exs.tex

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\section{EXAMPLES}
%%%%%%%%%%%%%%%% 
In this section, I consider several particular examples of stacking
operators used in seismic data processing and derive their asymptotic
pseudo-unitary versions.

\subsection{Datuming}
%%%%%%%%%%%%%%%%
Let $x$ denote a point on the surface at which the propagating
wavefield is recorded. Let $y$ denote a point on another surface, to
which the wavefield is propagating. Then the summation path of the
stacking operator for the forward wavefield continuation is
\begin{equation}
\theta(x;t,y)  =  t - T(x,y)\;,
\label{eqn:datp}
\end{equation}
where $t$ is the time recorded at the $y$-surface, and $T(x,y)$ is the
traveltime along the ray connecting $x$ and $y$. The backward
propagation reverses the sign in (\ref{eqn:datp}), as follows:
\begin{equation}
\widehat{\theta}(y;z,x)  =  z + T(x,y)\;.
\label{eqn:datm}
\end{equation}
Substituting the summation path formulas (\ref{eqn:datp}) and (\ref{eqn:datm}) into
the general weighting function formulas (\ref{eqn:wp}) and (\ref{eqn:wm}), we
immediately obtain
\begin{equation}
w^{(+)}  =  w^{(-)}  =  {1\over{\left(2\,\pi\right)^{m/2}}} \, 
\left|{{\partial^2 T}\over{\partial x\,\partial y}}\right|^{1/2}\;.
\label{eqn:datw}
\end{equation}
Gritsenko's formula \cite[]{gritsenko,ig1} states that the second mixed
traveltime derivative ${{\partial^2 T}\over{\partial x\,\partial y}}$
is connected with the geometric spreading $R$ along the $x$-$y$ ray by
the equality
\begin{equation}
R(x,y) = {\sqrt{\cos{\alpha(x)}\,\cos{\alpha(y)}}\over v(x)}\,
\left|{{\partial^2 T}\over{\partial x\,\partial y}}\right|^{-1/2}\;,
\label{eqn:gritsenko}
\end{equation}
where $v(x)$ is the velocity at the point $x$, and $\alpha(x)$ and
$\alpha(y)$ are the angles formed by the ray with the $x$ and $y$
surfaces, respectively. In a constant-velocity medium,
\begin{equation}
R(x,y) = v^{m-1}\,T(x,y)^{m/2}\;.
\end{equation}
Gritsenko's formula (\ref{eqn:gritsenko}) allows us to
rewrite equation (\ref{eqn:datw}) in the form \cite[]{goldin}
\begin{eqnarray}
w^{(+)}(x;t,y) & = & {1\over{\left(2\,\pi\right)^{m/2}}} \,
{\sqrt{\cos{\alpha(x)}\,\cos{\alpha(y)}}\over {v(x)\,R(x,y)}}\;,
 \\
w^{(-)}(y;z,x) & = & {1\over{\left(2\,\pi\right)^{m/2}}} \,
{\sqrt{\cos{\alpha(x)}\,\cos{\alpha(y)}}\over {v(y)\,R(y,x)}}\;.
\end{eqnarray}
\par
The weighting functions commonly used in Kirchhoff datuming
\cite[]{GEO44-08-13291344,GEO49-08-12391248,svgdat} are defined as
\begin{eqnarray}
w(x;t,y) & = & {1\over{\left(2\,\pi\right)^{m/2}}} \,
{{\cos{\alpha(x)}}\over {v(x)\,R(x,y)}}\;,
 \\
\widehat{w}(y;z,x) & = & {1\over{\left(2\,\pi\right)^{m/2}}} \,
{{\cos{\alpha(y)}}\over {v(y)\,R(y,x)}}\;.
\label{eqn:datw2}
\end{eqnarray}
These two operators appear to be asymptotically inverse according to
formula (\ref{eqn:whatw}). They coincide with the asymptotic pseudo-unitary
operators if the velocity $v$ is constant ($v(x)=v(y)$), and the two
datum surfaces are parallel ($\alpha(x) = \alpha(y)$).


\subsection{Migration}
%%%%%%%%%%%%%%%%%
\emph{Least-squares} migration, envisioned by \cite{lailly} and 
\cite{taran}, has recently
become a practical method and gained a lot of attention in the
geophysical literature \cite[]{nemeth99,chavent,duquet,resol}. Using the
theory of asymptotic pseudo-unitary operators allows us to reconcile 
this approach with the method of \emph{asymptotic true-amplitude}
migration \cite[]{bleistein}.

As recognized by \cite{tygel}, true-amplitude migration
\cite[]{tam,vor} is the asymptotic inversion of seismic modeling
represented by the Kirchhoff high-frequency approximation. The
Kirchhoff approximation for a reflected wave \cite[]{haddon,norm}
belongs to the class of stacking-type operators (\ref{eqn:operator})
with the summation path
\begin{equation}
\theta(x;t,y)  =  t - T\left(s(y),x\right) - T\left(x,r(y)\right)\;,
\label{eqn:modt}
\end{equation}
the weighting function
\begin{equation}
w(x;t,y)  =  {1\over{\left(2\,\pi\right)^{m/2}}} \,
{{C\left(s(y),x,r(y)\right)} 
\over {R\left(s(y),x\right)\,R\left(x,r(y)\right)}}\;,
\label{eqn:modw}
\end{equation}
and the additional time filter $\left({\partial \over {\partial
z}}\right)^{m/2}$. Here $x$ denotes a point at the reflector surface,
$s$ is the source location, and $r$ is the receiver location at the
observation surface. The parameter $y$ corresponds to the
configuration of observation. That is, $s(y) = s\,,\;r(y) = y$ for the
common-shot configuration, $s(y) = r(y) = y$ for the zero-offset
configuration, and $s(y) = y - h\,,\;r(y) = y + h$ for the
common-offset configuration (where $h$ is the half-offset). The
functions $T$ and $R$ have the same meaning as in the datuming example,
representing the one-way traveltime and the one-way geometric
spreading, respectively. The function $C(s,x,r)$ is known as the {\em
obliquity factor}. Its definition is
\begin{equation}
C(s,x,r) = {1 \over 2}\,
\left({{\cos{\alpha_s(x)}} \over {v_s(x)}} +
      {{\cos{\alpha_r(x)}} \over {v_r(x)}}\right)\;, 
\label{eqn:obliquity}
\end{equation}
where the angles $\alpha_s(x)$ and $\alpha_r(x)$ are formed by the
incident and reflected waves with the normal to the reflector at the
point $x$, and $v_s(x)$ and $v_r(x)$ are the corresponding velocities
in the vicinity of this point. In this paper, I leave the case of
converted (e.g., P-SV) waves outside the scope of consideration and
assume that $v_s(x)$ equals $v_r(x)$ (e.g., in P-P reflection). In this
case, it is important to notice that at the stationary point of the
Kirchhoff integral, $\alpha_s(x) = \alpha_r(x) = \alpha(x)$ (the law of
reflection), and therefore
\begin{equation}
C(s,x,r) = {{\cos{\alpha(x)}} \over {v(x)}}\;.
\label{eqn:statpoint}
\end{equation}
The stationary point of the Kirchhoff integral is the point where the
stacking curve (\ref{eqn:modt}) is tangent to the actual reflection
traveltime curve. When our goal is asymptotic inversion, it is
appropriate to use equation (\ref{eqn:statpoint}) in place of
(\ref{eqn:obliquity}) to construct the inverse operator. The weighted
function (\ref{eqn:modw}) can include other factors affecting the
leading-order (WKBJ) ray amplitude, such as the source signature, 
caustics counter (the KMAH-index), and transmission coefficient
for the interfaces \cite[]{chapman,cerveny}. In the following analysis,
I neglect these factors for simplicity.
\par
The model $M$ implied by the Kirchhoff modeling integral is the
wavefield with the wavelet shape of the incident wave and the
amplitude proportional to the reflector coefficient along the
reflector surface. The goal of true-amplitude migration is to recover
$M$ from the observed seismic data. In order to obtain the image of
the reflectors, the reconstructed model is evaluated at the time $z$
equal to zero. The Kirchhoff modeling integral requires explicit
definition of the reflector surface. However, its inverse doesn't
require explicit specification of the reflector location. For each
point of the subsurface, one can find the normal to the hypothetical
reflector by bisecting the angle between the $s-x$ and $x-r$
rays. Born scattering approximation provides a different physical
model for the reflected waves. According to this approximation, the
recorded waves are viewed as scattered on smooth local inhomogeneities
rather than reflected from sharp reflector surfaces. The inversion of
Born modeling \cite[]{GEO52-07-09430964,GEO52-07-09310942} closely
corresponds with the result of Kirchhoff integral inversion. For an
unknown reflector and the correct macro-velocity model, the asymptotic
inversion reconstructs the signal located at the reflector surface
with the amplitude proportional to the reflector coefficient.
\par
As follows from the form of the summation path (\ref{eqn:modt}), the
integral migration operator must have the summation path
\begin{equation}
\widehat{\theta}(y;z,x)  =  z + T\left(s(y),x\right) + T\left(x,r(y)\right)
\end{equation}
to reconstruct the geometry of the reflector at the migrated section.
According to equation~(\ref{eqn:inverse}), the asymptotic
reconstruction of the wavelet requires, in addition, the derivative
filter $\left(- {\partial \over {\partial t}}\right)^{m/2}$. The
asymptotic reconstruction of the amplitude defines the true-amplitude
weighting function in accordance with equation~(\ref{eqn:whatw}), as
follows:
\begin{equation}
\widehat{w}(y;z,x)  =  
{{v(x)\,R\left(s(y),x\right)\,R\left(x,r(y)\right)}   
\over {\left(2\,\pi\right)^{m/2}\,\cos{\alpha(x)}}}\,
\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|\;.
\label{eqn:migw}
\end{equation}
The weighting function of the asymptotic pseudo-unitary migration is
found analogously to equation~(\ref{eqn:datw}) as
\begin{equation}
w^{(+)}  =  w^{(-)}  =  {1\over{\left(2\,\pi\right)^{m/2}}} \,
\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|^{1/2}\;.
\label{eqn:pumigw}
\end{equation}
Unlike true-amplitude migration, this type of migration operator
does not change the dimensionality of the input. Several specific cases exist
for different configurations of the input data.

\subsubsection{1. Common-shot migration}
In the case of common-shot migration, we can simplify equation (\ref{eqn:migw})
with the help of Gritsenko's formula (\ref{eqn:gritsenko}) to the form
\begin{equation}
\widehat{w}_{CS}(r;z,x)  =  {1\over{\left(2\,\pi\right)^{m/2}}} \,
{{\cos{\alpha(r)}} \over {v(x)}}\,
{{R(s,x)} \over {R(x,r)}} = {1\over{\left(2\,\pi\right)^{m/2}}} \,
{{\cos{\alpha(r)}} \over {v(r)}}\,
{{R(s,x)} \over {R(r,x)}} \;,
\label{eqn:csmigw}
\end{equation}
where the angle $\alpha(r)$ is measured between the reflected ray and
the normal to the observation surface at the reflector point $r$.
Formula (\ref{eqn:csmigw}) coincides with the analogous result of
\cite{GEO53-12-15401546}, derived directly from Claerbout's imaging
principle \cite[]{GEO35-03-04070418}. An alternative derivation is
given by \cite{goldin87}.  \cite{GEO56-08-11641169} points out a
remarkable correspondence between this formula and the classic results
of Born scattering inversion \cite[]{GEO52-07-09310942}.

For common-shot migration, pseudo-unitary weighting coincides with the
weighting of datuming and corresponds to the downward continuation of the
receivers.

\subsubsection{2. Zero-offset migration}
In the case of zero-offset mi\-gra\-tion, Grit\-senko's for\-mu\-la
sim\-pli\-fies the true-amp\-li\-tude mi\-gra\-tion weighting function
(\ref{eqn:migw}) to the form
\begin{equation}
\widehat{w}_{ZO}(y;z,x)  =  {{2^m}\over{\left(2\,\pi\right)^{m/2}}} \,
{{\cos{\alpha(y)}} \over {v(y)}}\;.
\label{eqn:zomigw}
\end{equation}
In a constant-velocity medium, one can accomplish the true-amplitude
zero-offset migration by premultiplying the recorded zero-offset
seismic section by the factor $\left(v \over 2 \right)^{m-1}\,\left(t
\over 2\right)^{m/2}$ [which corresponds at the stationary point to the
geometric spreading $R(x,y)$] and downward continuation according to
formula (\ref{eqn:datw2}) with the effective velocity $v/2$ 
\cite[]{goldin87,GEO56-01-00180026}. This conclusion is in
agreement with the analogous result of Born inversion
\cite[]{GEO50-08-12531265}, though derived from a different viewpoint. 

In the zero-offset case, the pseudo-unitary forward operator reduces to
downward pseudo-unitary continuation with a velocity of $v/2$.

\subsubsection{3. Common-offset migration}
In the case of common-offset migration in a general variable-velocity
medium, the weighting function (\ref{eqn:migw}) cannot be simplified to a
different form, and all its components need to be calculated
explicitly by dynamic ray tracing \cite[]{castro}.  In the