exs.tex

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simple asymptotic relationship
\begin{equation}
\left(\pm\,{d \over {d\,z}}\right)^{m/2} =  
\left(\pm\,{d \over {d\,t}}\right)^{m/2}\,
\left({{dt} \over {dz}}\right)^{m/2} =
\left(\pm\,{d \over {d\,t}}\right)^{m/2}\,
\left({z\over t}\right)^{m/2}\;,
\end{equation}
which transforms the reversed velocity continuation operator to the
familiar form (\ref{eqn:backward}) with the weighting function equal to
(\ref{eqn:rmwp}). According to formula (\ref{eqn:zomww}), these two operators are
seen to be asymptotically inverse. 
\par
To obtain the velocity continuation operator completely equivalent to
residual migration with the weighting function (\ref{eqn:zomw}), we can
divide the continued wavefield by the time $t$, which is equivalent to
transforming equation (\ref{eqn:VCequation}) to the form
\begin{equation}
{{\partial^2 P}\over{\partial v\,\partial t}} +
v\,t\,{{\partial^2 P}\over{\partial x^2}} +
{1 \over t}\,{\partial P \over {\partial v}} = 0\;.
\label{eqn:VCequation1}
\end{equation}
The reverse continuation in this case has the weighting function
(\ref{eqn:invzomw}).  
\par
Analogously, one can obtain the pseudo-unitary residual migration with the weighting
functions (\ref{eqn:zomwp}) and (\ref{eqn:zomwm}) by dividing the
wavefield by $\sqrt{t}$. This leads to the equation
\begin{equation}
{{\partial^2 P}\over{\partial v\,\partial t}} +
v\,t\,{{\partial^2 P}\over{\partial x^2}} +
{1 \over {2\,t}}\,{\partial P \over {\partial v}} = 0\;.
\end{equation}
\par
It is apparent that the operators of forward and reverse continuation
with equation (\ref{eqn:VCequation}) become adjoint to each other if the
definition of the dot product is changed according to formulas
(\ref{eqn:wsproduct}) and (\ref{eqn:wmproduct}) with the model weight $W_M(z)=z$
and the data weight $W_S(t)=t$. Analogously, the solutions of equation
(\ref{eqn:VCequation1}) are adjoint if $W_M(z)={1 \over z}$ and $W_S(t)={1
\over t}$. This is a simple example of how the arbitrarily chosen definition of
the dot product can affect the basic properties of the
inverted operators.
\end{comment}

\subsection{Velocity Transform}
%%%%%%%%%%%%%%%%%%%%%%%%%% 
\inputdir{velinv}

Velocity transform is another form of hyperbolic stacking with the
summation path
\begin{equation}
\widehat{\theta}(h;t_0,s)  =  \sqrt{t_0^2 + s^2\,h^2}\;,
\label{eqn:vtt}
\end{equation}
where $h$ corresponds to the offset, $s$ is the stacking slowness, and
$t_0$ is the estimated zero-offset traveltime. Hyperbolic stacking is
routinely applied for scanning velocity analysis in common-midpoint
stacking. Velocity transform inversion has proved to be a powerful
tool for data interpolation and amplitude-preserving multiple
suppression \cite[]{Thorson.sepphd.39,Ji.sepphd.90,SEG-1995-1460}.
\par
Solving equation (\ref{eqn:vtt}) for $t_0$, we find that the asymptotic
inverse and adjoint operators have the elliptic summation path
\begin{equation}
\theta(s;t,h)  =  \sqrt{t^2 - s^2\,h^2}\;.
\end{equation}
The weighting functions of the asymptotic pseudo-unitary velocity
transform are found using formulas (\ref{eqn:wp}) and (\ref{eqn:wm})
to have the form
\begin{eqnarray}
  \label{eqn:vtpsun1}
w^{(+)} & = & {1\over{\left(2\,\pi\right)^{1/2}}} \, 
\left|F\,\widehat{F}\right|^{1/4}\,
\left|\partial \widehat{\theta} \over \partial t_0\right|^{- 1/4} =
{1\over{\sqrt{\pi}}}\, 
{{\sqrt{s\,h}\,\sqrt{t/t_0}} \over {\sqrt{t}}}\;.
 \\
 \label{eqn:vtpsun2}
w^{(-)} & = & {1\over{\left(2\,\pi\right)^{1/2}}} \, 
\left|F\,\widehat{F}\right|^{1/4}\,
\left|\partial \widehat{\theta} \over \partial t_0\right|^{3/4} =
{1\over{\sqrt{\pi}}} \, 
{{\sqrt{s\,h}\,\sqrt{t_0/t}} \over {\sqrt{t}}}\;.
\end{eqnarray}
The factor $\sqrt{s\,h}$ for pseudo-unitary velocity transform
weighting has been discovered empirically by \cite{Claerbout.bei.95}.

Figure~\ref{fig:cgiter} shows the output of a numerical test of the
least-squares velocity transform inversion using a CMP gather from the
Mobil AVO dataset \cite[]{SEG-1995-1460}. The input CMP gather (shown in
the left plot of Figure~\ref{fig:dircvv}) is inverted using an
iterative conjugate-gradient method and two different weighting
scheme: the uniform weighting and the asymptotic pseudo-unitary
weights~(\ref{eqn:vtpsun1}-\ref{eqn:vtpsun2}). Analogously to
Figure~\ref{fig:migiter}, the iterative convergence is measured by the
least-squares norm of the data residual error at different iterations.
Figure~\ref{fig:cgiter} shows that the pseudo-unitary weighting
provides a noticeably faster convergence at the first three
iterations. At later iterations, the residual errors of the two
methods are very close to each other. The use of a pseudo-unitary
weighting will be justified in this case if only three iterations are
practically affordable.  The results of inversion after 10
conjugate-gradient iterations are plotted in Figures \ref{fig:dircvv}
and \ref{fig:dirrst}. The right plot in Figure~\ref{fig:dircvv} shows
the output of the velocity transform inversion: an optimized velocity
scan. Figure~\ref{fig:dirrst} shows the corresponding modeled CMP
gather and the residual error. The error is negligible which
indicates a successful inversion.

\sideplot{cgiter}{height=3in}{Comparison of convergence of the
  iterative velocity transform inversion. The dashed line corresponds
  to the unweighted (uniformly weighted) operator. The solid line
  corresponds to the asymptotic pseudo-unitary operator.  The latter
  provides a faster convergence at early iterations.}

\plot{dircvv}{width=6in,height=3in}{Input CMP gather (left) and its
  velocity transform counterpart (right) after 10 iterations of
  iterative least-squares inversion.}

\plot{dirrst}{width=6in,height=3in}{The modeled CMP gather (left) and
  the residual error (right) plotted at the same scale.}


\subsection{Offset Continuation and DMO}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Offset continuation is the operator that transforms seismic reflection
data from one offset to another
\cite[]{GPR30-06-08130828,GPR30-06-08290849}. If the data are continued
from half-offset $h_1$ to a larger offset $h_2$, the summation path of
the post-NMO integral offset continuation has the following form
\cite[]{Biondi.sep.80.125,stovas,Fomel.sepphd.107}:
\begin{equation}
\theta(x;t,y)  =  {t \over h_2}\,\sqrt{{U+V} \over 2}\;,
\label{eqn:ttOC12}
\end{equation}
where $U = h_1^2 + h_2^2 - (x - y)^2$, $V = \sqrt{U^2 -
4\,h_1^2\,h_2^2}$, and $x$ and $y$ are the midpoint coordinates before and
after the continuation. The summation path of the reverse continuation
is found from inverting (\ref{eqn:ttOC12}) to be
\begin{equation}
\widehat{\theta}(y;z,x)  =  {z \,h_2}\,\sqrt{2 \over {U+V}} = 
{z \over h_1}\,\sqrt{{U-V} \over 2}\;.
\label{eqn:ttOC21}
\end{equation}
The Jacobian of the time coordinate transformation in this case is simply
\begin{equation}
\left|\partial \widehat{\theta} \over \partial z\right| = {t \over z}\;.
\label{eqn:OCj}
\end{equation}
Differentiating summation paths (\ref{eqn:ttOC12}) and (\ref{eqn:ttOC21}), we
can define the product of the weighting functions according to formula
(\ref{eqn:whatw}), as follows:
\begin{equation}
w\,\widehat{w}={1\over{2\,\pi}} \, 
{\sqrt{\left|F\,\widehat{F}\right|\,
\left|\partial \widehat{\theta} \over \partial z\right|}} =
{t\over{2\,\pi}} \,{{\left(h_2^2-h_1^2\right)^2 - (x-y)^4} \over 
V^3}\;.
\label{eqn:OCww}
\end{equation}
The weighting functions of the amplitude-preserving offset
continuation have the form \cite[]{Fomel.sepphd.107}
\begin{eqnarray}
w(x;t,y) & = & \sqrt{z \over {2\,\pi}}\;
{{h_2^2-h_1^2-(x-y)^2} \over {V^{3/2}}}\;, 
\label{eqn:wOC12} \\
\widehat{w}(y;z,x) & = & {{t/\sqrt{z}} \over \sqrt{2\,\pi}}\;
{{h_2^2-h_1^2 + (x-y)^2} \over {V^{3/2}}}\;. 
\label{eqn:wOC21} 
\end{eqnarray}
It easy to verify that they satisfy relationship (\ref{eqn:OCww});
therefore, they appear to be asymptotically inverse to each other. 
\par
The weighting functions of the asymptotic pseudo-unitary offset
continuation are defined from formulas (\ref{eqn:wp}) and (\ref{eqn:wm}), as follows:
\begin{eqnarray}
w^{(+)} & = & {1\over{\left(2\,\pi\right)^{1/2}}} \; 
\left|F\,\widehat{F}\right|^{1/4}\,
\left|\partial \widehat{\theta} \over \partial t_0\right|^{- 1/4} =
\sqrt{z \over {2\,\pi}}\,
{{\left(\left(h_2^2-h_1^2\right)^2 - (x-y)^4\right)^{1/2}} 
\over {V^{3/2}}}\;, 
\label{eqn:puwOC12} \\
w^{(-)} & = & {1\over{\left(2\,\pi\right)^{1/2}}} \; 
\left|F\,\widehat{F}\right|^{1/4}\,
\left|\partial \widehat{\theta} \over \partial t_0\right|^{3/4} =
{{t/\sqrt{z}} \over \sqrt{2\,\pi}}\,
{{\left(\left(h_2^2-h_1^2\right)^2 - (x-y)^4\right)^{1/2}} 
\over {V^{3/2}}}\;.
\label{eqn:puwOC21} 
\end{eqnarray}
\par
The most important case of offset continuation is the continuation
to zero offset. This type of continuation is known as {\em dip moveout
(DMO)}. Setting the initial offset $h_1$ equal to zero in the general
offset continuation formulas, we deduce that the inverse and forward
DMO operators have the summation paths
\begin{eqnarray}
\theta(x;t,y)  & = & {t \over h_2}\,\sqrt{h_2^2-(x-y)^2}\;,
\label{eqn:ttDMO12} \\
\widehat{\theta}(y;z,x) &  = & {{z \,h_2} \over \sqrt{h_2^2-(x-y)^2}}\;. 
\label{eqn:ttDMO21}
\end{eqnarray}
The weighting functions of the amplitude-preserving inverse and
forward DMO are
\begin{eqnarray}
w(x;t,y) & = & \sqrt{z \over {2\,\pi}}\;{1 \over h_2}\;,
\label{eqn:wDMO12} \\
\widehat{w}(y;z,x) & = & {{t/\sqrt{z}} \over \sqrt{2\,\pi}}\;
{{h_2\,\left(h_2^2 + (x-y)^2\right)} \over
{\left(h_2^2-(x-y)^2\right)^2}}\;,
\label{eqn:wDMO21} 
\end{eqnarray}
and the weighting functions of the asymptotic pseudo-unitary DMO are
\begin{eqnarray}
w^{(+)} & = & 
\sqrt{z \over {2\,\pi}}\;
{\sqrt{h_2^2 + (x-y)^2} \over {h_2^2-(x-y)^2}}\;,
\label{eqn:puDMO12} \\
w^{(-)} & = & 
{{t/\sqrt{z}} \over \sqrt{2\,\pi}}\;
{\sqrt{h_2^2 + (x-y)^2} \over {h_2^2-(x-y)^2}}\;.
\label{eqn:puDMO21} 
\end{eqnarray}
Equations similar to~(\ref{eqn:wDMO12}) and~(\ref{eqn:wDMO21}) have
been published by \cite{stovas}. Equation~(\ref{eqn:wDMO21}) differs
from the similar result of \cite{black} by a simple time
multiplication factor. This difference corresponds to the difference
in definition of the amplitude preservation criterion.
Equation~(\ref{eqn:wDMO21}) agrees asymptotically with the
frequency-domain Born DMO operators
\cite[]{Born,GEO56-02-01820189,cwp}. Likewise, the stacking operator
with the weighting function (\ref{eqn:wDMO12}) corresponds to Ronen's
inverse DMO \cite[]{GEO52-07-09730984}, as discussed by
\cite{Fomel.sepphd.107}.  Its adjoint, which has the weighting
function
\begin{equation}
\widetilde{w}(x;t,y)  =  {{t/\sqrt{z}} \over {2\,\pi}}\;{1 \over h_2}\;,
\label{eqn:hDMO21}
\end{equation}
corresponds to Hale's DMO \cite[]{GEO49-06-07410757}.

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