The Exponentially Weighted Recursive Least Squares (EWRLS) method is adopted to estimate adaptive prediction filters for F-X seismic interpolation. Adaptive prediction filters are able to model signals where the dominant wave-numbers are varying in space. This concept leads to a F-X interpolation method that does not require windowing strategies for optimal results. Synthetic and real data examples are used to illustrate the performance of the proposed adaptive F-X interpolation method.
Spitz (1991) introduced a seismic trace interpolation method that utilizes prediction filters in the frequency-space (F-X) domain. Spitz''s algorithm is based on the fact that linear events in time-space (T-X) domain map to a superposition of complex sinusoids in the F-X domain. Complex sinusoids can be reconstructed via prediction filters (autoregressive operators); this property is used to establish a signal model for F-X interpolation (Spitz, 1991) and F-X random noise attenuation (Canales, 1984; Soubaras, 1994; Sacchi and Kuehl, 2000). Spitz (1991) showed that prediction filters obtained at frequency f can be used to interpolate data at temporal frequency 2 f . Prediction filters estimated from the low-frequency (alias-free) portion of the data are used to interpolate the high-frequency (aliased) data components. Several modifications to Spitz''s prediction filtering interpolation have been proposed. For instance, Porsani (1999) proposed a half-step prediction filter scheme that makes the interpolation process more efficient. Gulunay (2003) introduced an algorithm with similarities to F-X prediction filtering with a very elegant representation in the frequencywavenumber F-K domain. Recently, Naghizadeh and Sacchi (2007) proposed a modification of F-X interpolation that allows to reconstruct data with gaps.
Seismic interpolation algorithms depend on a signal model. F-X interpolation methods are not an exception to the preceding statement; they assume data composed of a finite number of waveforms with constant dip. This assumption can be validated via windowing. Interpolation methods driven by, for instance, local Radon transforms (Sacchi et al., 2004) and Curvelet frames (Herrmann and Hennenfent, 2008) assume a signal model that consists of events with constant local dip. In addition, they implicitly define operators that are local without the necessity of windowing. This is an attractive property, in particular, when compared to non-local interpolation methods (operators defined on a large spatial aperture) where optimal results are only achievable when seismic events match the kinematic signature of the operator. Examples of the latter are interpolation methods based on the hyperbolic/ parabolic Radon transforms (Darche, 1990; Trad et al., 2002) and migration operators (Trad, 2003).
As we have already pointed out, F-X methods require windowing strategies to cope with continuous changes in dominant wave-numbers (or dips in T-X). In this article we propose a method that avoids the necessity of spatial windows. The proposed interpolation automatically updates prediction filters as lateral variations of dip are encountered. This concepts can be implemented in a somehow cumbersome process that requires classical F-X interpolation in a rolling window. In this paper we have preferred to use the framework of recursive least squares (Honig and Messerschmidt, 1984; Marple, 1987) to update prediction filters in a recursive fashion.