An algorithm for interpolation in the pyramid domain^‡

Abstract

A B S T R A C TWith the pyramid transform, 2D dip spectra can be characterized by 1D predictionerror filters (pefs) and 3D dip spectra by 2D pefs. These filters, contrary to pefs estimated in the frequency-space domain (ω, x), are frequency independent. Therefore, one pef can be used to interpolate all frequencies. Similarly, one pef can be computed from all frequencies, thus yielding robust estimation of the filter in the presence of noise. This transform takes data from the frequency-space domain (ω, x) to data in a frequency-velocity domain (ω, u = ω · x) using a simple mapping procedure that leaves locations in the pyramid domain empty. Missing data in (ω, x)-space create even more empty bins in (ω, u)-space. We propose a multi-stage least-squares approach where both unknown pefs and missing data are estimated. This approach is tested on synthetic and field data examples where aliasing and irregular spacing are present.

Multidimensional de-aliased Cadzow reconstruction of seismic records

“…The idea is to use alias-free lowfrequency spatial data to properly estimate prediction error filters that are used to interpolate high-frequency aliased spatial data. An abundance of literature describes beyond alias interpolation methods based on Spitz's principle (Porsani, 1999;Gulunay, 2003;Guitton and Claerbout, 2010). Modifications of Spitz's method have also been proposed to address problems of irregularities in the data (Naghizadeh and Sacchi, 2007) and nonstationary events (Naghizadeh and Sacchi, 2010a).…”

Achieving Robust Compressive Sensing Seismic Acquisition with a Two-Step Sampling Approach

“…This type of noise is usually represented by the surface waves that are suffering the most from the sparse aliasing sampling intervals. On the data processing side, different algorithms can be applied to mitigate the aliasing in the data, for instance, reconstruction methods based on the prediction error filter [5,6]. In addition to a long-standing dilemma of shooting a dense survey or working with aliased seismic gathers, the ideal regular seismic layout can be compromised by field deployment circumstances: obstacles and areas without access, equipment malfunction, zones with poor recording conditions, logistical constraints, and surface topography.…”

Seismic data interpolation and denoising in the frequency-wavenumber domain

“…Recently, Wang et al (2009) also introduced a greedy least-squares method which utilizes local high resolution Radon transform for interpolation of aliased seismic records. In addition, Guitton and Claerbout (2010) have utilized the pyramid transform for beyond alias interpolation of seismic records by searching for a small collection of plane waves across all frequencies.…”