exs.tex

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constant-velocity case, we can differentiate the explicit expression
for the summation path
\begin{equation}
\widehat{\theta}(y;z,x)  =  z + 
{{\rho_s(x,y) + \rho_r(x,y)} \over v}\;,
\end{equation}
where $\rho_s$ and
$\rho_r$ are the lengths of the incident and reflected rays:
\begin{eqnarray}
\rho_s(y,x) & = & 
\sqrt{x_3^2 + (x_1 - y_1 + h_1)^2 + (x_2 - y_2 + h_2)^2}\;,
 \\
\rho_r(y,x) & = & 
\sqrt{x_3^2 + (x_1 - y_1 - h_1)^ 2+ (x_2 - y_2 - h_2)^2}\;.
\end{eqnarray}
For simplicity, the vertical component of the midpoint $y_3$ is set here to zero. Evaluating the second derivative term in formula
(\ref{eqn:migw}) for the common-offset geometry leads, after some heavy
algebra, to the expression 
\begin{equation}
\left|{{\partial^2 T\left(s(y),x\right)} 
\over {\partial x\,\partial y}} +     
      {{\partial^2 T\left(x,r(y)\right)}
\over {\partial x\,\partial y}}\right| = 
{{x_3\,(\rho_s^2 + \rho_r^2)} \over {v\,(\rho_s\,\rho_r)^2}}\,
\left({{\rho_s + \rho_r} \over {v\,\rho_s\,\rho_r}}\right)^{m-1}
\,\cos{\alpha(x)}\;.
\label{eqn:cobeilk}
\end{equation}
Substituting (\ref{eqn:cobeilk}) into the general formula (\ref{eqn:migw}) yields
the weighting function for the common-offset true-amplitude
constant-velocity migration:
\begin{equation}
\widehat{w}_{CO}(y;z,x)  =  {1\over{\left(2\,\pi\right)^{m/2}}} \,
{{x_3\,(\rho_s + \rho_r)^{m-1}\,(\rho_s^2 + \rho_r^2)} \over 
{v\,(\rho_s\,\rho_r)^{m/2+1}}}\;.
\label{eqn:comigw}
\end{equation}
Equation~(\ref{eqn:comigw}) is similar to the result obtained by
\cite{GEO52-06-07450754}. In the case of zero offset $h=0$,
it reduces to equation~(\ref{eqn:zomigw}). Note that the value
of $m=1$ in (\ref{eqn:comigw}) corresponds to the two-dimensional (cylindric)
waves recorded on the seismic line. A special case is the 2.5-D inversion,
when the waves are assumed to be spherical, while the recording is on a line,
and the medium has cylindric symmetry. In this case, the modeling weighting
function (\ref{eqn:modw}) transforms to
\cite[]{GPR31-02-02930333,GPR34-05-06860703}
\begin{equation}
w(x;t,y)  =  {1\over{\left(2\,\pi\right)^{1/2}}} \,
{\sqrt{v}\,{C\left(s(y),x,r(y)\right)} 
\over {\sqrt{\rho_s\,\rho_r\,(\rho_s + \rho_r)}}}\;,
\end{equation}
and the time filter is $\left({\partial \over {\partial
z}}\right)^{1/2}$. Combining this result with formula (\ref{eqn:cobeilk})
for $m=1$, we obtain the weighting function for the 2.5-D
common-offset migration in a constant velocity medium
\cite[]{GEO52-06-07450754}:
\begin{equation}
\widehat{w}_{CO;2.5D}(y;z,x)  =  {1\over{\left(2\,\pi\right)^{1/2}}} \,
{{x_3\,\sqrt{\rho_s + \rho_r}\,(\rho_s^2 + \rho_r^2)} \over 
{\sqrt{v}\,(\rho_s\,\rho_r)^{3/2}}}\;.
\end{equation}
The corresponding time filter for 2.5-D migration is $\left(-
{\partial \over {\partial t}}\right)^{1/2}$.
\par

In the
common-offset case, the pseudo-unitary weighting is defined from
(\ref{eqn:pumigw}) and (\ref{eqn:cobeilk}) as follows:
\begin{equation}
w^{(-)}_{CO}(y;z,x)  =  {1\over{\left(2\,\pi\,v\right)^{m/2}}} \,
{{\sqrt{x_3\,\cos{\alpha}}\,(\rho_s + \rho_r)^{{m-1} \over 2}\,
\sqrt{\rho_s^2 + \rho_r^2}} \over 
{(\rho_s\,\rho_r)^{{m+1} \over 2}}}\;,
\end{equation}
where
\begin{equation}
\cos{\alpha} = \left({
{(x - y)^2 + \rho_s\,\rho_r - h^2} \over {2\,\rho_s\,\rho_r}}
\right)^{1/2}\;.
\end{equation}

\subsection{Post-Stack Time Migration}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\inputdir{miginv}

An interesting example of a stacking operator is the hyperbola summation
used for time migration in the post-stack domain. In this case, the
summation path is defined as
\begin{equation}
\widehat{\theta}(y;z,x)  =  \sqrt{z^2+{{(x-y)^2}\over {v^2}}}\;,
\label{eqn:zomt}
\end{equation}
where $z$ denotes the vertical traveltime, $x$ and $y$ are the
horizontal coordinates on the migrated and unmigrated sections
respectively, and $v$ stands for the effectively constant
root-mean-square velocity \cite[]{Claerbout.bei.95}.  The summation path
for the reverse transformation (demigration) is found from solving
equation (\ref{eqn:zomt}) for $z$. It has the well-known elliptic form
\begin{equation}
\theta(x;t,y)  =  \sqrt{t^2-{{(x-y)^2}\over {v^2}}}\;.
\end{equation}
The Jacobian of transforming $z$ to $t$ is
\begin{equation}
\left|\partial \widehat{\theta} \over \partial z\right| = {z \over t}\;.
\label{eqn:zomj}
\end{equation}
If the migration weighting function is defined by conventional
downward continuation \cite[]{GEO43-01-00490076}, it takes the following form,
which is equivalent to equation (\ref{eqn:datw2}):
\begin{equation}
\widehat{w}(y;z,x)  =  {1\over{\left(2\,\pi\right)^{m/2}}} \,
{{\cos{\alpha(y)}}\over {v\,R(y,x)}} = 
{1\over{\left(2\,\pi\right)^{m/2}}} \,
{\cos{\alpha} \over {v^m\,t^{m/2}}}\;.
\label{eqn:zomw}
\end{equation}
The simple trigonometry of the reflected ray suggests that the cosine
factor in formula (\ref{eqn:zomw}) is equal to the simple ratio between the
vertical traveltime $z$ and the zero-offset reflected traveltime $t$:
\begin{equation}
\cos{\alpha} =  {z \over t}\;.
\label{eqn:cos}
\end{equation}
The equivalence of the Jacobian (\ref{eqn:zomj}) and the cosine factor
(\ref{eqn:cos}) has important interpretations in the theory of Stolt
frequency-domain migration
\cite[]{GEO43-01-00230048,GEO46-05-07170733,Levin.sep.48.147}.
According to equation (\ref{eqn:tildew}), the weighting function of the
adjoint operator is the ratio of (\ref{eqn:zomw}) and (\ref{eqn:zomj}):
\begin{equation}
\widetilde{w}(x;t,y) = {1\over{\left(2\,\pi\right)^{m/2}}} \,
{1 \over {v^m\,t^{m/2}}}\;.
\label{eqn:adjzomw}
\end{equation}
We can see that the cosine factor $z/t$ disappears from the adjoint
weighting. This is completely analogous to the known effect of
``dropping the Jacobian'' in Stolt migration
\cite[]{Harlan.sep.35.181,Levin.sep.80.513}.  The product of the
weighting functions for the time migration and its asymptotic inverse
is defined according to formula (\ref{eqn:whatw}) as
\begin{equation}
w\,\widehat{w}={1\over{\left(2\,\pi\right)^m}} \, 
{\sqrt{\left|F\,\widehat{F}\right|\,
\left|\partial \widehat{\theta} \over \partial z\right|^m}} =
{1 \over {(v^2\,t)^m}}\;.
\label{eqn:zomww}
\end{equation}
Thus, the asymptotic inverse of the conventional time migration has
the weighting function determined from equations (\ref{eqn:whatw}) and
(\ref{eqn:zomw}) as
\begin{equation}
w(x;t,y) = {1\over{\left(2\,\pi\right)^{m/2}}} \, {{t/z} \over
{v^m\,t^{m/2}}}\;.
\label{eqn:invzomw}
\end{equation}
The weighting functions of the asymptotic pseudo-unitary operators are
obtained from formulas (\ref{eqn:wp}) and (\ref{eqn:wm}). They have the form
 \begin{eqnarray} 
w^{(+)}(x;t,y) & = &
{1\over{\left(2\,\pi\right)^{m/2}}} \, 
{\sqrt{t/z} \over {v^m\,t^{m/2}}}\;.
\label{eqn:zomwp} \\
w^{(-)}(y;z,x) & = & {1\over{\left(2\,\pi\right)^{m/2}}} \, 
{\sqrt{z/t} \over {v^m\,t^{m/2}}}\;.
\label{eqn:zomwm}
\end{eqnarray}
The square roots of the cosine factor appearing in formulas
(\ref{eqn:zomwp}) and (\ref{eqn:zomwm}) correspond to the analogous
terms in the pseudo-unitary Stolt migration proposed by
\cite{Harlan.sep.48.127}.

Figure~\ref{fig:migiter} shows the output of a simple numerical test.
The synthetic zero-offset section used in this test is shown in the
left plot of Figure~\ref{fig:migcvv}. The data are taken from
\cite{Claerbout.bei.95} and correspond to a synthetic reflectivity
model, which contains several dipping layers, a fault, and an
unconformity. The input zero-offset section is inverted using an
iterative conjugate-gradient method and two different weighting
schemes: the uniform weighting and the asymptotic pseudo-unitary
weighting~(\ref{eqn:zomwp}-\ref{eqn:zomwm}). I compare the iterative
convergence by measuring the least-squares norm of the data residual
error at different iterations.  Figure~\ref{fig:migiter} shows that
the pseudo-unitary weighting provides a significantly faster
convergence.  The result of inversion after 10 conjugate-gradient
iterations is shown in Figures \ref{fig:migcvv} and \ref{fig:migrst}.
The right plot in Figure~\ref{fig:migcvv} shows the output of the
least-squares migration. Figure~\ref{fig:migrst} shows the
corresponding modeled data and the residual error. The latter is very
close to zero. Although this example has only a pedagogical value, it
clearly demonstrates possible advantages of using asymptotic
pseudo-unitary operators in least-squares migration.

\sideplot{migiter}{height=3in}{Comparison of convergence
  of the iterative least-squares migration. The dashed line
  corresponds to the unweighted (uniformly weighted) operator. The
  solid line corresponds to the asymptotic pseudo-unitary operator.
  The latter provides a noticeably faster convergence.}

\plot{migcvv}{width=6in,height=3in}{Input zero-offset
  section (left) and the corresponding least-squares image (right)
  after 10 iterations of iterative inversion.}

\plot{migrst}{width=6in,height=3in}{The modeled
  zero-offset (left) and the residual error (right) plotted at the
  same scale.}

\begin{comment}
\subsection{Post-Stack Residual Migration}
%%%%%%%%%%%%%%%%%%%%%%%%%%
In an earlier article \cite[]{vc}, I found the integral solution of the boundary
problem for the velocity continuation partial differential equation
\cite[]{Claerbout.sep.48.79}
\begin{equation}
{{\partial^2 P}\over{\partial v\,\partial z}} +
v\,t\,{{\partial^2 P}\over{\partial x^2}} = 0
\label{eqn:VCequation}
\end{equation}
with the boundary conditions $\left.P\right|_{v=v_0} = P_0$ and
$\left.P\right|_{z \rightarrow \infty} = 0$. The solution has the form
of the stacking operator (\ref{eqn:operator}), with the model $M$ replaced by
$P_0$, the summation path
\begin{equation}
\widehat{\theta}(y;z,x)  =  \sqrt{z^2+{{(x-y)^2}\over {v^2-v_0^2}}}\;,
\label{eqn:rmt}
\end{equation}
the weighting function 
\begin{equation}
w_{(-)}(y;z,x) = {1\over{\left(2\,\pi\right)^{m/2}}} \,
{1 \over {v^m\,t^{m/2}}}\;,
\label{eqn:rmwp}
\end{equation}
which is coincident with (\ref{eqn:adjzomw}), and the correction filter
$\left(\mbox{sign}(v_0-v)\,{d \over {d t}}\right)^{m/2}$. Comparing equations
(\ref{eqn:rmt}) and (\ref{eqn:zomt}), we can see that this solution is equivalent
kinematically to residual migration with the velocity
$v_r=\sqrt{v^2-v_0^2}$ \cite[]{GEO50-01-01100126}. The reverse operator
is the solution of equation (\ref{eqn:VCequation}) with the boundary
condition on $v$ and has the reciprocal form of the summation path
\begin{equation}
\theta(x;t,y)  =  
\sqrt{t^2+{{(x-y)^2}\over {v_0^2-v^2}}} = 
\sqrt{t^2-{{(x-y)^2}\over {v^2-v_0^2}}}\;,
\end{equation}
the weighting function
\begin{equation}
w_{(+)}(x;t,y) = {1\over{\left(2\,\pi\right)^{m/2}}} \,
{1 \over {v^m\,z^{m/2}}}\;,
\end{equation}
and the correction filter $\left(\mbox{sign}(v-v_0)\,{d \over
{d\,z}}\right)^{m/2}$. The derivative filters are connected by the