Some remarks on surface multiple attenuation

The surface multiple attenuation algorithm discussed in this paper is a prestack inversion of a surface-recorded, 2-D wavefield that aims to remove all orders of all surface multiples present within the wavefield. Although the algorithm requires no assumptions or modeling regarding the positions and reflection coefficients of the multiple-causing reflectors, it does require complete internal physical consistency between primary and multiple events-something that exists only in ideal 2-D data sets. In field data sets the physical consistency between primaries and multiples is disturbed by phenomena such as variations in the acquisition wavelet, cable feathering, cross-line dip, a finite near offset, and unequal or too coarse spatial sampling in source and receiver coordinates. Careful survey design can minimize the impact of those phenomena on surface multiple attenuation. If it is not too large, trace extrapolation can solve the finite near-offset problem. Minor adjustments to the algorithm allow processing of data for which the source and receiver intervals differ by an integer multiple, although for those and other acquisition geometries, trace interpolation may be preferred. In the f-x domain, surface multiple attenuation can be formulated as an equation whose straightforward solution involves the inversion of a large matrix that is a function of the acquisition wavelet. Since that wavelet is generally unknown, solving this matrix equation becomes an optimization problem. Many matrix inversions are needed to estimate the acquisition wavelet that leads to the best multiple suppression, rendering the straightforward solution to the surface multiple attenuation equation quite costly. We offer two alternative approaches. In our first approach we compute an eigenvalue decomposition of the large matrix, allowing the equation to be recast so that the wavelet dependency appears in a diagonal matrix for which repetitive inversion is trivial. In our second approach we begin by using the surface multiple attenuation algorithm with a fixed, approximately correct wavelet to compute the surface multiple wavefield. We then filter the predicted multiples adaptively to match the actual multiples in the original wavefield and subtract these filtered multiples from the original wavefield. The second approach is relatively inexpensive and to some extent can cope with physical inconsistencies between primaries and multiples caused by field data set imperfections.