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\mhead{Three-dimensional filtering}
\mhead{{\it Jon Claerbout }}
\mhead{{\it Matthias Schwab}}
\mhead{{\it Curt Holden}}
\mhead{{\rm Stanford University}}
\def\eq{=}
\par\noindent


\newpage\mhead{Prediction-error filter in 2-D}
The PEF output is white because:
\par\noindent
\begin{tabular}{lll}
residual     & $\perp$ &  fitting function \\
output at A  & $\perp$ &  each input to A \\
output at A  & $\perp$ &  each input to B \\
output at A  & $\perp$ &  linear combination of each input to B \\
output at A  & $\perp$ &  output at B
\end{tabular}
\par  For fuller explanation, see my free book:
\par  http://sepwww.stanford.edu/sep/prof/tdf


\newpage\mhead{ curl grad =0 and LOMOPLAN}

\par\noindent   LOMOPLAN:  LOcal MOnoplane ANnihilation
$$
{\rm curl}
      = \left[
	\left(  {\partial \over\partial x} - {\partial \over\partial y}
	\right),
	\left(  {\partial \over\partial z} - {\partial \over\partial x}
	\right),
	\left(  {\partial \over\partial y} - {\partial \over\partial z}
	\right)
	\right]
$$
\par\noindent  
${\partial \over\partial y}$ is filter coefficients on a line.
\par\noindent  
A component of curl is filter coefficients in a plane.
\par\noindent  
${\rm curl} {\rm grad} =0$ means a travel time exists:
waves are not going round in circles like vortices.