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# Introduction

The problem of interpolation arises because the geometry and spacing of the data acquisition grid differs from that required ideally for seismic imaging. In most cases, either the data has some holes (missing data), or sampling is too coarse for high-resolution seismic imaging. By the very nature of the problem we can say that no interpolation scheme can be perfect in all respects, because a proxy can never substitute for the real data. The criterion normally used to judge the performance of any interpolation scheme is how well it resembles the real data. Hence most interpolation schemes exploit the information present in the spectrum of the data and make the inserted values conform to the same spectrum. Prediction Error Filters (PEF) exactly rely on this principle Claerbout (2005). Covariance in the data space is analogous to the spectrum, and in the method presented in this article I use this covariance structure to estimate the filter coefficients and subsequently fill in the missing values.

A general problem with various interpolation schemes is that they do not perform well in presence of sharp features (high-frequency components), and the image after interpolation gets smeared. A common solution is to estimate edges and dips explicitly and use this information while interpolating; this ensures good resolution along the estimated features. Two important drawbacks of this approach are first, that we are restricted to a finite number of choices, and second, that it leads to increased computational complexity. Li and Orchard (2001) proposed a covariance-based interpolation scheme especially for digital images. Their claim is that covariance has enough information that an interpolation scheme based on covariance structure can preserve the edges and sharp features without a need to explicitly estimate them.

In this article I explore the scope of application for such an algorithm in the context of seismic data, which must be handled somewhat differently from digital images. I extend the algorithm to incorporate the idea of intermediate proxy data (to handle aliasing) and model-styling. Interpolation of 3D seismic data is challenging because of the increased computational cost and complexity in estimating covariance matrices and filter coefficients. I also extend the method to work for 3D data sets, results obtained for post-stack 3D data from Gulf of Mexico are presented at the end.

Next: Theory Up: Madhav Vyas: Covariance based Previous: Madhav Vyas: Covariance based
Stanford Exploration Project
1/16/2007