paper.tex

国外免费地震资料处理软件包

\plot{vlcvrs}{width=6in}{Kinematic
velocity continuation in the post-stack migration domain. Solid lines
denote wavefronts: reflector images for different migration
velocities; dashed lines denote velocity rays. a: the case of a
point diffractor. b: the case of a dipping plane reflector.}

Another important example is the case of a dipping plane reflector. For
simplicity, let us put the origin of the midpoint coordinate $x$ at the point
of the plane intersection with the surface of observations. In this case, the
depth of the plane reflector corresponding to the surface point $x$ has the
simple expression
\begin{equation}
z_p(x) = x\,\tan{\alpha}\;,
\label{eq:plane}
\end{equation}
where $\alpha$ is the dip angle. The zero-offset reflection traveltime
$\tau_0(x_0)$ 
is the plane with a changed angle. It can be expressed as
\begin{equation}
\tau_0(x_0) = p\,x_0\;,
\label{eq:plt0}
\end{equation}
where $p = {{\sin{\alpha}}\over v_p}$, and $v_p$ is half of the actual
velocity. Applying formulas (\ref{eq:velrayg1}) leads to the following
parametric expression for the velocity rays:
\begin{eqnarray}
x(v) & = & x_0\,(1 -  p^2\,v^2)\;,
\label{eq:plrayg1} \\ 
\tau(v) & = & p\,x_0\,\sqrt{1 -  p^2\,v^2}\;.
\label{eq:plrayg2} 
\end{eqnarray}
Eliminating $x_0$ from the system of equations (\ref{eq:plrayg1}) and
(\ref{eq:plrayg2}) shows that the velocity continuation wavefronts are
planes with a modified angle:
\begin{equation}
\tau(x)={{p\,x} \over {\sqrt{1 -  p^2\,v^2}}}\;.
\label{eq:plfront}
\end{equation}
The right plot of Figure \ref{fig:vlcvrs} shows the geometry of the
kinematic velocity continuation for the case of a plane reflector.

\subsection{Kinematics of Residual NMO}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The residual NMO differential equation is the second term in
equation~(\ref{eq:eikonal}):
\begin{equation}
{{\partial \tau} \over {\partial v}} = 
{{h^2} \over {v^3\,\tau}}\;.
\label{eq:ResNMOeikonal} 
\end{equation}
Equation (\ref{eq:ResNMOeikonal}) does not depend on the midpoint
$x$. This fact indicates the one-dimensional nature of normal
moveout. The general solution of equation (\ref{eq:ResNMOeikonal}) is
obtained by simple integration. It takes the form
\begin{equation}
\tau^2(v) = C - {h^2 \over v^2} = \tau_1^2 + 
h^2\,\left({1 \over v_1^2} - {1 \over v^2}\right)\;,
\label{eq:ResNMO} 
\end{equation}
where $C$ is an arbitrary velocity-independent constant, and I have chosen the
constants $\tau_1$ and $v_1$ so that $\tau(v_1) = \tau_1$.
\new{Equation~(\ref{eq:ResNMO}) is applicable only for $v$ different from zero.}

For the case of a point diffractor, equation (\ref{eq:ResNMO}) easily
combines with the zero-offset solution (\ref{eq:diffront}). The result
is a simplified approximate version of the prestack residual migration
summation path:
\begin{equation}
\tau(x)=\sqrt{\tau_d^2 + 
{{(x - x_d)^2} \over {v_d^2 - v^2}} +
h^2\,\left({1 \over v_d^2} - {1 \over v^2}\right)}\;.
\label{eq:PSRMfront}
\end{equation}
Summation paths of the form (\ref{eq:PSRMfront}) for a set of diffractors
with different depths are plotted in Figures \ref{fig:vlcve1} and
\ref{fig:vlcve2}. The parameters chosen in these plots allow a direct
comparison with Etgen's Figures 2.4 and 2.5 \cite[]{Etgen.sepphd.68},
based on the exact solution and reproduced in Figures \ref{fig:vlcve3} and
\ref{fig:vlcve4}. The comparison shows that the approximate
solution (\ref{eq:PSRMfront}) captures the main features of the prestack
residual migration operator, except for the residual DMO cusps
appearing in the exact solution when the diffractor depth is smaller
than the offset.

\sideplot{vlcve1}{width=\textwidth}{Summation paths of the simplified
  prestack residual migration for a series of depth diffractors.
  Residual slowness $v/v_d$ is 1.2; half-offset $h$ is 1 km. This
  figure is to be compared with Etgen's Figure 2.4, reproduced in
  Figure~\ref{fig:vlcve3}.}

\sideplot{vlcve2}{width=\textwidth}{Summation paths of the simplified
  prestack residual migration for a series of depth diffractors.
  Residual slowness $v/v_d$ is 0.8; offset $h$ is 1 km. This figure is
  to be compared with Etgen's Figure 2.5, reproduced in
  Figure~\ref{fig:vlcve4}.}

Neglecting the residual DMO term in residual migration is
approximately equivalent in accuracy to neglecting the DMO step in
conventional processing. Indeed, as follows from the geometric analog
of equation (\ref{eq:eikonal}) derived in Appendix A
[equation~(\ref{eq:zeikonal})], dropping the residual
DMO term corresponds to the condition
 \begin{equation}
\tan^2{\alpha}\,\tan^2{\theta} \ll 1\;,
\label{eq:tatg}
\end{equation}
where $\alpha$ is the dip angle, and $\theta$ is the reflection angle.
As shown by \cite{GEO45-12-17531779}, the conventional processing
sequence without the DMO step corresponds to the separable
approximation of the double-square-root equation (\ref{eq:DSR}):
\begin{equation}
\sqrt{1 -  v^2\,\left({{\partial t} \over {\partial s}}\right)^2} +
\sqrt{1 -  v^2\,\left({{\partial t} \over {\partial r}}\right)^2}
\approx
2\,\sqrt{1 -  v^2\,\left({{\partial t} \over {\partial x}}\right)^2} +
2\,\sqrt{1 -  v^2\,\left({{\partial t} \over {\partial h}}\right)^2} -
2\;,
\label{eq:Sep}
\end{equation}
where $t$ is the reflection traveltime, and $s$ and $r$ are the source and
receiver coordinates: $s=x-h$, $r=x+h$.
In geometric terms, approximation (\ref{eq:Sep}) transforms to
\begin{equation}
\cos{\alpha}\,\cos{\theta}
\approx
\sqrt{1 -  \sin^2{\alpha}\,\cos^2{\theta}} +
\sqrt{1 -  \sin^2{\theta}\,\cos^2{\alpha}} - 1\;.
\label{eq:Sepgeom}
\end{equation}
Taking the difference of the two sides of
equation~(\ref{eq:Sepgeom}), one can estimate its accuracy by the
first term of the Taylor series for small $\alpha$ and $\theta$. The
estimate is ${3 \over 4}\,\tan^2{\alpha}\,\tan^2{\theta}$
\cite[]{GEO45-12-17531779}, which agrees qualitatively with
(\ref{eq:tatg}). Although approximation (\ref{eq:PSRMfront}) fails in situations
where the dip moveout correction is necessary, it is significantly
more accurate than the 15-degree approximation of the
double-square-root equation, implied in the migration velocity
analysis method of \cite{GEO49-10-16641674} and \cite{abma}.  The
15-degree approximation
\begin{equation}
\sqrt{1 -  v^2\,\left({{\partial t} \over {\partial s}}\right)^2} +
\sqrt{1 -  v^2\,\left({{\partial t} \over {\partial r}}\right)^2}
\approx
2 - {v^2 \over 2}\,
\left(  \left({{\partial t} \over {\partial s}}\right)^2 +
        \left({{\partial t} \over {\partial r}}\right)^2\right)
\label{eq:FDeg}
\end{equation}
corresponds geometrically to the equation 
\begin{equation}
2\,\cos{\alpha}\,\cos{\theta}
\approx
{{3 + \cos{2\alpha}\,\cos{2\theta}} \over 2}\;.
\label{eq:FDgeom}
\end{equation} 
Its estimated accuracy (from the first term of the Taylor series)
is ${1 \over 8}\,\tan^2{\alpha} + {1 \over
8}\,\tan^2{\theta}$.  Unlike the separable approximation, which is
accurate separately for zero offset and zero dip, the 15-degree
approximation fails at zero offset in the case of a steep dip and at zero
dip in the case of a large offset.

\subsection{Kinematics of Residual DMO}   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The partial differential equation for kinematic residual DMO is the
third term in equation~(\ref{eq:eikonal}):
\begin{equation}
{{\partial \tau} \over {\partial v}} = 
- {{h^2 v} \over {\tau}}\,
\left({{\partial \tau} \over {\partial x}}\right)^2\,
\left({{\partial \tau} \over {\partial h}}\right)^2\;.
\label{eq:ResDMOeikonal} 
\end{equation}
It is more convenient to consider the residual dip-moveout process
coupled with residual normal moveout. \cite{Etgen.sepphd.68} describes
this procedure as the cascade of inverse DMO with the initial velocity
$v_0$, residual NMO, and DMO with the updated velocity $v_1$. The
kinematic equation for residual NMO+DMO is the sum of the two terms in
(\ref{eq:eikonal}):
\begin{equation}
{{\partial \tau} \over {\partial v}} = 
{{h^2} \over {v^3\,\tau}}\,\left(1-v^4\,
\left({{\partial \tau} \over {\partial x}}\right)^2\,
\left({{\partial \tau} \over {\partial h}}\right)^2\right)\;.
\label{eq:DMONMOeikonal} 
\end{equation}
\begin{comment}
If the boundary data for equation (\ref{eq:DMONMOeikonal}) are on a
common-offset gather, it is appropriate to rewrite this equation
purely in terms of the midpoint derivative ${{\partial \tau} \over
{\partial x}}$, eliminating the offset-derivative term ${{\partial
\tau} \over {\partial h}}$. The resultant expression, derived in
Appendix A, has the form
\begin{equation}
v^3\,{{\partial \tau} \over {\partial v}} = 
{{2\,h^2} \over
{\sqrt{\tau^2 + 4\,h^2\,
Q\left(v,{{\partial \tau} \over {\partial x}}\right)} + \tau}}\;,
\label{eq:noth} 
\end{equation}
where 
\begin{equation}
Q(v,\tau_x) = {{\tau_x^2} \over 
{\left(1 + v^2\,\tau_x^2\right)^2}}\;.   
\label{eq:qtx} 
\end{equation}
\end{comment}

\new{The derivation of the residual DMO+NMO kinematics is detailed in
  Appendix B.}  Figure \ref{fig:vlcvcp} illustrates it with the
theoretical impulse response curves. Figure \ref{fig:vlccps} compares the
theoretical curves with the result of an actual cascade of the inverse
DMO, residual NMO, and DMO operators.

\plot{vlcvcp}{width=6in,height=3.5in}{Theoretical
kinematics of the residual NMO+DMO impulse responses for three
impulses. Left plot: the velocity ratio $v_1/v_0$ is $1.333$. Right
plot: the velocity ratio $v_1/v_0$ is $0.833$. In both cases the
half-offset $h$ is 1 km.}

\inputdir{resdmo}

\plot{vlccps}{width=6in,height=3.5in}{The result of
residual NMO+DMO (cascading inverse DMO, residual NMO, and DMO) for
three impulses. Left plot: the velocity ratio $v_1/v_0$ is
$1.333$. Right plot: the velocity ratio $v_1/v_0$ is $0.833$. In both
cases the half-offset $h$ is 1 km.}

\inputdir{Math}

Figure \ref{fig:vlcvrd} illustrates the residual NMO+DMO velocity
continuation for two particularly interesting cases. The left plot
shows the continuation for a point diffractor. One can see that when
the velocity error is large, focusing of the velocity rays forms a
distinctive loop on the zero-offset hyperbola. The right plot illustrates
the case of a plane dipping reflector. The image of the reflector
shifts both vertically and laterally with the change in NMO
velocity.

\plot{vlcvrd}{width=6in}{Kinematic
velocity continuation for residual NMO+DMO. Solid lines denote
wavefronts: zero-offset traveltime curves; dashed lines denote
velocity rays. a: the case of a point diffractor; the velocity
ratio $v_1/v_0$ changes from $0.9$ to $1.1$. b:
the case of a dipping plane reflector; the velocity
ratio $v_1/v_0$ changes from $0.8$ to $1.2$. In both cases, the
half-offset $h$ is 2 km.}

The full residual migration operator is the chain of
residual zero-offset migration and residual NMO+DMO. I illustrate the
kinematics of this operator in Figures \ref{fig:vlcve3} and \ref{fig:vlcve4},
which are designed to match Etgen's Figures 2.4 and 2.5
\cite[]{Etgen.sepphd.68}. A comparison with Figures \ref{fig:vlcve1} and
\ref{fig:vlcve2} shows that including the residual DMO term affects
the images of objects with the depth smaller than the half-offset
$h$. This term complicates the residual migration operator with cusps.

\sideplot{vlcve3}{width=\textwidth}{Summation paths of prestack
  residual migration for a series of depth diffractors. Residual
  slowness $v/v_d$ is 1.2; half-offset $h$ is 1 km. This figure
  reproduces Etgen's Figure 2.4.}

\sideplot{vlcve4}{width=\textwidth}{Summation paths of prestack
  residual migration for a series of depth diffractors. Residual
  slowness $v/v_d$ is 0.8; half-offset $h$ is 1 km. This figure
  reproduces Etgen's Figure 2.5.}

\section{FROM KINEMATICS TO DYNAMICS}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The theory of characteristics \cite[]{kurant2} states that if a partial
differential equation has the form
\begin{equation}
\sum_{i,j=1}^{n}\,\Lambda_{ij}(\xi_1,\ldots,\xi_n)\,
{{\partial^2 P} \over {\partial \xi_i\,\partial \xi_j}} +
F\left(\xi_1,\ldots,\xi_n,P,
{{\partial P} \over {\partial \xi_1}},\ldots,
{{\partial P} \over {\partial \xi_n}}\right) = 0\;,
\label{eq:gequation}
\end{equation}
where F is some arbitrary function, and if the eigenvalues of the
matrix $\Lambda$ are nonzero, and one of them is different in sign
from the others, then equation (\ref{eq:gequation}) describes a
wave-type process, and its kinematic counterpart is the characteristic
equation
\begin{equation}
\sum_{i,j=1}^{n}\,\Lambda_{ij}(\xi_1,\ldots,\xi_n)\,
{{\partial \psi} \over {\partial \xi_i}}\, 
{{\partial \psi} \over {\partial \xi_j}} = 0
\label{eq:charequation}
\end{equation}
with the characteristic surface 
\begin{equation}
\psi(\xi_1,\ldots,\xi_n) = 0
\label{eq:charsurface}
\end{equation}
corresponding to the wavefront. In velocity continuation problems,
it is appropriate to choose the variable $\xi_1$ to denote the time
$t$, $\xi_2$ to denote the velocity $v$, and the rest of the
$\xi$-variables to denote one or two lateral coordinates $x$. Without
loss of generality, let us set the characteristic surface to be
\begin{equation}
\psi = t - \tau(x;v) = 0\;,
\label{eq:chartau}
\end{equation}
and use the theory of characteristics to reconstruct the main
(second-order) part of the dynamic differential equation from the
corresponding kinematic equations. As in the preceding section, it is
convenient to consider separately the three different components of the prestack
velocity continuation process.

\subsection{Dynamics of Zero-Offset Velocity Continuation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the case of zero-offset velocity continuation, the characteristic
equation is reconstructed from equation (\ref{eq:POMeikonal}) to have
the form
\begin{equation}
{{\partial \psi} \over {\partial v}}\,
{{\partial \psi} \over {\partial t}} +
v\,t\,\left({{\partial \psi} \over {\partial x}}\right)^2 = 0\;,
\label{eq:POMchar} 
\end{equation}
where $\tau$ is replaced by $t$ according to
equation~(\ref{eq:chartau}).  According to
equation~(\ref{eq:gequation}), the corresponding dynamic equation is
\begin{equation}
{{\partial^2 P} \over {\partial v\, \partial t}} +
v\,t\,{{\partial^2 P} \over {\partial x^2}} +
F\left(x,t,v,P,
{{\partial P} \over {\partial t}},
{{\partial P} \over {\partial v}},
{{\partial P} \over {\partial x}}
\right) = 0\;,
\label{eq:POMequation} 
\end{equation}
where the function $F$ remains to be defined. The simplest case of $F$
equal to zero corresponds to Claerbout's velocity continuation
equation \cite[]{Claerbout.sep.48.79}, derived in a different way.
\cite{Levin.sep.48.101} provides the dispersion-relation derivation,
conceptually analogous to applying the method of characteristics.

In high-frequency asymptotics, the wavefield $P$ can be
represented by the ray-theoretical (WKBJ) approximation,
\begin{equation}
P(t,x,v) \approx A(x,v)\,f\left(t - \tau(x,v)\right)\;, 
\label{eq:WKB} 
\end{equation}
where $A$ is the amplitude, $f$ is the short (high-frequency) wavelet,