\email{bill@sep.stanford.edu}
\nokeywords
\title{Interpolating with data-space prediction-error filters}
\author{William Curry}
\lefthead{Curry}
\righthead{Data-space PEFs}
\footer{SEP--120}
\ABS{

The Madagascar sea elevation dataset presents a problem where data are
collected along crossing tracks.  These tracks are not straight, and
are therefore 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 with promising results.  I
then show how this strategy can be applied to 2D and 3D land surveys
when data predicted by reciprocity is included.
}

\section{INTRODUCTION}

The Madagascar seasat sea level dataset is a collection of two passes
of the GEOSAT satellite (ascending and descending) 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,
as shown in Figure \ref{fig:data}.
\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
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}. In the
latter, it is assumed that the statistics of the data are stationary over both regions.  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, 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.  Here, a different approach is presented to dealing with the
sparse track problem, where a pair of prediction-error filters (PEFs) are estimated
directly on the two sets of tracks.  This pair of filters is estimated
in the data space, so that the issues of missing data and irregular
geometry are no longer present.  Once these filters have been
estimated, they can be used in tandem to regularize the missing
portions of the model space.
\par
Extension of this method to incorporate non-stationary PEFs is quite
straightforward.  The similarities between the Madagascar data and a
Colombian 2D seismic data line are noteworthy enough that this method
should be applicable to 2D land data, where the two sets of tracks
correspond to positive and negative offsets.  The geometry of a 3D
cross-swath land seismic survey also has similarities to
the Madagascar data, where when data predicted by reciprocity is added irregular
crossing tracks are present in cmp_x, offset_x space.

\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

\plot{data}{width=6.0in,height=4.0in}{(a) sparse tracks.  (b)
  dense tracks.}

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 a
PEF.  This result provides an upper bound to what an ideal
interpolation of the data would be if we already knew the answer.  Any
new result should be much better than the Laplacian regularization
shown in Figure \ref{fig:best}b.

\plot{best}{width=6.0in,height=6.0in}{(a) dense data regularized with
  a Laplacian.  (b) sparse data regularized with a
  Laplacian. (c) sparse data regularized with a PEF trained on the
  top.}

\section{PEFs in the data space}

Once we attempt to interpolate this dataset using only the sparse
tracks, the above method no longer works, as no region would have
enough contiguous data on which to estimate a PEF.  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 (\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{sptracks}{width=6.0in,height=7.0in}{Sparse tracks, (a) ascending
  and (b) descending in model space, and (c) ascending and (d)
  descending tracks in data space.}

\plot{detracks}{width=6.0in,height=7.0in}{Dense tracks, (a) ascending
  and (b) descending in model space, and (c) ascending and (d)
  descending tracks in data 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 top halfs 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}.

\plot{maps}{width=7.0in,height=2.0in}{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}
   \  {\bf W} \frac{d}{dt}[{\bf L}{\bf m} -{\bf d}] &\approx& {\bf 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 data fitting and model styling goals.

\par
The 1D PEFs can also be replaced by 2D PEFs that are estimated by
scaling the filter so that it covers multiple sparse tracks.  If this
approach is taken, the interpolation can occur in the data space where
the PEFs are estimated (using a single PEF for each of the two track
spaces), or in the model space (using both PEFs simultaneously).  The
more straightforward data-space interpolation is shown in Figure \ref{fig:dataspinterp}.
 
\plot{dataspinterp}{width=6.0in,height=8.5in}{Interpolation of the two
    different track spaces (a) and (b).  (c) and (d) contain the same
    results mapped into model space.  The correct strike of the ridge
    is identified by the ascending tracks but not the descending tracks.}

Figures \ref{fig:dataspinterp}a and b are simply the interpolation of
the data space with 2D PEFs estimated on the sparse tracks.  Figures
\ref{fig:dataspinterp}c and d are those interpolated results mapped
back to the model space by using fitting goals
(\ref{eq:mad1}), where the input data are now the interpolated sets of tracks in the
first two panels.  Since fitting goals (\ref{eq:mad1}) were applied
when generating the new tracks, the track derivative is no longer
necessary.
\par
The results are mixed, as Figure \ref{fig:dataspinterp}c shows that
the trend of the ridge was correctly identified by the PEF estimated
on the ascending tracks.  The PEF estimated on the descending tracks
did not fare so well, as the direction of the ridge in the interpolated tracks of Figure
\ref{fig:dataspinterp}d does not match the densely sampled tracks in
Figure \ref{fig:detracks}d.  This is because the descending tracks are
oblique to the structure, so the structure is aliased beyond the point
where a spaced PEF can interpolate accurately.  In either case, the
result is better than that obtained with a Laplacian, and in the case
of the ascending tracks is not that far from the PEF estiamted on a
fully-sampled model space shown in Figure \ref{fig:best}c.

\section{Conclusions and Future Work}

By estimating a pair of PEFs in the data space, the problem of
irregular data acquistion has been avoided. The actual interpolation
can be performed in either the data space (track coordinates) or the
model space (latitude and longitude).  The results turned out to be
much better for the ascending tracks than for the descending tracks.
\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 as shown in
Figure \ref{fig:hulia}, and in 3D this also happens, but in cmp_x -
offset_x and cmp_y - offset_y spaces.

\plot{hulia}{height=4.0in,width=6.0in}{Fold map of an
  irregularly-sampled 2D land survey in cmp and absolute offset.
  Similar patterns of crossing tracks are also present in this survey
  due to reciprocity.}

\bibliographystyle{sep}
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