%  filters. This could be an appropriate approach for interpolating%  both primary and multiple reflections.  %\item Missing data interpolation problems can be greatly accelerated%  by preconditioning \cite[]{Fomel.sep.94.sergey1,Fomel.sep.95.sergey1}.%  Finding an appropriate preconditioning for the offset continuation%  method is an open problem.  The non-stationary nature of the%  continuation filter make this problem particularly challenging.%\end{itemize}Analogously to integral azimuth moveout operator\cite[]{GEO63-02-05740588}, differential offset continuation can beapplied in 3-D for regularizing seismic data prior to prestack imaging.    In the next section, I return to the 2-D case to consider an  important problem of shot gather interpolation.  \section{Shot continuation}    Missing or under-sampled shot records are a common example of data  irregularity \cite[]{Crawley.sepphd.104}. The offset continuation  approach can be easily modified to work in the shot record domain.  With the change of variables $s = y - h$, where $s$ is the shot  location, the frequency-domain equation~(\ref{eqn:OC}) transforms to  the equation  \begin{equation}    h \, \left( 2\,{\partial^2 \tilde{P} \over \partial s \partial h} -       {\partial^2 \tilde{P} \over \partial h^2} \right) -     i\,\Omega \, \left({\partial \tilde{P} \over   {\partial h}} -        {\partial \tilde{P} \over   {\partial s}}\right) = 0 \;.    \label{eqn:SC}   \end{equation}  Unlike equation~(\ref{eqn:OC}), which is second-order in the  propagation variable $h$, equation~(\ref{eqn:SC}) contains only  first-order derivatives in $s$. We can formally write its solution  for the initial conditions at $s=s_1$ in the form of a phase-shift  operator:    \begin{equation}           \widehat{\widehat{P}}(s_2) = \widehat{\widehat{P}}(s_1)\,      \exp{\left[i\,k_h\,\left(s_2-s_1\right)\,          \frac{k_h\,h-\Omega}{2\,k_h\,h-\Omega}\right]}\;,    \label{eqn:SCshift}   \end{equation}  where the wavenumber~$k_h$ corresponds to the half-offset~$h$.  Equation~(\ref{eqn:SCshift}) is in the mixed offset-wavenumber  domain and, therefore, not directly applicable in practice. However,  we can use it as an intermediate step in designing a  finite-difference shot continuation filter. Analogously to the cases  of plane-wave destruction and offset continuation, shot continuation  leads us to the rational filter  \begin{equation}    \label{eqn:SCpass}    \hat{P}_{s+1}(Z_h) =     \hat{P}_{s} (Z_h) \frac{S(Z_h)}{\bar{S}(1/Z_h)}\;,  \end{equation}  The filter is non-stationary, because the coefficients of $S(Z_h)$  depend on the half-offset $h$. We can find them by the Taylor  expansion of the phase-shift equation~(\ref{eqn:SCshift}) around  zero wavenumber $k_h$. For the case of the half-offset sampling  equal to the shot sampling, the simplest three-point filter is  constructed with three terms of the Taylor expansion. It takes the  form  \begin{equation}    \label{eqn:SCfilt}    S(Z_h) = - \left(\frac{1}{12} + i\,\frac{h}{2\,\Omega}\right)\,Z_h^{-1} +     \left(\frac{2}{3} - i\,\frac{\Omega^2 + 12\,h^2}{12\,\Omega\,h}\right) +    \left(\frac{5}{12} + i\,\frac{\Omega^2 + 18\,h^2}{12\,\Omega\,h}\right)\,    Z_h\;.  \end{equation}    Let us consider the problem of doubling the shot density.  If we use  two neighboring shot records to find the missing record between  them, the problem reduces to the least-squares system  \begin{equation}    \label{eqn:SCls}    \left[\begin{array}{c}        \mathbf{S} \\        \mathbf{\bar{S}}      \end{array}\right]\,\mathbf{p}_s \approx    \left[\begin{array}{c}        \mathbf{\bar{S}}\,\mathbf{p}_{s-1} \\        \mathbf{S}\,\mathbf{p}_{s+1}      \end{array}\right]\;,  \end{equation}  where $\mathbf{S}$ denotes convolution with the numerator of  equation~(\ref{eqn:SCpass}), $\mathbf{\bar{S}}$ denotes convolution  with the corresponding denominator, $\mathbf{p}_{s-1}$ and  $\mathbf{p}_{s+1}$ represent the known shot gathers, and $\mathbf{p}_s$  represents the gather that we want to estimate. The least-squares  solution of system~(\ref{eqn:SCls}) takes the form  \begin{equation}    \label{eqn:SCsol}    \mathbf{p}_s = \left(      \mathbf{S}^T\,\mathbf{S} +      \mathbf{\bar{S}}^T\,\mathbf{\bar{S}}    \right)^{-1}\,    \left(\mathbf{S}^T\,\mathbf{\bar{S}}\,\mathbf{p}_{s-1} +      \mathbf{\bar{S}}^T\,\mathbf{S}\,\mathbf{p}_{s+1}\right)\;.  \end{equation}  If we choose the three-point filter~(\ref{eqn:SCfilt}) to construct  the operators~$\mathbf{S}$ and $\mathbf{\bar{S}}$, then the inverted  matrix in equation~(\ref{eqn:SCsol}) will have five non-zero  diagonals. It can be efficiently inverted with a direct banded  matrix solver using the $LDL^T$ decomposition \cite[]{golub}. Since  the matrix does not depend on the shot location, we can perform the  decomposition once for every frequency so that only a triangular  matrix inversion will be needed for interpolating each new shot.  This leads to an extremely efficient algorithm for interpolating  intermediate shot records.  Sometimes, two neighboring shot gathers do not fully constrain the  intermediate shot. In order to add an additional constraint, I  include a regularization term in equation~(\ref{eqn:SCsol}), as  follows:  \begin{equation}    \label{eqn:SCreg}    \mathbf{p}_s = \left(      \mathbf{S}^T\,\mathbf{S} +      \mathbf{\bar{S}}^T\,\mathbf{\bar{S}} +         \epsilon^2\,\mathbf{A}^T\mathbf{A}    \right)^{-1}\,    \left(\mathbf{S}^T\,\mathbf{\bar{S}}\,\mathbf{p}_{s-1} +      \mathbf{\bar{S}}^T\,\mathbf{S}\,\mathbf{p}_{s+1}\right)\;,  \end{equation}  where $\mathbf{A}$ represents convolution with a three-point  prediction-error filter (PEF), and $\epsilon$ is a scaling  coefficient. The appropriate PEF can be estimated from  $\mathbf{p}_{s-1}$ and $\mathbf{p}_{s+1}$ using Burg's algorithm  \cite[]{GEO37-02-03750376,Burg.sepphd.6,Claerbout.fgdp.76}. A  three-point filter does not break the five-diagonal structure  of the inverted matrix.  The PEF regularization attempts to preserve  offset dip spectrum in the under-constrained parts of the estimated  shot gather.    Figure~\ref{fig:shotin} shows the result of a shot interpolation  experiment using the constant-velocity synthetic from  Figure~\ref{fig:data}. In this experiment, I removed one of the  shot gathers from the original NMO-corrected data and interpolated  it back using equation~(\ref{eqn:SCreg}). Subtracting the true shot  gather from the reconstructed one shows a very insignificant error,  which is further reduced by using the PEF regularization (right  plots in Figure~\ref{fig:shotin}).  The two neighboring shot gathers  used in this experiment are shown in the top plots of  Figure~\ref{fig:shot3}.  For comparison, the bottom plots in  Figure~\ref{fig:shot3} show the simple average of the two shot  gathers and its corresponding prediction error. As expected, the  error is significantly larger than the error of shot  continuation. An interpolation scheme based on local dips in the  shot direction would probably achieve a better result, but it is  significantly more expensive than the shot continuation scheme  introduced above.    \plot{shot3}{width=6in,height=7in}{Top: Two synthetic shot gathers    used for the shot interpolation experiment. An NMO correction has    been applied. Bottom: simple average of the two shot gathers    (left) and its prediction error (right).}  \plot{shotin}{width=6in,height=3.5in}{Synthetic shot interpolation    results. Left: interpolated shot gathers. Right: prediction errors    (the differences between interpolated and true shot gathers),    plotted on the same scale.} \inputdir{elfshot}   A similar experiment with real data from a North Sea marine dataset  is reported in Figure~\ref{fig:elfshotin}. I removed and  reconstructed a shot gather from the two neighboring gathers shown  in Figure~\ref{fig:elfshot3}. The lower parts of the gathers are  complicated by salt dome reflections and diffractions with  conflicting dips. The simple average of the two input shot gathers  (bottom plots in Figure~\ref{fig:elfshotin}) works reasonably well  for nearly flat reflection events but fails to predict the position  of the back-scattered diffractions events. The shot continuation  method works well for both types of events (top plots in  Figure~\ref{fig:elfshotin}). There is some small and random residual  error, possibly caused by local amplitude variations.     \plot{elfshot3}{width=6in,height=3.5in}{Two real marine shot gathers    used for the shot interpolation experiment. An NMO correction has    been applied.}      \plot{elfshotin}{width=6in,height=7in}{Real-data shot interpolation    results.  Top: interpolated shot gather (left) and its prediction    error (right). Bottom: simple average of the two input shot    gathers (left) and its prediction error (right).}    Analogously to the case of offset continuation, it is possible to  extend the shot continuation method to three dimensions. A simple  modification of the proposed technique would also allow us to use  more than two shot gathers in the input or to extrapolate missing  shot gathers at the end of survey lines.%  \section{Stack and partial stack as a data regularization problem}\section{Conclusions}Differential offset continuation provides a valuable tool forinterpolation and regularization of seismic data. Starting fromanalytical frequency-domain solutions of the offset continuationdifferential equation, I have designed accurate finite-differencefilters for implementing offset continuation as a local convolutionaloperator. A similar technique works for shot continuation acrossdifferent shot gathers. Missing data are efficiently interpolated byan iterative least-squares optimization. The differential filters havean optimally small size, which assures high efficiency.Differential offset continuation serves as a bridge between integraland convolutional approaches to data interpolation. It shares thetheoretical grounds with the integral approach but is applied in amanner similar to that of prediction-error filters in theconvolutional approach.Tests with synthetic and real data demonstrate that the proposedinterpolation method can succeed in complex structural situationswhere more simplistic methods fail.\section{Acknowledgments}The financial support for this work was provided by the sponsors ofthe Stanford Exploration Project (SEP).%The 3-D North Sea dataset was released to SEP by Conoco and its%partners, BP and Mobil. For the shot continuation test, I used aNorth Sea dataset, released to SEP by Elf Aquitaine.I thank Jon Claerbout and Biondo Biondi for helpful discussions aboutthe practical application of differential offset continuation.\bibliographystyle{seg}\bibliography{SEG,SEP2,oc}%%% Local Variables: %%% mode: latex%%% TeX-master: t%%% End: