Linearized tomography relies on some measure of moveout errors.
Our goal is to create the best migrated image possible.
Therefore some measure of moveout errors in migrated common reflection
point (CRP) gathers
seems an ideal choice.
\par
\subsection{Wave equation angle gathers}
When doing velocity analysis, general practice is to measure moveout
from a relatively sparse set of CRP gathers.  
Kirchhoff depth migration is the preferred construction method
because it can produce the sparse set of CRP gathers without needing to image
the entire volume.
In addition, if our Green's function table is constructed correctly,
Kirchhoff methods do not
suffer from the velocity approximations needed by  frequency domain methods.
Kirchhoff methods also have some deficiencies. The most glaring weakness
of Kirchhoff methods is the difficulty in constructing the  Green's function
table. 
To construct an accurate  Green's function table we must account for,
and weight correctly, the multiple arrivals that occur in complex geology.
Calculating and accounting for multiple arrivals adds significantly to both
coding complexity and memory requirements. As a result
a single arrival is often all that is used.  Eikonal methods 
\cite{GEO55.05.05210526,GEO56.06.08120821,podvin.gji.91,Fomel.sep.95.sergey3}
can efficiently
produce first arrivals, but in areas of complex geology the first arrival
is not always the most important arrival \cite{GEO62-05-15331543}. 
\longcite{Nichols.sep.81}
proposed a band-limited method that
gave the maximum amplitude arrival, but the method is computationally
impractical in 3-D.  As a result, people usually go to  expensive ray based 
methods but still face the difficult tasks of choosing the most
important arrival and to correctly and efficiently interpolate the 
traveltime field \cite{Sava.sep.95.paul1}.
\par

The most computationally
attractive alternative to Kirchhoff methods is frequency 
domain downward continuation.
Downward continuation has its own weaknesses.
Its primary weakness is speed.  Downward continuation 
can not be target oriented, so the  full volume imaging is required.
In addition, frequency domains methods in their purest form 
can not handle lateral variations in velocity.
By migrating with multiple velocities and applying a space domain correction
to the wavefield
they can do a fairly good job handling lateral variations
(this migration is normally
referred to as PSPI, Phase-shift plus interpolation)\cite{SEG-1993-0986}. 
Finally, downward continuation focuses the wavefield towards zero offset,
making conventional moveout analysis impossible. 
\par
We can create CRP
gathers where moveout analysis is possible by changing our imaging condition
\cite{GEO46.11.15591567,SEG-1999-824}.
Given a wave-field $P$ as a function 
of frequency $\freq$, midpoint $\mv$,  and offset $x_h$
we  follow the normal procedure of downward continuing the data
and extracting the image at the surface $z=0$.  The conventional approach is to
then extract the values at zero time,
\beqa
P\left(\freq,\mv,\hx;z=0\right) &
\stackrel{\DSR}{\Longrightarrow}&
P\left(\freq,\mv,\hx;z\right) \\ \nonumber
P\left(\freq,\mv,\hx;z\right) &
\stackrel{Imaging}{\Longrightarrow} &
P\left(t=0,\mv,\hx;z\right) .
\eeqa
Instead we  will perform a slant stack
before extracting the image at zero time,
\beqa
P\left(\freq,\mv,\hx;z=0\right) &
\stackrel{\DSR}{\Longrightarrow} &
P\left(\freq,\mv,\hx;z\right) \nonumber \\
P\left(\freq,\mv,\hx;z\right) &
\stackrel{Slant\;stack}{\Longrightarrow} &
P\left(\tau,\mv,p\hx;z\right) \\
P\left(\tau,\mv,p\hx;z\right) &
\stackrel{Imaging}{\Longrightarrow} &
P\left(\tau=0,\mv,p\hx;z\right) \nonumber, 
\eeqa
producing a CRP gather with a ray parameter $p\hx$ axis.  
\par
The topic of this thesis is not migration, but tomography. The tomography 
method could be applied with either Kirchhoff or PSPI.  For me,
PSPI proved to be a more attractive choice.   A 2-D and 3-D PSPI algorithm
was already available through Biondi's Gendown package \cite{Biondi.sep.98.biondo}, where
a Kirchhoff approach would have required the coding of the migration algorithm
along with a suitable traveltime computation method.

\subsection{Characterizing moveout errors}
Tomography requires us to provide moveout errors.
Of course, it is  unreasonable to hand pick every
reflector at every CRP gather  in 2-D and inconceivable
in 3-D.
As a result, people have tried to find alternate methods to
pick moveout errors.
\longcite{SEG-1996-0735} used a neural network approach to pick CRP gathers
and many people have suggested seeding-based approaches
to pick the gathers.
Both approaches describe
complicated moveouts, but they suffer from cycle skipping
and have problems in areas where the S/N ratio is not very high.
An alternative approach is to characterize the moveout in 
CRP gathers by a single parameter  \cite{Etgen.sepphd.68,Biondi.sepphd.64}.  A
 single parameter
is a much more robust estimator. It requires less human involvement
(less picking  and/or QA is necessary) and is less sensitive to signal
to noise problems.
\par
At early iterations a single parameter is especially valuable.
All that can be resolved at early iterations is gross features.
A single parameter can capture these where picking the entire
CRP gather the inversion is likely to be overwhelmed small features that 
are not resolvable at early iterations. When we were close to the 
correct velocity allowing freedom in moveout behavior is desirable
and benificial.

%Unfortunately,
%single parameter  may not fully characterize the residual moveout.
%As we move closer to the correct velocity function, the residual moveout
%should become less complicated, and a single parameter will become
%more accurate. As  long as the parameter can handle the gross features
%at early iterations it should be okay. For the  simple 
%synthetic dataset, this isn't
%as much of a concern.
\par

For the tomography problem I will begin with
a migrated image  $d$ at a depth $z$, 
ray parameter $p$, at CRP location $x$.  
For estimating the residual moveout in the CRP gathers I will begin
by calculating semblance $s$ in terms of some curvature parameter $c$,
\beq
s(z,c,x)={\left[ \sum_{p} d(z+p c^2,p,x)\right]^2   \over
n(z,x) \sum_{j=0} d(z+p c^2,x)^2 } ,
\eeq
where $n(z,x)$ is the number non-zero samples summed over for each
semblance calculation.
Figure~\ref{fig:semb-mig0} shows a ray parameter CRP gather, overlaid with
the curves corresponding to the picked $c$ value for each
reflector. For this synthetic, the $c$ does a good job characterizing
the moveout.

\sideplot{semb-mig0}{width=3.0in,height=3.0in}
{A ray parameter CRP gather  at 8 km.  Overlaid are the projection of the
maximum $c$ values for each reflection. Note that the general
moveout is fairly well described by $c$.}



\par
For this small synthetic hand picking the semblance along each reflector
would not be too tedious, but in 3-D it would quickly become so. As a result,
I wanted to come up with a simple way for the computer to do most of the
work.
One option would be to just pick the maximum semblance at
each location, but we can get an unrealistic, high spatial frequency
behavior for  $c(x)$.
When doing convention semblance analysis we are confronted with a similar 
problem, that picking the maximum semblance at each time could result in
an unreasonable velocity function.
\longcite{Clapp.sep.97.bob1} proposed a method to avoid hand picking that
still led to a reasonable velocity model. We can adapt that work by
starting with
the maximum curvature value at each CRP ${\bf c_{max}}$,
the semblance at the maximum curvature value $\bold W$,
and a derivative operator $\bold D$.
We can find a smooth curvature function ${\bf c}$
by setting up a simple set of equations 
\beqa
\zero \pox \bold W( {\bf c_{max}}  - {\bf c})  \nonumber \\
\zero \pox \epsilon \bold D {\bf c}  .
\eeqa
By increasing $\epsilon$ we get smoother ${\bf c}$ values  while small
$\epsilon$ values honor more our maximum semblance picks.
Figure~\ref{fig:semb-mig0-ref} shows the semblance panel for each 
reflector with the smooth pick overlaid. As we move non-linear
iteration to non-linear iteration are picks more reliable, smoothing is
not as necessary, and we can reduce $\epsilon$.

\plot{semb-mig0-ref}{width=6.0in,height=3.0in}
{Semblance along  the six reflectors and the automatically
selected $c$ values. }



\subsection{Endpoints, edge effects,  and errors}
To set up our tomography problem I need to cover
some final details. First, our inversion is based on traveltime
errors, not residual moveout.  To convert between the two I
apply the approximation
\beqa
\Delta t &=& {\Delta z \over v(z,{\bf x}) } \\ \nonumber
\Delta t &=& { c  p_x^3 } \over v(z,{\bf x}) 
\eeqa
where $v(z,{\bf x})$ is the velocity at the reflection point.
\par
In constructing our raypaths we benefit from having our
CRP gathers in terms of ray parameter.  If errors were in
terms of offset we would have to either
\begin{itemize}
\item shoot rays from every source and receiver location, and find
ray pairs that obey Snell's law at the position on the reflector imaged
at the offset dictated by the ray pair.
\item or shoot rays from our CRP locations along where our reflector is
imaged at every offset and then interpolate the ray field
to our source and receiver locations. 
\end{itemize}
Both options require significant additional ray-tracing in 2-D, and
even more in 3-D. In addition, we are always faced with the tradeoff of
how much should we interpolate our rays versus how many additional rays
should we shoot.
\par
With our errors in terms of  ray parameter  we can 
shoot a single ray-pair up
from our imaging point at the angle given by:
\beq
{\partial t \over \partial h}=p\hx=\frac{2 \sin \theta \cos\phi}{V\left(z,\mv\right)} ,
\eeq
where $\theta$ is one-half
the aperture angle,  $\phi$ is the geologic dip, and $V\left(z,\mv\right)$ is 
the velocity at the CRP location (Figure~\ref{fig:sketch}).
If  the rays emerge at surface locations corresponding to an
offset and CMP location inside our acquisition geometry we have a valid ray pair.
\sideplot{sketch}{width=2.0in,height=2.0in}{How the takeoff angle for a ray-pair
 are defined.}
