paper.tex

国外免费地震资料处理软件包

\sqrt{\rho_s^2 + \rho_0^2 - 2\,\rho_s\,\rho_0\,\cos{\gamma}} \,
+\, \sqrt{\rho_r^2 + \rho_0^2 - 2\,\rho_r\,\rho_0\,\cos{\gamma}}
 = 
\nonumber \\
= \sqrt{\rho_s^2 + \rho_r^2 - 2\,\rho_s\,\rho_r\,\cos{2\gamma}} \;,
\label{eqn:cos}
\end{eqnarray} 
where $\gamma$ is the reflection angle, as shown in the figure.
After straightforward
algebraic transformations of equation (\ref{eqn:cos}), we arrive at the
explicit relationship between the ray lengths:
\begin{equation}
{{(\rho_s + \rho_r)\,\rho_0} \over {2\,\rho_s\,\rho_r}} = 
\cos{\gamma}\;.
\label{eqn:r1r2}
\end{equation}
Substituting (\ref{eqn:r1r2}) into (\ref{eqn:ratio}) yields
\begin{equation} 
{A_{sr} \over A_0} = {\tau_0 \over \tau_{sr}}\,\cos{\gamma}\;,
\label{eqn:ratrat}
\end{equation}
where $\tau_0$ is the zero-offset two-way
traveltime ($\tau_0 = 2\,\rho_0/v$).
\par
What we have done is rewrite the finite-offset amplitude in the
Kirchhoff integral in terms of a particular zero-offset amplitude.
That zero-offset amplitude would arise as the geometric spreading
effect if there were a reflector whose dip was such that the
finite-offset pair would be specular at the scattering point.  Of
course, the zero-offset ray would also be specular in this case.
 
\section{THE OFFSET CONTINUATION EQUATION}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we introduce the offset continuation partial
differential equation.  We then develop its WKBJ, or ray theoretic,
solution for phase and leading-order amplitude.  We explain how we
verify that the traveltime and amplitude of the integrand of the
Kirchhoff representation (\ref{eqn:KIT}) satisfy the ``eikonal'' and
``transport'' equations of the OC partial differential equation.  To
do so, we make use of relationship (\ref{eqn:ratrat}), derived from
the Kirchhoff integral.  \par The offset continuation differential
equation derived in earlier papers
\cite[]{me,GEO68-02-07180732}\footnote{To our knowledge, the first
derivation of the revised offset continuation equation was
accomplished by Joseph Higginbotham of Texaco in 1989.  Unfortunately,
Higginbotham's derivation never appeared in the open literature.} is
\begin{equation}
h \, \left( {\partial^2 P \over \partial y^2} - 
{\partial^2 P \over \partial h^2} \right) \, = \, 
t_n \, {\partial^2 P \over {\partial t_n \,\partial h}} \;. 
\label{eqn:OCequation} 
\end{equation}
In this equation, $h$ is the half-offset ($h = l/2$), $y$ is the midpoint
(${\bf y = (s + r)}/ 2$) [hence, $y = (r + s)/2$],
and $t_n$ is the NMO-corrected traveltime
\begin{equation}
t_n = \sqrt{t^2 - {{l^2} \over {v^2}}}\;.
\label{eqn:NMO}
\end{equation}
Equation (\ref{eqn:OCequation}) describes the process of
seismogram transformation in the time-midpoint-offset domain. One can
obtain the high-frequency asymptotics of its solution by standard
methods, as follows.
We introduce a trial asymptotic solution of the form
\begin{equation}
P\left(y,h,t_n\right) =
A_n(y,h)\,f\left(t_n-\tau_n(y,h)\right) \;.
\label{eqn:raymethod} 
\end{equation}
It is important to remember the assumption that $f$ is a
``rapidly varying function,'' for example, a bandlimited delta
function.
We substitute this solution
into equation (\ref{eqn:OCequation}) and collect the terms
in order of derivatives of $f$.
This is the direct counterpart of collecting terms in powers of
frequency when applying WKBJ in the frequency domain.
From the leading
asymptotic order (the second derivative of the function $f$), we
obtain the eikonal equation describing the kinematics of the OC
transformation:
\begin{equation}
h \, \left[     {\left( \partial \tau_n \over \partial y \right)}^2 - 
                {\left( \partial \tau_n \over \partial h \right)}^2
     \right] = \, - \, \tau_n \, {\partial \tau_n \over \partial h} \;.  
\label{eqn:eikonal} 
\end{equation}
In this equation, we have replaced a multiplier of $t_n$ by
$\tau_n$ on the right side of the equation.  This is consistent with
our assumption that $f$ is a bandlimited delta function or some
equivalent impulsive source.
Analogously, collecting the terms containing the first derivative of
$f$ leads to the transport equation describing the transformation
of the amplitudes:
\begin{equation}
\left( \tau_n - 2h \, {\partial \tau_n \over {\partial h}} \right)
\, {\partial A_n \over \partial h} + 2h {\partial \tau_n \over \partial
y}   {\partial A_n \over \partial y} + h A_n \left( {\partial^2 \tau_n
\over {\partial y^2}} - {\partial^2 \tau_n \over {\partial h^2}} 
\right) \, = \, 0 \;.
\label{eqn:transport} 
\end{equation}
\par
We then rewrite the eikonal equation (\ref{eqn:eikonal}) in the
time-source-receiver coordinate system, as follows:
\begin{equation}
\left( \tau_{sr}^2 + {{l^2} \over {v^2}} \right) \left( {\partial \tau_{sr}
\over \partial r} -   {\partial \tau_{sr} \over \partial s} \right) = 2 \,l\,
\tau_{sr} \left( {1 \over {v^2}} - {\partial \tau_{sr} \over \partial r}
{\partial \tau_{sr} \over \partial s} \right) \;,
\label{eqn:SCeikonal} 
\end{equation}
which makes it easy (using Mathematica) to verify that the explicit
expression for the phase of the Kirchhoff integral kernel (\ref{eqn:time})
satisfies the eikonal equation for any scattering point\footnote{Note
that the scattering point $\bf{x}$ plays the role of a set of
parameters in the partial differential equation for $\tau_{sr}$.  To
pass from a two-dimensional in-plane traveltime to a three-dimensional
traveltime, one need only replace $z^2$ with $x_2^2 + z^2$. The role
of $x = x_1$ remains unchanged in the solution.}  ${\bf x} = (x_1 ,
x_2 , z)$.  Here, $\tau_{sr}$ is related to $\tau_n$ as $t$ is related
to $t_n$ in equation (\ref{eqn:NMO}).
\par
The general solution of the amplitude equation (\ref{eqn:transport})
has the form \cite[]{GEO68-02-07180732}
\begin{equation}
A_n = A_0\,{{\tau_0\,\cos{\gamma}}\over{\tau_n}}\,
\left({1+\rho_0\,K}\over{\cos^2{\gamma}+\rho_0\,K}\right)^{1/2}\;,
\label{eqn:ampOC} 
\end{equation}
where $K$ is the reflector curvature at the reflection point. The
kernel (\ref{eqn:kernel}) of the Kirchhoff integral (\ref{eqn:KIT})
corresponds to the reflection from a point diffractor: the integral
realizes the superposition of Huygens secondary source contributions.
We can obtain the solution of the amplitude equation for this case by
formally setting the curvature $K$ to infinity (setting the radius of
curvature to zero). The infinite curvature transforms formula
(\ref{eqn:ampOC}) to the relationship
\begin{equation}
{A_n \over A_0} = {\tau_0 \over \tau_n}\,\cos{\gamma}\;.
\label{eqn:diffampOC} 
\end{equation}
\par
Again, we exploit the assumption that
the signal $f$ has the form of the delta function.
In this case, the amplitudes
before and after the NMO correction are connected according to the
known properties of the delta function, as follows:
\begin{equation}
A_{sr}\,\delta\left(t - \tau_{sr}({\bf s,r,x})\right)=
\left|{{\partial t_n} \over {\partial t}}\right|_{t=\tau_{sr}}\,
A_{sr}\,\delta\left(t_n - \tau_n({\bf s,r,x})\right)=
A_n\,\delta\left(t_n - \tau_n({\bf s,r,x})\right)\;,
\label{eqn:NMOkernel}
\end{equation}
with 
\begin{equation}
A_n = {\tau_{sr} \over \tau_n}\,A_{sr}\;.
\label{eqn:asr2an} 
\end{equation}
Combining equations (\ref{eqn:asr2an}) and (\ref{eqn:diffampOC}) yields
\begin{equation}
{A_{sr} \over A_0} = {\tau_0 \over \tau_{sr}}\,\cos{\gamma}\;,
\label{eqn:final} 
\end{equation}
which coincides exactly with the previously found formula
(\ref{eqn:ratrat}).  As with the solution of the eikonal equation, we pass
from an in-plane solution in two dimensions to a solution for a
scattering point in three dimensions by replacing $z^2$ with $x_2^2 +
z^2$.
\par
Although the presented equations pertain to the case of offset
continuation that starts from $h=0$, i.e., inverse DMO, this is
sufficient, since every other continuation can be obtained as a chain
of DMO and inverse DMO.
\par
Thus, it is apparent that the OC differential equation
(\ref{eqn:OCequation}) relates to the Kirchhoff representation of
reflection data. We see that the amplitude and phase of the Kirchhoff
representation for arbitrary offset is the point diffractor WKBJ
solution of the offset continuation differential equation.  Hence, the
Kirchhoff approximation is a solution of the OC differential equation
when we hold the reflection coefficient constant.  This means that the
solution of the OC differential equation has all the features of
amplitude preservation, as does the Kirchhoff representation,
including geometrical spreading, curvature effects, and phase shift
effects.  Furthermore, in the Kirchhoff representation and the
solution of the OC partial differential equation by WKBJ, we have not
used the 2.5-D assumption.  Therefore the preservation of amplitude is
not restricted to cylindrical surfaces as it is in the true-amplitude
proof for DMO \cite[]{cwp}.  This is what we sought to confirm.



\section{DISCUSSION}
%%%%%%%%%%%%%%%%%% 

We have proved that the offset continuation equation correctly
transforms common-offset seismic data modeled by the Kirchhoff
integral approximation. The kinematic and dynamic equivalence of the
OC equation has been proved previously by different methods
\cite[]{GEO68-02-07180732}. However, connecting this equation with
Kirchhoff modeling opens new insights into the theoretical basis of
DMO and offset continuation: 
\begin{enumerate} 

\item 
The Kirchhoff integral can serve as a link between the wave-equation
theory, conventionally used in seismic data processing, and the
kinematically derived OC equation. Though the analysis in this paper
follows the constant-velocity model, this link can be extended in
principle to handle the case of a variable background velocity.

\item 
The OC equation operates on the kernel of the Kirchhoff integral,
which is independent of the local dip and curvature of the
reflector. This proves that the true-amplitude OC and DMO operators
can properly transform reflections from curved reflectors.
Moreover, this result does not imply any special
orientation of the reflector curvature matrix. Therefore, it does
not require a commonly made 2.5-D assumption
\cite[]{cwp}. Implicitly, this fact proves the amplitude
preservation property of the three-dimensional azimuth moveout (AMO)
operator \cite[]{GEO63-02-05740588}, based on cascading the true-amplitude DMO
and inverse DMO operators.
\end{enumerate}

\section{Acknowledgments}
We thank the Center for Wave Phenomena Consortium Project at the
Colorado School of Mines and the Stanford Exploration Project at
Stanford University. We appreciate the encouragement of Fabio Rocca
regarding the analysis of the offset continuation partial differential
equation. Finally, we thank the anonymous reviewer for the 
helpful suggestions.

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