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Abstract
To ensure comparable land cross-spread 3D to 2D data quality, many authors advocate dense in-line receiver and in-line shot station sampling but ignore the receiver and shot line dimensions that are generally 10 to 20 times larger than station spacings. Consequently land 3Ds are routinely acquired with coarse cross-lines with a severe asymmetry between line and station dimensions resulting in extremely erratic offset and azimuth sampling within and between bins. This asymmetry causes severe statistical and aliasing limitations for key processing and inversion steps such as surface consistent statics, pre-stack migration (PSTM) and even AVO due to a lack of near offsets, as all of these applications have strict requirements for dense regularly sampled shot, receiver and common reflection point domains. By contrast, 2D linear spatial sampling is regular and dense with fold that is higher by the same line to station spacing ratio of cross-spread 3D's i.e. by factors of 10 to 20 times. The reason for this is that 2D fold is a function of shot spacing that is routinely acquired to be equal to the receiver spacing, while 3D fold is a function of line spacing (shot and receiver) that is considerably coarser than station spacings. This irregularly sampled lower fold aliases high wavenumber surface noise and leaks interbed multiple interference that contaminates signal in the standard gather domains of common shot, receiver, offset, azimuth and mid-point, thereby distorting the final 3D stack in subtle ways resulting directly in drilling failure for stratigraphic targets. However reducing line spacings to station dimensions incurs significant additional cost and may be environmentally prohibitive.
One solution or compromise to this dilemma is to acquire a wide azimuth parallel geometry with coincident shots and receivers that naturally reduces the line dimension to the station spacing in the in-line direction and efficiently allows for a finer cross-line spacing that may be regular or staggered (see following discussion on random sampling). However the compromise is that the cross-line spacing creates an asymmetric coarse cross-line bin that needs to be reduced by factors of 2 to 6 times in order to create the desired symmetric bin of the cross-spread 3D design. A design variously termed MegaBin or SlimBin that achieves this goal was tested through an oversampled decimation experiment and introduced in the early 90's by Goodway and Ragan while at PanCanadian. Ever since its introduction MegaBin has been vociferously denounced by various authors notably Vermeer, Cary and Perz to name a few, with good reason as this regular coarse cross-line bin spacing leads to aliasing that violates the Nyquist limit.
This brings the discussion to spatial interpolation. At the time of its introduction in the 90's the MegaBin design purposefully incorporated and aimed to recover the coarse cross-line bin size through interpolation both pre- and post-stack. At that time many authors rejected the ability of interpolation to successfully de-alias coarse sampling as the commonly quoted statement "you can't recover what you did not pay for" clearly indicates. However notable exceptions to this thinking included Claerbout and Spitz (1991) who showed that the spatial Nyquist limit when viewed as a wavenumber bandwidth can be overcome, i.e. de-aliased, by assuming that the un-aliased, low temporal frequencies share the same linear velocity or plane as the aliased, higher temporal frequencies. In addition this spatial bandwidth need not be contiguous, i.e. the wavefield may be comprised of a few planes with wavenumbers both below and well above the Nyquist limit as long as the sum of these wavenumbers did not exceed Nyquist. Both authors concluded that the assumption of a smoothly varying linear model for the wavefield (or a plane wave decomposition in a limiting sense), permits a reasonable reconstruction of the pre-migrated wavefield at finer spatial sampling, by using FX spatial prediction. The MegaBin patent awarded to PanCanadian required recovery beyond Nyquist of the coarse cross-line dimension through processes such as FX interpolation that primarily de-aliases signal as pre-conditioning for PSTM, but in some cases for linear noise attenuation that is more challenging due to its spatially sporadic high wavenumbers. In the last few years interpolation has finally gained wide acceptance through the research and implementation of pre-stack 5D Minimum Weighted Norm Interpolation (MWNI) by Sacchi and Trad that has allowed previously Nyquist challenged processes such as PSTM to succeed. However unlike the model driven FX de-aliasing methods championed by Claerbout and Spitz, MWNI does not explicitly de-alias data beyond the Nyquist limit. The reason for this is that MWNI as its name implies, relies on iteratively reinforcing the correct un-aliased wavenumbers in the presence of aliased wavenumbers. In order to achieve this, the un-aliased data must have a stronger representation in FK space than the contaminating aliased wavenumbers. This misunderstanding of MWNI's ability to de-alias was recently expounded at a lunchbox talk by Cary last year where the suggestion was to resort to random sampling, as this reduces the strength of aliased wavenumbers by smearing the strong regularly sampled aliases across the full range of low signal to high noise wavenumbers. However this randomness is difficult to achieve and not acquired in practice by cross-spread 3D's due to logistical field limitations. An easier alternative mentioned in the second paragraph above, is to simulate randomness or introduce irregularity by staggering or dithering the cross-line spacings in parallel geometries such as MegaBin.
So far the discussion for land 3D has focused exclusively on the Nyquist limit as formulated by Shannon's (1949) sinc interpolation that forms the basis for MWNI. What has not appeared in the land 3D debate is the concept of Seismic Wavefield Gradiometry to accurately reconstruct both the aliased signal and noise wavefields beyond Nyquist by factors of 2 to 6 times according to the multichannel sampling theorem of Linden (1959). This theorem accurately recovers aliased wavenumbers by simultaneously exploiting point sampling and its various higher order derivatives or gradients using sinc interpolation combined with a Taylor series expansion.
The presentation will cover both the past and future solutions described above, to accurately recover an inherent and severe acquisition sampling limitation that arises from coarse land 3D line spacings whether they be cross-spread or linear in design.
