Deconvolving the ghost effect of the water surface in marine seismics

The ghost filters arising from the effect of the water surface on both source and receiver sides have a common time domain representation that consists of a unit impulse followed by its ghost, which is a delayed, negative unit impulse. The origin of the difficulties of deghosting lies in the zeroes in the spectrum of the ghost filter, which render incorrect any deghosting through least‐squares inverse filtering in the time domain. Another shortcoming of the time domain approach is that the digital description of the ghost filter is inexact when a sampling instant does not coincide with the instant of the onset of the ghost impulse. A frequency domain approach, on the other hand, is straightforward and accurate because it can avoid the zeroes of the filter either by explicitly choosing a recording band that excludes the zeroes or by recording at two depths. These two depths should be selected according to the criterion that their highest common measure is small enough to prevent zeroes at a common frequency of the two recordings. As the source‐side and the receiver‐side ghost filters have the same form, the criterion derived for the selection of the depths of the receivers would also hold for the selection of the depths of two sources whose aggregate signature is desired to have no zeroes in the spectrum, within the operative band. An important ramification of the analysis consists of the disproof of a prevalent conjecture that the zeroes in the spectrum of a wavelet make its autocorrelation matrix singular; actually, the zeroes cause an inexact and unacceptable least‐squares inverse, although the matrix itself is well conditioned.