Seismic data interpolation beyond aliasing using regularized nonstationary autoregression

Introduction

The regular and fine sampling along the time axis is common, whereas good spatial sampling is often more expensive or prohibitive and therefore is the main bottleneck for seismic resolution. Too large a spatial sampling interval may lead to aliasing problems that adversely affect the resolution of subsurface images. An alternative to expensive dense spatial sampling is interpolation of seismic traces. One important approach to trace interpolation is prediction interpolating methods (Spitz, 1991), which use low-frequency non-aliased data to extract antialiasing prediction-error filters (PEFs) and then interpolates high frequencies beyond aliasing. Claerbout (1992) extends Spitz's method using PEFs in the $t$ -$x$ domain. Porsani (1999) proposes a half-step PEF scheme that makes the interpolation process more efficient. Huard et al. (1996) and Wang (2002) extend $f$ -$x$ trace interpolation to higher spatial dimensions. Gulunay (2003) introduces an algorithm similar to $f$ -$x$ prediction filtering, which has an elegant representation in the $f$ -$k$ domain. Curry (2006) uses multi-dimensional nonstationary PEFs to interpolate diffracted multiples. Naghizadeh and Sacchi (2009) propose an adaptive $f$ -$x$ interpolation using exponentially weighted recursive least squares. More recently, Naghizadeh and Sacchi (2010a) propose a prediction approach similar to Gulunay's method but using the curvelet transform instead of the Fourier transform. Abma and Kabir (2005) compare the performance of several different interpolation methods.

Correcting irregular spatial sampling is another application for seismic data interpolation algorithms. A variety of interpolation methods have been published in the recent years. One approach is to estimate the PEF on multiple rescaled copies of the irregular data (Curry, 2003), where the data are rescaled with a number of progressively-larger bin sizes. Curry (2004) further improves the rescaling method by introducing multiple scales of the data where the location of the grid cells are varied in addition to the size of the cells. Curry and Shan (2008) use pseudo-primary data by crosscorrelating multiples and primaries to estimate nonstationary PEF and then interpolated missing near offsets. Naghizadeh and Sacchi (2010b) propose autoregressive spectral estimates to reconstruct aliased data and data with gaps.

Seismic data are nonstationary. The standard PEF is designed under the assumption of stationary data and becomes less effective when this assumption is violated (Claerbout, 1992). Cutting data into overlapping windows (patching) is a common method to handle nonstationarity (Claerbout, 2010), although it occasionally fails in the presence of variable dips. Crawley et al. (1999) propose smoothly-varying nonstationary PEFs with ``micropatches'' and radial smoothing, which typically produces better results than the rectangular patching approach. Fomel (2002) develops a nonstationary plane-wave destruction (PWD) filter as an alternative to $t$ -$x$ PEF (Claerbout, 1992) and applies the PWD operator to trace interpolation. The PWD method depends on the assumption of a small number of smoothly variable seismic dips. Curry (2003) uses Laplacian and radial rougheners to ensure a nonstationary PEF that varies smoothly in space, which specifies an appropriate regularization operator.

In this paper, we use the two-step strategy, similar to that of Claerbout (1992) and Crawley et al. (1999), but calculate the adaptive PEF by using regularized nonstationary autoregression (Fomel, 2009) to handle both nonstationarity and aliasing. The key idea is the use of shaping regularization (Fomel, 2007) to constrain the spatial smoothness of filter coefficients. We provide an approach to nonstationary data interpolation, which has an intuitive selection of parameters and fast iteration convergence. We test the new method by using several benchmark synthetic examples. Results of applying the proposed method to a field data example demonstrate that it can be effective in trace interpolation problems, even in the presence of multiple strongly variable slopes.

Seismic data interpolation beyond aliasing using regularized nonstationary autoregression