Multiple Attenuation Using an Apex Shift Radon Transform
Authors: Trad Daniel , Hargreaves Neil , verWest Bruce , Wombell Richard 
Published: October 2008

Abstract
Multiples from sea-floor scatterers and peg-leg multiples in complex geology are often resistant to conventional multiple removal techniques such as Radon demultiple. They have a complicated moveout behaviour in prestack gathers which can only be approximately represented by a conventional parabolic or hyperbolic Radon decomposition. Such multiples split into pairs of events, one for each of the shot or receiver side of the multiple. They are approximately parabolic after NMO correction with primary velocities but have their minimum travel times shifted to either side of zero-offset.
It is possible to extend the Radon multiple model by including apex-shifted parabolas or hyperbolas in the model space. In the first case we call this an "?-Radon transform" since the apex shift is indicated by ??in our notation. In the second approach we call it Stolt-Radon because we use the Stolt migration operator. In order to compute the transform it is then necessary to perform a large and somewhat costly constrained inversion. Nonetheless both techniques have the potential to attenuate multiple diffractions and other similarly complicated multiple events in complex geology.

Introduction - description of the problem
Figure 1(a) shows a portion of stack data from a North Sea survey. The water-bottom arrival is the event at 0.5 s and the first-order multiple of the water-bottom is at approximately 1.0s. A scatterer on the water-bottom produces the event that is dipping from left to right just below the water-bottom. What appears to be a multiple of the scatterer sits just below the water-bottom multiple, cutting across events from left to right.
Figure 1(b) shows the result of applying prestack multiple removal on these data using a high-resolution parabolic Radon transform (Hargreaves & Cooper, 2001). The flat portion of the water-bottom multiple and other multiples (e.g. the event at just above 1.2s) are reasonably well suppressed. However, the dipping event, which we interpreted as the multiple of the scatterer, remains relatively untouched.
We can obtain some additional insight into these results by looking at the prestack data. Figure 2(a) shows a time-window from a CDP gather in the region of the scatterer, after moveout correction. The first and second-order multiples of the waterbottom and the multiples of other events are easily identified, in addition to some slightly over-corrected primary events. There are, however, some other rather unusual events in this gather. Just below the first-order water-bottom multiple
Figure 1. (a) A portion of North Sea stack data. The water-bottom arrival is the event at 0.5s and the first-order multiple of the water-bottom is the event at approximately 1.0s. A water-bottom scatter produces the event dipping from left to right just below the water-bottom. (b) The result of applying prestack highresolution parabolic Radon demultiple to the data of the previous panel. (c) The result of applying prestack apex-shifted parabolic Radon demultiple.(Available in full paper)
Figure 2. (a) An input gather from the center of the stack panel in Figure 1. (b) The multiple model for this gather generated by high-resolution parabolic Radon demultiple. (c) The multiple model generated by apex-shifted parabolic Radon demultiple.(Available in full paper)