Application of Helix Filters to Pattern-Based Multiple Suppression
Authors: Bednar J. Bee , Brown M.P. , Clapp R.G. 
Published: October 2008

Abstract
Modern approaches to multiple suppression begin by producing corrupted estimates of recorded multiple energy. Popular methods include downward continuation [Refs. 2, 3, 4, 5, 6, 7, 8, 9, 19] and the Neumann series methods [Refs. 13, 17, 18]. Errors in the downward continuation model contribute to incorrect arrival times. Inaccuracies in the backpropagation algorithm result in poor amplitudes and phases. While the arrival times from a Neumann series approach are exact, wavefield phase and character are not. The limitations inherent in these estimation processes are generally overcome either through pattern capturing, or energy minimization techniques that seek to optimally remove the multiple energy from the recorded wavefield. In the pattern approach, the idea [Refs. 15, 16] is to use the recorded data and the estimated noise to characterize the primary and multiple patterns separately. Separating the signal from the noise is then accomplished by statistical techniques. The problem was originally formulated in the frequency domain. A significant advantage of this approach is that the prediction error filters that capture the various patterns are one-dimensional and easy to compute. The helix filter [10] offers a new and robust approach to these methods with the same advantage. Because they convert multi-dimensional filters to one-dimensional equivalents, helix filters provide the same advantages in space-time that frequency domain filters provide in f-x. This paper reviews the helix concepts and shows that they provide an effective approach to pattern-based multiple suppression.

Helix Filters
Helix filters [10] are a deceptively easy concept to master. Figure 1 explains the entire process. In this case a two-dimensional filter shown in (a) is wrapped around a coil (b) and then unwound (c) on a one-dimensional data vector (d). The result, within helix boundary constraints, is a two-dimensional convolution of a two-dimensional filter with a two-dimensional data set.
Traditionally, we think of one-dimensional objects as vectors, and multi-dimensional objects as either rows or columns of vectors. In the memory of most modern computers, the FORTRAN programming language stores multi-dimensional arrays as fixed length columns in contiguous memory locations. Treating this memory block as a one-dimensional array produces a super data vector. Storing a similar array of filter coefficients produces a one-dimensional filter operator that can be convolved in one-dimension. The resulting array is exactly equivalent to the result of a twodimensional convolution.
Fig. 1. Winding a two-dimensional filer onto a helix. (a) Twodimensional filter. (b) Two-dimensional filter as a helix. (c) A helix about to be unwound. (d) Helix unwound on a one-dimensional impulse. (Available in full paper)

Seismic Data Series
Using the appropriate prediction technology, seismic data can be expressed in the form
(Mathematical equation (1) available in full paper)
Where U0 is a suitably modified form of the recorded wavefield, UP is the data without coherent noise energy, and W is a correction factor chosen to optimally suppress (or add) desired coherent events. Each term, UI, in the series is generally estimated from the previous term through formulas of the form
(Mathematical equation (2) available in full paper)