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Interpolating with nonstationary filters in $ t$-$ x$ or in $ f$-$ x$

Interpolation can be phrased in two steps, where a nonstationary prediction-error filter is first estimated from fully-sampled training data and is then used to interpolate missing data. The training data required by this method should have the same local autocorrelation as that of the desired output interpolated data. Pseudo-primaries, having roughly the same dip as the missing data, serve as training data for a PEF. Some of the problems with the pseudo-primary data, such as the different phase and amplitude, do not influence the PEF.

To estimate a nonstationary PEF, we solve,

\begin{displaymath}\begin{array}{rcccl} & & \underset{{\bf f}_{\textrm{ns}}}{\mi...
... { \bf r}_{\textrm{f}}& =& { \bf Rf}_{\textrm{ns}} \end{array}.\end{displaymath} (3)

Here, the unknown nonstationary PEF coefficients, $ { \bf
K}_{\textrm{ns}}{ \bf f}_{\textrm{ns}}$, are estimated from the pseudo-primary data, $ { \bf d}$, a convolutional matrix, $ { \bf
D}_{\textrm{ns}}$, that is a function of $ { \bf d}$, and a regularization operator, $ { \bf R}$, that applies a Laplacian filter across the spatial axes of the PEF coefficients. This system of equations is solved to estimate a multi-dimensional, nonstationary prediction-error filter from a set of fully-sampled pseudo-primaries, such as was generated in the previous section.

Once this filter has been estimated from the pseudo-primary data, the filter is used in a second least-squares problem. In this problem, we estimate an interpolated output, $ { \bf m}$, composed of missing data, $ { \bf m}_{\textrm{unknown}}$, and known data, $ { \bf
m}_{\textrm{known}}$, that, when convolved with the nonstationary PEF, produces a minimized output,

$\displaystyle { \bf r}_{\textrm{m}} = { \bf F}_{\textrm{ns}} { \bf m}_{\textrm{unknown}} + { \bf F}_{\textrm{ns}}{ \bf m}_{\textrm{known}}.$ (4)

Here, the known nonstationary PEF convolution matrix, $ { \bf
F}_{\textrm{ns}}$, derived from $ { \bf f}_{\textrm{ns}}$ obtained in the previous step, is multiplied with both the known and unknown data values, $ { \bf
m}_{\textrm{known}}$ and $ { \bf m}_{\textrm{unknown}}$, to create a known quantity and a term with the unknown values to be interpolated. These terms are summed to form the residual $ { \bf
r}_{\textrm{m}}$. We minimize the $ L_2$ norm of this residual to estimate the interpolated data values.

Subsections
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Next: Interpolation in time and Up: Curry and Shan: Near-offset Previous: Generating Pseudo-primaries

2009-04-13