% Started 03/01/98
%\def\figdir{./../bob2/Figs}
\
\sepinput{macros}

\chapter{2-D field tests}
\label{chap:field2}

\section{Introduction and summary}
\sepinput{intro}
%\section{Data}
\sepinput{data}
\section{Initial errors}
\sepinput{errors}
\section{Building the steering filters}
\sepinput{steer}
\section{First iteration}
\sepinput{first}
\section{Final result}
\sepinput{best}
\section{Conclusions}
\sepinput{con}

%Seismic tomography is a non-linear problem with a significant
%null-space.  Our estimation problem often
%converges slowly,
%to a geologically unreasonable model, or not at all.
%One reason for slow or non-convergence is that we are attempting
%to simultaneously estimate reflector position (mapping velocity)
%and image our data (focusing velocity).
%By performing tomography in vertical travel-time space, we avoid
%estimating mapping velocity, instead concentrating on
%focusing velocity. By introducing anisotropic preconditioning
%oriented along bedding planes, we can quickly guide the inversion towards 
%a geologically reasonable model. We illustrate the benefits of
%our tomography method by  comparing it to more traditional methods
%on a synthetic anticline model. In addition, we demonstrate
%the method's ability to improve the velocity estimate, and the resulting
%migrated image of a real 2-D  dataset.
%

%Standard tomography schemes suffer from slow convergence and
%tend to create isotropic features in  velocity that are inconsistent
%with geology.
%These  isotropic features are due in large part to
%using  symmetric regularization operators.   
%By replacing these
%symmetric operators with operators that tend to spread information along
%structural dips we can generate more geologically reasonable velocity models.
%We can find the inverse of  these space-varying, anisotropic operators by 
%forming them in helix 1-D space and performing polynomial division.
%These inverse operators can be used as a preconditioner to a standard
%tomography problem, significantly improving convergence speed compared
%to the typical, regularized inversion problem.
%}
 

%background
%Velocity estimation, like most geophysical problems, is 
%under-determined. It has the added the problem of being
%inherently non-linear. 
%Tomography is inherently non-linear, therefore  a standard technique
%is to linearize the problem by assuming a stationary ray 
%field \cite{GEO56.04.04830495}.
%Unfortunately, we must still deal with
%the coupled relationship between reflector position and
%velocity \cite{GEO62-03-09700979,GEO60-01-01640175}.
%As a result, the back projection operator must
%attempt to handle both repositioning of the reflector {\it and} updating
%the velocity model \cite{Trier.sepphd.66} .  
%The resulting back projection operator is sensitive
%to our current guess at velocity and reflector position.
%%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.
%\par
%In addition to non-linearity, tomography problems are often under-determined.
%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 are 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}.
%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{Toldi.sep.43}.   In theory, this regularization
%term should be the inverse model covariance matrix \cite{tarantola-ipt}.
%The question is how to obtain an estimate of the model covariance matrix.
%The obvious answer is through {\it a priori} information sources such
%a geologist's structural model of the area, well log information,
%or preliminary stack or migration results. Incorporating
%these varied information sources into our objective function has always been
%problematic.
%For years, geostatisticians 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. 
%
%%claim
%In this paper we follow the course outlined in Clapp and Biondi \shortcite{Clapp.sep.97.bob2} to address both the velocity-depth ambiguity and  the
%problem of adding geologic
%constraints.
%We formulate our tomography problem in vertical travel-time ($\tau$) coordinates
%rather than depth.  In this coordinate system, reflectors are significantly
%less sensitive to velocity \cite{Biondi.sep.95.biondo1} and the resulting
%back projection operator is less sensitive to the background velocity
%model \cite{Clapp.sep.100.bob2}.  
%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},
%or {\it steering filters} oriented along
%geologic dip \cite{Clapp.sep.95.bob1, Clapp.sep.100.bob3}.  
%To speed up convergence, we reformulate our regularization
%problem to a preconditioned problem \cite{gee} using the helix
%transform and polynomial division \cite{helix}.
%\par
%We create a synthetic anticline velocity model and compare the
%inversion result using a symmetric regularization operator  in depth,
%steering filter in depth, and finally steering filter in vertical
%travel time space.  We study the speed and quality of our tomographic estimate
%using two different synthetic models.
%We conclude
%with some preliminary tests on a 2-D marine dataset with  gas hydrates.
%Preliminary migration results are encouraging.
%
%
%%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,
%
%
%
%%\sepinput{theory}
%
%%\sepinput{synthetic}
%%\sepinput{real}
%
%
%
%%\section{CONCLUSIONS}
%%By performing tomography in tau space we are able to quickly converge
%%to geologically realistic velocity models.  The results on synthetics
%%indicate that the method when velocity is not the smooth function
%%that the a Laplacian regularize will attempt to create.
%%%%%%Early tests on field data are encouraging.
\cleardoublepage

