paper.tex

国外免费地震资料处理软件包

\title{Theory of differential offset continuation}
\author{Sergey Fomel}

\maketitle

\begin{abstract}
 I introduce a partial differential equation to describe the
  process of prestack reflection data transformation in the offset,
  midpoint, and time coordinates. The equation is proved theoretically
  to provide correct kinematics and amplitudes on the transformed
  constant-offset sections. Solving an initial-value problem with the
  proposed equation leads to integral and frequency-domain offset
  continuation operators, which reduce to the known forms of dip
  moveout operators in the case of continuation to zero offset.  
\end{abstract}

\section{Introduction}
  
The Earth subsurface is three-dimensional, while seismic reflection
data from a multi-coverage acquisition belong to a five-dimensional
space (time, 2-D offset, and 2-D midpoint coordinates). This fact
alone indicates the additional connection that exists
in the data space. I show in this paper that it is possible, under
certain assumptions, to express this connection in a concise
mathematical form of a partial differential equation. The theoretical
analysis of this equation allows us to explain and predict the data
transformation between different offsets.
  
The partial differential equation, introduced in this
paper\footnote{To my knowledge, the first derivation of the revised
offset continuation equation was accomplished by Joseph Higginbotham
of Texaco in 1989.  Unfortunately, Higginbotham's derivation never
appeared in the open literature.}, describes the process of
\emph{offset continuation}, which is a transformation of common-offset
seismic gathers from one constant offset to another
\cite[]{GPR30-06-08130828}. \cite{GEO61-06-18461858} identified offset
continuation (OC) with a whole family of prestack continuation
operators, such as shot continuation \cite[]{SEG-1993-0673}, dip
moveout as a continuation to zero offset \cite[]{DMObook}, and
three-dimensional azimuth moveout \cite[]{GEO63-02-05740588}. An
intuitive introduction to the concept of offset continuation is
presented by \cite{TLE20-01-02100213}. A general data mapping
prospective is developed by \cite{GPR48-01-01350162}.
   
As early as in 1982, Bolondi et al.  came up with the idea of
describing offset continuation and dip moveout (DMO) as a continuous
process by means of a partial differential equation
\cite[]{GPR30-06-08130828}.  However, their approximate differential
operator, built on the results of Deregowski and Rocca's classic paper
\cite[]{GPR29-03-03740406}, failed in the cases of steep reflector dips
or large offsets.  \cite{Hale.sepphd.36} writes:
\begin{quote}
  The differences between this algorithm [DMO by Fourier transform]
  and previously published finite-difference DMO algorithms are
  analogous to the differen\-ces be\-t\-ween
  fre\-qu\-en\-cy-wave\-num\-ber
  \cite[]{GEO43-01-00230048,GEO43-07-13421351} and
  fi\-nite-dif\-fe\-rence \cite[]{Claerbout.fgdp.76} algorithms for
  migration. For example, just as finite-difference migration
  algorithms require approximations that break down at steep dips,
  finite-difference DMO algorithms are inaccurate for large offsets
  and steep dips, even for constant velocity.
\end{quote}
Continuing this analogy, we can observe that both finite-difference
and frequency-domain migration algorithms share a common origin: the
wave equation. The new OC equation, presented in this paper and valid
for all offsets and dips, plays a role analogous to that of the wave
equation for offset continuation and dip moveout algorithms. A
multitude of seismic migration algorithms emerged from the fundamental
wave-propagation theory that is embedded in the wave equation.
Likewise, the fundamentals of DMO algorithms can be traced to the OC
differential equation.

In the first part of the paper, I prove that the revised equation
is, under certain assumptions, kinematically valid. This means that
wavefronts of the offset continuation process correspond to the
reflection wave traveltimes and correctly transform between different
offsets.  Moreover, the wave amplitudes are also propagated correctly
according to the \emph{true-amplitude} criterion \cite[]{GEO58-01-00470066}. 

In the second part of the paper, I relate the offset continuation
equation to different methods of dip moveout. Considering DMO as a
continuation to zero offset, I show that DMO operators can be obtained
by solving a special initial value problem for the OC
equation.  Different known forms of DMO \cite[]{DMObook} appear as special
cases of more general offset continuation operators.

The companion paper \cite[]{GEO68-02-07330744} demonstrates a
practical application of differential offset continuation to seismic
data interpolation.

\section{Introducing the offset continuation equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Most of the contents of this paper refer to the following linear
partial differential equation:
\begin{equation}
h \, \left( {\partial^2 P \over \partial y^2} - {\partial^2 P \over \partial
h^2} \right) \, = \, t_n \, {\partial^2 P \over {\partial t_n \,
\partial h}} \,\,\, . 
\label{eqn:OCequation} 
\end{equation}
Equation~(\ref{eqn:OCequation}) describes an {\em artificial}
(non-physical) process of transforming reflection seismic data
$P(y,h,t_n)$ in the offset-midpoint-time domain. In
equation~(\ref{eqn:OCequation}), $h$ stands for the half-offset
($h=(r-s)/2$, where $s$ and $r$ are the source and the receiver
surface coordinates), $y$ is the midpoint ($y=(r+s)/2$), and $t_n$ is the time
coordinate after normal moveout correction is applied:
\begin{equation}
\label{eqn:tnmo}
t_n=\sqrt{t^2-{4 \, h^2 \over v^2}}\;.
\end{equation}
The velocity $v$ is assumed to be known a priori.
Equation~(\ref{eqn:OCequation}) belongs to the class of linear
hyperbolic equations, with the offset $h$ acting as a time-like
variable. It describes a wave-like propagation in the offset
direction.

\subsection{Proof of validity}

A simplified version of the ray method technique \cite[]{cerveny,babich}
can allow us to prove the theoretical validity of
equation~(\ref{eqn:OCequation}) for all offsets and reflector dips by
deriving two equations that describe separately wavefront (traveltime)
and amplitude transformation.  According to the formal ray theory,
the leading term of the high-frequency asymptotics for a reflected
wave recorded on a seismogram takes the form
\begin{equation}
   P\left(y,h,t_n\right) \approx
A_n(y,h)\,R_n\left(t_n-\tau_n(y,h)\right) \;,
\label{eqn:raymethod} 
\end{equation}  
where $A_n$ stands for the amplitude, $R_n$ is the wavelet shape of
the leading high-frequency term, and $\tau_n$ is the traveltime curve
after normal moveout. Inserting~(\ref{eqn:raymethod}) as a trial
solution for~(\ref{eqn:OCequation}), collecting terms that have the
same asymptotic order (correspond to the same-order derivatives of the
wavelet $R_n$), and neglecting low-order terms, we arrive at the set of
two first-order partial differential equations:
\begin{equation}
h \, \left[     {\left( \partial \tau_n \over \partial y \right)}^2 - 
                {\left( \partial \tau_n \over \partial h \right)}^2
     \right] = \, - \, \tau_n \, {\partial \tau_n \over \partial h} \,\,\,,  
\label{eqn:eikonal} 
\end{equation}   
\begin{equation}
\left( \tau_n - 2h \, {\partial \tau_n \over {\partial h}} \right)
\, {\partial A_n \over \partial h} + 2h {\partial \tau_n \over \partial
y}   {\partial A_n \over \partial y} + h A_n \left( {\partial^2 \tau_n
\over {\partial y^2}} - {\partial^2 \tau_n \over {\partial h^2}} 
\right) \, = \, 0 \,\,\,.
\label{eqn:transport} 
\end{equation}

Equation~(\ref{eqn:eikonal}) describes the transformation of
traveltime curve geometry in the OC process analogously to how the
eikonal equation describes the front propagation in the classic wave
theory.  What appear to be wavefronts of the wave motion described by
equation~(\ref{eqn:OCequation}) are traveltime curves of reflected
waves recorded on seismic sections.  The law of amplitude
transformation for high-frequency wave components related to those
wavefronts is given by equation~(\ref{eqn:transport}).  In terms of
the theory of partial differential equations,
equation~(\ref{eqn:eikonal}) is the characteristic equation
for~(\ref{eqn:OCequation}).

\subsection{Proof of kinematic equivalence}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In order to prove the validity of equation~(\ref{eqn:eikonal}), it is
convenient to transform it to the coordinates of the initial shot
gathers: $s=y-h$, $r=y+h$, and $\tau = \sqrt{\tau_n^2+{{4h^2} \over
    {v^2}}}$. The transformed equation takes the form
\begin{equation}
\left( \tau^2 + {{(r-s)^2} \over {v^2}} \right) \left( {\partial \tau
\over \partial r} -   {\partial \tau \over \partial s} \right) = 2 \, (r-s) \,
\tau \left( {1 \over {v^2}} - {\partial \tau \over \partial r}
{\partial \tau \over \partial s} \right) \,\,\,.
\label{eqn:SCeikonal} 
\end{equation}
Now the goal is to prove that any reflection traveltime function
$\tau(r,s)$ in a constant velocity medium satisfies 
equation~(\ref{eqn:SCeikonal}). 

Let $S$ and $R$ be the source and the receiver locations, and $O$ be a
reflection point for that pair.  Note that the incident ray $SO$ and
the reflected ray $OR$ form a triangle with the basis on the offset
$SR$ ($l=|SR|=|r-s|$).  Let $\alpha_1$ be the angle of $SO$ from the
vertical axis, and $\alpha_2$ be the analogous angle of $RO$ (Figure
\ref{fig:ocoray}). The law of sines gives us the following explicit
relationships between the sides and the angles of the triangle $SOR$:
\begin{eqnarray}
|SO|\,=\,|SR|\, {\cos{\alpha_2} \over
\sin{\left(\alpha_2-\alpha_1\right)}} \,\,\,, 
\label{eqn:triangle1} \\ 
|RO|\,=\,|SR|\, {\cos{\alpha_1} \over
\sin{\left(\alpha_2-\alpha_1\right)}} \,\,\,.
\label{eqn:triangle2} 
\end{eqnarray} 
Hence, the total length of the reflected ray satisfies
\begin{equation}
v \tau = |SO|+|RO|=|SR|\,  {{\cos{\alpha_1}+ \cos{\alpha_2}} \over
\sin{\left(\alpha_2-\alpha_1\right)}} = |r-s|\,{\cos{\alpha} \over
\sin{\gamma}} \,\,\,.
\label{eqn:length} 
\end{equation}
Here $\gamma$ is the reflection angle ($\gamma = (\alpha_2 -
\alpha_1)/2$), and $\alpha$ is the central ray angle ($\alpha =
(\alpha_2 + \alpha_1)/2$), which coincides with the local dip angle of
the reflector at the reflection point.  Recalling the well-known
relationships between the ray angles and the first-order traveltime
derivatives
\begin{eqnarray}
{{\partial \tau} \over {\partial s}} \,=\,{ {\sin{\alpha_1}} \over
{v}} \,\,\,,
\label{eqn:snell1}\\
{{\partial \tau} \over {\partial r}} \,=\, {{\sin{\alpha_2}} \over {v}}  
\label{eqn:snell2}\,\,\,,
\end{eqnarray}  
we can substitute~(\ref{eqn:length}),~(\ref{eqn:snell1}), and~(\ref{eqn:snell2}) into 
(\ref{eqn:SCeikonal}), which  leads to the simple trigonometric equality
\begin{equation}
\cos^2{\left( {\alpha_1 + \alpha_2} \over 2 \right)} +  
\sin^2{\left( {\alpha_1 - \alpha_2} \over 2 \right)}\, = \, 1 -
\sin{\alpha_1} \sin{\alpha_2} \,\,\,. 
\label{eqn:equality} 
\end{equation}
It is now easy to show that equality~(\ref{eqn:equality}) is true for any
$\alpha_1$ and $\alpha_2$, since
\[
\sin^2{a} - \sin^2{b} = \sin{(a+b)}\,\sin{(a-b)}\;.
\]

\inputdir{XFig}

\sideplot{ocoray}{height=2.5in}{Reflection rays in a constant
velocity medium (a scheme).}

Thus we have proved that equation
(\ref{eqn:SCeikonal}), equivalent to~(\ref{eqn:eikonal}), is valid in constant
velocity media independently of the reflector geometry and the offset.
This means that high-frequency asymptotic components of the waves,
described by the OC equation,
are located on the true reflection traveltime curves.  

The theory of characteristics can provide other ways to prove the
kinematic validity of equation~(\ref{eqn:eikonal}), as described by
\cite{me} and \cite{plag}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Comparison with Bolondi's OC equation}
Equation~(\ref{eqn:OCequation}) and the previously published OC
equation \cite[]{GPR30-06-08130828} differ only with respect to the
single term $\partial^2 P \over {\partial h^2}$. However, this
difference is substantial.  

From the offset continuation characteristic equation
(\ref{eqn:eikonal}), we can conclude that the first-order traveltime
derivative with respect to offset decreases with decreasing
offset. The derivative equals zero at the zero offset, as predicted by the
principle of reciprocity (the reflection traveltime has to be an {\em
  even} function of offset). Neglecting $\left({\partial \tau_n} \over
  {\partial h}\right)^2$ in (\ref{eqn:eikonal}) leads to the
characteristic equation
\begin{equation}
 h \,   {\left( \partial \tau_n \over \partial y \right)}^2  
 = \, - \, \tau_n \, {\partial \tau_n \over \partial h}\;, 
\label{eqn:ITeikonal} 
\end{equation}
which corresponds to the approximate OC equation of
\cite{GPR30-06-08130828}. The approximate equation has the form
\begin{equation}
h \, {\partial^2 P \over \partial y^2} \, = \, t_n \, {\partial^2 P
\over {\partial t_n \, \partial h}}\;.
\label{eqn:bolondi} 
\end{equation}
Comparing equations~(\ref{eqn:ITeikonal}) and (\ref{eqn:eikonal}), we
can note that approximation (\ref{eqn:ITeikonal}) is valid only if
\begin{equation}
{\left( \partial \tau_n \over \partial h \right)}^2 \, \ll\, {\left(
\partial \tau_n \over \partial y \right)}^2 \,\,\,. 
\label{eqn:condition} 
\end{equation}
To find the geometric constraints implied by inequality
(\ref{eqn:condition}), we can express the traveltime derivatives in
geometric terms. As follows from expressions (\ref{eqn:snell1}) and
(\ref{eqn:snell2}),
\begin{eqnarray}
\label{eqn:snells1}
{{\partial \tau} \over {\partial y}} & = & {{\partial \tau} \over
{\partial r}} + {{\partial \tau} \over {\partial s}} \,=\, { {2
\sin{\alpha} \cos{\gamma}} \over {v}}\;, \\
{{\partial \tau} \over
{\partial h}} & = & {{\partial \tau} \over {\partial r}} - {{\partial
\tau} \over {\partial s}} \,=\, { {2 \cos{\alpha} \sin{\gamma}} \over
{v}}\;.
\label{eqn:snells2}
\end{eqnarray}
Expression (\ref{eqn:length}) allows transforming