paper.tex

国外免费地震资料处理软件包

\def\cakedir{.}
\def\figdir{./Fig}
\lefthead{Fomel \& Bleistein}
\righthead{Offset continuation}
\footer{SEP--92}

\title{Amplitude preservation for offset continuation:\\
Confirmation for Kirchhoff data}
%\keywords{amplitudes, continuation, DMO, approximation }


%\noemailaddress
\author{Sergey Fomel\/\footnotemark[1] 
and Norman Bleistein\/\footnotemark[2]}

\footnotetext[1]{\emph{Lawrence Berkeley National Laboratory, 
   1 Cyclotron Road, Mail Stop 50A-1148, 
   Berkeley, CA 94720, USA}}
\footnotetext[2]{\emph{Center for Wave Phenomena,
       Colorado School of Mines, 
       Golden, CO 80401, USA}}

\maketitle
 
\begin{abstract}
Offset continuation (OC) is the operator that transforms common-offset
     seismic reflection data from one offset to another.  Earlier
     papers by the first author presented a partial differential
     equation in midpoint and offset to achieve this transformation.
     The equation was derived from the kinematics of the continuation
     process with no reference to amplitudes. We present here a proof
     that the solution of the OC partial differential equation does
     propagate amplitude properly at all offsets, at least to the same
     order of accuracy as the Kirchhoff approximation.  That is, the
     OC equation provides a solution with the correct traveltime and
     correct leading-order amplitude.  ``Correct amplitude'' in this
     case means that the transformed amplitude exhibits the right
     geometrical spreading and reflection-surface-curvature effects
     for the new offset.  The reflection coefficient of the original
     offset is preserved in this transformation.  This result is more
     general than the earlier results in that it does not rely on the
     two-and-one-half dimensional assumption.
\end{abstract}

\section{Introduction}
%%%%
Offset continuation (OC) is the operator that transforms common-offset
seismic reflection data to data with a different offset. Following the
classic results of \cite{GPR29-03-03740406},
\cite{GPR30-06-08130828,GPR32-06-10451073} described OC as a
continuous process of gradual change of the offset by means of a
partial differential equation. Because it is based on the small-offset
small-dip approximation, Bolondi's equation failed at large offsets or
steep reflector dips. Nevertheless, the OC concept inspired a flood of
research on dip moveout (DMO) correction
\cite[]{DMP00-00-01300130}. Since one can view DMO as a particular
case of OC (continuation to zero offset), the offset continuation
theory can serve as a natural basis for DMO theory. Its immediate
application is in interpolating data undersampled in the offset
dimension.  \par \cite{me,GEO68-02-07180732} introduced a
revised version of the OC differential equation and proved that it
provides the correct kinematics of the continued wavefield for any
offset and reflector dip under the assumption of constant effective
velocity. The equation is interpreted as an ``image wave equation'' by
\cite{hubral}.  Studying the laws of amplitude transformation shows
that in 2.5-D media the amplitudes of continued seismic gathers
transform according to the rules of geometric seismics, except for the
reflection coefficient, which remains unchanged
\cite[]{joint,GEO68-02-07180732}. The solution of the boundary problem
on the OC equation for the DMO case \cite[]{GEO68-02-07180732}
coincides in high-frequency asymptotics with the amplitude-preserving
DMO, also known as {\em Born DMO}
\cite[]{GEO56-02-01820189,SEG-1990-1366}.  However, for the purposes
of verifying that the amplitude is correct for any offset, this
derivation is incomplete.  \par In this paper, we perform a direct
test on the amplitude properties of the OC equation.  We describe the
input common-offset data by the Kirchhoff modeling integral, which
represents the high-frequency approximation of a reflected (scattered)
wavefield, recorded at the surface at nonzero offset \cite[]{norm}.
For reflected waves, the Kirchhoff approximation is accurate up to the
two orders in the high-frequency series (the ray series) for the
differential operator applied to the solution, with the first order
describing the phase function alone and the second order describing
the amplitude.  We prove that both orders of accuracy are satisfied
when the offset continuation equation is applied to Kirchhoff data.
Thus, this differential equation is the ``right'' equation to two
orders, producing the correct amplitude as well as the correct phase
for offset continuation.  That is, the geometric spreading effects and
curvature effects of the reflected data are properly transformed.  The
angularly dependent reflection coefficient of the original offset is
preserved.  \par This proof relates the OC equation with
``wave-equation'' processing.  It also provides additional
confirmation of the fact that the true-amplitude OC and DMO operators
\cite[]{GEO58-01-00470066,joint,santos,GEO63-02-05570573} do not
depend on the reflector curvature and can properly transform
reflections from arbitrarily shaped reflectors
\cite[]{Goldin.sep.67.171,GEO61-03-07590775,cwp}.  The latter result
was specifically a 2.5-D result, whereas the result of this paper does
not depend on the 2.5-D assumption.  That is, the result presented
here remains valid when the reflector has out-of-plane variation.
\par Our method of proof is indirect.  We first write the Kirchhoff
representation for the reflected wave in a form that can be easily
matched to the solution of the OC differential equation.  We then
present the analogues of the eikonal and transport equations for the
OC equation and show that the amplitude and phase of the Kirchhoff
representation satisfy those two equations.

\section{THE KIRCHHOFF MODELING APPROXIMATION}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we introduce the Kirchhoff approximate integral
representation of the upward propagating response to a single
reflector, with separated source and receiver points.  We then show
how the amplitude of this integrand is related to the zero-offset
amplitude at the source receiver point on the ray, making equal angles
at the scattering point with the rays from the separated source and
receiver.  The Kirchhoff integral representation \cite[]{haddon,norm}
describes the wavefield scattered from a single reflector. This
representation is applicable in situations where the high-frequency
assumption is valid (the wavelength is smaller than the characteristic
dimensions of the model) and corresponds in accuracy to the WKBJ
approximation for reflected waves, including phase shifts through
buried foci.  The general form of the Kirchhoff modeling integral is
\begin{equation}
U_S({\bf r,s, \omega}) = \int_\Sigma\,R ({\bf x;r,s})\,
{\partial \over {\partial n}}\,\left[ 
U_I({\bf s, x ,\omega})\,G({\bf x,r,\omega})\right]\,
d\Sigma \; ,
\label{eqn:KI}
\end{equation}  
where ${\bf s} = (s,0,0)$ and ${\bf r} = (r,0,0)$ stand for the source
and the receiver location vectors at the surface of observation; ${\bf
x}$ denotes a point on the reflector surface $\Sigma$; $R $ is the
reflection coefficient at $\Sigma$; $n$ is the upward normal to the
reflector at the point ${\bf x}$; and $U_I$ and $G$ are the incident
wavefield and Green's function, respectively represented by their WKBJ
approximation,
\begin{equation}
U_I({\bf s,x,\omega}) = F(\omega)\,
A_s({\bf s,x})\,e^{i\omega\,\tau_s({\bf s,x})}\;,
\label{eqn:ui}
\end{equation}  
\begin{equation}
G({\bf x,r,\omega}) = A_r({\bf x,r})\,e^{i\omega\,\tau_r({\bf x,r})}\;.
\label{eqn:g}
\end{equation}
In this equation,
$\tau_s({\bf s,x})$ and $A_s({\bf s,x})$ are the traveltime and the
amplitude of the wave propagating from ${\bf s}$ to ${\bf x}$;
$\tau_r({\bf x,r})$ and $A_r({\bf x,r})$ are the corresponding
quantities for the wave propagating from ${\bf x}$ to ${\bf r}$; and
$F(\omega)$ is the spectrum of the input signal, assumed to be
the transform of a bandlimited impulsive source.  
In the time domain,
the Kirchhoff modeling integral transforms to
\begin{equation}
u_S({\bf r,s},t) = 
\int_\Sigma\,R ({\bf x;r,s})\,{\partial \over {\partial n}}
\left[A_s({\bf s,x})\,A_r({\bf x,r})\,
f\left(t- \tau_s({\bf s,x}) - \tau_r({\bf x,r})\right)\right]\,
d{\bf x}\;,
\label{eqn:KIT}
\end{equation} 
with $f$ denoting the inverse temporal transform of $F$.  The
reflection traveltime $\tau_{sr}$ corresponds physically to the
diffraction from a point diffractor located at the point ${\bf x}$ on
the surface $\Sigma$, and the amplitudes $A_s$ and $A_r$ are point
diffractor amplitudes, as well.
\par
The main goal of this paper is to test the compliance of
representation (\ref{eqn:KIT}) with the offset continuation differential
equation.  The OC equation contains the derivatives of the wavefield
with respect to the parameters of observation (${\bf s, r}$, and
$t$). According to the rules of classic calculus, these derivatives
can be taken under the sign of integration in formula
(\ref{eqn:KIT}). Furthermore, since we do not assume that the true-amplitude
OC operator affects the reflection coefficient $R $, the
offset-dependence of this coefficient is outside the scope of
consideration. Therefore, the only term to be considered as a trial
solution to the OC equation is the kernel of the Kirchhoff integral,
which is contained in the square brackets in equations (\ref{eqn:KI}) and
(\ref{eqn:KIT}) and has the form
\begin{equation}
k({\bf s,r,x},t) = A_{sr}({\bf s,r,x})\,
f\left(t - \tau_{sr}({\bf s,r,x})\right)\;,
\label{eqn:kernel}
\end{equation}
where
\begin{equation}
\tau_{sr}({\bf s,r,x}) =  \tau_s({\bf s,x}) + \tau_r({\bf x,r})\;,
\label{eqn:time}
\end{equation} 
\begin{equation}
A_{sr}({\bf s,r,x}) =  A_s({\bf s,x})\,A_r({\bf x,r})\;.
\label{eqn:amplitude}
\end{equation}
\par
In a 3-D medium with a constant velocity $v$, the traveltimes and
amplitudes have the simple explicit expressions
\begin{equation}
\tau_s({\bf s,x}) = {{\rho_s({\bf s,x})} \over v}\;,\;\;
\tau_r({\bf x,r}) = {{\rho_r({\bf x,r})} \over v}\;,
\label{eqn:taur}
\end{equation}
\begin{equation}
A_s({\bf s,x}) = {1 \over {4 \pi\,\rho_s({\bf s,x})}}\;,\;\;
A_r({\bf x,r}) = {1 \over {4 \pi\,\rho_r({\bf x,r})}}\;,
\label{eqn:ampr}
\end{equation}
where $\rho_s$ and $\rho_r$ are the lengths of the incident and reflected
rays, respectively (Figure \ref{fig:cwpgen}). If the reflector surface
$\Sigma$ is explicitly defined by some function $z=z({\bf x})$, then
\begin{equation}
\rho_s({\bf s,x}) = \sqrt{({\bf x-s})^2 + z^2({\bf x})}\;,\;\; 
\rho_r({\bf x,r}) = \sqrt{({\bf r-x})^2 + z^2({\bf x})}\;.
\label{eqn:lr}
\end{equation}

\inputdir{XFig}

\plot{cwpgen}{width=\textwidth}{Geometry of diffraction in a
constant velocity medium: view in the reflection plane.}

\par We then introduce a particular zero-offset amplitude, namely the
amplitude along the zero offset ray that bisects the angle between the
incident and reflected ray in this plane, as shown in Figure
\ref{fig:cwpgen}.  We denote the square of this amplitude as $A_0$.
That is,
\begin{equation}
A_0 = {1 \over {(4 \pi \rho_0 )^2}}\;.
\label{eqn:azero}
\end{equation}
$A_0$ is the amplitude factor that appears in the Kirchhoff integral
set up for a zero-offset reflection along the ray $\rho_0$. It is,
thus, the desired output factor inside the Kirchhoff integral after
DMO. As follows from formulas (\ref{eqn:amplitude}) and
(\ref{eqn:ampr}), the amplitude transformation in DMO (continuation to
zero offset) is characterized by the dimensionless ratio
\begin{equation}
{A_{sr} \over A_0} = {{\rho_0^2}\over {\rho_s\,\rho_r}}\;,
\label{eqn:ratio}
\end{equation}
where $\rho_0$ is the length of the zero-offset ray (Figure \ref{fig:cwpgen}).
\par
As follows from the simple trigonometry of the triangles, formed by
the incident and reflected rays (the law of cosines),
\begin{eqnarray}