\def\figdir{./Figs}
In general, geophysical inverse problems (inverting for some
model $\model$, given some $\data$,
while applying some operator $\lop$)
are ill-posed.
A classic example of this is the missing data problem \cite{gee,geostat}.
The goal of
the missing data problem is to interpolate intelligently
between a sparse set of known points. For example let us take
a synthetic velocity model with an upper horizontal reflector,
an anticline between two unconformities, and updipping layer
at the bottom of the model. Suppose
we have velocity measurements at several wells and
you would like to interpolate it  onto a regular 2-D mesh,
Figure~\ref{fig:well-logs}.
\activeplot{well-logs}{width=6.0in,height=3.0in}{ER}{Left panel shows a
synthetic velocity model, right panel shows a subset of  that data
chosen to simulate well log data.}
\par
The geophysicist  might follow the approach described by
\longcite{gee},
first interpolating  the irregular data onto a regular mesh
by applying some type of binning operator, $\bf B$,
then defining a fitting goal that requires the model to fit
the data exactly at the known points $\bold J$.
\begin{equation}
{\bf J} {\bf B}  \data \approx{\bf J}   \model \label{eq:missing} .\end{equation}
At model locations where  there are  no data  values
we want the model to be `smooth', therefore
we will use Tikhonov regularization\cite{tikhonov}
to minimize the output of a  roughening operator applied to
the model,
\begin{equation}
\zero \approx \reg \model  .
\label{eq:reg-basic}\end{equation}
%\cite{tikhonov}.
For the remainder of the thesis I will refer to (\ref{eq:reg-basic}) as
the {\it model styling} goal.

If
we don't have any other knowledge about our model, 
an
isotropic operator like the Laplacian might be a logical choice for $\bf A$. 
If I apply the fitting goals implied by (\ref{eq:missing}) and
(\ref{eq:reg-basic}) for 200 iterations
using the Laplacian for $\reg$, I get Figure~\ref{fig:qdome-lap}.
The result is what
has been   euphemistically referred to as
the `ice cream cone result'  \cite{Brown.sep.97.morgan1}. 
By spreading information
isotropically the model goes smoothly from
our known points to some local average.
We see little to  no continuation of layers,
generally a thoroughly unsatisfactory result.

\activesideplot[t]{qdome-lap}{height=3.0in,width=3.0in}{ER}{Interpolation result after 200
iterations using an inverse Laplacian regularization operator. Note
the edges effects at the top and bottom of the model due to using an
internal convolution operator.}

