Rigorous absorbing boundary conditions for 3-D one‐way wave extrapolation

Geophysics ◽  
2000 ◽  
Vol 65 (2) ◽  
pp. 638-645 ◽  
Author(s):  
Hongbo Zhou ◽  
George A. McMechan
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Frequency Domain ◽  
Internal Consistency ◽  
Absorbing Boundary Conditions ◽  
Extra Cost ◽  
Three Dimensions ◽  
Scalar Wave ◽  
Numerical Schemes ◽  
Absorbing Boundary

Absorbing boundary conditions play an important role in one‐way wave extrapolations by reducing reflections at grid edges. Clayton and Engquist’s 2-D absorbing boundary conditions for one‐way wave extrapolation by depth stepping in the frequency domain are extended to three dimensions using paraxial approximations of the scalar wave equation. Internal consistency is retained by incorporating the interior extrapolation equation with the absorbing boundary conditions. Numerical schemes are designed to make the proposed absorbing boundary conditions both mathematically correct and efficient with negligible extra cost. Synthetic examples illustrate the effectiveness of the algorithm for extrapolation with the 3-D 45° one‐way wave equation.

3D scalar-wave absorbing boundary conditions with optimal coefficients in the frequency-space domain

Geophysics ◽  
2012 ◽  
Vol 77 (3) ◽  
pp. T83-T96 ◽  
Author(s):  
Kun Xu ◽  
George A. McMechan
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Absorbing Boundary Conditions ◽  
Grid Size ◽  
Impedance Matrix ◽  
Scalar Wave ◽  
Absorbing Boundary ◽  
Space Domain ◽  
Frequency Space

To improve the computational efficiency for the solution of the 3D Helmholtz equation in the frequency-space domain, high-order compact forms of finite differences are preferred. We applied a pointwise Padé approximation to develop a 3D 27-point fourth-order compact finite-difference (FD) stencil in the grid interior, with a space-differentiated source term, for the scalar-wave equation; this has similar high-accuracy (4–5 grid points per the shortest wavelength) to another 27-point fourth-order FD stencil using a parsimonious mixed-grid and staggered-grid combination, but is much simpler. For absorbing boundary conditions (ABCs), a damping zone is expensive, and a perfectly matched layer can not be straightforwardly introduced into the compact FD form for the second-order wave equation. Thus, we developed 3D one-way wave equation (OWWE) ABCs with adjustable coefficients. They have different angle approximations and FD forms for the six faces, twelve edges, and eight corners in 3D models to fit with the interior compact FD form. By adjusting the coefficients to the optimum, the OWWE ABCs have wider-angle absorbing ability than those without optimal coefficients. Finally, all the interior and boundary FD forms were combined into a sparse complex-valued impedance matrix of the frequency-space modeling equation, and solved for each frequency. Because the storage of the sparse impedance matrix was determined by the 3D discrete grid size, the OWWE ABCs with only one outer layer needed the minimum grid size compared with other ABCs, thus were the most efficient for the solution of the impedance matrix. The modeling algorithm was performed on multicore processors using a MPI parallel direct solver. Numerical tests on homogeneous and heterogeneous models gave satisfactory absorbing effects.

Stability of wide‐angle absorbing boundary conditions for the wave equation

Geophysics ◽  
1989 ◽  
Vol 54 (9) ◽  
pp. 1153-1163 ◽  
Author(s):  
R. A. Renaut ◽  
J. Petersen
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Least Squares ◽  
Absorbing Boundary Conditions ◽  
Computational Domain ◽  
Wide Angle ◽  
Courant Number ◽  
Absorbing Boundary ◽  
Wide Range ◽  
Artificial Boundaries

Numerical solution of the two‐dimensional wave equation requires mapping from a physical domain without boundaries to a computational domain with artificial boundaries. For realistic solutions, the artificial boundaries should cause waves to pass directly through and thus mimic total absorption of energy. An artificial boundary which propagates waves in one direction only is derived from approximations to the one‐way wave equation and is commonly called an absorbing boundary. Here we investigate order 2 absorbing boundary conditions which include the standard paraxial approximation. Absorption properties are compared analytically and numerically. Our numerical results confirm that the [Formula: see text] or Chebychev‐Padé approximations are best for wide‐angle absorption and that the Chebychev or least‐squares approximations are best for uniform absorption over a wide range of incident angles. Our results also demonstrate, however, that the boundary conditions are stable for varying ranges of Courant number (ratio of time step to grid size). We prove that there is a stability barrier on the Courant number specified by the coefficients of the boundary conditions. Thus, proving stability of the interior scheme is not sufficient. Furthermore, waves may radiate spontaneously from the boundary, causing instability, even if the stability bound on the Courant number is satisfied. Consequently, the Chebychev and least‐squares conditions may be preferred for wide‐angle absorption also.

scholarly journals Absorbing boundary conditions for wave‐equation migration

Geophysics ◽  
1980 ◽  
Vol 45 (5) ◽  
pp. 895-904 ◽  
Author(s):  
Robert W. Clayton ◽  
Björn Engquist
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Final Section ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary ◽  
Seismic Section ◽  
Difference Wave ◽  
Different Types ◽  
Almost All ◽  
The Cost

The standard boundary conditions used at the sides of a seismic section in wave‐equation migration generate artificial reflections. These reflections from the edges of the computational grid appear as artifacts in the final section. Padding the section with zero traces on either side adds to the cost of migration and simply delays the inevitable reflections. We develop stable absorbing boundary conditions that annihilate almost all of the artificial reflections. This is demonstrated analytically and with synthetic examples. The absorbing boundary conditions presented can be used with any of the different types of finite‐difference wave‐equation migration, at essentially no extra cost.

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