paper.tex

国外免费地震资料处理软件包

and at far offset.
You can see that reciprocal seismograms usually have the same polarity,
and often have nearly equal amplitudes.
(The figure shown is the best of three such figures I prepared).
\par
Each constant time slice in Figure~\ref{fig:recipslice}
shows the reciprocity of many seismogram pairs.
\plot{recipslice}{width=6in,height=2.5in}{
	Constant time slices after NMO at 1 second and 2.5 seconds.
	}
Midpoint runs horizontally over the same range as in Figure~\ref{fig:toldi}.
Offset is vertical.
Data is not recorded near the vibrators
leaving a gap in the middle.
To minimize irrelevant variations,
moveout correction was done before making the time slices.
(There is a missing source that shows up on the left side of the figure).
A movie of panels like Figure~\ref{fig:recipslice} shows that
the bilateral symmetry you see in the individual panels
is characteristic of all times.
On these slices, you notice that the long wavelengths
have the expected bilateral symmetry
whereas the short wavelengths do not.

\par
In the laboratory,
reciprocity can be established
to within the accuracy of measurement.
This can be excellent.
(See White's example in FGDP).
In the field,
the validity of reciprocity will be dependent on the degree
that the required conditions are fulfilled.
A marine air gun should be reciprocal to a hydrophone.
A land-surface weight-drop source should be reciprocal to a vertical geophone.
But a buried explosive shot need not be reciprocal to
a surface vertical geophone
because the radiation patterns are different
and the positions are slightly different.
Under varying field conditions Fenati and Rocca found
that small positioning errors
in the placement of sources and receivers
can easily create discrepancies
much larger than the apparent reciprocity discrepancy.

\par
Geometrical complexity within the earth
does not diminish the applicability of the principle of linearity.
Likewise,
geometrical complexity does not reduce the applicability of reciprocity.
Reciprocity does not apply to sound waves in the presence of \bx{wind}.
Sound goes slower upwind than downwind.
But this effect of wind is much less than
the mundane irregularities of field work.
Just the weakening of echoes with time leaves noises that are not reciprocal.
Henceforth we will presume that reciprocity is generally applicable
to the analysis of reflection seismic data.

\section{SURVEY SINKING WITH THE DSR EQUATION}
\par
Exploding-reflector imaging will be replaced
by a broader imaging concept, 
{\em \bx{survey sinking}.}
A new equation called the
double-square-root (DSR) equation will be developed
to implement survey-sinking imaging. 
The function of the \bx{DSR equation}
is to downward continue an entire seismic survey,
not just the geophones but also the shots.
Peek ahead at equation (\ref{eqn:3p9})
and you will see an equation with two square roots.
One represents the cosine of the wave
{\em  arrival}
angle.
The other represents the
{\em  takeoff}
angle at the shot.
One cosine is expressed in terms of  $k_g$, the Fourier component
along the geophone axis of the data volume in $(s,g,t)$-space.
The other cosine, with  $k_s$, is the Fourier component
along the shot axis.
%\par
%Our field seismograms lie in the $(s,g)$-plane.
%To move onto
%the $(y,h)$-plane inhabited by seismic interpreters
%requires only a simple rotation.
%The data could be Fourier transformed with 
%respect to  $y$  and  $h$, for example.
%Then downward continuation would proceed
%with equation (\ref{eqn:3p17}) instead of equation (\ref{eqn:3p9}).
\subsection{The survey-sinking concept}
\par
The exploding-reflector concept has great utility because it
enables us to associate the seismic waves
observed at zero offset in many experiments
(say 1000 shot points) with the wave of a single thought experiment,
the exploding-reflector experiment.
The exploding-reflector analogy has a
few tolerable limitations connected with lateral velocity
variations and multiple reflections,
and one major limitation:
it gives us no clue as to
how to migrate data recorded at nonzero offset.
A broader imaging concept is needed.
\par
Start from field data where a 
survey line has been run along the  $x$-axis.
Assume there has been an infinite number of experiments,
a single experiment consisting of placing  a point
source or shot at  $s$  on the $x$-axis and 
recording echoes with geophones
at each possible location  $g$  on the $x$-axis.
So the observed data is an upcoming wave that is a two-dimensional
function of  $s$  and  $g$,  say  $P(s,g,t)$.
\par
Previous chapters have shown how to downward continue the
{\em  upcoming}
wave.
Downward continuation of the upcoming wave is really the
same thing as downward continuation of the geophones.
It is irrelevant for the continuation procedures where the
wave originates.
It could begin from an exploding reflector,
or it could begin at the surface, go down, and then be reflected back upward.
\par
To apply the imaging concept of survey sinking,
it is necessary to downward continue the sources as well as the geophones.
We already know how to downward continue geophones.
Since reciprocity permits interchanging geophones with shots,
we really know how to downward continue shots too.
\par
Shots and geophones may be downward continued to different levels,
and they may be at different levels during the process,
but for the final result they are only required to be at the same level.
That is, taking  $ z_s $  to be the depth of the shots
and  $ z_g $  to be the depth of the
geophones, the downward-continued survey will be required at all
levels  $z = z_s = z_g $.  
\par
The image of a reflector at  $(x,z)$  is
defined to be the strength and
polarity of the echo seen by the
closest possible source-geophone pair.
Taking the mathematical limit, this 
closest pair is a source and geophone located
together on the reflector.
The travel time for the echo is zero.
This survey-sinking concept of imaging is summarized by
\begin{equation}
\hbox{Image} (x,z)   \ \ \  = \ \ \ \hbox{Wave} (s=x,g=x,z,t=0)
\label{eqn:3p1}
\end{equation}
For good quality data, i.e. data that fits the
assumptions of the downward-continuation method,
energy should migrate to zero offset at zero travel time.
Study of the energy that doesn't do so should enable improvement of the model.
Model improvement usually amounts to improving the
spatial distribution of velocity.

\subsection{Survey sinking with the double-square-root equation}
\par
An equation was derived for paraxial waves.
The assumption of a
{\em  single}
plane wave means that the arrival time
of the wave is given by a single-valued  $t(x,z)$.
On a plane of constant  $z$, such as the earth's surface,
Snell's parameter  $p$  is measurable.
It is
\begin{equation}
{ \partial t   \over  \partial x } \  \  \eq \ 
{ \sin \, \theta   \over v }\  \eq \  p
\label{eqn:3p2a}
\end{equation}
In a borehole there is the constraint that measurements 
must be made
at a constant  $x$,  where the relevant measurement from an
{\em  upcoming}
wave would be
\begin{equation}
{ \partial t   \over  \partial z } \   \  \eq \ 
-\ { \cos \, \theta   \over v }\ \ \ \ = \quad
- \  \sqrt{ {1 \over  v^2 } \ -\ 
\left( {\partial t  \over \partial x} \  \right)^2 \ } \ 
\label{eqn:3p2b}
\end{equation}
Recall the time-shifting partial-differential equation and its
solution  $U$  as some arbitrary functional form  $f$:
\begin{eqnarray}
{ \partial U   \over  \partial z } \ \  \ \ &=&\ \ \ \  - \ 
{ \partial t   \over  \partial z } \ 
{ \partial U   \over  \partial t }
\label{eqn:3p3a}
\\
U \ \ \ \ &=&\ \ \ \ 
f \left( \ t \ -\  \int_0^z \  {\partial t  \over \partial z} \  dz \right)
\label{eqn:3p3b}
\end{eqnarray}
The partial derivatives
in equation (\ref{eqn:3p3a}) are taken to be at constant $x$,
just as is equation (\ref{eqn:3p2b}).
After inserting (\ref{eqn:3p2b}) into (\ref{eqn:3p3a}) we have
\begin{equation}
{ \partial U   \over  \partial z } \quad = \quad \sqrt{ {1 \over  v^2 } \ -\ 
\left( {\partial t  \over \partial x} \  \right)^2
 \ }\  { \partial U  
\over  \partial t }
\label{eqn:3p4a}
\end{equation}
Fourier transforming the wavefield over  $(x,t)$,   we
replace  $ \partial / \partial t $  by  $ -\,i \omega $.
Likewise, for the traveling wave
of the Fourier kernel  $ \exp (-\,i \omega t \ +\  ik_x x )$,
constant phase means that  ${\partial t}/{\partial x} \,=\, k_x / \omega $.
With this, (\ref{eqn:3p4a}) becomes
\begin{equation}
{ \partial U   \over  \partial z } \  \eq \  - \, i \omega  \  
\sqrt{
{1 \over  v^2 } \ -\  { k_x^2   \over \omega^2} \  } \  U
\label{eqn:3p4b}
\end{equation}
The solutions to (\ref{eqn:3p4b}) agree with those to the scalar wave equation
unless  $v$  is a function of  $z$,  in which case
the scalar wave equation has both upcoming and downgoing solutions,
whereas (\ref{eqn:3p4b}) has only upcoming solutions.
We
go into the lateral space
domain by replacing  $ i k_x $  by  $ \partial / \partial x $.
The resulting equation is useful for superpositions of many local plane waves
and for lateral velocity variations  $v(x)$.
\subsection{The DSR equation in shot-geophone space}
\par
Let the geophones descend a distance  $ dz_g $  into the earth.
The change of the travel time of the observed upcoming wave will be  
\begin{equation}
{{\partial t}\   \over {\partial z}_g} \  \eq \ 
- \   \sqrt{\  {1 \over  v^2 } \ -\ 
\left( {\partial t  \over \partial g} \, \right)^2 \  }
\label{eqn:3p5a}
\end{equation}
Suppose the shots had been let off at depth  $ dz_s $  instead of at  $z = 0$.
Likewise then,
\begin{equation}
{{\partial t}\   \over {\partial z}_s} \  \eq \ 
- \  \sqrt{\  {1 \over  v^2 } \ -\ 
\left( {\partial t  \over \partial s} \, \right)^2 \  }
\label{eqn:3p5b}
\end{equation}
Both (\ref{eqn:3p5a}) and (\ref{eqn:3p5b})
require minus signs because the travel time
decreases as either geophones or shots move down.
\par
Simultaneously downward project both the shots and
geophones by an identical vertical amount  $dz=dz_g = dz_s$.
The travel-time change is the sum
of (\ref{eqn:3p5a}) and (\ref{eqn:3p5b}), namely,
\begin{equation}
dt \  \eq \ 
{{\partial t}\ \over {\partial z}_g } \  dz_g \ +\  
{ {\partial t} \   \over {\partial z}_s } \  dz_s
\  \eq \ 
\left( {{\partial t}\   \over  {\partial z}_g } \ +\  { {\partial t} \ 
\over  {\partial z}_s} \right) \  dz \ \ \ \ \ \ \ 
\label{eqn:3p6}
\end{equation}
or
\begin{equation}
{\partial t  \over \partial z}  \  \eq \ 
- \  \left( \  \sqrt{ {1 \over  v^2 } \ -\ 
\left( {\partial t  \over \partial g} \, \right)^2 \  } \ +\ 
\sqrt{ {1 \over  v^2 } \ -\ 
\  {\left( {\partial t  \over \partial s} \, \right)^2} 
\  } \  \right)
\label{eqn:3p7}
\end{equation}
This expression for  ${\partial t}/{\partial z} $  may be substituted
into equation (\ref{eqn:3p3a}):
\begin{equation}
{ \partial U   \over  \partial z } \ \  =\ \ \ \  + \  \left(\   \sqrt{
{1 \over  v^2 } \ -\  \left( {\partial t  \over \partial g} \, \right)^2
\  } \ +\  \sqrt{ {1
\over  v^2 } \ -\  \left( {\partial t  \over \partial s} \, \right)^2 \ 
} \  \right) \  { \partial U   \over  \partial t }
\label{eqn:3p8}
\end{equation}
\par
Three-dimensional Fourier transformation converts
upcoming wave data  $u(t,s,g)$  to  $U( \omega , k_s , k_g )$.
Expressing equation (\ref{eqn:3p8}) in Fourier space gives
\begin{equation}
{ \partial U   \over  \partial z } \eq -\,i\,\omega \  \left[ \  
\sqrt{ {1 \over  v^2 } \ -\  \left( { k_g 
\over \omega }\, \right)^2 \  } \ +\  \sqrt{{1 \over  v^2 } \ -\  \left(
{ k_s   \over \omega }\, \right)^2 \  } \ \right] \  U
\label{eqn:3p9}
\end{equation}
Recall the origin of the two square roots in equation (\ref{eqn:3p9}).
One is the cosine of the arrival angle at the geophones
divided by the velocity at the geophones.
The other is the cosine of the takeoff angle at the shots
divided by the velocity at the shots.
With the wisdom of previous chapters
we know how to go into the lateral space domain by
replacing  $i k_g$  by  $\partial / \partial g$  and
$i k_s$  by  $\partial / \partial s$.
To incorporate lateral velocity variation  $v(x)$,
the velocity at the shot location must be distinguished
from the velocity at the geophone location.
Thus,
\par
\boxit{
	\begin{equation}
	{ \partial U   \over  \partial z }  \eq  
	\left[ \   \sqrt{\   \left( \