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	{-\,i \omega\   \over v (g)} \, \right)^2 -\ 
	{\partial^2 \   \over \partial g^2} \, } \ +\  
	 \sqrt{\  \left( {-\,i \omega\   \over v (s)} \, \right)^2 -\ 
	{\partial^2 \   \over \partial s^2} \, } \  \right] \  U
	\label{eqn:3p10}
	\end{equation}
	}

\par
Equation (\ref{eqn:3p10}) is known as
the double-square-root (DSR) equation in shot-geophone space.
It might be more descriptive to call it the survey-sinking equation
since it pushes geophones and shots downward together.
Recalling the section on splitting and full separation
we realize that the two square-root operators are commutative
($v(s)$  commutes with  $ \partial / \partial g $),
so it is completely equivalent
to downward continue shots and geophones separately or together.
This equation will produce waves for the rays
that are found on zero-offset sections
but are absent
from the exploding-reflector model.
\subsection{The DSR equation in midpoint-offset space}
\par
By converting the DSR equation to midpoint-offset space
we will be able to identify the familiar zero-offset migration part
along with corrections for offset.
The transformation between  $(g,s)$  recording parameters
and  $(y,h)$  interpretation parameters is
\begin{eqnarray}
y \ \ \ \ &=&\ \ \ \  { g \ +\  s   \over 2 }
\label{eqn:3p11a}
\\
h \ \ \ \ &=&\ \ \ \  { g \ -\  s   \over 2 }
\label{eqn:3p11b}
\end{eqnarray}
Travel time  $t$  may be parameterized in $(g,s)$-space or $(y,h)$-space.
Differential relations for this 
conversion are given by the chain rule for derivatives:
\begin{eqnarray}
{\partial t  \over \partial g} \ \ \ \ &=&\ \ \ \ 
{\partial t  \over \partial y} \  {\partial y  \over \partial g} \ +\ 
{\partial t  \over \partial h} \  {\partial h   \over \partial g} \  \eq \ 
{1 \over 2 }\   \left( {\partial t  \over \partial y} \ +\ 
{\partial t  \over \partial h } \, \right)
\label{eqn:3p12a}
\\
{\partial t  \over \partial s} \ \ \ \ &=&\ \ \ \ 
{\partial t  \over \partial y} \  {\partial y  \over \partial s} \ +\ 
{\partial t  \over \partial h} \  {\partial h   \over \partial s} \  \eq \ 
{1 \over 2 }\   \left( {\partial t  \over \partial y} \ -\ 
{\partial t  \over \partial h } \, \right)
\label{eqn:3p12b}
\end{eqnarray}
\par
Having seen how stepouts transform from shot-geophone space
to midpoint-offset space,
let us next see that spatial frequencies transform in much the same way.
Clearly, data could be transformed from $(s,g)$-space
to $(y,h)$-space with (\ref{eqn:3p11a}) and (\ref{eqn:3p11b})
and then Fourier transformed to $ ( k_y , k_h ) $-space.
The question is then,
what form would the double-square-root equation (\ref{eqn:3p9})
take in terms of the spatial frequencies  $ ( k_y , k_h ) $?
Define the seismic data field in either coordinate system as
\begin{equation}
U ( s, g )\  \eq \ U'  ( y , h )
\label{eqn:3p13}
\end{equation}
This introduces a new mathematical function  $ U'  $  with the same
physical meaning as  $U$  but,
like a computer subroutine or function call,
with a different subscript look-up procedure
for  $(y,h)$  than for  $(s,g)$.
Applying the chain rule for partial differentiation to (\ref{eqn:3p13}) gives
\begin{eqnarray}
{ \partial U \over \partial s} \ \ \ \ &=&\ \ \ \ 
{ \partial y \over \partial s} \  { \partial U'    \over  \partial y \ } \ +\ 
{ \partial h \over \partial s} \  { \partial U'    \over  \partial h \ }
\label{eqn:3p14a}
\\
{ \partial U \over \partial g}\ \ \ \ &=&\ \ \ \ 
{ \partial y \over \partial g} \  { \partial U'    \over  \partial y \ }\ +\ 
{ \partial h \over \partial g}\  { \partial U'    \over  \partial h \ }
\label{eqn:3p14b}
\end{eqnarray}
and utilizing (\ref{eqn:3p11a}) and (\ref{eqn:3p11b}) gives
\begin{eqnarray}
{ \partial U   \over  \partial s }\ \ \ \ &=&\ \ \ \ 
{1 \over 2 }\ \left( { \partial U'    \over  \partial y \, }\ -\ 
{ \partial U'    \over  \partial h \, } \, \right)
\label{eqn:3p15a}
\\
{ \partial U   \over  \partial g }\ \ \ \ &=&\ \ \ \ 
{1 \over 2 }\  \left( { \partial U'    \over  \partial y \, }\ +\ 
{ \partial U'    \over  \partial h \, } \, \right)
\label{eqn:3p15b}
\end{eqnarray}
In Fourier transform space
where  $ \partial / \partial x $  transforms to  $ i k_x $,
equations (\ref{eqn:3p15a}) and (\ref{eqn:3p15b}),
when  $i$  and $U =  U'  $  are cancelled, become
\begin{eqnarray}
k_s\ \ \ \ &=&\ \ \ \ {1 \over 2 }\ ( k_y\ -\ k_h )
\label{eqn:3p16a}
\\
k_g\ \ \ \ &=&\ \ \ \ {1 \over 2 }\  ( k_y\ +\ k_h )
\label{eqn:3p16b}
\end{eqnarray}
Equations~(\ref{eqn:3p16a})
and (\ref{eqn:3p16b})
are Fourier representations of (\ref{eqn:3p15a}) and (\ref{eqn:3p15b}).
Substituting (\ref{eqn:3p16a}) and (\ref{eqn:3p16b})
into (\ref{eqn:3p9}) achieves the main purpose of this section,
which is to get the double-square-root migration equation
into midpoint-offset coordinates:
\begin{equation}
{\partial\   \over \partial z} \ U\ \ =\ \  -\,i \, 
{\omega \over v }\  \left[ \  \sqrt{\ 
1\,-\,\left( { v k_y \,+\, v k_h   \over  2\,\omega } \, \right)^2
\  } + 
\sqrt{\ 
1\,-\,\left( { v k_y \,-\, v k_h   \over  2\,\omega } \, \right)^2
\  } \  \right] \ U
\label{eqn:3p17}
\end{equation}
\par
Equation (\ref{eqn:3p17}) is the takeoff point
for many kinds of common-midpoint seismogram analyses.
Some convenient definitions that simplify its appearance are
\begin{eqnarray}
G\ \ \ \ &=&\ \ \ \  { v\ k_g   \over \omega }
\label{eqn:3p18a}
\\
S\ \ \ \ &=&\ \ \ \ { v\ k_s   \over \omega }
\label{eqn:3p18b}
\\
Y\ \ \ \ &=&\ \ \ \ { v\ k_y   \over  2\ \omega }
\label{eqn:3p18c}
\\
H\ \ \ \ &=&\ \ \ \ { v\ k_h   \over  2\ \omega }
\label{eqn:3p18d}
\end{eqnarray}
%Chapter \ref{omk/paper:omk} showed that the quantity  $v \, k_x / \omega $  can
%be interpreted as the angle of a wave.
The new definitions  $S$  and  $G$  are the sines
of the takeoff angle and of the arrival angle of a ray.
When these sines are at their limits of  $ \pm 1 $  they refer
to the steepest possible slopes in $(s,t)$- or $(g,t)$-space.
Likewise, $Y$ may be interpreted as the dip of the data as seen
on a seismic section.
The quantity  $H$  refers to stepout observed on a common-midpoint gather.
With these definitions (\ref{eqn:3p17}) becomes slightly less cluttered:
\begin{equation}
\begin{tabular}{|c|}  \hline
 \\  $\displaystyle {\strut\partial\over\partial z}U
      \ =\ -\displaystyle {\strut i\omega\over v} 
                    \left(\sqrt{1-(Y+H)^2} +
 \sqrt{1-(Y-H)^2} \ \right) U$  \\
 \\    \hline
\end{tabular}
\label{eqn:3p19}
\end{equation}

%\par
%Most present-day before-stack migration procedures
%can be interpreted through
%equation (\ref{eqn:3p19}).
%Further analysis of it should explain
%the limitations of conventional processing procedures
%as well as suggest improvements in the procedures.

\begin{exer}

\item
Adapt equation (\ref{eqn:3p17}) to allow for a difference in velocity
between the shot and the geophone.
\item
Adapt equation (\ref{eqn:3p17}) to allow for downgoing pressure waves
and upcoming shear waves.
\end{exer}

\section{THE MEANING OF THE DSR EQUATION}
\par
The double-square-root equation
%contains most nonstatistical aspects of seismic data
%processing for petroleum prospecting.
%This equation, which was derived in the previous section,
is not easy to understand because it is an operator in a
four-dimensional space, namely,  $(z,s,g,t)$.
We will approach it through various applications, each of which is like a
picture in a space of lower dimension.
In this section lateral velocity variation will be neglected
(things are bad enough already!).
%Begin with
%\begin{eqnarray}
%{dU \over dz }\ \  &=&\ \  { -\,i \omega  \over v }\  \left( \  \sqrt
%{ 1\ -\ G^2 } \ \ +\ \  \sqrt { 1\ -\ S^2  } \  \right) \ U
%\label{eqn:4p1a}
%\\
%{dU \over dz }\ \  &=&\ \  { -\,i \omega   \over v }\   \left( \   \sqrt
%{1\ -\ (Y+H)^2 } \ \ +\ \  \sqrt { 1\ -\ (Y-H)^2 }\  \right) \ U
%\label{eqn:4p1b}
%\end{eqnarray}
%\subsection{Zero-offset migration (H = 0)}
\par
One way to reduce the dimensionality of (\ref{eqn:3p10})
is simply to set  $H \,=\, 0$.
Then the two square roots become the same, so that they can be
combined to give the familiar paraxial equation:
\begin{equation}
{dU \over dz }\  \eq 
{ -\,i  \omega } \ {2 \over v }\ \sqrt  { 1 \ -\ 
\  { v^2 \, k_y^2   \over  4 \, \omega^2 } } \ \  U
\label{eqn:4p2}
\end{equation}
In both places in equation (\ref{eqn:4p2}) where the rock velocity occurs,
the rock velocity is divided by 2.
Recall that the rock velocity needed to be halved in order for field
data to correspond to the exploding-reflector model.
So whatever we did by setting  $H \,=\, 0$,
gave us the same migration equation we used in chapter \ref{dwnc/paper:dwnc}.
Setting  $H\,=\,0$  had the effect of making the survey-sinking concept
functionally equivalent to the exploding-reflector concept.

\subsection{Zero-dip stacking (Y = 0)}
\inputdir{.}
\par
When dealing with the offset  $h$  it is common to assume that
the earth is horizontally layered so that experimental results will be
independent of the midpoint  $y$.
With such an earth the Fourier transform of all data over  $y$  will
vanish except for  $ k_y = 0 $,
or, in other words, for  $Y  =  0$.
The two square roots in (\ref{eqn:3p10})
again become identical,
and the resulting equation is once more the paraxial equation:
\begin{equation}
{dU \over dz }\  \eq 
{ - \, i \omega } \ {2 \over v }\ \sqrt  { 1 \ -\ 
\  { v^2 \, k_h^2   \over  4\,\omega^2 } } \  \  U
\label{eqn:4p3}
\end{equation}
Using this equation to downward continue hyperboloids from the
earth's surface, we find the hyperboloids shrinking with depth, until the
correct depth where best focus occurs is reached.
This is shown in Figure~\ref{fig:dc2}.
\plot{dc2}{width=4in}{
	With an earth model of three layers,
	the common-midpoint gathers are three hyperboloids.
	Successive frames show downward continuation
	to successive depths where best focus occurs.
	}
\par
The waves focus best at zero offset.
The focus represents a downward-continued experiment,
in which the downward continuation has gone just to a reflector.
The reflection is strongest at zero travel time for
a coincident source-receiver pair just above the reflector.
Extracting the zero-offset value at  $t = 0$  and
abandoning the other offsets amounts
to the conventional procedure of summation along
a hyperbolic trajectory on the original data.
Naturally the summation can be expected to be best
when the velocity used for downward continuation
comes closest to the velocity of the earth.

\par
Actually, the seismic energy will not all go precisely to zero offset;
it goes to a focal region near zero offset.
A further analysis (not begun here) can analyze the focal region
to upgrade the velocity estimation.   Dissection of this focal region
can also provide information about reflection strength versus angle.


\subsection{Giving up on the DSR}
\par
The DSR operator
defined by (\ref{eqn:3p19})
is fun to think about,
but it doesn't really go
to any very popular place very easily.
There is a serious problem with it.
It is {\em not separable }
into a sum of an offset operator and a midpoint operator.
{\em  Nonseparable}
means that a Taylor series for (\ref{eqn:3p10})
contains terms like $ Y^2 \, H^2 $.
Such terms cannot be expressed as a function of  $Y$  plus a function of  $H$.
Nonseparability is a data-processing disaster.
It implies that migration and stacking must be done simultaneously,
not sequentially.
The only way to recover pure separability would be
to return to the space of  $S$  and  $G$.

\par
This chapter tells us that lateral velocity variation
is very important.
Where the velocity is known,
we have the DSR equation in shot-geophone
space to use for migration.
A popular test data set is called the Marmousi data set.
The DSR equation is particularly popular with it
because with synthetic data, the velocity really is known.
Estimating velocity $v(x,z)$ with
real data is a more difficult task,
one that is only crudely handled by
by methods in this book.
In fact, it is not easily done by the even best
of current industrial practice.