Amplitude Preserving Radon Demultiple
Authors: Herrmann Philippe , Mojesky Todd , Magesan Mag 
Published: October 2008

Abstract
Multiple elimination methods based on the move-out discrimination between primaries and multiples rely heavily on the focusing of seismic events in the parabolic Radon domain. This focusing, however, is affected both by the finite spatial aperture and sampling of the data. As a consequence of the resulting smearing, multiple energy may be mapped into the primary model and conversely primary energy may be mapped into the multiple model. This leads to poor multiple removal and to the non-preservation of the primary amplitudes. To overcome these pitfalls one has to make use of De-aliased, High-Resolution Radon transforms. Highresolution Radon transforms have already been proposed by some authors. Here we present a novel approach that simulatenously tackles the aliasing and resolution issues in a non-iterative way.

Introduction
The widespread use of multiple removal in the parabolic Radon domain (Hampson, 1986, Kostov, 1990) is related to its effectiveness and efficiency for most situations. However, when applied under severe conditions: reduced spatial aperture (Figures 1a-f) and/or coarse offset sampling (Figure 2a-d), we observe (Figure 1d) a poor focusing of the events in the parabolic Radon domain, combined with severe aliasing artifacts (Figure 2d). The limited aperture (Figure 1a) reduces the effective moveout difference between the parabolas and thus the resolution of the parabolic decomposition. At high frequencies (Figure 2c), the finite offset sampling makes ambiguous the apparent moveout of the parabolas, leading to a non-unique parabolic decomposition of the data (Figure 2d). As a consequence of these combined effects, the multiple model loses some multiple energy and includes significant primary energy. Once the multiple model is subtracted from the input data, this leads to poor multiple removal and deteriorates primaries (see Figures 3a-d for illustration).

De-aliased High-Resolution (DHR) Radon transform
Finite spatial aperture limits the resolution of the Radon transform, while finite spatial sampling introduces aliasing artifacts. To overcome these limitations one has to constrain the parabolic decomposition of the data to its most significant parabolic components. This issue was first investigated by Thorson and Claerbout, 1985. More recently Sacchi and Ulrych, 1995, Hugonnet and Canadas, 1997, and Cary, 1998 have developed high-resolution Radon transforms (in the frequency-space or time-space domain). These constrain the Radon spectra to be sparse in q (curvature) and ?intercept), using a re-weighted iterative approach. At a given iteration, the Radon decomposition is focalized onto the most significant spectral components highlighted at the previous iteration.
We now describe how De-aliased, High-Resolution (DHR) Radon spectra may be constructed with a direct approach avoiding the iterative process.
Going back to the previous synthetic example (Figures 3a-d), the following observations can be made:Due to the curvature range involved in this example and the parabolic sampling rate chosen, 550 parabolas are used to perform the parabolic decomposition of 15 traces. That is an obvious underdetermined Least-squares problem.Among the 550 parabolas, only 4 are actually needed to properly decompose the data. Constraining the parabolic decomposition of the data onto these four parabolas will lead to well-focused parabolic Radon spectra.