Shape-based geophysical parameter inversion

1. A method for imaging a subsurface region, comprising:

providing at least one geophysical data set containing data pertaining to a subsurface region to an objective function, wherein the at least one geophysical data set includes one of seismic data, time-lapse seismic data, electromagnetic data, gravity data, gradiometry data, well log data, and well pressure data;

determining (i) an implicitly represented geologic structure of the subsurface region, wherein the implicitly represented geological structure is a portion of the subsurface which includes one or more facies or stratigraphic layers and is described through the use of one or more auxiliary functions defined in the subsurface region, and (ii) at least one geophysical parameter of the subsurface region, wherein the at least one geophysical parameter is structurally consistent with the geologic structure;

by optimizing the objective function using gradient-based techniques, wherein the objection function is Mumford-Shah or phase-field; and

using a computer system to display geologic structures and geophysical parameters of the subsurface region produced by the objective function optimization.

2. The method of claim 1 , wherein the geologic structure is a boundary set including at least one interface within the subsurface region.

3. The method of claim 1 , wherein the geologic structure is a set of domains within the subsurface region.

4. The method of claim 1 , further comprising adjusting a β coefficient of the objective function to adjust a complexity of the geologic structure.

5. The method of claim 1 , further comprising decreasing a coefficient α of the objective function to decrease a smoothness of the at least one geophysical parameter.

6. The method of claim 1 , wherein the least one geophysical parameter is one of a wave speed, a permeability, a porosity, an electrical conductivity, and a density.

7. A method for imaging a subsurface region, comprising:

providing at least one geophysical data set containing data pertaining to a subsurface region to a Mumford-Shah-based objective function, wherein the at least one geophysical data set includes one of seismic data, time-lapse seismic data, electromagnetic data, gravity data, gradiometry data, well log data, and well pressure data;

optimizing the Mumford-Shah-based objective function using gradient-based techniques to determine at least one of the following: a geologic structure of the subsurface region and at least one geophysical parameter of the subsurface region wherein the at least one geophysical parameter is structurally consistent with the geologic structure; and

using a computer system to display the geological structures and the geophysical parameters of the subsurface region produced by the Mumford-Shah-based objective function optimization.

8. The method of claim 7 , wherein the geologic structure is represented by an implicit method.

9. The method of claim 7 , wherein the geologic structure is a boundary set including at least one interface within the subsurface region.

10. The method of claim 7 , wherein the Mumford-Shah-based objective function includes a least-squares data misfit and a Mumford-Shah functional, which is a geometric regularization.

11. The method of claim 7 , further comprising:

fixing the at least one geophysical parameter; and

optimizing the Mumford-Shah-based objective function to invert for the geologic structure.

12. The method of claim 7 , further comprising:

fixing the geologic structure; and

optimizing the Mumford-Shah-based objective function to invert for the at least one geophysical parameter.

13. The method of claim 7 , wherein the Mumford-Shah functional is approximated by an Ambrosio-Tortorelli approximation.

14. The method of claim 7 , wherein the Mumford-Shah functional is approximated by a Chan-Vese level-set approach.

15. A method for imaging a subsurface region, comprising:

providing at least one geophysical data set containing data pertaining to a subsurface region to an objective function, wherein the at least one geophysical data set includes one of seismic data, time-lapse seismic data, electromagnetic data, gravity data, gradiometry data, well log data, and well pressure data;

applying a phase-field functional to optimize the objective function using gradient based techniques and determine at least one of the following: a geologic structure of the subsurface region and at least one geophysical parameter of the subsurface region wherein the at least one geophysical parameter is structurally consistent with the geologic structure; and

using a computer system to display the geological structures and the geophysical parameters of the subsurface region produced by the objective function optimization.

16. The method of claim 15 , wherein the geologic structure is a set of domains within the subsurface region.

17. The method of claim 15 , wherein

the objective function includes a least-squares data misfit, and

the phase-field functional is a geometric regularization.

18. The method of claim 15 , further comprising:

fixing the at least one geophysical parameter; and

optimizing the objective function to invert for the geologic structure.

19. The method of claim 15 , further comprising:

fixing the geologic structure; and

optimizing the objective function to invert for the at least one geophysical parameter.