Spitz's $\omega$ stretching method

To find the missing traces, we need to invert the constrained linear system defined in equations (7) and (8), which requires that we have the prediction filter $\hat{P}(\omega,\hat{k}_x)$. Spitz showed that the coefficients of the prediction filter $\hat{P}(\omega,\hat{k}_x)$ are related to the coefficients of prediction filter $P(\omega,k_x)$ as follows:

 

$$\hat{P}_l(\omega)=P_l({\omega \over M}),$$ (9)

whereas $P_l(\omega)$ can be computed from the known traces with equation (4). Spitz also proved that the coefficients of the prediction filters are predictable along the frequency axis. Thus, if the known traces have a limited frequency bandwidth, the prediction filters at frequencies outside the signal band can be computed from the prediction filters at frequencies inside the signal band.

Spitz's derivations of equation (9) and his proof of the predictability of the prediction filters assume that the amplitude functions of events are constant functions and that the dip structure of data is invariant with respect to frequencies. The first assumption is approximately valid if the window sizes of subsections are small enough. The second one is probably valid if all the events on the subsection have the same frequency content. However, it may be invalid if the seismic waves these events represent are initiated from independent sources or have multi-propagation modes. In these cases, the dip structure within one frequency band may be different from that within another frequency band. Hence, Spitz's $\omega$ stretching method does not give correct prediction filters in these cases. Furthermore, Spitz's algorithm does not work when events are completely aliased.