Figure 4 The prediction-error filter for suppressing the regions of the velocity semblance spectrum that are associated with primary reflections.
Figure 5 The estimated prediction-error filter for the synthetic data of Figure -b.
Once the filter coefficients are defined, the weighting operator is obtained by
Because of the large size of the problem to be solved in equation (6) and since the semblance is a smooth function of time, it is convenient to subsample S in time before solving equation (6) and applying equation (7). The resulting operator is then interpolated to the original time sampling interval. The prediction-error filter estimated for the synthetic data of Figure -b is shown in Figure .
Figure compares the original semblance with the result of applying the prediction operator to it for the synthetic dataset of Figure . It is impressive that the filter was able to correctly predict the different multiples patterns present in the data. None of the primaries are visible on the output. Figure -a shows the window function constructed from , and Figure -b shows the semblance spectrum of the synthetic data after 15 iterations of the optimization process using as the window function. In -b the events associated with reverberations are barely visible. Most importantly, the parts of the spectrum associated with the converted waves were preserved, as show the events at 2.05, 2.5, and 4.3 seconds.
A comparison of the output of the optimization process with the ``ideal" multiple free data (Figure ) shows that not only have the multiples been substantially attenuated, but also the amplitude and phase of the primaries were correctly preserved, including the weak, low-velocity converted waves.