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

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:
• ∂ q ∂ t + α ⁢ ⁢ q = - ⁢ ∂ p ∂ N ∂ U L1 ∂ t + β ⁢ ⁢ U L1 = - ⁢ ∂ p ∂ L 2 ∂ U L2 ∂ t + β ⁢ ⁢ U L2 = - ⁢ ∂ p ∂ L 2 ∂ p ∂ t = - C 2 ⁡ ( L 1 , L 2 ) ⁢ ( ∂ U L1 ∂ L 1 + ∂ U L2 ∂ L 2 + ∂ q ∂ N ) .

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.

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.

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.

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:
• ∂ q ∂ t + α ⁢ ⁢ q = - ⁢ ∂ p ∂ N ∂ U L1 ∂ t + β ⁢ ⁢ U L1 = - ⁢ ∂ p ∂ L 2 ∂ U L2 ∂ t + β ⁢ ⁢ U L2 = - ⁢ ∂ p ∂ L 2 ∂ p ∂ t = - C 2 ⁡ ( L 1 , L 2 ) ⁢ ( ∂ U L1 ∂ L 1 + ∂ U L2 ∂ L 2 + ∂ q ∂ N ) .