Method and apparatus for absorbing boundary conditions in numerical finite-difference acoustic applications

• US 6,687,659 B1
• Filed: 03/24/2000
• Issued: 02/03/2004
• Est. Priority Date: 03/24/2000
• Status: Expired due to Term

• Current Assignee: Conocophillips Company (ConocoPhillips)
• Original Assignee: Conocophillips Company (ConocoPhillips)
• Inventors: Shen, Yunqing
• Primary Examiner(s): FREJD, RUSSELL WARREN
• Application Number:
  US09/535,232
• Time in Patent Office: 1,411 Days
• Field of Search: 703/2, 703/10, 703/22, 367/50, 367/51, 367/52, 367/53, 367/72, 367/73, 702/14, 702/18, 711/170, 711/171, 711/172, 711/173
• US Class Current: 703/2
• CPC Class Codes: G01V 1/28 Processing seismic data, e.g. for interpretation or for event detection G01V1/48 takes precedence
• CPC Class Codes: G01V 2210/67 Wave propagation modeling
• CPC Class Codes: G06F 17/13 Differential equations using digital differential analysers G06F7/64

Abstract

A hybrid approach is disclosed for implementing highly efficient absorbing boundary conditions in three dimensional (3D) finite difference (FD) acoustic applications such as post-stack and pre-stack seismic migration, and forward modeling. The “Perfectly matched layer” (PML) absorbing medium is able to absorb waves arriving at all incidence angle and at any frequency. An optimized PML medium is disclosed including its formulation and implementation to reduce its memory usage and associated computational cost. To avoid costly and cumbersomely implementation of PML medium at the edges and corners of truncated 3D numerical models the PML medium is combined with a “One-way Wave Equation” (1WWE) absorbing boundary to eliminate reflections and diffraction from the edges and corners. The hybrid approach significantly reduces the memory usage and computational cost for multiprocessor applications when large 3D applications are partitioned among multiple processors and artificial boundaries are created between multiple processors.

Claims

1. A method of eliminating reflections resulting from an advancing simulated seismic wave contacting an artificial boundary in a computer implemented migration of seismic data, said method comprising:

(a) storing an earth velocity model having a wavefield propagation region in a computer memory, wherein said propagation region is truncated by limited capacity of said computer memory;

(b) surrounding said computer memory contacting truncated boundaries of said propagation region with a wave absorbing region, wherein said absorbing region comprises two types of absorbing conditions;

(c) wherein a first absorbing condition is applied on exterior surfaces of memory volumes contacting said truncated propagation region, and a second absorbing condition is applied at exterior edges and corners of said truncated propagation region in said computer memory;

(d) imposing said second absorbing condition on the exterior surface of said first absorbing condition; and

(e) computing seismic wave propagation in said earth velocity model and in said absorbing region, wherein said advancing simulated seismic wave is essentially absorbed in said absorbing region.

2. A method in accordance with claim 1, wherein said wave absorbing region presents an essentially reflectionless interface for outgoing waves advancing through said propagation region.

3. A method in accordance with claim 1, wherein said first absorbing condition is effective for attenuating waves at essentially all incident angles, and said second absorbing condition is effective for attenuating waves at or near a selective incidence angle.

4. A method in accordance with claim 1, wherein said memory comprises a rectangularly shape volume having four generally vertical side faces, and wherein said first absorbing condition is applied at the exterior of said four vertical side faces and at the exterior of a generally horizontal bottom surface.

5. A method in accordance with claim 1, wherein said wave propagation region comprises finite-difference computations, said first absorbing condition comprises a perfectly matched layer (PML) absorbing condition, and said second absorbing condition comprises a one-way wave equation (1WWE) absorbing condition.

6. A method of eliminating reflections resulting from an advancing simulated seismic wave contacting an artificial boundary in a computer implemented migration of seismic data, said method comprising:

(a) storing an earth velocity model having a wavefield propagation region in a computer memory, wherein said propagation region is truncated by limited capacity of said computer memory;

(b) surrounding said computer memory contacting truncated boundaries of said propagation region with a wave absorbing region, wherein said absorbing region comprises two types of absorbing conditions;

(c) wherein a perfectly matched layer (PML) absorbing condition is applied on exterior surfaces of memory volumes contacting said truncated propagation region, and a second one-way wave equation (1WWE) absorbing condition is applied at exterior edges and corners of said truncated propagation region in said computer memory;

(d) imposing said 1WWE absorbing condition on the exterior surface of said first absorbing condition; and

(e) computing seismic wave propagation in said earth velocity model and in said absorbing region, wherein said advancing simulated seismic wave is essentially absorbed in said absorbing region;

wherein said PML absorbing conditions are governed by four variables expressed in the set of equations comprising;

$\frac{\partial <br><br>q}{\partial <br><br>t}+\mathrm{\alpha <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q=-\text{\hspace{1em}<br><br>}<br><br>\frac{\partial <br><br>p}{\partial <br><br>N}$$\frac{\partial <br><br>U_{\mathrm{L1}}}{\partial <br><br>t}+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>U_{\mathrm{L1}}=-\text{\hspace{1em}<br><br>}<br><br>\frac{\partial <br><br>p}{\partial <br><br>L_2}$$\frac{\partial <br><br>U_{\mathrm{L2}}}{\partial <br><br>t}+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>U_{\mathrm{L2}}=-\text{\hspace{1em}<br><br>}<br><br>\frac{\partial <br><br>p}{\partial <br><br>L_2}$$\frac{\partial <br><br>p}{\partial <br><br>t}=-C^2<br><br>\left(L_1,L_2\right)<br><br>\left(\frac{\partial <br><br>U_{\mathrm{L1}}}{\partial <br><br>L_1}+\frac{\partial <br><br>U_{\mathrm{L2}}}{\partial <br><br>L_2}+\frac{\partial <br><br>q}{\partial <br><br>N}\right).$

7. A method of eliminating reflections resulting from an advancing simulated seismic wave contacting an artificial boundary in a computer implemented migration of seismic data, said method comprising:

(a) storing an earth velocity model having a wavefield propagation region in a computer memory, wherein said propagation region is truncated by limited capacity of said computer memory;

(b) surrounding said computer memory contacting truncated boundaries of said propagation region with a wave absorbing region, wherein said absorbing region comprises two types of absorbing conditions;

(c) wherein a first absorbing condition is applied on exterior surfaces of memory volumes contacting said truncated propagation region, and a second absorbing condition is applied at exterior edges and corners of said truncated propagation region in said computer memory;

(d) imposing said second absorbing condition on the exterior surface of said first absorbing condition; and

(e) computing seismic wave propagation in said earth velocity model and in said absorbing region, wherein said advancing simulated seismic wave is essentially absorbed in said absorbing region, by the following steps performed in the indicated sequence;

(i) updating time and space variables in the finite-difference computations in said truncated wavefield propagation region;

(ii) applying said second absorbing boundary conditions at the vertical edges and corners of said truncated propagation region;

(iii) updating particle velocity variables and the q variable in the interior of said first absorbing region;

(iv) applying said second boundary condition on the exterior of said first absorbing condition;

(v) updating the pressure variable in the interior of said absorbing region; and

(vi) updating the pressure variable on two layers in said wavefield propagating region immediately above said absorbing region.

8. A method in accordance with claim 7, wherein said wavefield propagation region comprises a finite-difference computation region, said first absorbing condition comprises a perfectly matched layer (PML) absorbing condition, and said second absorbing condition comprises a one-way wave equation (1WWE) absorbing condition.

9. A method of establishing an absorbing region for terminating a wavefield propagation region of an earth velocity model stored in a computer memory, wherein said propagation region is truncated by limited memory capacity, said method comprising:

(a) surrounding said computer memory contacting truncated boundaries of said propagation region with a wave absorbing region, wherein said absorbing region comprises two types of absorbing conditions; and

(b) wherein a first absorbing condition is applied on the exterior surfaces of memory volumes contacting said truncated propagation region, and a second absorbing condition is applied at the exterior edges and corners of said truncated propagation region in said computer memory.

10. A method in accordance with claim 9, additionally comprising:

imposing said second absorbing condition on the exterior surface of said first absorbing condition.

11. A method in accordance with claim 9, wherein said first absorbing condition is effective for attenuating advancing waves at essentially all incidence angles, and said second absorbing condition is effective for attenuating advancing waves at or near a selective incidence angle.

12. A method in accordance with claim 9, wherein said propagation region comprises a finite-difference computation region, said first absorbing region comprises perfectly matched layer (PML) absorbing condition, and said second absorbing condition comprises a one-way wave equation (1WWE) absorbing condition.

13. A method in accordance with claim 9, wherein said memory comprises a rectangularly shaped volume having four generally vertical side faces, and wherein said first absorbing condition is applied at the exterior of said four vertical side faces, and at the exterior of a generally horizontal bottom surface.

14. Apparatus for eliminating reflections resulting from an advancing simulated wave contacting an artificial boundary in a computer implemented migration of seismic data, said method comprising:

a computer programmed for;

(a) storing an earth velocity model having a wavefield propagation region in said computer memory;

wherein said propagation region is truncated by limited capacity in said computer memory;

(b) surrounding said computer memory contacting truncated boundaries of said propagation region with a wave absorbing region, wherein said absorbing region comprises two types of absorbing conditions;

(c) wherein a first absorbing condition is applied on exterior surfaces of memory volumes contacting said truncated propagation region, and a second absorbing condition is applied at exterior edges and corners of said truncated propagation region in said computer memory;

(d) imposing said second absorbing condition on the exterior surface of said first absorbing condition; and

(e) computing seismic wave propagation in said earth velocity model and in said absorbing region, wherein said simulated advancing seismic wave is essentially absorbed in said absorbing region.

15. Apparatus in accordance with claim 14, wherein said memory comprises a rectangularly shape volume having four generally vertical side faces, and wherein said first absorbing condition is applied at the exterior of said four vertical side faces and at the exterior of a generally horizontal bottom surface.

16. Apparatus in accordance with claim 14, wherein said wave propagation region comprises finite-difference computations, said first absorbing condition comprises a perfectly matched layer (PML) absorbing condition, and said second absorbing condition comprises a one-way wave equation (1WWE) absorbing condition.

17. Apparatus in accordance with claim 14, wherein said first absorbing condition is effective for attenuating waves at essentially all incident angles, and said second absorbing condition is effective for attenuating waves at or near a selective incidence angle.

18. Apparatus for eliminating reflections resulting from an advancing simulated wave contacting an artificial boundary in a computer implemented migration of seismic data, said method comprising:

a computer programmed for;

(a) storing an earth velocity model having a wavefield propagation region in said computer memory;

wherein said propagation region is truncated by limited capacity in said computer memory;

(b) surrounding said computer memory contacting truncated boundaries of said propagation region with a wave absorbing region, wherein said absorbing region comprises two types of absorbing conditions;

(c) wherein a first absorbing condition is applied on exterior surfaces of memory volumes contacting said truncated propagation region, and a second absorbing condition is applied at exterior edges and corners of said truncated propagation region in said computer memory;

(d) imposing said second absorbing condition on the exterior surface of said first absorbing condition; and

(e) computing seismic wave propagation in said earth velocity model and in said absorbing region, wherein said simulated advancing seismic wave is essentially absorbed in said absorbing region, by the following steps performed in the indicated sequence;

(i) updating time and space variables in the finite-difference computations in said truncated wavefield propagation region;

(ii) applying said second absorbing boundary conditions at the vertical edges and corners of said truncated propagation region;

(iii) updating particle velocity variables and the q variable in the interior of said first absorbing region;

(iv) applying said second boundary condition on the exterior of said first absorbing condition;

(v) updating the pressure variable in the interior of said absorbing region; and

(vi) updating the pressure variable on two layers in said wavefield propagating region immediately above said absorbing region.

19. Apparatus for eliminating reflections resulting from an advancing simulated wave contacting an artificial boundary in a computer implemented migration of seismic data, said method comprising:

a computer programmed for;

(a) storing an earth velocity model having a wavefield propagation region in said computer memory;

wherein said propagation region is truncated by limited capacity in said computer memory;

(b) surrounding said computer memory contacting truncated boundaries of said propagation region with a wave absorbing region, wherein said absorbing region comprises two types of absorbing conditions;

(c) wherein a perfectly matched layer (PML) absorbing condition is applied on exterior surfaces of memory volumes contacting said truncated propagation region, and a one-way wave equation (1WWE) absorbing condition is applied at exterior edges and corners of said truncated propagation region in said computer memory;

(d) imposing said 1WWE absorbing condition on the exterior surface of said first absorbing condition; and

(e) computing seismic wave propagation in said earth velocity model and in said absorbing region, wherein said simulated advancing seismic wave is essentially absorbed in said absorbing region, wherein said PML absorbing conditions are governed by the set of equations comprising;

$\frac{\partial <br><br>q}{\partial <br><br>t}+\mathrm{\alpha <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>q=-\text{\hspace{1em}<br><br>}<br><br>\frac{\partial <br><br>p}{\partial <br><br>N}$$\frac{\partial <br><br>U_{\mathrm{L1}}}{\partial <br><br>t}+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>U_{\mathrm{L1}}=-\text{\hspace{1em}<br><br>}<br><br>\frac{\partial <br><br>p}{\partial <br><br>L_2}$$\frac{\partial <br><br>U_{\mathrm{L2}}}{\partial <br><br>t}+\mathrm{\beta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>U_{\mathrm{L2}}=-\text{\hspace{1em}<br><br>}<br><br>\frac{\partial <br><br>p}{\partial <br><br>L_2}$$\frac{\partial <br><br>p}{\partial <br><br>t}=-C^2<br><br>\left(L_1,L_2\right)<br><br>\left(\frac{\partial <br><br>U_{\mathrm{L1}}}{\partial <br><br>L_1}+\frac{\partial <br><br>U_{\mathrm{L2}}}{\partial <br><br>L_2}+\frac{\partial <br><br>q}{\partial <br><br>N}\right).$