Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

4.2. ANALYSIS OF OPTIMUM FILTER LENGTH 51
I first define the observed data as (D), the noise free data or clean data (Dc) and
the predicted data (Dp) . I now define the following two measures of goodness of
fit:
RMSE1 =
RMSE2 =
||D − Dp|| 2
nx × nt
||Dc − Dp|| 2
nx × nt
(4.9)
(4.10)
where nx × nt denotes the dimensions of the data: length of time series (nx) by
number of traces (nt), respectively. RMSE2 is not computable for real cases but
it can be used gain understanding about the problem of order selection. The best
(optimum) operator length is given for the operator that minimizes RMSE2 (Fig-
ure 4.1(a)). On the other hand, RMSE1 shows that increasing the filter length leads
to a decrease of error that is only accounted by for fitting the noise (Figure 4.1(b)).
In Figure 4.2(a), 2-D synthetic data consisting of three linear events yields pre-
dictions for different filter lengths. The signal Dc is contaminated with additive
noise (Figure 4.2(b)) and the signal-to-noise ratio (SNR) has been taken as 4. Figure
4.3(a) corresponds to the prediction of data presented in Figure 4.2(b) with param-
eter p = 3. Noise is rejected but the data are also poorly modeled (Figure 4.3(a)).
The prediction of linear events in the prediction panel is not satisfactory because
a large amount of energy leaks to the noise panel (Figure 4.3(b)). The prediction
with a filter length p = 6 provides good noise rejection and the data is properly
modeled (Figure 4.4(a)). The noise panel contains a small amount of coherent en-