paper.tex

国外免费地震资料处理软件包

emerging from the point $\{t_n,h,y\}=\{t_1,h_1,y_1\}$. The
common-offset slices of the characteristic conoid are shown in the
left plot of Figure \ref{fig:cont}.

\plot{cont}{width=6in}{
Constant-offset sections of the characteristic conoid - ``offset
continuation fronts'' (left), and branches of the conoid used in the
integral OC operator (right). The upper part of the plots
(small times) corresponds to continuation to smaller offsets; the 
lower part (large times) corresponds to larger offsets.}

As a second-order differential equation of the hyperbolic type,
equation (\ref{eqn:OCequation}) describes two different processes. The
first process is ``forward'' continuation from smaller to larger
offsets, the second one is ``reverse'' continuation in the opposite
direction.  These two processes are clearly separated in the
high-frequency asymptotics of operator~(\ref{eqn:integral}). To obtain
the asymptotic representation, it is sufficient to note that ${1
  \over \sqrt{\pi}}\, {H(t) \over \sqrt{t}}$ is the impulse response
of the causal half-order integration operator and that $H(t^2-a^2)
\over \sqrt{t^2-a^2}$ is asymptotically equivalent to $H(t-a) \over
{\sqrt{2a}\,\sqrt{t-a}}$ $(t, a >0)$.  Thus, the asymptotical form of
the integral offset-continuation operator becomes
\begin{eqnarray}
P^{(\pm)}(t_n,h,y) & = &
{\bf D}^{1/2}_{\pm\,t_n}\,\int w^{(\pm)}_0(\xi;h_1,h,t_n)\,
P^{(0)}_1(\theta^{(\pm)}(\xi;h_1,h,t_n),y_1-\xi)\,d\xi
\nonumber \\ 
& \pm  & 
{\bf I}^{1/2}_{\pm\,t_n}\,\int w^{(\pm)}_1(\xi;h_1,h,t_n)\,
P^{(1)}_1(\theta^{(\pm)}(\xi;h_1,h,t_n),y_1-\xi)\,d\xi\;.
\label{eqn:asintegral} 
\end{eqnarray}
Here the signs ``$+$'' and ``$-$'' correspond to the type of
continuation (the sign of ${h-h_1}$), ${\bf D}^{1/2}_{\pm\,t_n}$ and
${\bf I}^{1/2}_{\pm\,t_n}$ stand for the operators of causal and
anticausal half-order differentiation and integration applied with
respect to the time variable $t_n$, the summation paths
$\theta^{(\pm)}(\xi;h_1,h,t_n)$ correspond to the two non-negative
sections of the characteristic conoid~(\ref{eqn:conoid}) (Figure
\ref{fig:cont}):
\begin{equation}
t_1=\theta^{(\pm)}(\xi;h_1,h,t_n)=
{t_n \over h}\,\sqrt{{U \pm V} \over 2 }\;,
\label{eqn:summation}
\end{equation}
where $U=h^2+h_1^2-\xi^2$, and $V=\sqrt{U^2-4\,h^2\,h_1^2}$; $\xi$ is
the midpoint separation (the integration parameter), and $w^{(\pm)}_0$
and $w^{(\pm)}_1$ are the following weighting functions:
\begin{eqnarray}
w^{(\pm)}_0 & = & {1 \over \sqrt{2\,\pi}}\,
{\theta^{(\pm)}(\xi;h_1,h,t_n) \over \sqrt{t_n\,V}}\;,
\label{eqn:w0} \\
w^{(\pm)}_1 & = & {1 \over \sqrt{2\,\pi}}\,
{{\sqrt{t_n}\, h_1} \over {\sqrt{V}\,\theta^{(\pm)}(\xi;h_1,h,t_n)}}\;. 
\label{eqn:w1} 
\end{eqnarray}  
Expression~(\ref{eqn:summation}) for the summation path of the OC
operator was obtained previously by \cite{stovas} and
\cite{SEG-1994-1541}. A somewhat different form of it is proposed by
\cite{GEO61-06-18461858}. I describe the kinematic interpretation
of formula~(\ref{eqn:summation}) in Appendix~B.

In the high-frequency asymptotics, it is possible to replace the two
terms in equation~(\ref{eqn:asintegral}) with a single term
\cite[]{GEO68-03-10321042}. The single-term expression is
\begin{equation}
P^{(\pm)}(t_n,h,y)  = 
{\bf D}^{1/2}_{\pm\,t_n}\,\int w^{(\pm)}(\xi;h_1,h,t_n)\,
P^{(0)}_1(\theta^{(\pm)}(\xi;h_1,h,t_n),y_1-\xi)\,d\xi\;,
\end{equation}
where
\begin{eqnarray}
w^{(+)} & = & \sqrt{\theta^{(+)}(\xi;h_1,h,t_n) \over {2\,\pi}}\;
{{h^2-h_1^2-\xi^2} \over {V^{3/2}}}\;, 
\label{eqn:wOC12} \\
w^{(-)} & = & {{\theta^{(-)}(\xi;h_1,h,t_n)} \over \sqrt{2\,\pi t_n}}\;
{{h_1^2-h^2 +\xi^2} \over {V^{3/2}}}\;. 
\label{eqn:wOC21} 
\end{eqnarray}
A more general approach to true-amplitude asymptotic offset
continuation is developed by \cite{tygel}.

The limit of expression~(\ref{eqn:summation}) for the output offset $h$
approaching zero can be evaluated by L'Hospitale's rule. As one would
expect, it coincides with the well-known expression for the summation
path of the integral DMO operator
\cite[]{GPR29-03-03740406}
\begin{equation}
t_1=\theta^{(-)}(\xi;h_1,0,t_n)=
\lim_{h \rightarrow 0} {{t_n \over h}\,\sqrt{{U - V} \over 2 }}=
{{t_n\,h_1} \over \sqrt{h_1^2-\xi^2}}\;.
\label{eqn:DMOsummation}
\end{equation}
I discuss the connection between offset continuation and DMO in the
next section.

\section{Offset continuation and DMO}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Dip moveout represents a particular case of offset continuation for
the output offset equal to zero. In this section, I consider the DMO
case separately in order to compare the solutions of equation
(\ref{eqn:OCequation}) with the Fourier-domain DMO operators, which
have been the standard for DMO processing since Hale's outstanding
work \cite[]{Hale.sepphd.36,GEO49-06-07410757}.

Equation~(\ref{eqn:integral}) transforms to the time-wavenumber domain
with the help of integral tables:
\begin{equation}
\widetilde{P}(t_n,h,k)=
H(t_n)\,\left(\widetilde{P}_0(t_n,h,k) +
t_n\,\widetilde{P}_1(t_n,h,k)\right)\;,
\label{eqn:pp0p1} 
\end{equation}
where
\begin{eqnarray}
\widetilde{P}_0  & = & 
{\partial \over {\partial t_n}}\,
\int_{\left(h_1/h\right)\,t_n}^{t_n}
\widetilde{P}^{(0)}_1\left(\left|t_1\right|,k\right)\,
J_0\left(k\,\sqrt{\left({h^2 \over t_n^2}-{h_1^2 \over t_1^2}\right)\,
\left(t_n^2-t_1^2\right)}\right)\,dt_1\;,
\label{eqn:p0} \\
\widetilde{P}_1  & = & 
\int_{\left(h_1/h\right)\,t_n}^{t_n}
h_1\,\widetilde{P}^{(1)}_1\left(\left|t_1\right|,k\right)\,
J_0\left(k\,\sqrt{\left({h^2 \over t_n^2}-{h_1^2 \over t_1^2}\right)\,
\left(t_n^2-t_1^2\right)}\right)\,{dt_1 \over t_1^2}\;,
\label{eqn:p1} 
\end{eqnarray}
\begin{eqnarray}
\widetilde{P}^{(j)}_1(t_1,k) & = & 
\int\,P\,^{(j)}_1(t_1,y_1)\exp (-iky_1)\,dy_1\;(j=0,1)\;,
\label{eqn:tildej} \\
\widetilde{P}(t_n,h,k)  & = & 
\int\,P(t_n,h,y)\exp (-iky)\,dy\;(j=0,1)\;.
\label{eqn:tilde}
\end{eqnarray}

Setting the output offset to zero, we obtain the following DMO-like
integral operators in the $t$--$k$ domain:
\begin{equation}
\widetilde{P}(t_0,0,k)=
H(t_0)\,\left(\widetilde{P}_0(t_0,k) +
t_0\,\widetilde{P}_1(t_0,k)\right)\;,
\label{eqn:dp0p1} 
\end{equation}
where
\begin{equation}
\widetilde{P}_0(t_0,k)  = 
- {\partial \over {\partial t_0}}\,
\int_{t_0}^{\infty}
\widetilde{P}^{(0)}_1\left(\left|t_1\right|,k\right)\,
J_0\left({{k\,h_1}\over t_1}\,
\sqrt{t_1^2-t_0^2}\right)\,dt_1\;,
\label{eqn:d0} 
\end{equation}
\begin{equation}
\widetilde{P}_1(t_0,k)  = 
- \int_{t_0}^{\infty}
h_1\,\widetilde{P}^{(1)}_1\left(\left|t_1\right|,k\right)\,
J_0\left({{k\,h_1}\over t_1}\,
\sqrt{t_1^2-t_0^2}\right)\,{dt_1 \over t_1^2}\;,
\label{eqn:d1} 
\end{equation}
the wavenumber $k$ corresponds to the midpoint axis $y$, and $J_0$ is
the zeroth-order Bessel function.  The Fourier transform
of~(\ref{eqn:d0}) and (\ref{eqn:d1}) with respect to the time variable
$t_0$ reduces to known integrals \cite[]{table} and creates explicit
DMO-type operators in the frequency-wavenumber domain, as follows:
\begin{equation}
\widetilde{\widetilde{P}}_0(\omega_0,k)  = 
i\,
\int_{-\infty}^{\infty}
\widetilde{P}^{(0)}_1\left(\left|t_1\right|,k\right)\,
{\sin{\left(\omega_0\,|t_1|\,A\right)} \over A}
\,dt_1\;,
\label{eqn:dw0} 
\end{equation}
\begin{equation}
\widetilde{\widetilde{P}}_1(\omega_0,k)  = 
i\, \int_{-\infty}^{\infty}
h_1\,\widetilde{P}^{(1)}_1\left(\left|t_1\right|,k\right)\,
{\sin{\left(\omega_0\,|t_1|\,A\right)} \over A}
{dt_1 \over t_1^2}\;,
\label{eqn:dw1} 
\end{equation}
where 
\begin{equation}
A=\sqrt{1+{(k\,h_1)^2 \over (\omega_0\,t_1)^2}}\;,
\label{eqn:a} 
\end{equation}
\begin{equation}
\widetilde{\widetilde{P}}_j(\omega_0,k)=
\int\,\widetilde{P}_j(t_0,k)\,\exp (i\omega_0 t_0)\,dt_0\;.
\label{eqn:FFT} 
\end{equation}

It is interesting to note that the first term of the continuation to
zero offset~(\ref{eqn:dw0}) coincides exactly with the imaginary part
of Hale's DMO operator \cite[]{GEO49-06-07410757}. However, unlike Hale's,
operator~(\ref{eqn:dp0p1}) is causal, which means that its impulse
response does not continue to negative times. The non-causality of
Hale's DMO and related issues are discussed in more detail by
\cite{stovas}.
%I include a brief summary of this discussion in Appendix~\ref{chapter:hale}.

Though Hale's DMO is known to provide correct reconstruction of the
geometry of zero-offset reflections, it does not account properly for
the amplitude changes \cite[]{GEO58-01-00470066}. The preceding section of this
paper shows that the additional contribution to the amplitude
is contained in the second term of the OC operator
(\ref{eqn:integral}), which transforms to the second term in the DMO
operator~(\ref{eqn:dp0p1}). Note that this term vanishes at the input offset
equal to zero, which represents the case of the inverse DMO operator.

Considering the inverse DMO operator as the continuation from zero
offset to a non-zero offset, we can obtain its representation in the
$t$-$k$ domain from equations~(\ref{eqn:pp0p1}-\ref{eqn:p1}) as
\begin{equation}
\widetilde{P}(t_n,h,k)  = 
H(t_n) {\partial \over {\partial t_n}}\,
\int_{0}^{t_n}
\widetilde{P}_0\left(\left|t_0\right|,k\right)\,
J_0\left({{k\,h}\over t_n}\,
\sqrt{t_n^2-t_0^2}\right)\,dt_0\;,
\label{eqn:i0} 
\end{equation}
Fourier transforming equation~(\ref{eqn:i0}) with respect to the time
variable $t_0$ according to equation~(\ref{eqn:FFT}), we get the
Fourier-domain version of the ``amplitude-preserving'' inverse DMO:
\begin{equation}
\widetilde{P}(t_n,h,k) = 
{H(t_n) \over {2\,\pi}}\,{\partial \over \partial t_n}\,
\int_{-\infty}^{\infty}
\widetilde{\widetilde{P}}_0(\omega_0,k)\, 
{{\sin{\left(\omega_0\,|t_n|\,A\right)} \over {\omega_0\,A}}}
\,d\omega_0\;,
\label{eqn:IDMO} 
\end{equation}
\begin{equation}
A=\sqrt{1+{(k\,h)^2 \over (\omega_0\,t_n)^2}}\;.
\label{eqn:a1} 
\end{equation}

Comparing operator~(\ref{eqn:IDMO}) with Ronen's version of inverse DMO
\cite[]{GEO52-07-09730984}, one can see that if Hale's DMO is denoted
by ${\bf D}_{t_0}\,{\bf H}$, then Ronen's inverse DMO is ${\bf
H^{T}\,D}_{-t_0}$, while the amplitude-preserving inverse~(\ref{eqn:IDMO})
is ${\bf D}_{t_n}\,{\bf H^T}$. Here ${\bf D}_t$ is the derivative
operator $\left( \partial \over \partial t\right)$, and ${\bf H^T}$
stands for the adjoint operator defined by the dot-product test
\begin{equation}
{\bf (Hm,d)=(m,H^{T}d)},
\label{eqn:dot}
\end{equation}
where the parentheses denote the dot product:
\begin{equation}
{\bf (m_1,m_2)}=\int\!\int\,m_1(t_n,y)\,m_2(t_n,y)\,dt_n\,dy\;.
\nonumber
\end{equation}
  
In high-frequency asymptotics, the difference between the amplitudes
of the two inverses is simply the Jacobian term ${d\,t_0 \over
  d\,t_n}$, asymptotically equal to ${t_0 \over t_n}$. This difference
corresponds exactly to the difference between Black's definition of
amplitude preservation \cite[]{GEO58-01-00470066} and the definition used in Born
DMO \cite[]{born,GEO56-02-01820189}, as discussed above. While operator
(\ref{eqn:IDMO}) preserves amplitudes in the Born DMO sense, Ronen's
inverse satisfies Black's amplitude preservation criteria. This means
Ronen's operator implies that the ``geometric spreading'' correction
(multiplication by time) has been performed on the data prior to DMO.

To construct a one-term DMO operator, thus avoiding the estimation of
the offset derivative in~(\ref{eqn:w1}), let us consider the problem
of inverting the inverse DMO operator~(\ref{eqn:IDMO}). One of the
possible approaches to this problem is the least-squares iterative
inversion, as proposed by \cite{GEO52-07-09730984}. This
requires constructing the adjoint operator, which is Hale's DMO (or
its analog) in the case of Ronen's method. The iterative least-squares
approach can account for irregularities in the data geometry
\cite[]{IZO,SEG-1994-1545} and boundary effects, but it is computationally
expensive because of the multiple application of the operators. An
alternative approach is the asymptotic inversion, which can be viewed
as a special case of preconditioning the adjoint operator
\cite[]{SEG-1988-S17.5,SEG-1996-0032}. The goal of the asymptotic inverse is to
reconstruct the geometry and the amplitudes of the reflection events
in the high-frequency asymptotic limit.

According to Beylkin's theory of asymptotic inversion, also known as
the {\em generalized Radon transform} \cite[]{beylkin}, two operators of the
form
\begin{equation}
D(\omega)=\int X(t,\omega)\,M(t)\,
\exp\left[i\omega \phi (t,\omega)\right]\,dt
\label{eqn:b0}
\end{equation}
and