app.tex

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%\APPENDIX{A}
%\section{Review of the asymptotic inverse theory}

%\newpage

\append{Least-squares Radon transform inversion}

This appendix exemplifies the application of adjoint operators by
reviewing the analytical least-squares inversion of the classic Radon
transform (slant stack operator).

Forming the product ${\bf A^{T}\,A}$ for this case leads
to the double integral
\begin{eqnarray}
H(z,x) & = & {\bf (A^{T}\,A)}[M(z,x)] = 
\nonumber \\
& = & \iint\,\widehat{w}(y;z,x)\,
w\left(\xi;\widehat{\theta}(y;z,x),y\right)\,
M\left(\theta\left(\xi;\widehat{\theta}(y;z,x),y\right),\xi\right)\,
d\xi\,dy =
\nonumber \\
& = & \iint\,M\left(z + y\,(\xi - x)\right)\,d\xi\,dy\;.
\label{eqn:LSradon2}
\end{eqnarray}
Applying Fourier transform with respect to $z$, we can rewrite
equation (\ref{eqn:LSradon2}) in the frequency domain as
\begin{equation}
\check{H}(\omega,x) = \int\,\check{M}(\omega,\xi)\,\int\,
e^{i\omega\,y\,(\xi-x)}\,dy\,d\xi\;,
\label{eqn:Fourier}
\end{equation}
where
\begin{eqnarray}
\check{H}(\omega,x) & = & \int\,H(z,x)\,e^{-i\omega\,z}\,dz\;,
 \\
\check{M}(\omega,x) & = & \int\,M(z,x)\,e^{-i\omega\,z}\,dz\;.
\end{eqnarray}
The inner integral in equation (\ref{eqn:Fourier}) reduces to the $m$-dimensional
delta function:
\begin{equation}
\check{H}(\omega,x) = (2\,\pi)^m\,\int\,\check{M}(\omega,\xi)\,
\delta\left(\omega^m\,(\xi-x)\right)\,d\xi\;.
\end{equation}
As follows from the properties of delta function,
\begin{equation}
\check{H}(\omega,x) = {{(2\,\pi)^m} \over {|\omega|^m}}\,
\int\,\check{M}(\omega,\xi)\,
\delta(\xi-x)\,d\xi\; = {{(2\,\pi)^m} \over {|\omega|^m}}\,
\check{M}(\omega,x)\;.
\label{eqn:NoDelta}
\end{equation}
Inverting (\ref{eqn:NoDelta}) for $M$, we conclude that
\begin{equation}
{\bf (A^{T}\,A)^{-1}} = {{|{\bf D}|^m} \over {(2\,\pi)^m}}\;.
\label{eqn:LSradon}
\end{equation}
Substituting equation~(\ref{eqn:LSradon}) into~(\ref{eqn:LS}) produces
the result precisely equivalent to Radon's
inversion~(\ref{eqn:radon}).

%\APPENDIX{C}
%\section{Constant velocity migration}