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## Size of the reflection coefficient

Let us look at some of the details of the reflection coefficient calculation. A unit amplitude, monochromatic plane wave incident on the side boundary generates a reflected wave of magnitude c. The mathematical representation is:
 (3)
In equation (3) and kx are arbitrary, and kz is determined from and kx using the dispersion relation of the interior region, i.e., a semicircle approximation. Assuming this interior solution is applicable at the side boundary, you insert equation (3) into the differential equation (2), which represents the side boundary. As a result, is converted to on the incident wave, and is converted to on the reflected wave. Also, is converted to i kz. Thus the first term in (3) produces the dispersion relation times the amplitude P. The second term produces the reflection coefficient c times times P. So (2) with (3) inserted becomes:
 (4)

The case of zero reflection arises when the numerical value of kz selected by the interior equation at happens also to satisfy exactly the dispersion relation D of the side boundary condition. This explains why we try to match the quarter-circle as closely as possible. The straight-line dispersion relation does not correspond to the most general form of a side boundary condition, which is expressible on just two end points. A more general expression with adjustable parameters b1, b2, and b3, which fits even better, is
 (5)

The absolute stability of straight-line absorbing side boundaries for the 15 equation can be established, including the discretization of the x-axis. Unfortunately, an airtight analysis of stability seems to be outside the framework of the Muir impedance rules. As a consequence, I don't believe that stability has been established for the 45 equation.

Next: TUNING UP FOURIER MIGRATIONS Up: ABSORBING SIDES Previous: Engquist sides for the
Stanford Exploration Project
10/31/1997