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# Introduction

Interpolation of seismic data can be performed in many different domains. A simple band-limited interpolation is achieved by padding in the f-k domain. Other methods in the f-k domain involve a sparse inversion of a non-uniform (or uniform) Fourier transform Duijndam and Schonewille (1999); Liu and Sacchi (2004); Xu et al. (2005).

Interpolation beyond aliasing can be performed using prediction filters in the f-x domain Spitz (1991) as well as prediction-error filters in the t-x domain Claerbout (1999); Crawley (2000). Interpolating in the t-x domain involves estimating a single filter for the entire dataset while the f-x domain method involves estimating an independent filter for each frequency and separately interpolating each frequency. Interpolation in f-x appears to be much faster than t-x and requires a much smaller memory footprint, while the t-x approach allows for the introduction of a time-variable weighting function and is also less sensitive to noise.

Interpolation in t-x can also be performed using non-stationary filters Crawley (2000). These prediction-error filters vary in space and time, and solve a global problem instead of the more traditional approach of breaking the problem up into stationary regions and solving those problems independently. Interpolation in the f-x domain can also be performed using non-stationary filters, but due to the Fourier transform these filters do not vary as a function of time. This can lead to some issues when dealing with non-stationarity in time.

Non-stationary interpolation in the f-x domain provides a much less expensive alternative to the t-x domain. With this method it is now possible to perform higher-dimensional interpolation for large-scale applications such as surface-related multiple prediction. The drawbacks include acausal energy appearing in the result as well as the inability to weight in time.

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Stanford Exploration Project
5/6/2007