We propose a robust interpolation scheme for aliased regularly sampled seismic data that uses the curvelet transform. In a first pass, the curvelet transform is used to compute the curvelet coefficients of the aliased seismic data. The aforementioned coefficients are divided into two groups of scales: alias-free and alias-contaminated scales. The alias-free curvelet coefficients are upscaled to estimate a mask function that is used to constrain the inversion of the alias-contaminated scale coefficients. The mask function is incorporated into the inversion via a minimum norm least squares algorithm that determines the curvelet coefficients of the desired alias-free data. Once the alias-free coefficients are determined, the curvelet synthesis operator is used to reconstruct seismograms at new spatial positions. Synthetic and real data examples are used to illustrate the performance of the proposed curvelet interpolation method.
Interpolation and reconstruction of seismic data has become an important topic for the seismic data processing community. It is often the case that 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. This 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). Methods for seismic wave field reconstruction can be classified into two categories: wave-equation based methods and signal processing methods. Wave-equation methods utilize the physics of wave propagation to reconstruct seismic volumes. In general, the idea can be summarized as follows. An operator is used to map seismic wave fields to a physical domain. Then, the modeled physical domain is transformed back to data space to obtain the data we would have acquired with an ideal experiment. It is basically a regression approach where the regressors are built based on wave equation principles (in general, approximations to kinematic ray theoretical solutions of the wave equation). The methods proposed by Ronen (1987), Bagaini and Spagnolini (1999), Stolt (2002), Trad (2003), Fomel (2003), Malcolm et al. (2005), Clapp (2006) and Leggott et al. (2007) fall under this category. These methods require the knowledge of some sort of velocity distribution in the earth’s interior (migration velocities, root-meansquare velocities, stacking velocities). While reconstruction methods based on wave equation principles are very important, this paper will not investigate this category of reconstruction algorithms. Seismic data reconstruction via signal processing approaches is an ongoing research topic in exploration seismology.