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.}