Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

4.4. NOISE REMOVAL AND VOLTERRA SERIES 62
4.4 Is Noise Removal Possible with a Volterra Se-
ries?
I have presented a third- order Volterra model for the prediction of seismic signals.
The usual trade-off between noise reduction and signal preservation is controlled
by the length of the linear and the nonlinear prediction operators (p, q, and r). It
is important to stress that waveforms that exhibit linear moveout can be predicted
with a linear system. Conversely, when waveforms exhibit nonlinear moveout, which
translates in the f − x domain as chirps, the nonlinear part of the Volterra system
helps in modeling the signal.
When I started this project, the idea was to design a new noise reduction system.
The premise was to use nonlinear prediction to reconstruct signals in such a way that
the reconstruction misfit could be attributed to noise. The latter is the basis of f −x
deconvolution for signal-to-noise ratio enhancement. However, nonlinear filtering
requires too many coefficients for the quadratic and cubic parts of the operator,
therefore, signal and noise are simultaneously modeled (overfitting) (Figure 4.11).
The data in the Figure 4.6(a) ) and the prediction of this data via a third-order
Volterra series predicts the signal as well as the original data but noise is also
fitted with the prediction. A small amount of coherent energy leaks to the noise
panel. The third-order Volterra series can perfectly model the data, but for f − x
noise attenuation it is not useful as it produces many coefficients and therefore
models both signal and noise. The next step is to study how one can introduce
regularization (smoothing) to the estimation of Volterra kernels and avoid fitting
the noise.