Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

4.1. LINEAR PREDICTION IN THE F − X DOMAIN 48
An ARMA model can be approximated by a long autoregressive model (AR), which
turns out to be a representation of the linear prediction problem. In summary, linear
events in t−x space transform into complex sinusoids in the f −x domain and linear
prediction filtering can properly model the spatial variability of the waveforms at
any given monochromatic temporal frequency f.
The seismic signal is considered to be an AR model; let us assume a single
waveform in time domain. In addition, let’s assume that the signal has linear
moveout
s(x, t) = a(t − xθ) (4.1)
where x is offset of the trace, t is the time and θ is the slowness of the event. In
the frequency domain this signal becomes
S(x, f) = A(f)e −i2πfθx
(4.2)
where A(f) denotes the source spectrum and f is the temporal frequency for x. By
discretizing x = (j − 1)δx, f = fl
Sjl = Ale −i2πflθ(j−1)δx
(4.3)
I can develop this model as a function of wave number by fixing kl = 2πflθ equation
(4.3) becomes
Sjl = Ale −ikl(j−1)δx
(4.4)
I can define the problem by predicting data along the each trace to fix temporal