Simulating PEF using dips picked

In the case of randomly missing traces in regularly sampled data, a prediction-error filter cannot be found. This section explains an alternative way to find a filter whose spectrum is the inverse of the given data's spectrum.

If there exist several linear events in a data set, the zeros which were found by the prediction-error filter would locate along the dips in the spectrum. If we know the dips of the events, therefore, we can simulate the prediction-error filter by putting zeros along the dips in the spectrum.

Suppose we have N plane waves and each dip is p_j, j=1,...,N, respectively. For each frequency $\omega$, then, N zeros along the wavenumber are determined as follows :

$$k_j = p_j \omega$$

and the corresponding zeros are

$$z_j = \exp(-ik_j).$$

From those zeros, the prediction-error filter for each $\omega$ is determined in the form of the Z transform like

$$F(z) = (1-z/z_1) (1-z/z_2) \dots (1-z/z_N).$$