Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

6.1. FUTURE DIRECTIONS 93
seismic sections. I have satisfactory results for adaptive subtraction.
Limitations
All methods have limitations. Volterra series produces many coefficients and cal-
culation of these coefficients is computationally expensive. Also, a large number of
coefficients causes overfitting of the signal and this leads to modeling both signal
and noise.
6.1 Future Directions
Volterra series have been implemented in nonlinear systems in a range of problems
from biomedical research to communication applications. Adaptations of these
methods to seismic exploration problems are presented in Chapter 4 and Chapter
5 of this thesis. The Volterra filter I developed is time-invariant; the signal is sta-
tionary. However, in the future the Volterra filter presented here could be designed
as a time-variant filter.
The reconstruction of monochromatic complex sinusoids is a well studied prob-
lem in seismic exploration in the case of regularly spaced traces in seismic sections.
However, there are various interpolation methods in f − x and f − k domains;
these methods can not reconstruct missing traces with events that exhibit curva-
ture. There are examples of Volterra series implementation in the reconstruction
of nonlinear image interpolation. For instance, Collis et al. (1997) presented an
application of pixel interpolation in television images based on a nonlinear Volterra
filter. Particularly, a third-order Volterra filter will be useful to interpolate missing
data in seismic traces.