Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

4.5. SYNTHETIC AND REAL DATA EXAMPLES 70
predicted with quadratic and cubic nonlinear terms. Figure 4.17(b) is the prediction
using a third-order Volterra series with p = 7, q = 7, and r = 7. In Figure 4.17(c)
I portray the part of the prediction attributed to the linear kernel (p = 7). Figures
4.17(d) and 4.17(e) show the parts of the prediction associated with quadratic
(q = 7) and cubic (r = 7) terms in the third-order Volterra series. In addition,
Figure 4.17(f) illustrates the part of the prediction associated to both quadratic
and cubic terms (q = 7 and r = 7). These terms give good predictions, especially
for the apexes of events where linear terms cannot predict the data.
Finally, I test the performance of the Volterra series with a marine data set. The
data consist of 60 traces extracted from a marine common offset section acquired
over a salt body in the Gulf of Mexico (Figures 4.18(a) and 4.19(a)). The real data
set has a combination of diffractions, roughly linear events, and hyperbolic events.
I used filters with order p = 9, q = 9, and r = 9 for a third-order Volterra prediction
(Figures 4.18(b) and 4.19(b)). In this case the data is properly modeled. I also
compute the linear prediction filter with parameter p = 9 and attempt to model
the data (Figure 4.18(d)). The prediction error between the original data and the
prediction with a third-order Volterra series is small (Figure 4.18(c)), whereas the
difference between the original data and the predicted data via a linear prediction
method is large. In particular, the diffractions are leaking to the error panel as
a consequence of improper modeling (Figure 4.18(e)). It is clear that the linear
prediction was not able to properly model the data.
In Figures 4.19(c), 4.19(d) and 4.19(e) I examine the individual contributions
of the linear and nonlinear parts of the Volterra series to the prediction. Linear
terms mostly predict linear moveouts with parameter p = 9; nonlinear terms model