Robust reconstruction of aliased data using autoregressive spectral estimates

Autoregressive modeling is used to estimate the spectrum of aliased data. A region of spectral support is determined by identifying the location of peaks in the estimated spatial spectrum of the data. This information is used to pose a Fourier reconstruction problem that inverts for a few dominant wavenumbers that are required to model the data. Synthetic and real data examples are used to illustrate the method. In particular, we show that the proposed method can accurately reconstruct aliased data and data with gaps. INTRODUCTION During recent years, interpolation and reconstruction of seismic data has become an important topic for the seismic processing community. In general, logistic and economic constraints dictate the spatial sampling of seismic surveys. Wave fields are continuous, in other words, seismic energy reaches the surface of the earth everywhere in our area of study. The process of acquisition records a finite number of spatial samples of the continuous wave field generated by a finite number of sources. The latter leads to a regular or irregular distribution of sources and receivers. Many important techniques for removing coherent noise and imaging the earth interior have stringent sampling requirements which are often not met in real surveys. In order to avoid information losses, the data should be sampled according to the Nyquist criterion (Vermeer, 1990). When this criterion is not honored, reconstruction can be used to recover the data to a denser distribution of sources and receivers and mimic a properly sampled survey (Liu, 2004). Seismic data reconstruction via signal processing approaches is a topic of current research interest in exploration seismology. During the last decade, important advances have been made in this area. Nowadays, signal processing reconstruction algorithms based on Fourier synthesis operators can cope with multidimensional sampling as demonstrated by several authors (Duijndam et al. , 1999; Liu et al. , 2004; Zwartjes and Gisolf, 2006; Schonewille et al. , 2009). These methods are based on signal processing principles, they do not require information about the subsurface and, in addition, are quite robust in situations were the optimality condition under which they were designed are not completely satisfied (Trad, 2009). Signal processing methods for seismic data reconstruction often rely on transforming the data to other domains. The latter can be achieved via the Fourier transform (Sacchi and Ulrych, 1996; Sacchi et al. , 1998; Duijndam et al. , 1999; Liu et al. , 2004; Xu et al. , 2005; Zwartjes and Gisolf, 2006), the Radon transform (Darche, 1990; Verschuur and Kabir, 1995; Trad et al. , 2002), the local Radon transform (Sacchi et al. , 2004; Wang Robust reconstruction of aliased data