Theory and practice of interpolation in the pyramid domain

Introduction

For interpolation, it is often assumed that the seismic data are made from a superposition of locally planar events [Symes (1994); Claerbout (1992); Fomel (2002)]. Since plane-waves are highly predictable, interpolating seismic data with prediction-error filters (pefs) is both very effective and efficient, either in the Fourier [Spitz (1991)] or time domain [Crawley et al. (1999)].

In the Fourier domain and for aliased data, pefs from lower frequencies are used to de-alias higher frequencies [Spitz (1991)]. Therefore, many pefs need to be estimated for complete processing. Sun and Ronen (1996) introduce a frequency dependent sampling in the $(\omega ,x)$ domain such that a data vector $\bf {d}$ at each frequency in the $(\omega ,x)$ domain is mapped into a new vector $\bf {m}$ (pyramid domain) according to

$$m(\omega,u)=d(\omega,x= u /\omega),$$ (1)

where $u$ has units of velocities and $x$ is a spatial axis (offset or mid-point position). We demonstrate in this paper that this transform has the advantage of making the pefs frequency independent, thus offering useful properties (for stationary data):

• Only one pef is necessary for interpolation
• This single pef can be estimated from all the frequencies
• For noisy data, many regressions are available from all the frequencies, yielding robust pef estimation

This paper investigates the pyramid domain method and introduces a linear operator called the pyramid transform. First, we illustrate the properties of the pyramid transform and explain why, in theory, only one pef is necessary for interpolation. Second, we identify mapping artifacts from $(\omega ,x)$-space to $(\omega,u=\omega\cdot x)$-space and propose a strategy both to attenuate them and to interpolate seismic data. This strategy works for both aliased and irregularly-sampled data. Finally, we illustrate the proposed algorithm on synthetic and real data cases.

Theory and practice of interpolation in the pyramid domain