The convolutional model in the frequency domain
The convolutional model for the noise-free seismogram (assumption 4: the noise component n(t) is zero) is represented by equation (3a). Convolution in the time domain is equivalent to multiplication in the frequency domain (the 1-D Fourier transform). This means that the the amplitude spectrum of the seismogram equals the product of the amplitude spectra of the seismic wavelet and the earth’s impulse response (Synthetic Seismogram):
where Ax(ω), Aw(ω), and Ae(ω) are the amplitude spectra of x(t), w(t), and e(t), respectively.
Figure 2.1-6 shows the amplitude spectra (top row) of the impulse response e(t), the seismic wavelet w(t), and the seismogram x(t). The impulse response is the same as that shown in Figure 2.1-1d. The similarity in the overall shape between the amplitude spectrum of the wavelet and that of the seismogram is apparent. In fact, a smoothed version of the amplitude spectrum of the seismogram is nearly indistinguishable from the amplitude spectrum of the wavelet. It generally is thought that the rapid fluctuations observed in the amplitude spectrum of a seismogram are a manifestation of the earth’s impulse response, while the basic shape is associated primarily with the source wavelet.
Mathematically, the similarity between the amplitude spectra of the seismogram and the wavelet suggests that the amplitude spectrum of the earth’s impulse response must be nearly flat (Synthetic Seismogram). By examining the amplitude spectrum of the impulse response in Figure 2.1-6, we see that it spans virtually the entire spectral bandwidth. As seen in Figure 2.1-5, a time series that represents a random process has a flat (white) spectrum over the entire spectral bandwidth. From close examination of the amplitude spectrum of the impulse response in Figure 2.1-6, we see that it is not entirely flat — the high-frequency components have a tendency to strengthen gradually. Thus, reflectivity is not entirely a random process. In fact, this has been observed in the spectral properties of reflectivity functions derived from a worldwide selection of sonic logs .
We now study the autocorrelation functions (middle row, Figure 2.1-6) of the impulse response, seismic wavelet, and synthetic seismogram. Note that the autocorrelation functions of the basic wavelet and seismogram also are similar. This similarity is confined to lags for which the autocorrelation of the wavelet is nonzero. Mathematically, the similarity between the autocorrelogram of the wavelet and that of the seismogram suggests that the impulse response has an autocorrelation function that is small at all lags except the zero lag (Synthetic Seismogram). The autocorrelation function of the random series in Figure 2.1-5 also has similar characteristics. However, there is one subtle difference. When compared, Figures 2.1-5 and 2.1-6 show that autocorrelation of the impulse response has a significantly large negative lag value following the zero lag. This is not the case for the autocorrelation of random noise. The positive peak (zero lag) followed by the smaller negative peak in the autocorrelogram of the impulse response arises from the spectral behavior discussed above. In particular, the positive peak and the adjacent, smaller negative peak of the autocorrelogram together nearly act as a fractional derivative operator (the 1-D Fourier transform), which has a ramp effect on the amplitude spectrum of the impulse response as seen in Figure 2.1-6.
Figure 2.1-1 (a) A segment of a measured sonic log, (b) the reflection coefficient series derived from (a), (c) the series in (b) after converting the depth axis to two-way time axis, (d) the impulse response that includes the primaries (c) and multiples, (e) the synthetic seismogram derived from (d) convolved with the source wavelet in Figure 2.1-4. One-dimensional seismic modeling means getting (e) from (a). Deconvolution yields (d) from (e), while 1-D inversion means getting (a) from (d). Identify the event on (a) and (b) that corresponds to the big spike at 0.5 s in (c). Impulse response (d) is a composite of the primaries (c) and all types of multiples.
Figure 2.1-5 A random signal with infinite length has a flat amplitude spectrum and an autocorrelogram that is zero at all lags except the zero lag. The discrete random series with finite length shown here seems to satisfy these requirements. What distinguishes a random signal from a spike (1, 0, 0, …)?
Figure 2.1-6 Convolution of the earth’s impulse response (a) with the wavelet (b) (equation 2a) yields the seismogram (c) (bottom row). This process also is convolutional in terms of their autocorrelograms (middle row) and multiplicative in terms of their amplitude spectra (top row). Assumption 6 (white reflectivity) is based on the similarity between autocorrelograms and amplitude spectra of the impulse response and wavelet.
The above observations made on the amplitude spectra and autocorrelation functions (Figure 2.1-6) imply that reflectivity is not entirely a random process. Nonetheless, the following assumption almost always is made about reflectivity to replace the statement made in assumption 5 (the source waveform is known).
- Assumption 6. Reflectivity is a random process. This implies that the seismogram has the characteristics of the seismic wavelet in that their autocorrelations and amplitude spectra are similar.
This assumption is the key to implementing the predictive deconvolution. It allows the autocorrelation of the seismogram, which is known, to be substituted for the autocorrelation of the seismic wavelet, which is unknown. In optimum wiener filters, we shall see that as a result of assumption 6, an inverse filter can be estimated directly from the autocorrelation of the seismogram. For this type of deconvolution, Assumption 5, which is almost never met in reality, is not required. But first, we need to review the fundamentals of inverse filtering.
- Walden and Hosken, 1984, Walden, A. T. and Hosken, J. W. J., 1984, An investigation of the spectral properties of primary reflections coefficients: Presented at the 46th Ann. Mtg. Eur. Assoc. Expl. Geophys.