Methods of multinary inversion for imaging objects with discrete physical properties
A method of multinary inversion for imaging objects with discrete physical properties of the examined medium is described. The model parameters of the target area are parameterized in terms of a multinary function of the physical properties that accepts a finite number of discrete values from the continuum of at least one physical property. The multinary function is chosen such that the derivative of the multinary function with respect to the physical property is a continuous and known function. The imaging is based on solving the optimization problem for parametric functional of the multinary functions describing the target model parameters. The method can be applied for multi-modal imaging, such that at least one physical property representing the physical properties of the examined medium, may be derived to provide a reconstruction or classification of the physical properties of the examined medium.
1. A method of multinary inversion for imaging of physical properties of an examined medium characterized by discrete physical property values, the method comprising:
a. placing a data acquisition system in proximity of a target area of the examined medium, and acquiring and processing observed data related to the target area;
b. selecting a multinary function that accepts a finite number of discrete values from physical property values of the target area, and which characterizes a set of discrete values of the physical properties of the target area;
c. emulating the data acquisition system and its parameters by a forward modeling calculation of predicted data related to the target area;
d. calculating a misfit functional and a residual vector between the observed data and the predicted data;
e. calculating stabilizing functionals for model parameters and calculating Fréchet derivatives (sensitivities) of the observed data with respect to the multinary model parameters;
f. solving a multinary inverse problem by minimizing a parametric functional formed by a linear combination of the misfit and stabilizing functionals;
g. evaluating termination criteria of the multinary inverse problem; and
h. calculating physical property values to produce intermediate and/or final images of the examined medium.
2. The method of claim 1 , wherein the physical property comprises one of electrical conductivity, dielectric permittivity, induced polarization parameters, density, magnetization, susceptibility, compressional wave velocity, acoustic impedance, and/or shear wave velocity, and/or any combination thereof, representing the physical properties of the examined medium.
3. The method of claim 1 , wherein the examined medium is a geological formation.
4. The method of claim 1 , wherein the examined medium is a human and/or animal body.
5. The method of claim 1 , wherein the examined medium is a man-made object.
6. The method of claim 1 , wherein the target area is characterized by different physical properties including one or more of conductivity, permeability, induced polarization parameters, density, magnetization, magnetic susceptibility, compressional wave velocity, shear wave velocity, and/or any combination thereof; and wherein the data acquisition system collects multi-modal data representing different physical properties of the examined medium.
7. The method of claim 1 , wherein the stabilizing functionals for model parameters include any smooth or focusing stabilizing functions such as Laplacian, minimum gradient, minimum norm, minimum support, or minimum gradient support.
8. The method of claim 1 , wherein termination criteria of the multinary inverse problem includes a predetermined misfit being achieved.
9. A physical non-transitory computer readable medium having stored thereon computer executable instructions that when executed by a processor cause a computing system to perform a method of multinary inversion for imaging of physical properties of an examined medium characterized by discrete physical property values, the method comprising:
a. acquiring and processing observed data related to a target area of the examined medium;
b. selecting a multinary function that accepts a finite number of discrete values from physical property values of the target area, and which characterizes a set of discrete values of the physical properties of the target area;
c. emulating a data acquisition system and its parameters by a forward modeling calculation of predicted data related to the target area;
d. calculating a misfit functional and a residual vector between the observed data and the predicted data;
e. calculating stabilizing functionals for model parameters and calculating Fréchet derivatives (sensitivities) of the observed data with respect to the multinary model parameters;
f. solving a multinary inverse problem by minimizing a parametric functional formed by a linear combination of the misfit and stabilizing functionals;
g. evaluating termination criteria of the multinary inverse problem; and
h. calculating physical property values to produce intermediate and/or final images of the examined medium.
10. The computer readable medium of claim 9 , wherein the physical property comprises one of electrical conductivity, dielectric permittivity, induced polarization parameters, density, magnetization, susceptibility, compressional wave velocity, acoustic impedance, and/or shear wave velocity, and/or any combination thereof, representing the physical properties of the examined medium.
11. The computer readable medium of claim 9 , wherein the target area is characterized by different physical properties including one or more of conductivity, permeability, induced polarization parameters, density, magnetization, magnetic susceptibility, compressional wave velocity, shear wave velocity, and/or any combination thereof.
12. The computer readable medium of claim 9 , wherein the examined medium is one of a geological formation, a human and/or animal body, or a man-made object.
13. A system for multinary inversion for imaging of physical properties of an examined medium characterized by discrete physical property values comprising:
a data acquisition system configured to obtain data related to a target area of the examined medium; and
a computing system, the computing system comprising:
a processor; and
one or more physical non-transitory computer readable medium having computer executable instructions stored thereon that when executed by the processor, cause the computing system to perform:
a. acquiring and processing observed data related to the target area;
b. selecting a multinary function that accepts a finite number of discrete values from physical property values of the target area, and which characterizes a set of discrete values of the physical properties of the target area;
c. emulating a data acquisition system and its parameters by a forward modeling calculation of predicted data related to the target area;
d. calculating a misfit functional and a residual vector between the observed data and the predicted data;
e. calculating stabilizing functionals for model parameters and calculating Fréchet derivatives (sensitivities) of the observed data with respect to the multinary model parameters;
f. solving a multinary inverse problem by minimizing a parametric functional formed by a linear combination of the misfit and stabilizing functionals;
g. evaluating termination criteria of the multinary inverse problem; and
h. calculating physical property values to produce intermediate and/or final images of the examined medium.
14. The system of claim 13 , wherein the physical property comprises one of electrical conductivity, dielectric permittivity, induced polarization parameters, density, magnetization, susceptibility, compressional wave velocity, acoustic impedance, and/or shear wave velocity, and/or any combination thereof, representing the physical properties of the examined medium.
15. The system of claim 13 , wherein the examined medium is a geological formation.
16. The system of claim 13 , wherein the examined medium is a human and/or animal body.
17. The system of claim 13 , wherein the examined medium is a man-made object.
18. The system of claim 13 , wherein the target area is characterized by different physical properties including one or more of conductivity, permeability, induced polarization parameters, density, magnetization, magnetic susceptibility, compressional wave velocity, shear wave velocity, and/or any combination thereof; and wherein the data acquisition system collects multi-modal data representing different physical properties of the examined medium.
19. The system of claim 13 , wherein the stabilizing functionals for model parameters include any smooth or focusing stabilizing functions such as Laplacian, minimum gradient, minimum norm, minimum support, or minimum gradient support.
20. The system of claim 13 , wherein termination criteria of the multinary inverse problem includes a predetermined misfit being achieved.