Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

5.3. SYNTHETIC DATA EXAMPLES 78
Notice that the new operator corresponds to the filter that predicts the noise or
error and not to the signal (equation (3.14)). In addition, equation (6.1) is valid
for both linear prediction and nonlinear prediction. The operator f is then applied
to the data (primaries plus multiples). The concept is illustrated with examples.
5.3 Synthetic Data Examples
Figure 5.2(a) represents three linear events with one primary (horizontal event) and
two multiples (dipping events). The data are similar to the examples provided in
Abma et al. (2005). Figure 5.2(b) is an estimation of the multiples and Figure 5.2(c)
is the primary estimation via the f − x linear prediction filter (Spitz, 1999). The
removal of multiples and the preservation of the primary reflection is fairly good.
In Figure 5.3(a) I portray one primary (horizontal event) and two multiples
that cannot be approximated by linear events. Figure 5.3(b) shows the distorted
estimation of multiples. Figures 5.3(c) and 5.3(d) are the solution via the classical
linear prediction and the solution with the third- order Volterra series, respectively.
Linear prediction algorithm can not removed any multiples, the nonlinear prediction
error filter, on the other hand, was able to model the curved multiples and attenuate
multiples. Both methods were able to preserve the primary event without affecting
its amplitude response.
In Figure 5.4(a) I portray a similar synthetic data example with one primary and
two multiples. Figure 5.4(d) shows the distorted estimation of multiples. Figures
5.4(b) and 5.4(c) are the solution via classical linear prediction and the solution
with the third- order Volterra series, respectively. Nonlinear prediction error filter
subtracts more multiples than linear one. In this example, I illustrate the removed