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Introduction

Data interpolation is an important step in seismic data processing that can greatly affect the results of later processing steps, such as multiple removal, migration and inversion. There are many ways to interpolate data, including Fourier-transform-based approaches (e.g., Xu et al., 2005) and PEF-based approaches (e.g., Crawley, 2000; Spitz, 1991). A PEF is a filter that predicts one data sample from $ n$ previous samples, where $ n$ is the length of the PEF. One important feature of a PEF is that it has the inverse spectrum of the known data, so when it is convolved with known data, it minimizes the convolution result in the least-square sense. PEF estimation can be done in either time-space ($ t$-$ \bf x$) domain or frequency-space ($ f$-$ \bf x$) domain (e.g. Curry, 2007; Crawley, 2000; Claerbout, 1999), however, if PEF estimation is done in the $ f$-$ \bf x$ domain, every frequency needs one distinct PEF.

The pyramid domain was introduced by Ronen (Hung et al., 2005), and is a resampled representation of an ordinary $ f$-$ \bf x$ domain. Although it has frequency and space axes, the spatial sampling is different for different frequencies. This is attractive because we can use sparser sampling to adequately sample the data at lower frequencies, which makes uniform sampling for all frequencies unnecessary. Therefore in the pyramid domain, coarser grid spacing is used for lower frequencies, while finer spacing is used for higher frequencies. This makes it possible to capture the character of all frequency components of stationary events with only one PEF. So the information in the low frequency data can be better used to interpolate higher frequency data.

In this paper, I present a 3D version of data interpolation in the pyramid domain based on PEF estimation, which is based on Shen (2008). The paper is organized as follows: I first show the 3D pyramid transform and corresponding missing-data interpolation and PEF estimation. I then show synthetic data examples. Finally, I conclude with the advantages and disadvantages of this interpolation method.


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Next: Methodology Up: 3D pyramid interpolation Previous: 3D pyramid interpolation

2009-04-13