Fourier methods of seismic data regularisation

Introduction

Regularisation of input data is required, or at least assumed, for a variety of advanced seismic data processing methods. Consequently if the input data is irregularly sampled then the effectiveness of many methods are compromised. Particularly algorithms such as 3D surface related multiple elimination (SRME) and pre-stack wave-equation depth migration have been shown to suffer considerably when applied to irregular data, and often induce new processing artifacts (Canning and Gardner, 1996). Furthermore irregular sampling can also lead to poor repeatability between 4D seismic surveys (at least for conventional 4D processing). To help alleviate these problems a common seismic data processing step involves some form of regularisation and/or interpolation.

Whilst precise definitions can vary, the concept of regularisation involves transferring samples from their irregular input grid to new locations on a regular output grid. Often interpolation will follow regularisation, whereby missing samples in the output grid are filled in; this is also often used to interpolate to a finer grid. A regularisation operator that can handle aliased and non-stationary events is desired, as this will assist in the preservation of all signal.

Several advanced regularisation methods have been suggested to overcome the problem of irregularly sampled data. One such technique is to use convolution operators known as prediction error filters (PEFs) that can predict absent traces honouring the spectra of known traces, however this can be limited due to an assumption of local linearity (Abma, 1995). As discussed in Biondi (2005) the problem can also be solved by AMO regularisation, whereby data is reconstructed at an arbitrary (regular) azimuth.

Fourier based methods can also be used, where the regularisation is performed by preserving the spectrum of the data and inverse transforming to a desired grid; however this suffers from the phenomena known as spectral leakage. The problem arises due to the non-orthogonality of the global basis functions (the $sinc$ function) that are exhibited by an irregular grid (Xu and Pham, 2004a) and is discussed in detail in the next section. This phenomenon has been addressed by many disciplines that involve time series analysis, and several solutions to remove or reduce spectral leakage have been created.

A further possibility is to design an appropriate taper to minimise spectral leakage about a certain frequency or band of frequencies, and these are known as Slepian tapers, as discussed in Slepian (1978) and Percival and Walden (1993). The concept is that the taper parameters are designed to maximise spectral concentration within a predefined resolution bandwidth window, and so the use of these should help to greatly reduce the number of iterations required in other Fourier regularisation methods, or possibly replace them altogether. Such methods have been shown to work on random signals, synthetic earthquake signals and for bathymetry profiles (Prieto, 2007), and so the extension of this concept to active seismic data could help the solve the problem of Fourier regularisation.

Fourier methods of seismic data regularisation