characteristics estimate, and storing said region characteristics estimate in said computer memory.

A method of producing an image of an object from wavefield energy that has been transmitted into and scattered by the object, said image comprising a matrix of scattering parameters at points within the object, said image being stored in a memory of a central processing unit (CPU), and said method comprising the steps of.

• (a) electronically transmitting an electric signal at one or more frequencies and transducing said electric signal at each said frequency into wavefield energy propagated toward said object from one or more of transducer transmitter positions;
• (b) electronically processing said-electric signal to determine from said one or more transmitter positions an incident field corresponding to said propagated wavefield energy, said incident field being stored in the memory of the CPU in the form of digitized electric signals;
• (c) detecting at one or more of transducer receiver positions said wavefield energy transmitted into and scattered by said object;
• (d) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into one or more digitized electric signals stored in said memory of said CPU and corresponding to a scattered field detected at said one or more transducer receiver positions;
• (e) said CPU setting a present estimate of one or more scattering parameters for said object at each said frequency and storing each said present estimate of one or more scattering parameters in said memory;
• (f) said CPU performing a convergence step at each of said one ore more frequencies comprising the following steps:
• (1) said CPU preparing, using a first Fourier marching method, a composite field representing the incident field and the scattered field derived from:
• ((a)) said incident field, and
• ((b)) said present estimate of said one or more scattering parameters, where said composite field at each of said one or more frequencies comprises multiple orders of scattering;
• and
• (2) comparing said scattered field detected at said one or more transducer receiver positions to said composite field determined by said CPU to derive a comparator;
• (3) when said comparator is greater than a selected tolerance, said CPU determining, using a second Fourier marching method, and storing in said memory, a matrix of updated scattering parameters calculated from:
• ((a)) said composite field,
• ((b)) said present estimate of one or more scattering parameters,
• ((c)) said scattered field detected at said one or more transducer receiver positions;
• and
• ((d)) said CPU utilizing the Jacobian of the composite field with respect to the present estimate of one or more scattering parameters and the Hermitian conjugate thereof, and then setting the present estimate of one or more scattering parameters equal to the matrix of the updated scattering parameters;
• (g) repeating said CPU convergence step until said comparator is less than or equal to said selected tolerance, said CPU thereafter using the matrix of the updated scattering parameters to store in said CPU memory and to produce said image.