paper.tex

国外免费地震资料处理软件包

equations~(\ref{eqn:snells1}) and (\ref{eqn:snells2}) to the form
\begin{eqnarray}
\label{eqn:connection1}
\tau_n \, {{\partial \tau_n} \over {\partial y}} & = & \tau \, {{\partial
\tau} \over {\partial y}} \,=\, 4h\,{{\sin{\alpha} \cos{\alpha}
\cot{\gamma}} \over {v^2}}\;; \\ 
\tau_n \, {{\partial \tau_n} \over {\partial h}} & = &\tau \, {{\partial
\tau} \over {\partial h}} - {{4h} \over {v^2}} \,=\,-\,
4h\,{{\sin^2{\alpha}} \over {v^2}}\;.   
\label{eqn:connection2} 
\end{eqnarray}
Without loss of generality, we can assume $\alpha$ to be positive.
Consider a plane tangent to a true reflector at the reflection
point
(Figure \ref{fig:ocobol}).
The traveltime of a wave, reflected from the plane, has the
known explicit expression
\begin{equation}
\tau\,=\,{2 \over v}\,\sqrt{L^2+h^2\,\cos^2{\alpha}}\,\,\,,   
\label{eqn:CDP} 
\end{equation}
where $L$ is the length of the normal ray from the midpoint. As
follows from combining (\ref{eqn:CDP}) and (\ref{eqn:length}),
\begin{equation}
{\cos{\alpha} \cot{\gamma}} \,=\, {L \over h}   \,\,\,.
\label{eqn:ratio} 
\end{equation}
We can now combine equations~(\ref{eqn:ratio}),
(\ref{eqn:connection1}), and (\ref{eqn:connection2}) to transform
inequality (\ref{eqn:condition}) to the form
\begin{equation}
 h \ll {L \over {\sin{\alpha}}} \,=\, z\, \cot{\alpha}\,\,,   
\label{eqn:hm} 
\end{equation}
where $z$ is the depth of the plane reflector under the midpoint.  For
example, for a dip of 45 degrees, equation~(\ref{eqn:bolondi}) will be
satisfied only for offsets that are much smaller than the depth of the
reflector.

\sideplot{ocobol}{height=2.5in}{Reflection rays and
tangent to the reflector in a constant velocity medium (a scheme).}
  
\subsection{Offset continuation geometry: time rays}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To study the laws of traveltime curve transformation in the OC
process, it is convenient to apply the method of characteristics
\cite[]{kurant} to the eikonal-type equation~(\ref{eqn:eikonal}). The
characteristics of equation~(\ref{eqn:eikonal}) [{\em
  bi}-characteristics with respect to equation (\ref{eqn:OCequation})]
are the trajectories of the high-frequency energy propagation in the
imaginary OC process. Following the formal analogy with seismic rays,
I call those trajectories {\em time rays}, where the word {\em time}
refers to the fact that the trajectories describe the traveltime
transformation \cite[]{me}.  According to the theory of first-order
partial differential equations, time rays are determined by a set of
ordinary differential equations (characteristic equations) derived
from equation (\ref{eqn:eikonal}) :
\begin{eqnarray}
{{{dy} \over {dt_n}}   =  - {{2 h Y} \over {t_n H}}}\;,\; 
{{{dY} \over {dt_n}}  =  {Y \over t_n}}\;, 
\nonumber \\
{{{dh} \over {dt_n}}  =  {-{1 \over H}+{{2 h} \over t_n}}}\;,\;
{{{dH} \over {dt_n}} = {{Y^2} \over {t_n H}}}\;, 
\label{eqn:rays} 
\end{eqnarray}
where $Y$ corresponds to $\partial \tau_n \over \partial y$ along a
ray and $H$ corresponds to $\partial \tau_n \over \partial h$. In this
notation, equation~(\ref{eqn:eikonal}) takes the form
\begin{equation}
h\, (Y^2-H^2) = -\, t_n H 
\label{eqn:rayeikonal} 
\end{equation}
and serves as an additional constraint for the definition of time
rays.  System~(\ref{eqn:rays}) can be solved by standard mathematical
methods \cite[]{ode}. Its general solution takes the parametric form,
where the time variable $t_n$ is the parameter changing along a time
ray:
\begin{eqnarray}
y(t_n)  =  C_1-C_2\,t_n^2 \; & ; & \;h(t_n)=t_n \sqrt{C_2^2 t_n^2 + C_3}\;;
\\ 
Y(t_n)  =  {{C_2\,t_n}\over C_3}\; & ; & \;H(t_n)={h \over {C_3\,t_n}}
\label{eqn:ray} 
\end{eqnarray}
and $C_1$, $C_2$, and $C_3$ are independent coefficients, constant
along each time ray. To find the values of these coefficients, we can
pose an initial-value problem for the system of differential
equations~(\ref{eqn:rays}).  The traveltime curve $\tau_n(y;h)$ for a
given common offset $h$ and the first partial derivative $\partial
\tau_n \over \partial h$ along the same constant offset section
provide natural initial conditions. A particular case of those
conditions is the zero-offset traveltime curve. If the first partial
derivative of traveltime with respect to offset is continuous, it
vanishes at zero offset according to the reciprocity principle
(traveltime must be an even function of the offset):
\begin{math}
t_0\left(y_0\right)=\tau_n(y;0), 
\left. {\partial \tau_n \over \partial h} \right|_{h=0}=0\,.  
\end{math}
Applying the initial-value conditions to the general
solution~(\ref{eqn:ray}) generates the following expressions for the
ray invariants:
\begin{eqnarray}
C_1 & = & y+h\,{Y \over H}=y_0-{t_0\left(y_0\right) \over
t_0'\left(y_0\right)}\;;\;
C_2={{h\,Y} \over {\tau_n^2\,H}}=
-{1 \over t_0\left(y_0\right)\,t_0'\left(y_0\right)}\;;\;
\nonumber \\
C_3 & = & {h \over {\tau_n\,H}}=
-{1 \over \left(t_0'\left(y_0\right)\right)^2}\;,
\label{eqn:abc} 
\end{eqnarray}
where $t_0'\left(y_0\right)$ denotes the derivative
$\frac{d\,t_0}{d\,y_0}$.  Finally, substituting
equations~(\ref{eqn:abc})
into~(\ref{eqn:ray}), we obtain an explicit parametric form of the ray
trajectories:
\begin{eqnarray}
\label{eqn:yhray1}
y_1\left(t_1\right)  & = & \displaystyle{y+{{h\,Y} \over
{t_n^2\,H}}\,\left(t_n^2-t_1^2\right)= y_0+{{t_1^2-t_0^2\left(y_0\right)} \over
{t_0\left(y_0\right)\,t_0'\left(y_0\right)}}}\;;
\\
h_1^2\left(t_1\right) & = & \displaystyle{{{h\,t_1^2} \over {t_n^3\,H}}\,
\left(t_n^2+t_1^2\,{{h\,Y^2} \over {t_n\,H}}\right)=
t_1^2\,{{t_1^2-t_0^2\left(y_0\right)} \over
{\left(t_0\left(y_0\right)\,t_0'\left(y_0\right)\right)^2}}}\;.
\label{eqn:yhray2}
\end{eqnarray}
Here $y_1$, $h_1$, and $t_1$ are the coordinates of the continued
seismic section. Equations~(\ref{eqn:yhray1}) indicates
that the time ray projections to a common-offset section have a
parabolic form. Time rays do not exist for $t_0'\left(y_0\right)=0$ (a
locally horizontal reflector) because in this case post-NMO offset
continuation transform is not required.

The actual parameter that
determines a particular time ray is the reflection point location.
This important conclusion follows from the known parametric equations
\begin{eqnarray}
\label{eqn:gur1}
t_0(x) & = & \displaystyle{t_v \sec{\alpha}=
t_v(x)\,\sqrt{1+u^2\left(t_v'(x)\right)^2}}\;, 
\\
y_0(x) & = & \displaystyle{x+u t_v\tan{\alpha} =x+u^2\,t_v(x)t_v'(x)}\;,
\label{eqn:gur2}
\end{eqnarray}
where $x$ is the reflection point, $u$ is half of the wave velocity ($u=v/2$), 
$t_v$ is the vertical time (reflector depth divided by $u$), and
$\alpha$ is the 
local reflector dip. Taking into account that the derivative of the zero-offset
traveltime curve is
\begin{equation}
{{dt_0}\over{dy_0}}={{t_0'(x)}\over{y_0'(x)}}={{\sin{\alpha}}\over u}=
{{t_v'(x)} \over \sqrt{1+u^2\left(t_v'(x)\right)^2}}
\label{eqn:gurtx}
\end{equation}
and substituting equations~(\ref{eqn:gur1}) and~(\ref{eqn:gur2})
into~(\ref{eqn:yhray1}) and~(\ref{eqn:yhray2}), we get
\begin{eqnarray}
\label{eqn:rayrp1}
y_1\left(t_1\right) & = &
\displaystyle{x+{{t_1^2-t_v^2\left(x\right)} \over
{t_v\left(x\right)\,t_v'\left(x\right)}}}\;;
\\
u^2 t^2\left(t_1\right) & = & 
\displaystyle{t_1^2\,{{t_1^2-t_v^2\left(x\right)} \over
{\left(t_v\left(x\right)\,t_v'\left(x\right)\right)^2}}}\;,
\label{eqn:rayrp2}
\end{eqnarray}
where $t^2\left(t_1\right)=t_1^2+h_1^2\left(t_1\right)/u^2$.

To visualize the concept of time rays, let us consider some simple
analytic examples of its application to geometric analysis of the
offset-continuation process.

\subsubsection{Example 1: plane reflector}

\inputdir{Math}

The simplest and most important example is the case of a plane dipping
reflector. Putting the origin of the $y$ axis at the intersection of
the reflector plane with the surface, we can express the reflection
traveltime after NMO in the form
\begin{equation}
\tau_n(y,h)=p\,\sqrt{y^2-h^2}\;,
\label{eqn:planett}
\end{equation}
where $p=2\,{ \sin{\alpha} \over v}$, and $\alpha$ is the dip angle. 
The zero-offset traveltime in this case is a straight line:
\begin{equation}
t_0\left(y_0\right)=p\,y_0\;.
\label{eqn:planezott}
\end{equation}
According to equations~(\ref{eqn:yhray1}-\ref{eqn:yhray2}), the time
rays in this case are defined by
\begin{equation}
y_1\left(t_1\right)={t_1^2 \over {p^2\,y_0}}\;;\;
h_1^2\left(t_1\right)=t_1^2\,{{t_1^2-p^2\,y_0^2} \over
{p^4\,y_0^2}}\;;\;
y_0={{y^2-h^2} \over y}\;.
\label{eqn:planerays}
\end{equation}
The geometry of the OC transformation is shown in Figure
\ref{fig:ocopln}.

\plot{ocopln}{width=6in,height=3in}{Transformation of
  the reflection traveltime curves in the OC process: the case of a
  plane dipping reflector. Left: Time coordinate before the NMO
  correction. Right: Time coordinate after NMO. The solid lines
  indicate traveltime curves at different common-offset sections; the
  dashed lines indicate time rays.}

\subsubsection{Example 2: point diffractor}

  The second example is the case of a point diffractor (the left side
  of Figure \ref{fig:ococrv}).  Without loss of generality, the origin
  of the midpoint axis can be put above the diffraction point. In this
  case the zero-offset reflection traveltime curve has the well-known
  hyperbolic form
\begin{equation}
t_0\left(y_0\right)={\sqrt{z^2+y_0^2} \over u}\;,
\label{eqn:pointzott}
\end{equation}
where $z$ is the depth of the diffractor and $u=v/2$ is half of the
wave velocity. Time rays are defined according to
equations~(\ref{eqn:yhray1}-\ref{eqn:yhray2}), as follows:
\begin{equation}
y_1\left(t_1\right)={{u^2\,t_1^2-z^2} \over y_0}\;;\;
u^2\,t^2\left(t_1\right)=u^2\,t_1^2+h_1^2\left(t_1\right)=
u^2\,t_1^2\,{{u^2\,t_1^2-z^2} \over y_0^2}\;.
\label{eqn:pointrays}
\end{equation}


\plot{ococrv}{width=6in,height=3in}{Transformation of
  the reflection traveltime curves in the OC process. Left: the case
  of a diffraction point. Right: the case of an elliptic reflector.
  Solid lines indicate traveltime curves at different common-offset
  sections, dashed lines indicate time rays.}

\subsubsection{Example 3: elliptic reflector}

The third example (the right side of Figure \ref{fig:ococrv}) is the
curious case of a focusing elliptic reflector. Let $y$ be the center
of the ellipse and $h$ be half the distance between the foci of the
ellipse. If both foci are on the surface, the zero-offset
traveltime curve is defined by the so-called ``DMO smile''
\cite[]{GPR29-03-03740406}:
\begin{equation}
t_0\left(y_0\right)={t_n \over h}\,\sqrt{h^2-\left(y-y_0\right)^2}\;,
\label{eqn:smilezott}
\end{equation} 
where $t_n=2\,z/v$, and $z$ is the small semi-axis of the ellipse.
The time-ray equations are
\begin{equation}
y_1\left(t_1\right)=y+{h^2\over {y-y_0}}\,{{t_1^2-t_n^2} \over t_n^2}\;;\;
h_1^2\left(t_1\right)=h^2\,{t_1^2 \over t_n^2}\,
\left(1+
{h^2\over \left(y-y_0\right)^2}\,{{t_1^2-t_n^2} \over t_n^2}
\right)\;.
\label{eqn:smilerays}
\end{equation}
When $y_1$ coincides with $y$, and $h_1$ coincides with $h$, the
source and the receiver are in the foci of the elliptic reflector, and
the traveltime curve degenerates to a point $t_1=t_n$. This remarkable
fact is the actual basis of the geometric theory of dip moveout
\cite[]{GPR29-03-03740406}.

\subsection{Proof of amplitude equivalence} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us now consider the connection between the laws of traveltime
transformation and the laws of the corresponding amplitude
transformation.  The change of the wave amplitudes in the OC process
is described by the first-order partial differential transport
equation~(\ref{eqn:transport}). We can find the general solution of
this equation by applying the method of characteristics. The solution
takes the explicit integral form
\begin{equation}
A_n\left(t_n\right)=A_0\left(t_0\right)\,\exp{\left(\int_{t_o}^{t_n}
\left[h\,\left({\partial^2 \tau_n \over \partial y^2}-
{\partial^2 \tau_n \over \partial h^2}\right)\,
\left(\tau_n\,{\partial \tau_n \over \partial h} \right)^{-1}\right]\,
d\tau_n\right)}\;.
\label{eqn:ampint}
\end{equation}
The integral in equation~(\ref{eqn:ampint}) is defined on a curved
time ray, and $A_n(t_n)$ stands for the amplitude transported along
this ray. In the case of a plane dipping reflector, the ray amplitude
can be immediately evaluated by substituting the explicit traveltime
and time ray equations from the preceding section
into~(\ref{eqn:ampint}). The amplitude expression in this case takes
the simple form
\begin{equation}
A_n\left(t_n\right)=A_0\left(t_0\right)\,\exp{\left(-\int_{t_o}^{t_n}
\frac{d\tau_n}{\tau_n}\right)} = A_0\left(t_0\right)\,{t_0 \over t_n}\;.
\label{eqn:ampplane}
\end{equation}
In order to consider the more general case of a curvilinear reflector,