paper.tex

国外免费地震资料处理软件包

\begin{equation}
\widetilde{M}(t)=\int Y(t,\omega)\,D(\omega)\,
\exp\left[-i\omega \phi (t,\omega)\right]\,d\omega
\label{eqn:b1}
\end{equation}
constitute a pair of asymptotically inverse operators
($\widetilde{M}(t)$ matching $M(t)$ in the high-frequency asymptotics)
if
\begin{equation}
X(t,\omega)\,Y(t,\omega)={Z(t,\omega) \over {2\,\pi}}\;,
\label{eqn:belka}
\end{equation}
where $Z$ is the ``Beylkin determinant''
\begin{equation}
Z(t,\omega)=\left|\partial \omega \over \partial \hat{\omega}\right|\;
\mbox{for}\;\hat{\omega}=\omega\,{\partial \phi(t,\omega) \over \partial t}\;.
\label{eqn:det}
\end{equation}

With respect to the high-frequency asymptotic representation, we can
recast (\ref{eqn:IDMO}) in the equivalent form by moving the time
derivative under the integral sign:
\begin{equation}
\widetilde{P}(t_n,k) \approx 
{H(t_n) \over {2\,\pi}}\,\mbox{Re}\left[
\int_{-\infty}^{\infty}A^{-2}
\widetilde{\widetilde{P}}_0(\omega_0,k)\, 
\exp\left(-i \omega_0\,|t_n|\,A\right)
\,d\omega_0\right]
\label{eqn:IDMO2} 
\end{equation}
Now the asymptotic inverse of~(\ref{eqn:IDMO2}) is evaluated by
means of Beylkin's method (\ref{eqn:b0})-(\ref{eqn:b1}), which leads
to an amplitude-preserving one-term DMO operator of the form
\begin{equation}
\widetilde{\widetilde{P}}_0(\omega_0,k)  = 
\mbox{Im}\left[
\int_{-\infty}^{\infty} B
\widetilde{P}^{(0)}_1\left(\left|t_1\right|,k\right)\,
\exp\left(i \omega_0\,|t_1|\,A\right)
\,dt_1\right]\;,
\label{eqn:born} 
\end{equation}
where 
\begin{equation}
B = A^2 {\partial \over \partial \omega_0}\left(\omega_0\,
{{\partial (t_n\,A)} \over \partial t_n}\right) = 
A^{-1}\,(2\,A^2 - 1)\;.
\label{eqn:jack} 
\end{equation}
  
The amplitude factor~(\ref{eqn:jack}) corresponds exactly to that of
Born DMO \cite[]{born} in full accordance with the conclusions of the
asymptotic analysis of the offset-continuation amplitudes. An
analogous result can be obtained with the different definition of
amplitude preservation proposed by \cite{GEO58-01-00470066}. In
the time-and-space domain, the operator asymptotically analogous to
(\ref{eqn:born}) is found by applying either the stationary phase
technique \cite[]{GEO55-05-05950607,GEO58-01-00470066} or Goldin's method of
discontinuities \cite[]{goldintomo,Goldin.sep.67.171}, which is the
time-and-space analog of Beylkin's asymptotic inverse theory
\cite[]{stovas}. The time-and-space asymptotic DMO operator takes the
form
\begin{equation}
P_0(t_0,y) = {\bf D}^{1/2}_{-t_0}\,\int w_0(\xi;h_1,t_0)\,
P^{(0)}_1(\theta^{(-)}(\xi;h_1,0,t_0),y_1-\xi)\,d\xi\;,
\label{eqn:TADMO}
\end{equation} 
where the weighting function $w_0$ is defined as 
\begin{equation}
w_0(\xi;h_1,t_0)=\sqrt{t_0 \over {2\,\pi}}\, 
{{h_1\,(h_1^2+\xi^2)} \over (h_1^2-\xi^2)^2}\;.
\label{eqn:TAw}
\end{equation}

\section{Offset continuation in the log-stretch domain}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The log-stretch transform, proposed by \cite{GPR30-06-08130828}
and further developed by many other researchers, is a useful
tool in DMO and OC processing. Applying a log-stretch transform of the
form
\begin{equation}
\sigma = \ln\left|t_n \over t_* \right|\;, 
\label{eqn:log}
\end{equation}
where $t_*$ is an arbitrarily chosen time constant, eliminates the
time dependence of the coefficients in equation~(\ref{eqn:OCequation})
and therefore makes this equation invariant to time shifts. After the
double Fourier transform with respect to the midpoint coordinate $y$
and to the transformed (log-stretched) time coordinate $\sigma$, the
partial differential equation~(\ref{eqn:OCequation}) takes the form of
an ordinary differential equation,
\begin{equation}
h\,\left({{d^2 \widehat{\widehat{P}}} 
\over {dh^2}} + k^2\,\widehat{\widehat{P}}\right) =
i\Omega\,{{d \widehat{\widehat{P}}} \over {dh}}\;,
\label{eqn:LSequation}
\end{equation}
where
\begin{equation}
\widehat{\widehat{P}}(h) = \int\!\int P(t_n=t_*\,\exp(\sigma),h,y)\,
\exp(i\Omega\sigma - iky)\,d\sigma\,dy\;. 
\label{eqn:LSFT}
\end{equation}

Equation~(\ref{eqn:LSequation}) has the known general solution,
expressed in terms of cylinder functions of complex order $\lambda =
{{1+i\Omega} \over 2}$ \cite[]{watson}
\begin{equation}
\widehat{\widehat{P}}(h) = 
C_1(\lambda)\,(kh)^{\lambda}\,J_{-\lambda}(kh)+
C_2(\lambda)\,(kh)^{\lambda}\,J_{\lambda}(kh)\;,
\label{eqn:gensol}
\end{equation}
where $J_{-\lambda}$ and $J_{\lambda}$ are Bessel functions, and $C_1$
and $C_2$ stand for some arbitrary functions of $\lambda$ that do not
depend on $k$ and $h$.

In the general case of offset continuation, $C_1$ and $C_2$ are
constrained by the two initial conditions~(\ref{eqn:bound0}) and
(\ref{eqn:bound1}). In the special case of continuation from zero offset, we
can neglect the second term in~(\ref{eqn:gensol}) as vanishing at the zero
offset. The remaining term defines the following operator of inverse
DMO in the ${\Omega,k}$ domain:
\begin{equation}
\widehat{\widehat{P}}(h) = \widehat{\widehat{P}}(0)\,Z_{\lambda}(kh)\;,
\label{eqn:OKDMO}
\end{equation}
where $Z_{\lambda}$ is the analytic function
\begin{eqnarray}
\nonumber
Z_{\lambda}(x) & = & \Gamma(1-\lambda)\,\left(x \over 2\right)^{\lambda}\,
J_{-\lambda}(x)=
{}_0F_1\left(;1-\lambda;-\frac{x^2}{4}\right) \\
& = &
\sum_{n=0}^{\infty} {(-1)^n \over n!}\,
{\Gamma(1-\lambda) \over \Gamma(n+1-\lambda)}\,
\left(x \over 2\right)^{2n}\;,
\label{eqn:z}
\end{eqnarray}
$\Gamma$ is the gamma function and ${}_0F_1$ is the confluent
hypergeometric limit function \cite[]{ab}.

The DMO operator now can be derived as the inversion of operator
(\ref{eqn:OKDMO}), which is a simple multiplication by
$1/Z_{\lambda}(kh)$. Therefore, offset continuation becomes a
multiplication by $Z_{\lambda}(kh_2)/Z_{\lambda}(kh_1)$ (the cascade
of two operators). This fact demonstrates an important advantage of
moving to the log-stretch domain: both offset continuation and DMO are simple
filter multiplications in the Fourier domain of the log-stretched time
coordinate.

In order to compare operator~(\ref{eqn:OKDMO}) with the known versions
of log-stretch DMO, we need to derive its asymptotic representation
for high frequency $\Omega$. The required asymptotic expression
follows directly from the definition of function $Z_{\lambda}$ in
equation~(\ref{eqn:z}) and the known asymptotic representation for a Bessel
function of high order \cite[]{watson}:
\begin{equation}
J_{\lambda}(\lambda z) \stackrel{\lambda \rightarrow \infty}{\approx} 
{{(\lambda z)^{\lambda}\,\exp\left(\lambda\,\sqrt{1-z^2}\right)} \over
{e^{\lambda}\,\Gamma(\lambda+1)\,(1-z^2)^{1/4}\,
\left\{1+\sqrt{1-z^2}\right\}^{\sqrt{1-z^2}}}}\;.
\label{eqn:carlini}
\end{equation}
Substituting approximation~(\ref{eqn:carlini}) into~(\ref{eqn:z}) and
considering the high-frequency limit of the resultant expression
yields
\begin{equation}
Z_{\lambda}(kh) \approx 
\left\{{1+\sqrt{1-\left(kh \over \lambda\right)^2}} \over
2\right\}^{\lambda}\, 
{{\exp\left(\lambda\,\left[1 - \sqrt{1-\left(kh \over
\lambda\right)^2}\right]\right)} \over 
\left(1-\left(kh \over \lambda\right)^2\right)^{1/4}} \approx
{F(\epsilon)\,e^{i\Omega\,\psi(\epsilon)}}\;,
\label{eqn:AOKDMO}
\end{equation}
where $\epsilon$ denotes the ratio ${2\,k\,h} \over
\Omega$,
\begin{equation}
F(\epsilon)=\sqrt{{1+\sqrt{1+\epsilon^2}} \over
{2\,\sqrt{1+\epsilon^2}}}\,
\exp\left({1-\sqrt{1+\epsilon^2}} \over 2\right)\;,
\label{eqn:F}
\end{equation}
and
\begin{equation}
\psi(\epsilon)={1 \over 2}\,\left(1 - \sqrt{1+\epsilon^2} +
\ln\left({1 + \sqrt{1+\epsilon^2}} \over 2\right)\right)\;.
\label{eqn:psi}
\end{equation}

The asymptotic representation~(\ref{eqn:AOKDMO}) is valid for high
frequency $\Omega$ and $|\epsilon| \leq 1$. The
phase function $\psi$ defined in~(\ref{eqn:psi}) coincides precisely
with the analogous term in Liner's \emph{exact log DMO}
\cite[]{GEO55-05-05950607}, which provides the correct
geometric properties of DMO. Similar expressions for the log-stretch
phase factor $\psi$ were derived in different ways by
\cite{GEO61-03-08150820} and \cite{GEO61-04-11031114}.
However, the amplitude term $F(\epsilon)$ differs from the previously
published ones because of the difference in the amplitude preservation
properties.
  
A number of approximate log DMO operators have been proposed in the
literature. As shown by \cite{GEO55-05-05950607}, all of them but
exact log DMO distort the geometry of reflection effects at large
offsets. The distortion is caused by the implied approximations of the
true phase function $\psi$. Bolondi's OC operator
\cite[]{GPR30-06-08130828} implies $\psi(\epsilon) \approx -{\epsilon^2
  \over 8}$, Notfors' DMO \cite[]{GEO52-12-17181721} implies
$\psi(\epsilon) \approx 1 - \sqrt{1+(\epsilon /2)^2}$, and the ``full
DMO'' \cite[]{SEG-1987-S14.1} has $\psi(\epsilon) \approx {1 \over 2}
\ln\left[1-(\epsilon / 2)^2\right]$. All these approximations are
valid for small $\epsilon$ (small offsets or small reflector dips) and
have errors of the order of $\epsilon^4$ (Figure \ref{fig:pha}).  The
range of validity of Bolondi's operator is defined in
equation~(\ref{eqn:hm}).
  
  \sideplot{pha}{height=1.5in}{ Phase functions of the log
    DMO operators. Solid line: exact log DMO; dashed line: Bolondi's
    OC; dashed-dotted line: Bale's full DMO; dotted line: Notfors'
    DMO.  }
  
  In practice, seismic data are often irregularly sampled in space but
  regularly sampled in time. This makes it attractive to apply offset
  continuation and DMO operators in the $\{\Omega,y\}$ domain, where
  the frequency $\Omega$ corresponds to the log-stretched time and
  $y$ is the midpoint coordinate. Performing the inverse Fourier
  transform on the spatial frequency transforms the inverse DMO
  operator~(\ref{eqn:OKDMO}) to the $\{\Omega,y\}$ domain, where the
  filter multiplication becomes a convolutional operator:
\begin{equation}
\widehat{P}(\Omega,h,y) =
{\widehat{F}(\Omega) \over \sqrt{2\,\pi}}\,\int_{|\xi|<h}
{ h \over {h^2-\xi^2}}\,
\widehat{P_0}(\Omega,y-\xi)\,
\exp\left(-{i\Omega \over 2}\,\ln\left(1-{\xi^2 \over h_1^2}\right)\right) 
\,d\xi\;.
\label{eqn:OXDMO} 
\end{equation}
Here $\widehat{F}(\Omega)$ is a high-pass frequency filter:
\begin{equation}
\widehat{F}(\Omega)={{\Gamma(1/2-i\Omega/2)}
\over {\sqrt{1/2}\,\Gamma(-i\Omega/ 2)}}\;.
\label{eqn:hat} 
\end{equation}
At high frequencies $\widehat{F}(\Omega)$ is approximately equal to
$(- i \Omega)^{1/2}$, which corresponds to the half-derivative
operator $\left(\partial \over \partial \sigma \right)^{1/2}$, which,
in turn, is equal to the $\left(t_n {\partial \over \partial t_n}
\right)^{1/2}$ term of the asymptotic OC
operator~(\ref{eqn:asintegral}). The difference between the exact
filter $\widehat{F}$ and its approximation by the half-order
derivative operator is shown in Figure \ref{fig:flt}. This difference
is a measure of the validity of asymptotic OC operators.

\plot{flt}{width=6in}{ Amplitude (left) and phase (right) of
  the time filter in the log-stretch domain. The solid line is for the
  exact filter; the dashed line for its approximation by the
  half-order derivative filter. The horizontal axis corresponds to the
  dimensionless log-stretch frequency $\Omega$.}

Inverting operator~(\ref{eqn:OXDMO}), we can obtain the DMO operator in the
$\{\Omega,y\}$ domain.

\section{Discussion}

The differential model for offset continuation is based on several
assumptions. It is important to fully realize them in order to
understand the practical limitations of this model.
\begin{itemize}
\item The \emph{constant velocity} assumption is essential for
  theoretical derivations. In practice, this limitation is not too
  critical, because the operators act locally. DMO and
  offset continuation algorithms based on the constant-velocity
  assumptions are widely used in practice \cite[]{DMO00-00-04960496}.
\item The \emph{single-mode} assumption does not include multiple
  reflections in the model. If multiple events (with different
  apparent velocities) are present in the data, they might require
  extending the model. Convolving two (or more) differential offset
  continuation operators, corresponding to different velocities, we
  can obtain a higher-order differential operator for predicting
  multiple events.
\item The \emph{continuous AVO} assumption implies that the
  reflectivity variation with offset is continuous and can be
  neglected in a local neighborhood of a particular offset. While the
  offset continuation model correctly predicts the geometric spreading
  effects in the reflected wave amplitudes, it does not account for
  the variation of the reflection coefficient with offset.
\item The \emph{2.5-D} assumption was implicit in the derivation of
  the offset continuation equation. According to this assumption, the
  reflector does not change in the cross-line direction, and we can
  always consider the reflectio