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 CRP
gathers
seems an ideal choice.

\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.
%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. 
In addition,  even if our Green's function traveltime table is constructed correctly,
%Kirchhoff methods do not
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 usually 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, general practice is to 
use expensive ray based 
methods. These methods still face the difficult tasks of choosing the most
important arrival and correctly and efficiently interpolating 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 
cannot be target oriented, so  full volume imaging is required.
In addition, wave number domain methods 
cannot 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
I can create CRP
gathers where moveout analysis is possible using
wave equation migration methods  by changing the imaging condition
\cite{GEO46.11.15591567,SEG-1999-824}.
Given a wave-field
we  follow the normal procedure of downward continuing the sources and
receivers
and extracting the image at zero time at our new recording surface.  Instead
of extracting the image at zero offset, we note that reflection
angle $\theta$ can be evaluated by the differential equation:
\begin{equation}
\tan \theta = - \frac{\partial z}{\partial \hx}
\end{equation}
where $z$ is the depth, $\hx$ is half-offset \cite{Sava.sep.103.paul2}.
\par
The topic of this thesis is not migration, but tomography. The tomography 
method could be applied with either Kirchhoff or downward continuation
based migration.  For me,
PSPI proved to be a more attractive choice.   A 2-D and 3-D PSPI algorithm
was already available, 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.
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  high.
An alternative approach is to characterize the moveout in 
CRP gathers by a single parameter  \cite{Etgen.sepphd.68,Trier.sepphd.66}.  
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 are gross features.
A single parameter can capture these where picking the entire
CRP gather is likely 
to cause the inversion to be overwhelmed small features that
are not resolvable at early iterations. When we are close to the
correct velocity allowing freedom in moveout behavior is desirable
and beneficial.
\par
For the tomography problem I will begin with
a migrated image  $d$ at a depth $z$, 
angle $\theta$, at CRP location $x$.  
For estimating the residual moveout in the CRP gathers 
by calculating semblance $s$ in terms of some curvature parameter \footnote{\longcite{Ottolini.sepphd.33} demonstrated that residual  moveout is not 
hyperbolic, even in constant velocity, in angle domain gathers.  
For the purpose of moveout analysis for tomography the hyperbolic 
approximation proved to be acceptable.} 
$\gamma$
\begin{equation}
s(z,\gamma,x)={\left[ \sum_{\theta} d(z+|\theta\gamma|\gamma,\theta,x)\right]^2   \over
n(z,x) \sum_{\theta} d(z+|\theta\gamma|\gamma,x)^2 } \label{eq:semb-anal} ,
\end{equation}
where $n(z,x)$ is the number non-zero samples summed over for each
semblance calculation.
Figure~\ref{fig:semb-mig0} shows a parameter CRP gather, overlaid with
the curves corresponding to the picked $\gamma$ value for each
reflector. For this synthetic, the $\gamma$ does a good job characterizing
the moveout.

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



\par
For this dataset 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 I can get an unrealistic, high spatial wavenumber
behavior for  $\gamma(x)$.
When doing conventional 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. I can adapt that work by
starting with
a initial curvature  ${\boldsymbol \gamma_{\bf 0}}$  equal to
the curvature corresponding to the maximum semblance $\bold W$ at the CRP,
and a derivative operator $\bold D$.
I can find a smooth curvature function ${\boldsymbol \gamma}$
by setting up a simple set of regression equations 
\begin{eqnarray}
\zero \pox \bold W( {\boldsymbol \gamma_{\bf 0}}  - {\boldsymbol \gamma})  \nonumber \\
\zero \pox \epsilon \bold D {\bf \boldsymbol \gamma}  .
\end{eqnarray}
By increasing $\epsilon$ we get smoother ${\boldsymbol \gamma}$ 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.

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



\subsection{Endpoints, edge effects,  and errors}
To set up our tomography problem I need to cover
some final details. 
I can convert our semblance picks back into a $\Delta z$ shift by
applying
\begin{equation}
\Delta z  =   \gamma \theta^2 .
\end{equation}
From a simple geometric argument (Figure~\ref{fig:ztot}) we can approximate
errors in traveltime from our depth focusing errors \cite{stork92}.
We approximate the change in ray length caused by the positioning error
$\Delta z$ by multiplying by the cosine of the half aperture angle $\theta$
and the cosine of the geologic dip $\phi$ at the reflection point `A'.
We multiply by the local slowness $s_{\rm ref}$ and get the final expression
\begin{equation}
\Delta t = 2 \gamma  \theta^2  s_{\rm ref} \cos \theta \cos \phi  \label{eq:smooth}  .
\end{equation}

\sideplot{ztot}{width=3.0in,height=3.0in}{Geometric relation between
a positioning error $\Delta z$ and the change in ray length for a raypair
reflecting at `A' at the half aperture angle $\theta$ with a local
dip $\phi$.}


\par
In constructing our raypaths we benefit from having our
CRP gathers in terms of angle.  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. We are always faced with the tradeoff of
how much should we interpolate our rays versus how many additional rays
should we shoot. In addition handling triplications of the wavefield
can prove daunting.
\par
With our moveout errors in terms of  angle  we  only
need to shoot a single ray-pair up
from our imaging point at the angle $\alpha$ and $\beta$,
\begin{eqnarray}
\alpha &=& \phi +  \theta \\ \nonumber
\beta &=& \phi -  \theta
\end{eqnarray}
where $\theta$ is one-half
the aperture angle,  $\phi$ is the geologic dip
(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.}
