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The weighting function of the AMO operator can be determined from
cascading the DMO and inverse DMO operators by means of equation
(10). In the case of Hale's DMO Hale (1984) and
its adjoint Ronen (1987),
| ![\begin{displaymath}
w_{10}\left({\bf x_1;x_0, h_1},t_0\right) =
\sqrt{t_0 \over...
...\vert h_1\right\vert} \over {h_2^2-\left(x_1-x_0\right)^2}}}\;,\end{displaymath}](img66.gif) |
(22) |
| ![\begin{displaymath}
w_{02}\left({\bf x_0;x_2, h_2},t_2\right) =
\sqrt{t_2 \over...
...\vert h_2\right\vert} \over {h_2^2-\left(x_0-x_2\right)^2}}}\;.\end{displaymath}](img67.gif) |
(23) |
As follows from (A-1),(A-2), and (10),
| ![\begin{eqnarray}
\times\,
{{{\bf \left\vert h_1\right\vert\,\left\vert h_2\right...
...ft({\bf h_2^2}\,\sin{\varphi}^2-\left(y_2-y_1\right)^2\right)}}\;.\end{eqnarray}](img69.gif) |
(24) |
In the case of the so-called true-amplitude DMO
Black et al. (1993) and its asymptotic inverse,
| ![\begin{displaymath}
w_{10}\left({\bf x_1;x_0, h_1},t_0\right) =
\sqrt{t_0 \over...
...t h_1\right\vert\,\left(h_1^2-\left(x_1-x_0\right)^2\right)}\;,\end{displaymath}](img70.gif) |
(25) |
| ![\begin{displaymath}
w_{02}\left({\bf x_0;x_2, h_2},t_2\right) =
\sqrt{t_2 \over...
...\vert h_2\right\vert} \over {h_2^2-\left(x_0-x_2\right)^2}}}\;.\end{displaymath}](img67.gif) |
(26) |
Inserting (A-4) and (A-5) into (10) yields
| ![\begin{eqnarray}
\times\,
{{{\bf h_1^2}\,\sin{\varphi}^2+
\left(\left(x_2-x_1\ri...
...ft({\bf h_2^2}\,\sin{\varphi}^2-\left(y_2-y_1\right)^2\right)}}\;.\end{eqnarray}](img72.gif) |
(27) |
Next: DERIVING THE AMO APERTURE
Up: Fomel & Biondi: t-x
Previous: REFERENCES
Stanford Exploration Project
9/12/2000