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The 3-D cone dispersion relation of the two-dimensional
scalar wave equation is:
| |
(1) |

where *k*_{x} and *k*_{z} are the horizontal and vertical wave-numbers,
respectively, and is the temporal frequency. Decomposition of the
wavefield into leftgoing (-*x*) and rightgoing (+*x*) waves, requires equation (1)
to be changed into the following form:
| |
(2) |

where the positive sign is used for rightgoing, the negative sign for
leftgoing waves.
Our starting point is that the total wavefield *P*_{T} at the boundary is
the sum of the incident wavefield *P*_{I} from the interior and the reflected
wavefield *P*_{R} from the boundary, such that *P*_{T}=*P*_{I}+*P*_{R}.
If we let *P*_{T}=*P*_{I},
then *P*_{R}=0, we obtain a perfect nonreflecting boundary. So we use the
leftgoing (rightgoing) wave equation as a boundary condition for the left(right)
edge. To get difference equations in the space-time domain, we need
to rationally approximate equation (2).

Here we give two forms of rational approximation to (2):

| |
(3) |

and
| |
(4) |

In the space-time domain:
| |
(5) |

and
| |
(6) |

The criterion for the absorbing boundary condition is that it
is transparent to the incidence wavefield,
that is, the
boundary reflection coefficient is very small.
Our motivation is to find a boundary condition
with a reflection coefficient
as small as possible.

Consider a rightgoing plane wave:

| |
(7) |

Suppose the boundary reflection coefficient is *R*, then the right boundary
reflection is:
| |
(8) |

Locally near the boundary, the total wave field *P*_{T}=*P*_{I}+*P*_{R} must
satisfy both the boundary condition and the interior wave equation.
Applying *A*2 to (*P*_{I}+*P*_{R}), we obtain:

| |
(9) |

From the interior two-way wave equation dispersion relation, we have:
| |
(10) |

and
| |
(11) |

where is the angle of incidence measured from the normal to the
boundary. Then we have:
| |
(12) |

In the same way we obtain the reflection coefficient for A1:
| |
(13) |

We first give a reflection coefficient *R*_{0}=0.1, then by trial and error,
we will find
a, b, c, and d, so that for as many as possible. We have

*
A1: a=-0.55, b=1
*

*
A2: a=8, b=8, c=-7.5, d=10.
**
*

*
In Figure 1, the reflection coefficients for A1, A2, versus C2 from
Clayton et al. (1977) are plotted. The low angle reflection coefficients
for A1 and A2 can be decreased, but we have to sacrifice the high
angle counterparts. Figure 2 displays the finite difference modeled
shot records of different absorbing boundary conditions on the model
given by Reynolds (1978, p. 1103).
*

*
***fig1
**

Figure 1
Graph of reflection coefficients for boundary conditions,
A1, A2 versus C2 (Clayton et al. 1977).

*
***fig2
**

Figure 2
Modeled shot records by different absorbing boundary conditions,
(a) A1, (b) A2, (c) C2 by Clayton et al. (1977), (d) Reynolds (1978)

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** Up:** Introduction
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Stanford Exploration Project

12/18/1997