y Input k y2 k y1 Introduction

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1 A regularization workflow for the processing of cross-spread COV data Gordon Poole 1, Philippe Herrmann 1, Erika Angerer 2, and Stephane Perrier 1 1 : CGG-Veritas, 2 : OMV Data regularization is critical for the suppression of Kirchhoff migration noise and the production of a clean migration image. We introduce a workflow utilizing Fourier reconstruction in two spatial dimensions to fully regularize cross-spread COV data in midpoint, offset, and azimuth sampling. Midpoint regularization is achieved through processing in the cross-spread and COV volume domains whilst the COV cmp domain is used to regularize offset-x/y or map to the offset-azimuth domain. In all stages the source and receiver coordinates are updated in a way that is consistent with the acquisition of the data. Introduction New processing strategies are required to get the most out of high density cross-spread datasets which are now more frequently being acquired. The use of Common Offset Vector (COV) geometries (Vermeer, 1998) leads to more uniformly sampled data volumes. COV geometries include offset-x and offset-y bin definitions as well as bin sizes in the inline and crossline directions. The offset-x/y bin sizes relate directly to the cross-spread source and receiver line spacing. COV datasets provide data sampling in four spatial dimensions; inline, crossline, offset-x and offset-y. While regularization solutions to simultaneously regularize in all four dimensions are available (Liu, et al. 2004), it is also possible to utilize multiple passes of a 3D algorithm (two spatial dimensions). The use of a 3D algorithm will be faster and for this reason it is the focus of this paper. We introduce a Fourier reconstruction algorithm for 3D data and describe how it may be used in cross-spread, COV volume, and COV CMP domains to fully regularize high density crossspread data. Algorithm We utilize a Fourier reconstruction algorithm as described by Poole and Herrmann, The algorithm preserves the signal and data character and can handle irregular data sampling. Firstly we apply a forward Fourier decomposition that respects the irregular recording positions. The resulting Fourier spectrum (f-k x -K y domain) describes the input data with functions that can be evaluated at any position in the input data range. The reverse transform maps the energy back on to a regular grid. A schematic of the algorithm is given in Figure 1. Figure 1 Representation of the Fourier reconstruction algorithm y Input Spatial Frequency Domain y Output k y2 Sum Fwd FT k y1 Rev FT k y0 x k x0 k x1 k x2 Each sub-panel in the central display represents the 2D wavenumber function in the x-y domain. Every spatial frequency can be evaluated at any (x,y) coordinate position. To interpolate to a discrete output position we sum the contributions of all frequencies. x

2 The input coordinates are calculated from the irregular input source-receiver positions, for example the midpoint position. The output coordinates are defined by the processing grid. The workflow is split into four steps. Pre Migration Step 1 Bin centring in the cross-spread domain Step 2 Hole filling in the COV volume domain Step 3 Offset-x/y regularization in the COV CMP domain Post Migration Step 4 Offset-Azimuth mapping in the COV CMP domain The same Fourier reconstruction algorithm is used for all four steps but the way the source and receiver coordinates are updated changes. The next sections go through the workflow in more detail. Step 1 Bin centring in the cross-spread domain With cross-spread acquisition we aim to make regular recordings along shot and receiver lines. In practice obstructions mean that our recordings can be far from regular, resulting in the acquired traces not being at bin centre. In this first step we regularize the cross-spread data to bin centre. This is achieved with a forward Fourier transform with the irregular input midpoint positions and the reverse transform to map the data to bin centre. The offset and azimuth of the input data are not constant so we must be careful in the way we assign source and receiver coordinates to the output data. To be consistent with the acquisition we must locate source and receiver positions on the acquisition lines that also have their midpoint at bin centre. This is achieved with non-linear inversion. Figure 2 shows how the output source and receiver coordinates are constructed. Figure 2 Construction of the cross-spread gather output coordinates Input Coordinates Regularisation Coordinates Shot Receiver Midpoint The input coordinates relate to a number of discrete shot and receiver positions forming a crossspread gather. To be consistent with the input acquisition, the regularized data must have source coordinates on the source line and receiver coordinates on the receiver line. An example midpoint position before and after regularisation has been highlighted. It is important to apply demultiple and denoise processes before data regularization as the Fourier reconstruction algorithm can be affected by high amplitude multiple and ground roll energy.

3 Step 2 Hole filling in the COV volume domain Although step 1 has mapped the data to bin centre, we still observe empty bins after sorting the data to the COV volume domain. These empty bins relate to gaps in coverage from neighbouring cross-spread gathers. The forward Fourier transform uses the input midpoint positions and the reverse transform fills in bins where no traces were recorded. The missing traces are assigned midpoint positions at bin centre and the source and receiver positions are calculated using trigonometry with the offset and azimuth of the COV volume. Figure 3 shows data before and after regularization. Figure 3 Midpoints and crossline data before and after COV volume regularisation Input Midpoints Crossline Regularisation Midpoints Crossline Input Crossline Regularisation Crossline For a given COV volume the input bins show holes in coverage relating to gaps between neighbouring cross-spreads. After regularisation the holes have been filled. Data at the input positions are unchanged. We observe that the character of the interpolated data is consistent with the input traces. We now have fully populated COV volumes with all traces at the centre of the bins. While this regular sampling improves migration results, we still have variations in offset and azimuth that will be a source of migration noise. Step 3 Offset-x/y regularization in the COV CMP domain We regularize in offset-x/y by sorting the data into COV CMPs. All traces in a COV CMP now have midpoints at bin centre, but each trace has its own offset-x/y position. We perform a forward Fourier transform with the irregular offset-x/y input positions and the reverse transform to output data on a regular offset-x/y grid. Source and receiver positions are updated by modifying offset-x/y whilst leaving the midpoint unchanged.

4 Following this step all traces have a regular offset-x/y sampling. This means that when we sort the data to the COV volume domain for migration all traces will be at bin centre with the same offset and azimuth. This provides ideal sampling for the minimization of migration related noise (Poole and Lecerf, 2006). Step 4 Offset-Azimuth mapping in the COV CMP domain After migration we often perform azimuthal analyses directly on the COV data. To verify these results it is often useful to look at the data in the common offset-common azimuth (COCA) domain (Gray, 2007). As in step 3 we perform a forward Fourier decomposition in the offset-x/y domain, but for the reverse transform we choose polar coordinate sampling to map the data to the offset-azimuth domain. The offset-azimuth sampling is user defined and need not relate to the input sampling. Figure 4 shows how the offset-azimuth domain can highlight velocity variations with azimuth. Figure 4 Azimuthal velocity variations in the CMP offset-azimuth domain Azimuth (degrees) Offset (m) Data in the azimuth-offset domain highlights azimuthal velocity variations. The event at 1200ms exhibits an arrival time that varies with azimuth relating to fast and slow velocity directions. Summary 1500 We have presented a practical four-step workflow using multiple passes of Fourier reconstruction in two spatial dimensions to fully regularize cross-spread data. While all steps use the same Fourier reconstruction algorithm, it is necessary to vary how the source and receiver coordinates are updated. The first step bin centres the data in the cross-spread domain. The output traces are assigned shot and receiver coordinates that are consistent with the source and receiver lines. The second step fills in empty bins in the COV volume domain. The midpoint positions of the interpolated traces are set to bin centre and the source and receiver coordinates are constructed so as to be consistent with the offset and azimuth of the COV volume. The final regularization step before migration is applied in the COV CMP domain and reconstructs energy on the COV definition (regular offset-x/y sampling). The result of these first three steps is data that are optimally conditioned for the application of prestack time migration. The final stage is applied after migration and is used to map the energy to the offset-azimuth domain. This is particularly useful for the QC of AVA and velocity variations with azimuth. Acknowledgements We would like to thank OMV for permission to include the data examples and CGGVeritas for allowing us to publish this work.

5 References Gray, D., Observations of seismic anisotropy in prestack seismic data, 77 th Ann. Internat. Mtg: Soc of Expl. Geophys., Expanded Abstracts. Liu, B., Sacchi, M. D., and Trad, D., Simultaneous interpolation of 4 spatial dimensions, 74 th Ann. Internat. Mtg: Soc of Expl. Geophys., Expanded Abstracts (SP 3.6). Poole, G., and Lecerf. D Effect of regularisation in the migration of time-lapse data, First Break, 24, pages Poole, G. and Herrmann, P., Multi-dimensional data regularisation for modern acquisition geometries, 77 th Ann. Internat. Mtg: Soc of Expl. Geophys., Expanded Abstracts. Vermeer, G. J. O., 1998, 3-D symmetric sampling, Geophysics, 63,