Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

5.1. INTRODUCTION 76
ples remains as a problem in exploration seismology. In this Chapter, I present the
adaptive subtraction methods based on Volterra series that can be used to attenuate
the multiples.
In the adaptive multiple subtraction problem, one has access to a distorted
version of the multiples and the goal is to find an operator that eliminates the
distortion before subtracting them from the data. There are many methods to
annihilate multiple reflections such as inverse scattering and surface-related multiple
attenuation (SRME) (autoconvolution in space) (Weglein et al., 1992; Verschuur
et al., 1992; Berkhout and Verschuur, 1997; Spitz, 1999; Abma et al., 2005). I am
not going to explain here how a multiple model can be produced; this subject is well
understood and published in many studies (Verschuur et al., 1992; Weglein et al.,
1992; Berkhout and Verschuur, 1997; Verschuur and Berkhout, 1997; Verschuur and
Prein, 1999; Weglein, 1999). These methods are used to construct multiple models,
which are subsequently utilized by adaptive subtraction algorithms.
In this thesis I introduce a method for adaptive subtraction of multiples, using
a f − x linear prediction filter. A nonlinear prediction filter based on a Volterra
series for the removal of multiples from recorded data sets. The problem can also
be tackled using linear prediction error filters in the f − x domain as suggested
by Spitz (1999). The idea is to compute a f − x prediction error operator from
the estimate of the multiples. Then, the estimated prediction error filter is applied
to the data containing both primaries and multiples to annihilate the multiples.
The procedure is equivalent to finding a notch filter with the notch positioned at
the wave number representing the multiple. The procedure of Spitz (1999) can fail
when the multiple events in the window of analysis cannot be modeled as linear