% Started 03/01/98
%
\input{../steer/macros_act}

\chapter{Smoothing along geologic dip}
\label{chap:reg}

\section{Introduction and summary}
\input{../steer/intro_act}

\section{Why regularize?}
\input{../steer/regbasic_act}

\section{Approximating the covariance matrix}
\input{../steer/covar_act}

\section{Stationarity}
\input{../steer/continuous_act}

\section{Steering filter}
\input{../steer/steer_act}

\subsection{Constructing a filter}
\input{../steer/build_act}

\section{Regularizing vs. preconditioning}
\input{../steer/prec_act}
      
%\subsection{Missing data}
%\input{../steer/missing_act}

%\section{Regularizing Toldi}

%\section{Regularization vs. Parameterization}

%\section{Steering filters in 3-D}

\section{Regularizing tomography}
\input{../steer/reg_act}

\section{Synthetic comparisons}
\input{../steer/synth_act}

%\section{Conclusions}
%\input{../steer/con_act}

%\section{Introduction and summary}
%background
%Velocity estimation, like most geophysical problems, is 
%under-determined cite(something).  
%To obtain a pleasing result we impose some type of regularization
%criteria  such as
%diagonal scaling, limiting solutions to large
%singular values\cite{rowbatham}, or characterizing the model through
%a small number of spline coefficients\cite{jun}.
%These methods create velocity models that can correctly model
%the recorded travel times, but are unrealistic when applying 
%criteria such as geologic feasibility.
%
%To create more geologically feasible velocity models  and to speed
%up convergence
%Michelena\shortcite{GEO56.05.06350653} suggested using varying sized 
%grid cells.  Unfortunately, such a parameterization is prone to error
%when the wrong size blocks our chosen\cite{error}. Other authors have
%suggested locally clustering grid cells \cite{GEO56.09.13951404} or
%characterizing the velocity model as a series of layers\cite{GEO61.05.15111523}.
%Unfortunately, these methods are also 
%susceptible to errors when the wrong parameterization is chosen.
%
%An attractive alternative
%approach is to add an additional model regularization term to our
%objective function\cite{Trier.sep.66}.   In theory, this regularization
%term should be the inverse model covariance matrix\cite{tarantola-ipt}.
%The question is how do  we obtain an estimate of the model covariance matrix?
%The obvious answer is through a priori information sources such
%a geologists structural model of the area, well log information,
%or preliminary stack or migration results. How to incorporate
%these varied information sources into our objective function has always been
%problematic.
%For years, Geostatisticans have successfully
%combined these mixed types of information
%to produce variograms\cite{geostat}.
%Unfortunately, the geostatistical approach does not 
%easily fit within a standard global tomography problem. 
%
%%
%In this paper we  take a different approach to add geological constraints
%to our velocity estimation problem.
%We make the assumption that velocity
%follows geologic dip, or some other known
%trend. We then approximate the model covariance matrix by
%creating small, plane wave annihilation filters \cite{pvi} oriented along
%geologic dip.  To speed up convergence we reformulate our regularization
%problem into a preconditioned \cite{gee}  problem using the helix
%transform and polynomial division \cite{helix}.
%
%%agenda
%We show this methodology applied to three different problems of
%increasing complexity.
%We  begin by interpolating a velocity field starting from well-log type
%input.  Next, we precondition a linearized Dix inversion problem using
%the Marmousi dataset.  We conclude by applying the methodology to a
%tomography problem using synthetic anticline velocity model.
%
%%Often, velocity follows structural dip, you can ensure this by constraining
%%your model through a layered approach, but if your velocity layer boundaries
%%are incorrect, or in certain regions velocity is to some extent layer
%independent, you can quickly end up with models that are unreasonable.
%On the other hand,

%claim

%agenda

%\section{Theory}
%\input{../steer/steering_act}
%\input{../steer/dix_act}
%\input{../steer/tomography_act}
