PSEUDO-ANALYTICAL METHOD FOR THE SOLUTION OF WAVE EQUATIONS

• US 20100088035A1
• Filed: 10/06/2009
• Published: 04/08/2010
• Est. Priority Date: 10/06/2008
• Status: Active Grant

• Current Assignee: BP Corporation North America Incorporated (BP PLC)
• Original Assignee: BP Corporation North America Incorporated (BP PLC)
• Inventors: Brandsberg-Dahl, Sverre, Etgen, John T.
• Granted Patent: US 8,296,069 B2
• Time in Patent Office: Days
• US Class Current: 702/16
• CPC Class Codes: G01V 1/30 Analysis G01V1/50 takes precedence

Abstract

The instant invention is a method for numerically propagating waves or solving wave equations on a digital computer. This invention can be used to compute highly accurate solutions to the wave equation, in fact in some cases it computes the analytical solution, something previously considered impossible. The instant method can also propagate waves that are not described by differential equations, such as anisotropic scalar waves. The invention has the advantages that it is computationally efficient, accurate, and flexible. Of importance is the ability to propagate waves that simulate the P-wave arrivals in both isotropic and anisotropic media with a scalar as opposed to a vector equation.

Claims

1. A method of seismic exploration within a predetermined volume of the earth containing structural and stratigraphic features conducive to the generation, migration, accumulation, or presence of subsurface resources, comprising the steps of:

a. accessing a digital representation of a seismic survey that images at least a portion of said predetermined volume of the earth;

b. selecting at least a portion of said seismic survey, thereby selecting a plurality of seismic traces;

c. obtaining a velocity model at least approximately representative of at least a portion of said imaged portion of said predetermined volume of the earth, said velocity model having at least one velocity therein;

d. using at least said a portion of said velocity model to determine an error compensating pseudo-differential operator;

e. using said error compensating pseudo-differential operator to construct a time-marching wave equation solution for use in a migration program;

f. migrating said plurality of seismic traces using said migration program, thereby producing a plurality of migrated seismic traces; and

,

g. using at least a portion of said plurality of migrated seismic traces to explore for subsurface resources within the predetermined volume of the earth.

2. The method according to claim 1, wherein said velocity model consists of a single velocity.

3. The method according to claim 2, wherein said error compensating pseudo-differential operator F({right arrow over (k)}) is defined by the equation:

4. The method according to claim 1, wherein said time marching wave equation solution is a second order time marching wave equation solution.

5. The method of exploration for subsurface resources within a predetermined volume of the earth according to claim 1, wherein step (g) comprises the step of:

(g1) writing at least a portion of said plurality of migrated seismic traces to computer storage, and,

(g2) using at least a portion of said written portion of said plurality of migrated seismic traces to explore for subsurface resources within the predetermined volume of the earth.

6. The method according to claim 5, wherein said computer storage is selected from a group consisting of a magnetic disk, a magnetic tape, an optical disk, a magneto-optical disk, a DVD disk, RAM, a flash RAM drive, and non-volatile RAM.

7. The method according to claim 1, wherein step (g) comprises the step of:

(g1) viewing at least a portion of said plurality of migrated seismic traces on a display device, thereby using said at least a portion of said plurality of migrated seismic traces to explore for subsurface resources within the predetermined volume of the earth.

8. A method of seismic exploration within a predetermined volume of the earth containing structural and stratigraphic features conducive to the generation, migration, accumulation, or presence of subsurface resources, comprising the steps of:

a. accessing a digital representation of a seismic survey that images at least a portion of said predetermined volume of the earth;

b. selecting at least a portion of said seismic survey, thereby selecting a plurality of seismic traces;

c. obtaining a velocity model at least approximately representative of at least a portion of said imaged portion of said predetermined volume of the earth, said velocity model having at least one velocity therein;

d. using at least said a portion of said velocity model to determine an error compensating pseudo-differential operator;

e. using said error compensating pseudo-differential operator to construct time-marching wave equation solution for use in an inversion program;

f. inverting said plurality of seismic traces using said inversion program, thereby producing an earth parameter model; and

,

g. using at least a portion of said earth parameter model to explore for subsurface resources within the predetermined volume of the earth.

9. The method according to claim 8, wherein said velocity model consists of a single velocity.

10. The method according to claim 9, wherein said error compensating pseudo-differential operator F({right arrow over (k)}) is defined by the equation:

11. The method according to claim 8, wherein said time marching wave equation solution is a second order time arching wave equation solution.

12. The method according to claim 8, where said earth parameter model is chosen from a group consisting of an updated velocity model, an anisotropy model, a density model, and a porosity model.

13. The method of exploration for subsurface resources within a predetermined volume of the earth according to claim 8, wherein step (g) comprises the step of:

(g1) writing at least a portion of said plurality of migrated seismic traces to computer storage, and,

(g2) using at least a portion of said written portion of said plurality of migrated seismic traces to explore for subsurface resources within the predetermined volume of the earth.

14. The method according to claim 13, wherein said computer storage is selected from a group consisting of a magnetic disk, a magnetic tape, an optical disk, a magneto-optical disk, a DVD disk, RAM, a flash RAM drive, and non-volatile RAM.

15. A method of seismic exploration within a predetermined volume of the earth containing structural and stratigraphic features conducive to the generation, migration, accumulation, or presence of subsurface resources, comprising the steps of:

a. obtaining a velocity model at least approximately representative of at least a portion of said predetermined volume of the earth, said velocity model having at least one velocity therein;

b. using at least said a portion of said velocity model to determine an error compensating pseudo-differential operator;

c. using said error compensating pseudo-differential operator to construct a time-marching wave equation solution for use in a seismic modeling program;

d. using said seismic modeling program to create a plurality of synthetic seismic traces at least approximately representative of a portion of the predetermined volume of the earth; and

,

e. using said plurality of synthetic seismic traces to explore for subsurface resources within the predetermined volume of the earth.

16. The method according to claim 15, wherein said velocity model consists of a single velocity.

17. The method according to claim 16, wherein said error compensating pseudo-differential operator F({right arrow over (k)}) is defined by the equation:

18. The method according to claim 15, wherein said time marching wave equation solution is a second order time marching wave equation solution.

19. The method of exploration for subsurface resources within a predetermined volume of the earth according to claim 15, wherein step (e) comprises the step of:

(e1) accessing a seismic survey that images at least a part of said predetermined volume of the earth, said seismic survey comprising a plurality of seismic traces, and,

(e2) comparing at least one of said synthetic seismic traces with at least one of said seismic traces, thereby using at least a portion of said plurality of synthetic seismic traces to explore for subsurface resources within the predetermined volume of the earth.