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Background

Interpolation can be posed as a two-stage problem, where in the first stage some statistics of the data are gathered, and in the second stage this information is applied to fill in the missing data. In the case of transform-based interpolation methods, the initial transform corresponds to the gathering of information on existing data, and the second stage is the transform back to the original, more densely-sampled space.

In terms of the prediction-error filter based interpolation used in this paper, in the first stage an estimate of the data is made by creating a non-stationary prediction-error filter (PEF) from the data by solving a linear least-squares inverse problem,

$\begin{eqnarray} \bold{W(DKf} + \bold{d}) \approx \bold{0} \nonumber \ \epsilon \bf A f \approx 0 ,\end{eqnarray}$

(1)

where $\bf{D}$ represents non-stationary convolution with the data, $\bf{f}$ is a non-stationary PEF, $\bf{K}$ (a selector matrix) constrains the value of the first filter coefficient to 1, $\bf{d}$ is a copy of the data, $\bf{A}$ is a regularization operator (a Laplacian operating over space) and $\epsilon$ is a trade-off parameter for the regularization. Solving this system will create a smoothly varying non-stationary PEF that, when convolved with the data, will ideally remove all coherent energy from the input data.

Once the PEF has been estimated, we can use it to constrain the missing data by solving a second linear least-squares inverse problem,

$\begin{eqnarray} \bold{S(m} - \bold{d}) \approx \bold{0} \nonumber \ \epsilon \bf F m \approx 0 ,\end{eqnarray}$

(2)

where $\bf{S}$ is a selector matrix which is 1 where data is present and where it is not, $\bf{F}$ represents convolution with the non-stationary PEF, $\epsilon$ is now a trade-off parameter and $\bf{m}$ is the desired model.

In order to interpolate by a factor of two, the coefficients of the PEF are expanded so that the filter coefficients fall on known data. Once the PEF is estimated, the filter is shrunk down to its original size and then used to interpolate.

Next: Data Description Up: Curry: Interpolating diffracted multiples Previous: Introduction

Stanford Exploration Project
4/5/2006

	$\begin{eqnarray} \bold{W(DKf} + \bold{d}) \approx \bold{0} \nonumber \ \epsilon \bf A f \approx 0 ,\end{eqnarray}$
		(1)

	$\begin{eqnarray} \bold{S(m} - \bold{d}) \approx \bold{0} \nonumber \ \epsilon \bf F m \approx 0 ,\end{eqnarray}$
		(2)