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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:  
 \begin{displaymath}
P(x,z) \ \eq \ e^{ -\,i\, \omega t + i\, k_z z } \left(\ 
e^{ +\,i\, k_x x } \ +\ c \ e^{ -\,i\, k_x x } \right)\end{displaymath} (3)
In equation (3) $\omega$ and kx are arbitrary, and kz is determined from $\omega$ 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, $\partial / \partial x $ is converted to $ +\,i\, k_x $ on the incident wave, and $\partial / \partial x $ is converted to $ -\,i\, k_x $ on the reflected wave. Also, $ \partial / \partial z $ is converted to i kz. Thus the first term in (3) produces the dispersion relation $D( \omega , k_x , k_z ) $times the amplitude P. The second term produces the reflection coefficient c times $ D ( \omega , -\,k_x , k_z ) $times P. So (2) with (3) inserted becomes:  
 \begin{displaymath}
c \ \eq \ 
{ -\,D ( \omega , k_x , k_z ) \over D( \omega , -\,k_x , k_z ) }\end{displaymath} (4)

The case of zero reflection arises when the numerical value of kz selected by the interior equation at $ ( \omega , k_x ) $ 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
\begin{displaymath}
D( \omega , k_x , k_z ) \ \eq \ 
\left( \ 1 \ -\ b_3 \ { v k...
 ... }\ -\ 
\left( \ b_1 \ -\ b_2 \ { v k_x \over \omega }\,\right)\end{displaymath} (5)

The absolute stability of straight-line absorbing side boundaries for the 15$^\circ$ 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$^\circ$ equation.


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