Extended Subspace Method for Cross-Talk Mitigation in Multi-Parameter Inversion

• US 20140358504A1
• Filed: 05/08/2014
• Published: 12/04/2014
• Est. Priority Date: 06/03/2013
• Status: Active Grant

• Current Assignee: Anatoly Baumstein, Ivan Chikichev
• Original Assignee: Anatoly Baumstein, Ivan Chikichev
• Inventors: Baumstein, Anatoly, Chikichev, Ivan
• Granted Patent: US 10,459,117 B2
• Time in Patent Office: Days
• US Class Current: 703/2
• CPC Class Codes: G01V 20/00 Geomodelling in general

Abstract

An extended subspace method for inverting geophysical data to infer models for two or more subsurface physical properties, using gradients of an objective function as basis vectors for forming model updates. The extended set of basis vectors provides explicit mixing between gradient components corresponding to different medium parameters, for example P-wave velocity and an anisotropy parameter. In a preferred embodiment, off-diagonal elements of the mixing matrix may be scaled to adjust the degree of mixing between gradient components. Coefficients of the basis vector expansion are determined in a way that explicitly accounts for leakage or crosstalk between different physical parameters. The same extended subspace approach may be used to make further improvement to the model updates by incorporating well constraints, where well log data are available.

Claims

1. A computer-implemented method for iteratively inverting measured geophysical data to infer 3D subsurface models of N≧

2 physical properties, comprising;

providing an initial model for each physical property, wherein a subsurface region is subdivided into discrete cells, each cell having a value of the physical property;

for each physical property and for each of a plurality of the cells, representing a search direction, indicating whether the initial model needs to be updated positively or negatively, as a linear combination of M>

N basis vectors, wherein;

each basis vector has its own coefficient in the linear combination, said coefficient to be determined;

the basis vectors are chosen such that their coefficients account for cross-talk between the N physical properties during the inversion; and

optimal values of the coefficients are simultaneously solved for, using a computer, to minimize or maximize an objective function measuring misfit between model-simulated geophysical data and the measured geophysical data, wherein solving for the coefficients is performed at least by numerical computation constrained by the geophysical data; and

using the optimal coefficients to generate search directions, and using the search directions to generate an updated model for each physical property.

2. The method of claim 1, wherein each basis vector has a component that is, or is proportional to, a gradient, with respect to model parameters of one of the N physical properties, of the objective function.

3. The method of claim 1, wherein a mixing matrix is formed from the coefficients of the basis vectors, and degree of mixing between the model updates for the N physical properties is adjusted by scaling off-diagonal elements of the mixing matrix with scaling factors that are optimized in the inversion.

4. The method of claim 3, wherein the optimizing of the scaling factors is based on a line search.

5. The method of claim 4, wherein the line search is performed by steps comprising fitting a polynomial function of the scaling factors to the objective function and then finding values of the scaling factors that optimize the objective function.

6. The method of claim 4, further comprising performing a conventional line search to determine an optimal step size for the model update.

7. The method of claim 3, wherein the scaling factors are spatially dependent.

8. The method of claim 7, wherein the optimization comprises performing line searches, and a separate line search is performed for each source shot in a geophysical survey that generated the measured geophysical data, thereby providing the spatial dependence.

9. The method of claim 3, wherein the optimization of the coefficients and the scaling factors is based on one or more Hessians of the objective function.

10. The method of claim 3, wherein optimal values of the scaling factors are determined by a line search comprising evaluating the objective function E for an updated model m+Δ

m for a plurality of different values of the scaling factors, and selecting a combination of scaling factors giving a least value of E.

11. The method of claim 1, further comprising adjusting the search directions before the generating an updated model for each physical property, wherein each adjusted search direction is represented by a linear combination of a plurality of basis vectors, and coefficients of the linear combination are determined by minimizing a difference between one or more true parameters computed from well data or other known subsurface information and corresponding parameters predicted by the updated search direction.

12. The method of claim 11, wherein the plurality of basis vectors is N+1 in number, comprising a gradient of the objective function with respect to model parameters of each of the N physical properties, plus an additional basis vector whose coefficient allows matching to the well data or other known subsurface information.

13. The method of claim 12, wherein every component of the additional basis vector is unity.

14. The method of claim 1, wherein the geophysical data are seismic data, and the N physical properties are selected from the group consisting of compressional and shear-wave velocities, V_p and V_s, density ρ

, and Thompsen anisotropy parameters ε and

δ

.

15. The method of claim 1, wherein the optimization is based on one or more Hessians of the objective function.

19. The method of claim 1, wherein rock-physics-based or empirical relationships between physical property parameters, or well data constraints, or both, are used to reduce number of search direction coefficients to be solved for by the iterative numerical computation constrained by the geophysical data.

16. A computer-implemented method for iteratively inverting measured geophysical data to infer 3D subsurface models of N≧

2 physical properties, comprising;

providing an initial model for each physical property, wherein a subsurface region is subdivided into discrete cells, each cell having a value of the physical property;

for each physical property and for each of a plurality of the cells, representing a search direction, indicating whether the initial model needs to be updated positively or negatively, as a linear combination of a plurality of basis vectors;

determining coefficients of each linear combination by minimizing, using a computer, a difference between one or more true parameters computed from well data or other known subsurface information and corresponding parameters predicted by the updated search direction; and

using the determined coefficients to generate search directions, and using the search directions to generate an updated model for each physical property.

17. The method of claim 16, wherein the plurality of basis vectors is N+1 in number, comprising a gradient of the objective function with respect to model parameters of each of the N physical properties, plus an additional basis vector whose coefficient allows matching to the well data or other known subsurface information.

18. The method of claim 17, wherein every component of the additional basis vector is unity.