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. \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, evenif our Green's function traveltime 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 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 by changing our 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 hyberbolic, even in constant velocity, in angle domain gathers. For the purpose of this thesis the approximation is 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.}