Page 332 - Acquisition and Processing of Marine Seismic Data

6.2 ASSUMPTIONS FOR DECONVOLUTION 323

FIG. 6.10 Schematic display of the importance of Assumption 6. Components for the convolutional model to obtain the
seismogram are shown in the upper panel. Autocorrelation of a random function is the Dirac delta (δ(t)), and convolving
it with the wavelet’s autocorrelation yields again the wavelet’s autocorrelation (middle panel) since δ(t) is the identity element
for the convolution process. In this case, autocorrelation of the seismogram directly equals the autocorrelation of the wavelet.
In reality, autocorrelation of the reflectivity series does not equal to δ(t), and therefore only the initial parts of the autocorre-
lation of the seismogram resemble the characteristics of the autocorrelation of the source wavelet (white section in lower
panel). (*) and R{} denote convolution and autocorrelation, respectively; r(t) is reflectivity series, w(t) is source wavelet,
and s(t) is recorded seismogram.

autocorrelation of the wavelet, providing that In order to perform deconvolution, we
the reflectivity series is random. However, this definitely need Assumption 6 and it has key
is not the case in reality: autocorrelation of the importance in the theoretical foundation of
reflectivity series does not equal to the Dirac deconvolution, because it enables us to use the
delta because the earth’s reflectivity series does known autocorrelation function of the recorded
not have the characteristics of a random series seismogram instead of the unknown autocorrela-
(lower panel in Fig. 6.10). Even though it has a tion of the source wavelet as shown in Fig. 6.11.
maximum value at zero lag, it is actually non- As a result of this assumption, an inverse filter
zero at the remaining lags and has small ampli- or deconvolution operator can be obtained
tudes distributed along the whole axis, most of directly from the autocorrelation of the seismo-
which arise from short and long path multiples. gram. For such a deconvolution procedure, we
The autocorrelation of the seismogram obtained do not necessarily need to know the source wave-
by convolving the autocorrelation of the wavelet form to obtain its autocorrelation for the decon-
with this function does not exactly equal to the volution operator, and therefore Assumption 4
autocorrelation of the source wavelet. In fact, is no longer required.
only the initial part of this resultant autocorrela- Furthermore, the amplitude spectra of the
tion function reflects the characteristics of the recorded seismogram and the source wavelet
autocorrelation of the source wavelet (lower display a general similarity (Fig. 6.12).
panel in Fig. 6.10). According to the convolutional model, the