\email{bill@sep.stanford.edu}
\nokeywords
\title{interpolating with data-space prediction-error filters}
\author{William Curry}
\lefthead{Curry}
\righthead{Data-space PEFs}
\footer{Autumn 2004 Term Paper}
\ABS{

The Madagascar seasat dataset presents a problem where data are
collected along crossing tracks.  These tracks are not straight, and appear 
to be irregular in the model space.  Previous methods assumed that the
data were regularly sampled in the model space coordinate system, or
did not take into account the regularities in the acquisition of the data.
Instead of attempting to find a prediction-error filter in the model space,
I estimate two prediction-error filters in a coordinate system based
on the data's spatial distribution, and show how to
regularize the data with these filters.
}

\section{INTRODUCTION}

The Madagascar seasat sea level dataset is a collection of two passes (ascending and descending) of the GEOSAT satellite 
over a region of the Southwest Indian Ridge in the Indian Ocean.  There is a densely-acquired region of the dataset in the south, 
which ranges from $40$ to $70$ degrees (E) longitude and $30$ to $40$ degrees (S) latitude, while the latitude of the 
sparsely-acquired data ranges from $20$ to $40$ degrees (S) latitude.
\par
The satellite tracks are much like feathered marine geophone cables, sail lines, or shot lines in a 3D seismic survey.  Any method that hopes 
to succeed on 3D seismic data should be able to deal with this toy problem.
\par
In this paper, only the lower half of the dataset ($30$ to $40$
degrees (S) latitude) is examined, so that
interpolation of the sparse tracks can be compared with the dense tracks.
\par
Early work on this dataset at SEP \cite{Ecker.sep.84.295,Lomask.sep.97.lomask1,Lomask.sep.111.jesse2} has mainly dealt with 
the systematic errors present in the dense dataset \cite{Ecker.sep.84.295}, or with ways in which to use information in the dense portion of the 
data to regularize the missing bins in the northern, sparse portion of
the data \cite{Lomask.sep.97.lomask1,Lomask.sep.111.jesse2}.  More
recent work has started to deal with the interpolation of only the
sparse tracks \cite{Curry.sep.115.bill1,Lomask.sep.117.jesse1}.
\par
In this paper, a different approach is presented to dealing with this
problem, where a pair of prediction-error filters are estimated
directly on the two sets of tracks. 

\section{Background}
The Madagascar regularization problem has been approached using the following fitting goals \cite{Lomask.sep.111.jesse2}:
                        \begin{eqnarray}
                                \  {\bf W} \frac{d}{dt}[{\bf L}{\bf m} -{\bf d}] &\approx& {\bf 0} \nonumber \\
                                \ \epsilon {\bf A m} &\approx& {\bf 0} . \label{eq:mad1}
                        \end{eqnarray}
In these fitting goals: {\bf W} corresponds to a weight for ends of tracks and spikes in the data, $\frac{d}{dt}$ is a derivative along each track used to 
eliminate low frequency variations along each track, 
{\bf L} is a linear interpolation operator that moves from values on a regular grid to the data points, 
{\bf m} is the desired gridded model, {\bf d} are the data points along the tracks, {\bf A} is a regularization operator, and $\epsilon$ is a trade-off 
parameter between the two fitting goals.
\par
The regularization operator ({\bf A}) typically is a Laplacian, a prediction-error filter (PEF), or a non-stationary PEF \cite{Crawley.sepphd.104}.  When using a PEF, 
it first must be estimated on some training data, using a least-squares fitting goal,
\begin{equation}
\bold{W(DKa + d)} \approx \bold 0
\label{eqn:pefest},
\end{equation}
in which {\bf W} is a weight to exclude equations with missing data, {\bf D} is convolution with the data, {\bf K} constrains the 
first filter coefficient to $1$, {\bf a} is the unknown filter, and {\bf d} is a copy of the data.
\par
In order to set a benchmark for how effective any interpolation of the
sparse tracks is, 
a PEF is estimated using fitting goal \ref{eqn:pefest} on the output of fitting goals \ref{eq:mad1} when
using the dense tracks of the Madagascar dataset, and a Laplacian
regularization operator.  The PEF is then used to interpolate the sparse 
tracks in the same area, using the same fitting goals \ref{eq:mad1},
this time with the regularization being convolution with the
PEF.  This result provides an upper bound to what an ideal
interpolation of the data would be if we already knew the answer.  Any
result should perform much better than if {\bf A} in fitting goals
\ref{eq:mad1} were a Laplacian, which is 
This result is shown in Figure \ref{fig:best}.

\plot{best}{width=6.0in,height=6.0in}{Top: dense data regularized with
  a Laplacian.  Middle: sparse data regularized with a
  Laplacian. Bottom: sparse data regularized with a PEF trained on above.}

\section{PEFs in the data space}

Once we attempt to interpolate this dataset using only the sparse
tracks, the above method no longer works.  In this case we will pay
more attention to how our data are spatially distributed.  The known
data points in the model space are distributed along curved crossing
tracks, making it very difficult to estimate a PEF in this space, as
shown in Figure \ref{fig:data}.
However, in the data space of the fitting goals in equation
(\ref{eq:mad1}) of the previous section, the data are sampled in a
regular space: a series of regularly-sampled tracks, as shown in
Figures \ref{fig:sptracks} and \ref{fig:detracks}.

\plot{data}{width=6.0in,height=3.0in}{Left: sparse tracks.  Right:
  dense tracks.}

\plot{sptracks}{width=6.0in,height=6.0in}{Sparse tracks.  Top:
  ascending.  Bottom: descending.  Left: in data space.  Right: in
  model space}

\plot{detracks}{width=6.0in,height=6.0in}{Dense tracks.  Top:
  ascending.  Bottom: descending.  Left: in data space.  Right: in
  model space}

\par

Since these data are collected in two series of one-dimensional
tracks, it would be easiest to estimate a pair of one-dimensional PEFs
on these two sets of tracks, as shown in the left half of Figures
\ref{fig:sptracks} and \ref{fig:detracks}.

\par

We now have two PEFs which have been estimated in a data space, but
the model which we wish to regularize with these PEFs is in a
different space.  This requires the introduction of two additional
linear interpolation operators ${\bf L_1}$ and ${\bf L_2}$, which pull
bins from the model space into the ascending and descending track data
spaces, respectively.  The mappings used for these operators are shown
in Figure \ref{fig:maps}.

\sideplot{maps}{width=4.0in,height=9.0in}{Mappings from data space to
  model space, all shown in data space.  From the top down: ascending
  tracks latitude, ascending tracks longitude, descending tracks
  latitude, descending tracks longitude.}

Now that we have both two prediction-error filters for regularizations
operators as well as linear interpolation operators that pull model
points into the data space, we can put everthing together in the
following fitting goals,

\begin{eqnarray}
   \bold{Lm-d} &\approx& \bold 0 \nonumber \\
   \epsilon \bold{A_{1}L_{1}m} &\approx& \bold 0  \nonumber \\
   \epsilon \bold{A_{2}L_{2}m} &\approx& \bold 0,  
   \label{eq:madb1}
\end{eqnarray}
where \bold{L} pulls model points (\bold{m}) to where we have data
(\bold{d}), $\bold{A_{1}}$ and $\bold{A_{2}}$ are 1D PEFs that are
estimated on the ascending and descending tracks 
in the data space, respectively, and $\bold{L_{1}}$ and $\bold{L_{2}}$ are
linear interpolation operators that pull model points into the
ascending and descending track data spaces, respectively.  $\epsilon$ is
a tradeoff parameter between the two sets of goals.

\section{Conclusions and Future Work}
By estimating a pair of 1D PEFs in the data space, a method has been
shown where the problem of missing data in PEF estimation is avoided.
This method results in multiple lower-dimensional prediction-error
filters which can then be used in conjunction to fill in missing data.
\par
Crossing tracks are also present in land seismic data, where due to
reciprocity the negative offsets in the split-spread land experiment
cross: in 2D this happens in cmp - absolute offset space, and in 3D this
happens in cmp_x - offset_x and cmp_y - offset_y spaces.

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