paper.tex

国外免费地震资料处理软件包

and the function $\tau$ satisfies the kinematic equation
(\ref{eq:POMeikonal}). Substituting approximation (\ref{eq:WKB}) into
the dynamic velocity continuation equation (\ref{eq:POMequation}),
collecting the leading-order terms, and neglecting the $F$ function
leads to the partial differential equation for amplitude transport:
\begin{equation}
{\partial A \over \partial v} = v\,\tau\,\left(2\,
{\partial A \over \partial x}\,
{\partial \tau \over \partial x} + A\,
{\partial^2 \tau \over \partial x^2}\right)\;.
\label{eq:PAMPequation} 
\end{equation}
The general solution of equation (\ref{eq:PAMPequation}) follows from the
theory of characteristics. It takes the form
\begin{equation}
A(x,v) = A(x_0,0)\,\exp{\left(\int_0^{v}\,u\,\tau(x,u)\,
{\partial^2 \tau(x,u) \over \partial x^2}\,du\right)}\;,
\label{eq:PAMPsolution} 
\end{equation}
where the integral corresponds to the
curvilinear integration along the corresponding velocity ray, \new{and $x_0$
corresponds to the
starting point of the ray.}
In the case of a plane dipping reflector, the image of the reflector remains
plane in the velocity continuation process. Therefore, the second
traveltime derivative ${\partial^2 \tau(x,u) \over \partial x^2}$ in
(\ref{eq:PAMPsolution}) equals zero, and the exponential is equal to
one. This means that the amplitude of the image does not change with
the velocity along the velocity rays. This fact does not agree with the
theory of conventional post-stack migration, which suggests
downscaling the image by the ``cosine'' factor $\tau_0 \over
\tau$ \cite[]{GEO46-05-07170733,Levin.sep.48.147}. The simplest way to
include the cosine factor in the velocity continuation equation is to
set the function $F$ to be ${1 \over t}\,{\partial P \over \partial
v}$. The resulting differential equation
\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{eq:POMequation2} 
\end{equation}
has the amplitude transport
\begin{equation}
A(x,v) = {\tau_0 \over \tau}\,A(x_0,0)\,
\exp{\left(\int_0^{v}\,u\,\tau(x,u)\,
{\partial^2 \tau(x,u) \over \partial x^2}\,du\right)}\;,
\label{eq:PAMPsolution2} 
\end{equation}
corresponding to the differential equation
\begin{equation}
{\partial A \over \partial v} = v\,\tau\,\left(2\,
{\partial A \over \partial x}\,
{\partial \tau \over \partial x} + A\,
{\partial^2 \tau \over \partial x^2}\right) - 
A\,{1 \over \tau}\,{\partial \tau \over \partial v}\;.
\label{eq:PAMPequation2} 
\end{equation}
Appendix C proves that the time-and-space solution of the dynamic
velocity continuation equation (\ref{eq:POMequation2}) coincides with the
conventional Kirchhoff migration operator.

\begin{comment}
The finite-difference implementation of zero-offset velocity
continuation resembles the implementation of Claerbout's
15-degree equation in a retarded coordinate system
\cite[]{Claerbout.blackwell.76}. This implementation is discussed in
more detail in Appendix C. 
\end{comment}

\subsection{Dynamics of Residual NMO}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
According to the theory of characteristics, described in the beginning
of this section, the kinematic residual NMO equation
(\ref{eq:ResNMOeikonal}) corresponds to the dynamic equation of the form
\begin{equation}
{{\partial P} \over {\partial v}} + 
{{h^2} \over {v^3\,t}}\,{{\partial P} \over {\partial t}} 
+ F(h,t,v,P) = 0
\label{eq:ResNMOdyn} 
\end{equation}
\new{with the undetermined function $F$. In the case of $F=0$}, the 
general solution
is easily found to be
\begin{equation}
P(t,h,v) = \phi\left(t^2 + {h^2 \over v^2}\right)\;.
\label{eq:ResNMOsol} 
\end{equation}
where $\phi$ is an arbitrary smooth function.
The combination of dynamic equations (\ref{eq:ResNMOdyn}) and
(\ref{eq:POMequation2}) leads to an approximate prestack velocity
continuation with the residual DMO effect neglected. To accomplish the
combination, one can simply add the term ${{h^2} \over
{v^3\,t}}\,{{\partial^2 P} \over {\partial t^2}}$ from
equation~(\ref{eq:ResNMOdyn}) to the left-hand
side of equation (\ref{eq:POMequation2}). This addition changes the
kinematics of velocity continuation, but does not change the amplitude
properties embedded in the transport equation (\ref{eq:PAMPsolution2}).

\cite{GEO38-04-06350642} and \cite{Hale.sepphd.36} \new{advocate using
  an amplitude correction term in the NMO step. This term can be
  easily added by selecting an appropriate function $F$ in
  equation~(\ref{eq:ResNMOdyn}). The choice
  $F=\frac{h^2}{v^3\,t^2}\,P$ results in the equation}
\begin{equation}
{{\partial P} \over {\partial v}} + 
{{h^2} \over {v^3\,t^2}}\,\left(t\,{{\partial P} \over {\partial t}} 
+ P\right) = 0
\label{eq:ResNMOdyn2} 
\end{equation}
with the general solution
\begin{equation}
P(t,h,v) = \frac{1}{t}\,\phi\left(t^2 + {h^2 \over v^2}\right)\;,
\label{eq:ResNMOsol} 
\end{equation}
\new{which has the Dunkin-Levin amplitude correction term.}

\subsection{Dynamics of Residual DMO}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The case of residual DMO complicates the building of a dynamic
equation because of the essential nonlinearity of the kinematic
equation (\ref{eq:ResDMOeikonal}). One possible way to linearize the
problem is to increase the order of the equation. In this case, the
resultant dynamic equation would include a term that has the
second-order derivative with respect to velocity $v$. Such an equation
describes two different modes of wave propagation and requires
additional initial conditions to separate them. Another possible way
to linearize equation (\ref{eq:ResDMOeikonal}) is to approximate it at
small dip angles.
%For example, one
%can obtain a recursively accurate approximation by a continued
%fraction expansion of the square root in equation (\ref{eq:ResDMOeikonal}),
%analogously to Muir's method in conventional finite-difference
%migration \cite[]{Claerbout.blackwell.85}. 
In this case, the dynamic
equation would contain only the first-order derivative with respect to
the velocity and high-order derivatives with respect to the other
parameters. The third, and probably the most attractive, method is to
change the domain of consideration. For example, one could switch from
the common-offset domain to the domain of offset dip. This
method implies a transformation similar to slant stacking of
common-midpoint gathers in the post-migration domain in order to
obtain the local offset dip information. Equation (\ref{eq:ResDMOeikonal})
transforms, with the help of the results from Appendix A, to the form
\begin{equation}
v^3\,{{\partial \tau} \over {\partial v}} = 
{{\tau\,\sin^2{\theta}} \over
{\cos^2{\alpha} - \sin^2{\theta}}}\;,
\label{eq:noh} 
\end{equation}
with
\begin{equation}
\cos^2{\alpha} = \left(1 + v^2 \,
\left({{\partial \tau} \over {\partial x}}\right)^2\right)^{-1}\;,
\label{eq:cos2a} 
\end{equation}
and
\begin{equation}
\sin^2{\alpha} = v^2\,
\left({{\partial \tau} \over {\partial h}}\right)^2\,
\left(1 + v^2 \,
\left({{\partial \tau} \over {\partial h}}\right)^2\right)^{-1}\;.
\label{eq:sin2g} 
\end{equation}
For a constant offset dip $\tan{\theta} = v\,{{\partial \tau} \over
{\partial h}}$, the dynamic analog of equation (\ref{eq:noh}) is the
third-order partial differential equation
\begin{equation}
v\,  \cot^2{\theta}\,{{\partial^3 P} \over {\partial t^2\, \partial v}} -
v^3\,{{\partial^3 P} \over {\partial x^2\, \partial v}}
+ t\,{{\partial^3 P} \over {\partial t^2\, \partial v}} +
v^2\,t\,{{\partial^3 P} \over {\partial x^2\, \partial t}} = 0\;.
\label{eq:ResDMOdyn} 
\end{equation}
Equation (\ref{eq:ResDMOdyn}) does not strictly comply with the theory of
second-order linear differential equations. Its properties and
practical applicability require further research.

\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%
I have derived kinematic and dynamic equations for residual time migration
in the form of a continuous velocity continuation process. This
derivation explicitly decomposes prestack
velocity continuation into three parts corresponding to
zero-offset continuation, residual NMO, and residual DMO. These three
parts can be treated separately both for simplicity of theoretical
analysis and for practical purposes. It is important to note that in
the case of a three-dimensional migration, all three components of
velocity continuation have different dimensionality. Zero-offset
continuation is fully 3-D. It can be split into two 2-D continuations
in the in- and cross-line directions. Residual DMO is a
two-dimensional common-azimuth process. Residual NMO is a 1-D
single-trace procedure.

The dynamic properties of zero-offset velocity continuation are
precisely equivalent to those of conventional post-stack migration
methods such as Kirchhoff migration. Moreover, the Kirchhoff migration
operator coincides with the integral solution of the velocity
continuation differential equation for continuation from the zero
velocity plane.

This rigorous theory of velocity continuation gives us new insights into the
methods of prestack migration velocity analysis. Extensions to the case of
depth migration in a variable velocity background are developed by
\cite{hong} and \cite{adler}. \new{A practical application of
  velocity continuation to migration velocity analysis is demonstrated in the
  companion paper} \cite[]{second}, \new{where the general theory is used to
  design efficient and practical algorithms.}

\section{Acknowledgments}
%%%%%%%%%%%%%%%%%%%%%%%%%
This work was completed when the author was a member of the Stanford
Exploration Project (SEP) at Stanford University. The financial
support was provided by the SEP sponsors.

I thank Bee Bednar, Biondo Biondi, Jon Claerbout, Sergey Goldin, Bill Harlan,
David Lumley, and Bill Symes for useful and stimulating discussions.
\new{Paul Fowler, Hugh Geiger, Samuel Gray, and one anonymous reviewer
  provided valuable suggestions that improved the quality of the paper.}

\bibliographystyle{seg}
\bibliography{SEP2,paper,spec,velcon,SEG}

\append{DERIVING THE KINEMATIC EQUATIONS}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The main goal of this appendix is to derive the partial differential
equation describing the image surface in a
depth-midpoint-offset-velocity space.

\inputdir{XFig}

\sideplot{vlcray}{width=0.9\textwidth}{Reflection rays in a constant
  velocity medium (a scheme).}

The derivation starts with observing a simple geometry of reflection
in a constant-velocity medium, shown in Figure \ref{fig:vlcray}. The
well-known equations for the apparent slowness
\begin{equation}
{{\partial t} \over {\partial s}} \,=\,
{ {\sin{\alpha_1}} \over {v}}\;,
\label{eq:snell1}
\end{equation}
\begin{equation}
{{\partial t} \over {\partial r}} \,=\, 
{{\sin{\alpha_2}} \over {v}}  
\label{eq:snell2}
\end{equation} 
relate the first-order traveltime derivatives for the reflected waves
to the emergence angles of the incident and reflected rays. Here $s$
stands for the source location at the surface, $r$ is the receiver
location, $t$ is the reflection traveltime, $v$ is the constant
velocity, and $\alpha_1$ and $\alpha_2$ are the angles shown in Figure
\ref{fig:vlcray}. Considering the traveltime derivative with respect to
the depth of the observation surface $z$ shows that the
contributions of the two branches of the reflected ray, added
together, form the equation
\begin{equation}
- {{\partial t} \over {\partial z}} \,=\,
{{\cos{\alpha_1}} \over {v}} +
{{\cos{\alpha_2}} \over {v}}\;.
\label{eq:snell3}
\end{equation}
It is worth mentioning that the elimination of angles from equations
(\ref{eq:snell1}), (\ref{eq:snell2}), and (\ref{eq:snell3}) leads to
the famous {\em double-square-root equation,}
\begin{equation}
- v\,{{\partial t} \over {\partial z}} \,=\,
\sqrt{1 -  v^2\,\left({{\partial t} \over {\partial s}}\right)^2} +
\sqrt{1 -  v^2\,\left({{\partial t} \over {\partial r}}\right)^2}\;,
\label{eq:DSR}
\end{equation}
published in the Russian literature by \cite{alekseev} and commonly
used in the form of a pseudo-differential dispersion relation
\cite[]{Clayton.sep.14.21,Claerbout.blackwell.85} for prestack
migration \cite[]{Yilmaz.sepphd.18,Popovici.sep.84.53}. Considered
locally, equation (\ref{eq:DSR}) is independent of the constant velocity
assumption and enables \new{recursive} prestack downward
continuation of reflected waves in heterogeneous \new{isotropic}
media.

Introducing the midpoint coordinate $x = {{s+ r} \over 2}$ and half-offset
$h = {{r - s} \over 2}$, one can apply the chain rule and elementary
trigonometric equalities to formulas (\ref{eq:snell1}) and
(\ref{eq:snell2}) and transform these formulas to 
\begin{equation}
{{\partial t} \over {\partial x}} \,=\, 
{{\partial t} \over {\partial s}} + 
{{\partial t} \over {\partial r}} \,=\, 
{ {2 \sin{\alpha}\,\cos{\theta}} \over {v}}\;,
\label{eq:snells1}
\end{equation}
\begin{equation}
{{\partial t} \over {\partial h}} \,=\,
{{\partial t} \over {\partial r}} - 
{{\partial t} \over {\partial s}} \,=\, 
{ {2 \cos{\alpha}\,\sin{\theta}} \over {v}} \;,
\label{eq:snells2}
\end{equation}
where $\alpha = {{\alpha_1 + \alpha_2} \over 2}$ is the dip angle, and
$\theta = {{\alpha_2 - \alpha_1} \over 2}$ is the reflection angle
\cite[]{Clayton.sep.14.21,Claerbout.blackwell.85}. Equation
(\ref{eq:snell3}) transforms analogously to
\begin{equation}
- {{\partial t} \over {\partial z}} \,=\,
{{2 \cos{\alpha} \cos{\theta}} \over {v}}\;. 
\label{eq:snells3}
\end{equation}
This form of equation (\ref{eq:snell3}) is used to describe the stretching
factor of the waveform distortion in depth migration \cite[]{Tygel}.

Dividing (\ref{eq:snells1}) and (\ref{eq:snells2}) by
(\ref{eq:snells3}) leads to
\begin{equation}
{{\partial z} \over {\partial x}} \,=\,
- \tan{\alpha}\;, 
\label{eq:snellz1}
\end{equation}
\begin{equation}
{{\partial z} \over {\partial h}} \,=\,
- \tan{\theta}\;.
\label{eq:snellz2}
\end{equation}
Equation~(\ref{eq:snellz2}) is the basis of the angle-gather construction of
\cite{sandf}.
Substituting formulas (\ref{eq:snellz1}) and (\ref{eq:snellz2}) into equation
(\ref{eq:snells3}) yields yet another form of the double-square-root equation:
\begin{equation}
- {{\partial t} \over {\partial z}} \,=\, {2 \over {v}}\,
\left[\sqrt{1 + \left({\partial z} \over {\partial x}\right)^2}\,
\sqrt{1 + \left({\partial z} \over {\partial h}\right)^2}\right]^{-1}\;, 
\label{eq:snellz3}
\end{equation}
which is analogous to the dispersion relationship of Stolt prestack
migration \cite[]{GEO43-01-00230048}.
 
The law of sines in the triangle formed by the incident and reflected
ray leads to the explicit relationship between the traveltime and the
offset:
\begin{equation}
v\,t = 2\,h\,  {{\cos{\alpha_1}+ \cos{\alpha_2}} \over
\sin{\left(\alpha_2-\alpha_1\right)}} = 2\,h\,{\cos{\alpha} \over
\sin{\theta}} \;.
\label{eq:length} 
\end{equation}
An algebraic combination of formulas (\ref{eq:length}), (\ref{eq:snells1}), and
(\ref{eq:snells2}) forms the basic kinematic equation of the offset
continuation theory \cite[]{ofcon}:
\begin{equation}
{{\partial t} \over {\partial h}} \,
\left(t^2 + {{4\,h^2} \over {v^2}}\right)\,=\,